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1005.3189
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# Irreversibility in response to forces acting on graphene sheets
N. Abedpour Department of Physics, Sharif University of Technology,
11365-9161, Tehran, Iran School of Physics, Institute for Research in
Fundamental Sciences, (IPM) Tehran 19395-5531, Iran Reza Asgari School of
Physics, Institute for Research in Fundamental Sciences, (IPM) Tehran
19395-5531, Iran M. Reza Rahimi Tabar Department of Physics, Sharif
University of Technology, 11365-9161, Tehran, Iran Fachbereich Physik,
Universität Osnabrück, Barbarastraß, 49076 Osnabrück, Germany
###### Abstract
The amount of rippling in graphene sheets is related to the interactions with
the substrate or with the suspending structure. Here, we report on an
irreversibility in the response to forces that act on suspended graphene
sheets. This may explain why one always observes a ripple structure on
suspended graphene. We show that a compression-relaxation mechanism produces
static ripples on graphene sheets and determine a peculiar temperature
$T_{c}$, such that for $T<T_{c}$ the free-energy of the rippled graphene is
smaller than that of roughened graphene. We also show that $T_{c}$ depends on
the structural parameters and increases with increasing sample size.
###### pacs:
81.05.Uw, 71.15.Pd, 82.45.Mp, 05.70.Ce
Introduction— Two-dimensional graphene crystals novoselov have attracted
considerable attention both experimentally and theoretically, due to their
unusual electronic properties yafis . Ripples or undulations were first
observed in freely-suspended graphene flakes in experiments meyer . The
ripples affect the electronic properties juan , such as the conductivity and
the quantum transport properties. Charge inhomogeneities due to the ripples
have also been observed in graphene inhomo . A new method to fabricate
periodically rippled graphene on Ru(0001) under ultrahigh vacuum conditions
was reported in parga . The ripples in graphene were studied theoretically
flagg , and it was claimed that the ripples may be explained as a consequence
of absorbed molecules sitting on random sites.
There are two points of view on the physics of the ripples. Meyer et al. meyer
proposed that the reproducible appearance of the ripples across samples
indicates that it is an intrinsic effect. They emphasized that the homogeneity
and isotropy of the ripples are not compatible with the assumption of an
incompressible sheet. They estimated a local strain of up to 1% for a single-
layer flake. Using the generic free-energy for the long wavelength
deformations nelson , Castro Neto and Kim castro , on the other hand, argued
that graphene may be considered as an atomic thin membrane, with its physics
being also similar to a soft membrane. In this point of view the ripples are
not intrinsic and can be the results of the environment, such as the substrate
in the system.
Recently, direct observation and controlled creation of periodic ripples in
suspended graphene sheet was reported bao by using both spontaneously and
thermally generated strains via varying the substrate and annealing
conditions. The experimental measurements indicated that the ripples were
induced by the preexisting longitudinal strains in graphene. It was observed
at temperatures about 500 K that graphene sheets are flat, however, upon
cooling down to room temperature, the ripples invariably appear.
In this Letter we report on an irreversibility in response to forces that act
on the suspended graphene sheets that may explain why one always observes the
rippled structure on graphene in experiments. We show that a compression-
relaxation mechanism can produce static ripples on graphene sheets. We
determine a peculiar temperature $T_{c}$ such that for temperatures less than
$T_{c}$ the free-energy of the rippled graphene is smaller than that of
roughened graphene. We also show that for sample with size $400\times 200$
atoms, $T_{c}\simeq 90$ K and, moreover, $T_{c}$ is an increasing function of
the sample size.
Theory and Model— We used molecular dynamics (MD) simulations with the
empirical interatomic interaction potential due to Brenner brenner , i.e.,
carbon-carbon interaction in hydrocarbons that contains three-body
interaction. Many-body effects of electron system, on average, are considered
in the Bernner potential through the bond-order term. We employed the Nosé-
Hoover thermostat to control the temperature, when we used a canonical (NVT)
ensemble in the MD simulations. We note that although the Brenner potential is
not entirely a quantum mechanical potential, it predicts the correct
mechanical properties of the structures with carbon atoms by using the
classical MD simulations bee .
Here, we show that the compression and then relaxation in one or two
directions [$x$ (arm-chair) and $y$ (zigzag)] of graphene sheet can produce
static ripples, which means that there is an irreversibility in response to
forces acting on segments of the graphene sheets. Indeed, we have found that
if one compress the surface in one direction, say the $x-$direction (arm-
chair), after compressing about $0.13\%L$, where $L$ is the size of the
simulation sample, the ripples will appear. After the ripples emerge, we move
back the boundaries to their initial positions. We then observe that after
doing the compression-relaxation processes, the ripples survive, hence
implying that the compression process is not reversible. It is worthwhile
mentioning that we also simulated tethered membranes kantor and repeated the
compression-relaxation procedure. We obtained no any indication of
irreversibility in response to the forces acting on the membranes.
In the case of graphene, if the compression amount becomes larger than the
critical value (here, $0.13\%L$, which also depends on the temperature and the
size of simulation sample), the graphene sheet bends and, therefore, no ripple
appears. The typical height variance of the rippled graphene is about $5\;\AA$
at $T=50$ K. Our simulation results show that the wavelength of the static
ripples do not change with the size of the sample. Moreover, we observe that
the surface roughness (the variance of the height fluctuations) does not
change after the relaxation and, therefore, the ripples are static. Thus, we
might state that any primary stress on graphene sheet, for example in its
preparation in the experiments, can construct ripples that will survive during
the experimental measurements (see, for example, boukhvalov ).
Let us first determine the average wavelength of the ripples after relaxing
the system. To do so we calculate the two-dimensional Fourier components of
the height-height correlations, $G(|{\bf q}|)=<\left|h({\bf q})\right|^{2}>$.
Figure 1 shows $G({|\bf q|})$ as a function of ${|\bf q|}/q_{0}$, in
logarithmic scales, for both the roughened (no ripples) and relaxed states in
which we have stable ripples. Here $q_{0}=2\pi/L$, with $L$ being the length
of graphene in the $x-$direction. In the inset of Fig. 1 the same plot in
linear scale for ${|\bf q|}$ is shown to clarify a peak around ${|\bf
q|}\simeq 10q_{0}$ that corresponds to about $85\;\AA$ at 50 K. This is the
average wavelength of the ripples and is near to the value observed
experimentally meyer and calculated numerically Faso ; nima . In addition,
one can derive the wavelength of the ripples by calculating the first minimum
of the second moments of the height increments fluctuations
$<|h(x_{1})-h(x_{2})|^{2}>$ with respect to relative distance,
$|x_{1}-x_{2}|$.
The scale-dependence of $<\left|h({\bf q})\right|^{2}>$ is proportional to
$1/{|\bf q|}^{\alpha}$, where $\alpha\simeq 4$ at temperature 50 K.
Consequently, our results predict that the bending rigidity term prevails with
respect to the surface tension in graphene at short distances safran . Note
that the contribution of surface tension is a term like $T/\sigma{|\bf
q|}^{2}$, whereas the contribution of the bending rigidity is $T/\kappa{|\bf
q|}^{4}$, where $\sigma$ and $\kappa$ are the interfacial tension and bending
modulus, respectively safran . The exponent $\alpha$ might generally be
smaller than 4, due to thermal fluctuations, surface tension and anharmonic
corrections nelson . We estimate $\kappa$, the bending rigidity or bending
modulus, using the relation, $\kappa^{-1}\simeq|{\bf q}|^{4}<|h({\bf
q})|^{2}>/Nk_{B}T$. Plotting $\kappa$ vs $|{\bf q}|/q_{0}$ shows that the
bending rigidity is almost constant for $20<|{\bf q}|/q_{0}<100$, with
$\kappa\simeq 1$ eV-1.
Thus, for a given temperature, the compression-relaxation mechanism produces
the ripples, and graphene has at least two ”states” simultaneously, the
”rough” or normal sheet (no ripple) and the ”rippled” structure. The question
now is, which state is more stable? To answer this question one should
calculate the free energies of the two states and determine which state has a
smaller free-energy. In what follows we calculate the free-energy difference
of the rippled and roughened states of graphene sheets.
To compute the free-energy, we employed a well-known method (c.f., Haile allen
) in which one defines a continuous variable $\lambda$ for distinguishing two
different states Jarzynski . Suppose that by varying an external parameter,
such as slow compression and relaxation of the graphene, the system can go
from an initial state i (rough) to a final state f (rippled). When the
parameters are changed infinitely slowly along some path from i to f in the
parameter space, then the total work $W$ performed on the system is equal to
the Helmholtz free-energy difference between the initial and final
configurations. In contrast, when the parameters are switched along the path
at a finite rate, Jarzynski found that Jarzynski :
$\Delta A=-\frac{1}{\beta}\ln{\overline{{\exp(-\beta W)}}}$ (1)
where overbar denotes an average over an ensemble of measurements of $W$. We
ran the MD simulation to very long times in order to slowly pass the
intermediate quasistable states. In practice, for every step of compression-
relaxation, we checked whether the system was in equilibrium. We ensured the
existence of the true equilibrium condition by checking the stability of the
internal-energy fluctuations. Eventually, the problem of calculating $\Delta
A$ is the same as calculating the averaged $W$.
Figure 1: (Color online) Log-log plot of $<|h({\bf q})|^{2}>$ as a function of
${\bf q}/q_{0}$ (log scaled) both in the rough and ripple cases. In the inset,
$<|h({\bf q})|^{2}>$ is shown as a function of $|{\bf q}|/q_{0}$ to clarify a
peak around $|{\bf q}|\simeq 10q_{0}$. Graphene sheet incorporates $80000$
atoms at $50K$. Figure 2: (Color online) The dependence of the free-energy
and internal-energy (entropy) differences as a function of temperature. Figure
3: (Color online) Probability distribution function of the total energy $E$
for the rippled and rough states for $T=30,50,70,85,100$ and 300 K (from top
to bottom). The dashed curves are the Gaussian probability distribution
function. For clarity, the PDFs were shifted upward.
In Fig. 2 the free-energy differences $A_{\rm ripple}-A_{\rm rough}$ is given
as a function of $T$. We used the numerical results for a graphene sheet
incorporating $N=80000$ atoms $(400\times 200)$ at various temperatures. It
appears that the ripples are stable at low temperatures, namely, below
$T_{c}\approx 90$ K, such that above $T_{c}$ the rough state is more stable.
This feature is in good agreement with recent experimental observation bao .
Here, we would like to point out that the potential energy of the carbon-
carbon interaction in the compression process is different from that in the
relaxation process since the relative positions of the atoms in these two
configurations are different. Note that the morphology of the surface depends
strongly to the potential energy.
We also tested the dependence of $T_{c}$ on the size of the samples by
simulating the systems with $600\times 200$ and $800\times 200$ atoms, and
found their characteristic temperatures to be $T_{c}\simeq 115$ and
$T_{c}\simeq 140$, respectively. Moreover, we found that the wavelength of the
ripples depends on $T$ as $\lambda\simeq 35\ln(T)-55$, but it does not depend
on the system size. Accordingly, we calculated the entropy difference of the
two states and showed that for temperatures less than $T_{c}$, the rippled
state has a higher entropy and is stable. We plotted the internal energy
difference of the two states is shown in Fig. 2. The value of $T_{c}$ can be
also determined from the local stored stress on a graphene sheet; we will
report the results elsewhere mehdi .
We may expect that similar to second-order phase transitions the probability
distribution function (PDF) of the total internal energy possesses different
shapes for rough and ripple states, and exhibit non-Gaussian behavior. In Fig.
3 the PDF of the total internal energy $E$ for the ripple and rough states are
presented for $T=30,50,70,85,100$ and 300 K. To calculate the PDF, we used 200
ensembles of roughened and rippled graphenes, incorporating $400\times 200$
atoms. We observed that the PDF has a Gaussian form for both states indicating
that there is no longer second–order phase transition in the system. We also
checked the Gaussian nature of the PDF by using the $\chi^{2}$ test chi . The
dashed curves represent the Gaussian PDF.
As we argued earlier, there are at least two states for graphene sheets for a
given temperature. A question raised is, whether or not, there is any
possibility of a transition from one state to another? One possible way for
such a transition with fixed graphene sheet size is to increase the
temperature. For this purpose we simulated the graphene sheet with $80\times
40$ atoms and, after carrying out the compression-relaxation process, the
ripple structure appeared at $T=55$ K (see the upper figure of Fig. 4). We
then increased the temperature very slowly. As shown in Fig. 4, the ripples
begin to disappear at high temperatures. At $T=55$ K, we have almost two
wavelength of the ripples; however, at higher temperatures there is one
wavelength at $T=320$ K, and finally one half of the wavelength at $T=520$ K
remains. The final step may be called rough state. Accordingly, the energy
barrier of two states may be estimated by $465k_{B}=0.04$ eV (or $=0.0125$ meV
per particle) for a sample with 3200 atoms, where $k_{B}$ is the Boltzmann’s
constant. Such a transition has been observed experimentally in [10]. They
argued that the disappearing of ripples in high temperature is due to the fact
that graphene has negative thermal expansion coefficient.
As mentioned above, for temperatures less than $T_{c}$, the free-energy of the
rippled state is smaller than that of the free-energy of the roughened
graphene sheet. However, there is a possibility of having a transition from
the rippled state to the roughened state, due to a tunneling-type transition.
To detect this transition, we checked that the height fluctuations variance of
the rippled graphene sheet is stable with time. A sample size of $80\times 40$
atoms was used at $T=55$ K. The simulations showed that there is no transition
from the rippled to roughened state at a constant temperature $T$ less than
$T_{c}$, at least up to available time scales in the MD simulations. The
probability for such transition is about $\exp(-465/T)$ for a sample with size
3200 atoms. We remind that the variance of height fluctuations in graphene are
about $5\;\AA$ and $1$ nm, for rough and rippled states, respectively.
Figure 4: (Color online) Transition from rippled state (upper snapshot) to the
rough state, due to the increasing of the temperature from $55K$ to $520K$.
In summary, we have used a compression-relaxation mechanism to produce rippled
structures on graphene sheets. The constructed ripples survive even though the
system is relaxed to its initial position. In the closed-path loop, we
calculated the total work and, hence, the free-energy difference of the
rippled and roughened states. Our numerical results show that for sample with
$400\times 200$ atoms and below $T_{c}\approx 90$ K, the rippled surface is
stable and the entropy of the ripples should be larger than that of the rough
state. However, above $T_{c}$ the rough state is more stable. The rippled and
rough structures are also related to the morphology of such systems and we,
therefore, expect that the our simulations yield the correct and new results
for the free-energy of the rippled and roughened graphene. We have done
similar simulations for a bilayer graphene and observed that, for a given
temperature, the wavelength of the static ripples are larger than that for a
monolayer graphene. We will report the results for bilayer graphene elsewhere.
Acknowledgments— We thank A.K. Geim, , M. I. Katsnelson, P. Maaß, A. H.
MacDonald and M. Sahimi for very important comments and discussions. We also
thank M. Neek-Amal for early contributions to the numerical work.
## References
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* (2) Y. Barlas, et al., Phys. Rev. Lett.98, 236601 (2007) ; M. Polini, et al., Solid State Commun. 143, 58 (2007), M. Polini, et al., Phys. Rev. B 77, 081411(R) (2008); E. H. Hwang and S. Das Sarma, Phys. Rev. B 77, 081412(R) (2008) .
* (3) J. C. Meyer, et al., Nature 446, 60 (2008) .
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* (5) J. Martin, et al., Nature Physics 4, 144 (2007); Y. Zhang et al. Nature Physics 5, 722 (2009) .
* (6) A. L. V$\acute{a}$zquez de Parga, et al., Phys. Rev. Lett. 100, 056807 (2008) .
* (7) R. C. Thompson-Flagg, M. J. B. Moura and M. Marder, Europhys. Lett.85, 46002 (2009) .
* (8) D. R. Nelson, et al., Statistical Mechanics of Membranes and Surfaces, World Scientific, Singapore, (2004) .
* (9) A. H. Castro Neto and Eun-Ah Kim, Europhys. Lett. 84, 57007 (2008) .
* (10) W. Bao, et al. Nature Nanotechnology 4, 562 (2009) .
* (11) D. W. Brenner, Phys. Rev. B 42, 9458 (1990) .
* (12) Y. Chen, et al., Nanotechnology 20 035704, (2009); R. S. Ruoff, D. Qian, W. K. Liu, C. R. Physique 4 993 (2003); M. A. Osman and D. Srivastava, Nanotechnology 12 21, (2001) .
* (13) Y. Kantor, M. Kardar and D. R. Nelson, Phys. Rev. A 35, 3056 (1987); F. F. Abraham and W. E. Rudge, Phys. Rev. Lett. 62, 1757 (1989); F. F. Abraham and D. R. Nelson, J. Phys. France 51, 2653 (1990); F. F. Abraham and M. Kardar, Science 252, 419 (1991).
* (14) D. W. Boukhvalov and M. Katsnelson, J. Phys. Chem. C 113, 14176 (2009) .
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* (16) A. Fasolino, J. H. Los. and M. I. Katsnelson, Nature Materials 6, 858 (2007);
* (17) S. Safran, Statistical Thermodynamics of Surfaces, Interfaces, and membranes (Addison-Wesley Publishing Company) (1994) .
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|
arxiv-papers
| 2010-05-18T13:10:11 |
2024-09-04T02:49:10.468217
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "N. Abedpour, Reza Asgari, M. R. Rahim Tabar",
"submitter": "Reza Asgari",
"url": "https://arxiv.org/abs/1005.3189"
}
|
1005.3208
|
11institutetext: LESIA, Observatoire de Paris-Meudon, 5 place Jules Janssen,
92195 Meudon, France 11email: bruntt@phys.au.dk 22institutetext: LAM, UMR
6110, CNRS/Université de Provence, 38 rue F. Joliot-Curie, 13388 Marseille,
France 33institutetext: ESA, ESTEC, SRE-SA, Keplerlaan 1, NL2200AG, Noordwijk,
The Netherlands 44institutetext: Observatoire de Genève, Université de Genève,
51 Ch. des Maillettes, 1290 Sauverny, Switzerland 55institutetext: Institut
d’Astrophysique de Paris, UMR7095 CNRS, Université Pierre & Marie Curie, 98bis
Bd Arago, 75014 Paris, France 66institutetext: Observatoire de Haute-Provence,
CNRS/OAMP, 04870 St Michel l’Observatoire, France 77institutetext: Thüringer
Landessternwarte Tautenburg, Sternwarte 5, 07778 Tautenburg, Germany
# Improved stellar parameters of CoRoT-7††thanks: The CoRoT space mission,
launched on December 27, 2006, has been developed and is being operated by
CNES, with the contribution of Austria, Belgium, Brazil, ESA, The Research and
Scientific Support Department of ESA, Germany and Spain.
A star hosting two super Earths
H. Bruntt 11 M. Deleuil 22 M. Fridlund 33 R. Alonso 44 F. Bouchy 5566 A.
Hatzes 77 M. Mayor 44 C. Moutou 22 D. Queloz 44
(Received 27 January 2010; accepted 18 May 2010)
###### Abstract
Context. Accurate parameters of the host stars of exoplanets are important for
the interpretation of the new planet systems that continue to emerge. The
CoRoT satellite recently discovered a transiting rocky planet with a density
similar to the inner planets in our solar system, a so-called super Earth. The
mass was determined using ground-based follow-up spectroscopy, which also
revealed a second, non-transiting super Earth.
Aims. These planets are orbiting a relatively faint ($m_{V}=11.7$) G9V star
called CoRoT-7. We wish to refine the determination of the physical properties
of the host star, which are important for the interpretation of the properties
of the planet system.
Methods. We have used high-quality spectra from HARPS@ESO 3.6m and UVES@VLT
8.2m. We use various methods to analyse the spectra using 1D LTE atmospheric
models. From the analysis of Fe i and Fe ii lines we determine the effective
temperature, surface gravity and microturbulence. We use the Balmer lines to
constrain the effective temperature and pressure sensitive Mg 1b and Ca lines
to constrain the surface gravity. We analyse both single spectra and co-added
spectra to identify systematic errors. We determine the projected rotational
velocity and macroturbulence by fitting the line shapes of isolated lines. We
finally employ the Wilson-Bappu effect to determine the approximate absolute
magnitude.
Results. From the analysis of the three best spectra using several methods we
find $T_{\rm eff}=5250\pm 60$ K, $\log g=4.47\pm 0.05$, $[{\rm M/H}]=+0.12\pm
0.06$, and $v\sin i=1.1^{+1.0}_{-0.5}$ km s-1. The chemical composition of
$20$ analysed elements are consistent with a uniform scaling by the
metallicity $+0.12$ dex. We compared the $L/M$ ratio with isochrones to
constrain the evolutionary status. Using the age estimate of 1.2–2.3 Gyr based
on stellar activity, we determine the mass and radius $0.91\pm 0.03$ M⊙ and
$0.82\pm 0.04$ R⊙. With these updated constraints we fitted the CoRoT transit
light curve for CoRoT-7b. We revise the planet radius to be slightly smaller,
$R=1.58\pm 0.10$ R⊕, and using the planet mass the density becomes slightly
higher, $\rho=7.2\pm 1.8\,{\rm g\,cm}^{-3}$.
Conclusions. The host star CoRoT-7 is a slowly rotating, metal rich, unevolved
type G9V star. The star is relatively faint and its fundamental parameters can
only be determined through indirect methods. Our methods rely on detailed
spectral analyses that in turn depend on the adopted model atmospheres. From
the analysis of spectra of stars with well-known parameters with similar
parameters to CoRoT-7 (the Sun and $\alpha$ Cen B) we demonstrate that our
methods are robust within the claimed uncertainties. Therefore our methods can
be reliably used in subsequent analyses of similar exoplanet host stars.
###### Key Words.:
stars: fundamental parameters – stars: planetary systems – stars: individual:
TYCHO ID 4799-1733-1, $\alpha$ Cen B
## 1 Introduction
Figure 1: A section of the CoRoT 7/H1-7 spectrum illustrating how the spectrum
is normalised with rainbow. The top panels show a wide range and the lower
panels show a zoom near the edge of the same echelle order. The neighbouring
order is shown with a short-dashed line. The thick long-dashed line is a
spline fit to the continuum windows marked by circles. The normalised spectrum
agrees reasonably well with the template synthetic spectrum (smooth green
line). The agreement between the two overlapping orders is good and will
finally be merged to improve the S/N.
The discovery of the first super Earth planet outside of the Solar system with
a measured absolute mass and radius was recently announced, based on
photometric data from CoRoT (Convection, Rotation and planetary Transits;
Baglin et al. 2006). This planet, CoRoT-7b, has a radius of $1.68\pm 0.09$
$R_{\oplus}$ (Léger, Rouan, Schneider et al. 2009), mass $4.8\pm 0.8$
$M_{\oplus}$ (Queloz et al. 2009), and the orbital period is about 0.85 days
(Léger, Rouan, Schneider et al. 2009). The average density is $5.6\pm 1.3$ g
cm-3 which is similar to Mercury, Venus and the Earth (Queloz et al. 2009).
Furthermore, a second non-transiting super Earth has been found from radial-
velocity monitoring (Queloz et al. 2009). These results have only been
possible to achieve thanks to an extensive ground-based follow-up program of
the relatively faint star CoRoT-7 (TYCHO ID 4799-1733-1; $m_{V}=11.7$) over
more than one year.
In the derivation of the planetary parameters, one of the most important
factors is the correct identification of the host star’s fundamental
parameters and evolutionary stage. It is particularly important to estimate
the stellar radius which is imperative for determining the absolute planetary
radius. A first analysis by several of the CoRoT teams has been carried out in
Léger, Rouan, Schneider et al. (2009), based on a spectrum from the
“Ultraviolet and Visual Echelle Spectrograph” (UVES@VLT 8.2m) and a
preliminary analysis of 53 co-added spectra from the “High Accuracy Radial
velocity Planet Searcher” (HARPS@ESO 3.6m).
Since then, a total of 107 spectra from HARPS have become available (Queloz et
al. 2009). These spectra have higher spectral resolution and better signal-to-
noise (S/N) than the UVES spectrum analysed by Léger, Rouan, Schneider et al.
(2009). We can therefore now refine the analysis of CoRoT-7 and possibly
impose stronger constraints on the properties of the system. The methods we
employ have been developed during the analysis of other CoRoT targets (Deleuil
et al. 2008; Moutou et al. 2008; Rauer et al. 2009; Fridlund et al. 2010;
Bruntt 2009). In the current paper we have expanded these tools and we will
describe our approach in greater detail than previously done. These tools will
be the standard methods to be applied for the characterization of future CoRoT
targets.
## 2 Spectroscopic observations
We initially acquired one UVES spectrum which confirmed that the star is a
dwarf star, meaning the absolute radius of the planet must be small (Léger,
Rouan, Schneider et al. 2009). To constrain the mass of CoRoT-7b a series of
107 spectra were collected with the HARPS spectrograph between March 2008 and
February 2009 (Queloz et al. 2009).
The HARPS spectrograph has a spectral resolution of $115\,000$ (Mayor et al.
2003), covering the optical range from 3827 Å to 6911 Å. With exposure times
of 1800 or 2700 s, the signal-to-noise ratio of the individual spectra varies
from $\simeq 30$–$90$, depending on the conditions at the time of observation.
We used the data from the standard HARPS pipeline, and each order was divided
by the blaze function to get an approximately rectified spectrum. We shifted
each spectrum by the radial velocity to set it to the heliocentric rest frame,
using the values from Queloz et al. (2009). Each spectrum was rebinned to the
same wavelength grid with a constant step of $0.01$ Å.
We suspected that some of the exposures could be affected by reflected
Moonlight. While such data can be used for measurement of the radial velocity
variation, scattered light can potentially affect the relative line depths and
hence systematically affect the analysis. In order to identify such potential
systematic errors we made different combinations of the spectra as presented
in Table 1. We selected seven spectra acquired during dark time, and with the
highest S/N, computed in nearly line-free regions around 5830 Å. The co-
addition, order per order, of these 7 spectra gives the H1–7 combined
spectrum. We also analysed three individual HARPS spectra with the highest S/N
(H1, H2 and H3). We finally co-added all HARPS spectra using weights
$w\propto{\rm S/N}$ to get the H1-107 spectrum.
Table 1: List of the 10 spectra used for the spectroscopic analysis. Spec. | Date | Time | $t_{\rm exp}$ | |
---|---|---|---|---|---
ID | UT | UT | [s] | S/N |
H1 | 2008-12-26 | 04:56 | 2700 | 95 |
H2 | 2009-01-15 | 05:39 | 2700 | 90 |
H3 | 2009-01-17 | 01:45 | 2700 | 95 |
H1–7 | Combined spec. | | | 235 |
H1–107 | Combined spec. | | | 700 |
U1 | 2008-09-13 | 08:39 | 3600 | 470 |
Ceres | 2006-07-16 | 07:50 | 1800 | 220 |
Ganymede | 2007-04-13 | 09:40 | 300 | 340 |
Moon | 2008-08-09 | 02:39 | 300 | 400 |
$\alpha$ Cen B | Combined spec. | | | 1030 |
_Notes:_ H1 to H3 are individual HARPS spectra, H1–7 is 7 co-
added spectra, and H1–107 is the weighed sum of 107 spectra.
U1 is the UVES spectrum. Ceres, Ganymede and Moon
are solar spectra from HARPS. $\alpha$ Cen B is 25 co-added
HARPS spectra from 2004-05-15.
A preliminary analysis of the UVES spectrum of CoRoT-7 was described in Léger,
Rouan, Schneider et al. (2009). This spectrum has a lower resolution
($R=65000$) than HARPS. We include our updated analysis here for completion.
To calibrate our methods, we analysed three HARPS spectra of the Sun,
available from the ESO/HARPS intrumentation website111URL:
http://www.eso.org/sci/facilities/lasilla/instruments/harps/inst/
monitoring/sun.html. The spectra were acquired by observing Ceres, Ganymede
and the Moon, and have S/N around 250, 350 and 450, respectively. We note that
the “Moon” solar spectrum was observed in the high-efficiency EGGS mode which
has resolution $R=80000$. In addition, we analysed a co-added HARPS spectrum
of $\alpha$ Cen B, which has similar parameters to CoRoT-7. $\alpha$ Cen B has
been studied using direct, model-independent methods (interferometry, binary
orbit) and therefore its absolute parameters (mass, radius, luminosity and
$T_{\rm eff}$) are known with high accuracy (Porto de Mello et al. 2008). We
will use this to evaluate our indirect methods that rely on the validity of
the spectral analysis using 1D LTE atmospheres.
## 3 Versatile Wavelength Analysis (VWA)
We used the VWA package (Bruntt et al. 2004, 2008; Bruntt 2009) to analyse the
spectra listed in Table 1. It can perform several tasks from normalisation of
the spectrum, selection of isolated lines for abundance analysis, iterative
fitting of atmospheric parameters, and determination of the projected
rotational velocity ($v\sin i$). The basic tools of VWA have been described in
previous work (Bruntt et al. 2002) and here we shall give a more complete
description of some additional tools in relation to the results we determine
for CoRoT-7.
### 3.1 Normalisation of the spectra
In Fig. 1 we illustrate the principles of the rainbow program that we used to
normalise the spectra. The top panels show a wide wavelength range in a single
order and the bottom panels show a zoom near the edge of the same echelle
order. The user must manually identify continuum points by comparing the
observed spectrum with a template, which is usually a synthetic spectrum
calculated with the same approximate parameters as the star. The top panel in
Fig. 1 shows the spectrum before normalization where eight continuum points
have been identified and marked by circles. A spline function – optionally a
low-order polynomial – is fitted through these points and shown as the long-
dashed line. The spectrum from the adjacent echelle order is shown with the
short-dashed line. The lower panel shows the normalised spectrum along with
the template spectrum. The agreement with the adjacent order is very good and
there is acceptable agreement with the template. The overlapping orders are
finally merged to improve the S/N by up to 40%.
When the continuum points have been marked for all orders the normalised
spectrum is saved. When the first spectrum has been normalised the continuum
points are re-used for the other spectra. We then carefully check the
normalisation in each case since several readjustments are needed, especially
in the blue end of the spectrum.
The high S/N in the spectrum shown in Fig. 1 would indicate that the continuum
is determined to better than 0.5%. This is only true if the adopted synthetic
template spectrum is identical to that of the star, i.e. the atomic line list
is complete and the temperature and pressure structure of the atmosphere model
represents the real star. From comparison of the template and normalised
spectra in several regions (an example is given in Fig. 1), we estimate that
the continuum is correct to about 0.5%, while discrepancies of 1–2% may occur
in regions where the degree of blending is high and in the region of the wide
Balmer lines and the Mg i b lines.
Figure 2: Abundances of Fe i and Fe ii are shown as open and solid red circles, respectively, and plotted versus equivalent width and excitation potential (plot from the vwares program). The abundances are from the analysis of the H1-7 spectrum for four different sets of atmospheric models. The top panel is for the preferred model, the second panel is for a lower $T_{\rm eff}$, the third panel for lower $\log g$, and the bottom panel is for higher $v_{\rm micro}$. Also indicated is the solar Fe abundances (thin horizontal line) and a linear fit with 95% confidence limits indicated by the solid and dashed lines. Table 2: Determined atmospheric parameters for CoRoT-7, the Sun and $\alpha$ Cen B. | $\langle$Fe i-ii$\rangle$ | $\langle$Mg 1b$\rangle$ | $\langle$Ca6122$\rangle$ | $\langle$Ca6162$\rangle$ | $\langle$Ca6439$\rangle$ | $\langle$Isol. lines$\rangle$
---|---|---|---|---|---|---
Spec. | $T_{\rm eff}$ [K] | $\log g$ | [Fe/H] | $v_{\rm micro}$ [km s-1] | $\log g$ | $\log g$ | $\log g$ | $\log g$ | $v\sin i$ | $v_{\rm macro}$
H1 | $5180\pm 67$ | $4.30\pm 0.08$ | $+0.11\pm 0.13$ | $0.98\pm 0.07$ | $3.89\pm 0.46$ | $4.44\pm 0.08$ | $4.43\pm 0.10$ | $4.34\pm 0.19$ | 1.3 | 1.2
H1 SME | $5280\pm 44$ | $4.44\pm 0.06$ | $+0.13\pm 0.06$ | $0.80\pm 0.07$ | $4.62$ | $4.66$ | $4.53$ | | |
H2 | $5300\pm 41$ | $4.46\pm 0.08$ | $+0.14\pm 0.13$ | $0.80\pm 0.09$ | $4.02\pm 0.47$ | $4.43\pm 0.05$ | $4.56\pm 0.06$ | $4.13\pm 0.35$ | 1.1 | 1.2
H3 | $5350\pm 31$ | $4.57\pm 0.06$ | $+0.15\pm 0.09$ | $0.76\pm 0.07$ | $4.14\pm 0.44$ | $4.50\pm 0.09$ | $4.77\pm 0.07$ | $4.72\pm 0.26$ | 1.1 | 1.2
H1-7 | $5280\pm 35$ | $4.48\pm 0.06$ | $+0.13\pm 0.09$ | $0.80\pm 0.05$ | $3.94\pm 0.52$ | $4.47\pm 0.06$ | $4.60\pm 0.08$ | $4.32\pm 0.27$ | 0.9 | 1.4
H1-107 | $5300\pm 25$ | $4.54\pm 0.05$ | $+0.13\pm 0.08$ | $0.77\pm 0.05$ | $3.91\pm 0.55$ | $4.51\pm 0.06$ | $4.58\pm 0.05$ | $4.45\pm 0.18$ | |
H1-107 SME | $5290\pm 44$ | $4.49\pm 0.06$ | $+0.13\pm 0.06$ | $0.80\pm 0.05$ | $4.43$ | $4.49$ | $4.49$ | | |
U1 | $5300\pm 17$ | $4.50\pm 0.03$ | $+0.11\pm 0.06$ | $0.70\pm 0.08$ | $3.94\pm 0.49$ | $4.42\pm 0.05$ | $4.46\pm 0.06$ | $4.41\pm 0.19$ | |
Ceres | $5767\pm 17$ | $4.44\pm 0.03$ | $-0.01\pm 0.03$ | $1.01\pm 0.03$ | $4.50\pm 0.08$ | $4.46\pm 0.20$ | $4.43\pm 0.10$ | $4.43\pm 0.42$ | 1.4 | 2.1
Ganymede | $5757\pm 17$ | $4.43\pm 0.04$ | $-0.00\pm 0.04$ | $0.93\pm 0.03$ | $4.51\pm 0.10$ | $4.33\pm 0.17$ | $4.38\pm 0.08$ | $4.47\pm 0.37$ | 1.1 | 2.3
Moon | $5775\pm 25$ | $4.48\pm 0.03$ | $+0.02\pm 0.04$ | $0.91\pm 0.04$ | $4.41\pm 0.08$ | $4.53\pm 0.20$ | $4.41\pm 0.14$ | $4.50\pm 0.33$ | 2.1 | 2.2
$\alpha$ Cen B | $5185\pm 25$ | $4.50\pm 0.03$ | $+0.31\pm 0.05$ | $0.83\pm 0.04$ | $4.01\pm 0.50$ | $4.53\pm 0.07$ | $4.51\pm 0.05$ | $4.65\pm 0.16$ | 1.0 | 0.8
_Notes:_ Results are from VWA except H1 and H1-107 which are also given for
the SME analysis. The 1-$\sigma$ uncertainties are internal errors.
### 3.2 Determination of $T_{\rm eff}$ and $\log g$ from Fe i-ii lines
This part of the VWA program has been described in some detail by Bruntt et
al. (2002) and we will specify some updated details here. VWA uses 1D LTE
atmosphere models interpolated in grids from MARCS (Gustafsson et al. 2008) or
modified ATLAS9 models (Heiter et al. 2002). We have adopted MARCS models for
this study. The atomic line data are extracted from VALD (Kupka et al. 1999),
which is a collection from many different sources. The synthetic profiles are
computed with synth (Valenti & Piskunov 1996). The VWA abundances are measured
differentially with respect to a solar spectrum. We have used the FTS spectrum
by Kurucz et al. (1984) which was published in electronic form by Hinkle et
al. (2000). We found that making a differential abundance analysis
significantly improves the precision on the determined $T_{\rm eff}$ and $\log
g$ (see Bruntt et al. 2008). We assess the question of accuracy in Sects. 3.4
and 6.1.
VWA consists of three main programs written in IDL. Each program has a
graphical user interface, called vwaview, vwaexam and vwares. In vwaview the
user can inspect the observed spectrum and select a set of isolated lines.
They are fitted iteratively by computing synthetic profiles and adjusting the
abundance until the observed and synthetic profiles have identical equivalent
widths within a fixed wavelength range. The program vwaexam is used to inspect
how well the synthetic profiles fit the observed lines. The user can manually
reject lines or base the rejection on objective criteria like reduced
$\chi^{2}$ values and the relative line depths. It takes about one hour to fit
500 lines on a modern laptop. This is done using the program vwatask for fixed
values of $T_{\rm eff}$, $\log g$ and $v_{\rm micro}$. The process is then
repeated with various values of these parameters to measure the sensitivity of
each line. The user can then investigate this sensitivity in the program
vwares.
In Fig. 2 we show an example from vwares using the H1–7 spectrum of CoRoT-7.
The abundances from Fe i and Fe ii lines are plotted versus equivalent width
(EW; left panels) and excitation potential (EP; right panels) for four
different sets of atmospheric parameters. Open and solid red symbols are used
for neutral and ionised Fe lines, respectively. The top panels are for the
preferred parameters where we have minimised the correlation of Fe i with both
EW and EP and the mean abundances of Fe i and Fe ii agree. The second panel is
for $T_{\rm eff}$ decreased by 300 K, resulting in a clear correlation with
EP, and Fe i and Fe ii are no longer in agreement. For the third panel, $\log
g$ was decreased by 0.3 dex, leading to a low mean abundance of Fe ii. The
bottom panel is for microturbulence increased by 0.4 km s-1 which leads to
correlations of Fe i with both EW and EP. From such analyses we can determine
the “internal” uncertainty on the atmospheric parameters (see Bruntt et al.
2008 for a discussion).
In Fig. 3 we show an example of the abundances of six elements determined for
the H1–107 spectrum. The mean abundance and rms error is given in the right
panels. While Fe has the most lines, Ti, Cr and Ni also show no strong
correlation with equivalent with or excitation potential.
The atmospheric parameters for the six spectra of CoRoT-7 are summarised in
Table 2. The applied method is indicated in angled brackets in the first row.
There is good agreement between the results, although the H1 spectrum gives a
systematically lower $T_{\rm eff}$ and $\log g$. This is due to the
correlation between the two parameters as was also noted by Bruntt (2009).
They proposed that this degeneracy could be a problem for spectra with
relatively low S/N (H1 has ${\rm S/N}=57$). For our final value of $T_{\rm
eff}$ and $\log g$ of CoRoT-7 we adopt the weighted mean value of the analysis
of the three composite spectra: H1-7, H1-107 and U1: $T_{\rm eff}=5297\pm 13$
K, $\log g=4.51\pm 0.02$, $v_{\rm micro}=0.77\pm 0.03$ km s-1. The
uncertainties stated here are internal errors. We will assess the question of
“realistic uncertainties” in Sect. 6.
Figure 3: Abundances determined from the H-107 spectrum for six elements
plotted versus equivalent width and excitation potential. Solid and open
symbols are used for neutral and ionised Fe lines, respectively. There is no
correlation of the abundances and the line parameters, indicating that the
atmospheric model parameters are correct.
Figure 4: The sensitivity of synthetic Ca lines fitted to the observed spectra of the Sun (top panel; Ganymede spectrum) and CoRoT-7 (bottom panel; H1-7 spectrum). The synthetic profiles computed for the Sun have $\log g=4.04,4.44$ and $4.84$. For the CoRoT-7 spectrum synthetic profiles have $\log g=4.08$ and $4.48$. In each case a higher $\log g$ means the synthetic line becomes wider. The rectangles mark the areas used to compute the reduced $\chi^{2}$ and the hatched regions are used to normalise the spectrum. Table 3: The adjusted Van der Waals constants compared to the values from VALD. | | $\log\gamma$ [rad ${\rm cm}^{3}$/s]
---|---|---
Line | $\lambda$ [Å] | Adjusted | VALD
Mg i b | 5167.321 | $-7.42$ | $-7.267$
| 5172.684 | $-7.42$ | $-7.267$
| 5183.604 | $-7.42$ | $-7.267$
Na i D | 5889.951 | $-7.85$ | $-7.526$
| 5895.924 | $-7.85$ | $-7.527$
Ca i | 6122.217 | $-7.27$ | $-7.189$
| 6162.173 | $-7.27$ | $-7.189$
Ca i | 6439.075 | $-7.84$ | $-7.569$
### 3.3 Determination of $\log g$ from wide lines
The surface gravity of late-type stars can be determined from the Mg i b
lines, the Na i D and the Ca lines at $\lambda 6122$, $\lambda 6162$ and
$\lambda 6439$ Å. For Mg i b we used only the line at $\lambda$5184 Å because
the two lines around $\lambda$5170 Å are too blended. We followed the approach
of Fuhrmann et al. (1997) to adjust the van der Waals constants (pressure
broadening due to Hydrogen collisions) by requiring that our reference
spectrum of the Sun (Hinkle et al. 2000) produces the solar value $\log
g=4.437$. In Table 3 we list the adjusted van der Waals parameters along with
the values extracted from VALD (from Barklem et al. 2000). Following the
convention of VALD it is expressed as the logarithm (base 10) of the full-
width half-maximum per perturber number density at 10 000 K. The abundances of
the fitted lines are determined from weak lines with ${\rm EW}<120$ mÅ. The
broadening due to $v\sin i$ and $v_{\rm macro}$ is determined as described in
Sect. 3.6. Examples of fitting the $\lambda 6122$ Å and $\lambda 6162$ Å lines
are shown in Fig. 4 for the Sun (top panel) and CoRoT-7 (bottom panel). The
hatched regions are used to renormalise the spectrum by a linear fit. The
rectangles mark regions where reduced $\chi^{2}$ values are computed and they
are used to determine the best value of $\log g$ and the 1-$\sigma$
uncertainty.
We found that the Mg i b line in CoRoT-7 is not very sensitive and gave lower
values ($\log g\approx 4.0\pm 0.5$) than the Ca lines ($\log g\approx 4.5\pm
0.1$). The reason may be the high degree of blending with weaker lines for
such a late type star. Since the higher value of $\log g$ is in good agreement
with the result using Fe i-ii we neglect the results for the Mg i b lines.
There is good agreement for the $\log g$ from the individual spectra. For the
value of $\log g$ we adopt the weighted mean of the three composite spectra:
$\log g=4.50\pm 0.02$. The stated error does not include systematic errors,
see Sect. 3.4 and 6.1.
### 3.4 Results for the Sun and $\alpha$ Cen B
It is important to validate that the employed spectroscopic methods produce
trustworthy results. We therefore analysed two fundamental stars for which
$T_{\rm eff}$ and $\log g$ are known with very high accuracy: the Sun and
$\alpha$ Cen B. We analysed three single HARPS spectra of the Sun and one co-
added spectrum of $\alpha$ Cen B. The results are summarised in Table 2.
The parameters from the three solar spectra agree very well with the solar
values. The canonical value for $T_{\rm eff}$ is 5777 K (Cox 2000) and $\log
g$ calculated from the Solar mass and radius is $4.437$. The largest deviation
is 20 K for $T_{\rm eff}$ based on the analysis of Fe i-ii lines. The surface
gravity is constrained by several methods (Fe i-ii, Mg i, Ca lines) but the
largest deviation from the canonical value is only 0.1 dex. From Table 2 it is
seen that some lines are less useful for constraining $\log g$: Ca $\lambda
6439$ Å is the least sensitive line. For the weighted average, using the Mg
and three Ca lines, we find excellent agreement for the three Solar HARPS
spectra: $\log g=4.47\pm 0.06$, $4.42\pm 0.06$, and $4.43\pm 0.06$.
For $\alpha$ Cen B we find $T_{\rm eff}=5185\pm 25$ K, $\log g=4.50\pm 0.03$,
and ${\rm[Fe/H]}=+0.31\pm 0.05$ (the quoted uncertainties do not include
systematic errors). For this nearby binary star, $T_{\rm eff}$ and $\log g$
can be determined by direct methods, i.e. methods are only weakly dependent on
models. The angular diameter has been measured by Kervella et al. (2003).
Using the updated parallax from van Leeuwen (2007) we determine the radius
$R=0.864\pm 0.005\,{\rm R}_{\odot}$. The mass has been determined from the
binary orbit by Pourbaix et al. (2002): $M=0.934\pm 0.006\,{\rm M}_{\odot}$.
Coincidentally, this mass is nearly identical to that of CoRoT-7 (Léger,
Rouan, Schneider et al. 2009). Combing the mass and radius ($g\propto
M/R\,^{2}$) gives a very accurate value of the surface gravity for $\alpha$
Cen B: $\log g=4.538\pm 0.008$. This is in very good agreement with our
spectroscopic determination. We note that as for CoRoT-7, Mg i b is not useful
for constraining $\log g$. The $T_{\rm eff}$ can be determined from the
angular diameter and the bolometric flux: $T_{\rm eff}=5140\pm 56$ (Bruntt et
al. 2010). This is in excellent agreement with the result from VWA. Porto de
Mello et al. (2008) listed the results of 14 different analyses of $\alpha$
Cen B, based on different methods and quality of the data. Our value of
$T_{\rm eff}$ is in good agreement with previous determinations but our
metallicity is slightly higher than most previous estimates.
To conclude, our analysis of the spectra of the Sun and $\alpha$ Cen B show
that we can reliably determine $T_{\rm eff}$ and $\log g$. Since these two
stars bracket CoRoT-7 in terms of spectral type, we have confidence that the
spectroscopic results are robust and do not suffer from significant systematic
errors. We will discuss the uncertainties on the spectroscopic parameters in
Sect. 6.1.
Figure 5: Mean abundances of $20$ elements in CoRoT-7 determined from the H1-107 spectrum. Circle and box symbols are used for neutral and singly ionised lines, respectively. The horizontal bar indicates the mean metallicity and the $1$-$\sigma$ error range, $[{\rm M/H}]=0.12\pm 0.04$. The horizontal line at $0.0$ corresponds to the solar abundance. Table 4: Abundances relative to the Sun for $20$ elements in CoRoT-7. Also given is the number of lines used for each element. C i | $+0.06$ | 1 | Mn i | $+0.16$ | 2
---|---|---|---|---|---
Na i | $+0.11$ | 1 | Fe i | $+0.13\pm 0.04$ | 143
Mg i | $+0.13$ | 1 | Fe ii | $+0.13\pm 0.04$ | 16
Al i | $+0.12$ | 2 | Co i | $+0.10\pm 0.05$ | 6
Si i | $+0.15\pm 0.04$ | 6 | Ni i | $+0.12\pm 0.04$ | 40
Ca i | $+0.15\pm 0.04$ | 7 | Cu i | $+0.14$ | 1
Sc i | $+0.10$ | 1 | Zn i | $+0.10$ | 1
Sc ii | $+0.06\pm 0.05$ | 3 | Sr i | $+0.32$ | 1
Ti i | $+0.11\pm 0.04$ | 37 | Y ii | $+0.11\pm 0.09$ | 3
Ti ii | $+0.09\pm 0.04$ | 8 | Zr i | $-0.00$ | 2
V i | $+0.15\pm 0.05$ | 3 | Ba ii | $+0.24$ | 1
Cr i | $+0.09\pm 0.04$ | 8 | | |
Cr ii | $+0.10\pm 0.04$ | 2 | | |
### 3.5 The chemical composition of CoRoT-7
The abundance pattern of CoRoT-7 relative to the Sun is shown in Fig. 5 for
the H1-107 spectrum and in Table 4 we list the individual abundances of 20
elements. We adopted this spectrum since it has the highest S/N but we note
that the other spectra give very similar results. The mean metallicity is
computed from the mean of the metal abundances for species with at least 30
lines in the spectrum: Ti, Fe, and Ni. The mean value is ${\rm[M/H]}=+0.12\pm
0.04$ where the uncertainty includes the uncertainty on $T_{\rm eff}$, $\log
g$ and $v_{\rm micro}$. The horizontal bar in Fig. 5 marks the mean value and
the 1-$\sigma$ uncertainty range. It can be seen that all elements agree with
a scaling of $+0.12$ dex relative to the solar abundance. For elements with
few lines available ($n<3$) we assume an uncertainty of 0.1 dex.
Figure 6: Contours showing the reduced $\chi^{2}$ values computed for four
lines. The synthetic profiles have been convolved with different combinations
of $v\sin i$ and $v_{\rm macro}$. The minimum of the surface is marked by a
circle.
### 3.6 Determination of $v\sin i$ and $v_{\rm macro}$
From the detailed profile shapes of isolated lines one can ultimately extract
information about the granulation velocity fields (Dravins 2008). However,
this is not possible with our data where each single spectrum only has ${\rm
S/N}\approx 60$. The intrinsic shape of a spectral line is determined by
several factors (Gray 2008) but the broadening due to stellar rotation and
velocity fields in the atmosphere can to a good approximation be described by
two parameters: $v\sin i$ and macroturbulence ($v_{\rm macro}$). These two
parameters describe the projected velocity field due to rotation of a limb-
darkened sphere and the movement of granules due to convection, respectively.
To measure $v\sin i$ and $v_{\rm macro}$ we selected 64 isolated lines of
different metal species: Ni, Ca, Ti, Cr, and Fe. The lines lie in the range
5800–6450 Å with equivalent widths from 25–125 mÅ. For each line we determine
the small wavelength shifts needed so the observed line core fits the
synthetic spectrum. This was done by fitting a Gaussian to the line cores of
the observed and synthetic spectra. We then fitted the abundance of the line
for the adopted $T_{\rm eff}$, $\log g$ and $v_{\rm micro}$. We made a grid of
values for $v\sin i$ and $v_{\rm macro}$ from 0–6 km s-1 with steps of 0.15 km
s-1. For each grid point we convolved the synthetic spectrum and computed the
reduced $\chi^{2}$ of the fit to the observed line. In Fig. 6 we show examples
of the $\chi^{2}$ contours for four fitted lines. The circles mark the minimum
of the contour. The generally low reduced $\chi^{2}$ values indicate that our
simple representation of the line broadening is successful. It can be seen
that there is a strong correlation between the two parameters. The typical
$v_{\rm macro}$ value for a G9V star is about 1–2 km s-1 (Gray 2008). For this
range the $v\sin i$ values for $\chi^{2}<2$ is below 2.5 km s-1 for nearly all
lines. From this analysis we find mean values of $v\sin i=1.1^{+1.0}_{-0.5}$
km s-1 and $v_{\rm macro}=1.2^{+1.0}_{-0.5}$ km s-1. From the analysis of the
contours, as shown in Fig. 6, we can place a firm upper limit of $v\sin i<3$
km s-1.
From the transit light curve, Léger, Rouan, Schneider et al. (2009)
constrained the inclination angle to be $i=80.1\pm 0.3^{\circ}$ (see their
Fig. 19). Thus, the equatorial rotational velocity is $v_{\rm rot}\approx
v\sin i=1.1^{+1.0}_{-0.5}$ km s-1. This result is only valid if we assume that
the inclination of the rotation axis of the star is the same as the
inclination of the orbit. Léger, Rouan, Schneider et al. (2009) proposed that
the rotation period is 23 days222We adopt an uncertainty on the rotation
period of 2 days., based on the variation of the CoRoT light curve. Using the
radius determined in Sect. 6 we get $v_{\rm rot}$$=1.7\pm 0.2$ km s-1, in
agreement with value determined from the spectroscopy.
In Table 2 we list the mean values of $v\sin i$ and $v_{\rm macro}$ that we
have determined for several of the spectra. We did not use the U1 spectrum
since it has a lower resolution than the HARPS spectra. We also did not
consider the H1-107 spectrum since it is a combination of so many spectra,
which inevitably leads to less well-defined line shapes.
## 4 Spectroscopy Made Easy (SME)
Figure 7: The emission component of the Ca ii H & K line of CoRoT-7. The self-
reversal in the emission cores is shown in the insets.
In an independent analysis of the H1 and H1-107 spectra, we used the SME
package (version 2.1; Valenti & Piskunov 1996; Valenti & Fischer 2005). This
code uses a grid of stellar models (Kurucz models or MARCS models) to
iteratively determine the fundamental stellar parameters. This is done by
fitting the observed spectrum to a synthesised spectrum and minimizing the
discrepancies through a non-linear least-squares algorithm. SME is based on
the philosophy (Valenti & Piskunov 1996) that by matching synthetic spectra to
observed line profiles one can extract the information in the observed
spectrum and search among stellar and atomic parameters until the best fit is
achieved.
We use a large number of spectral lines, e.g. the Balmer lines (the extended
wings are used to constrain $T_{\rm eff}$), Na i D, Mg i b and Ca i (for
$T_{\rm eff}$ and $\log g$) and metal lines (to constrain the abundances).
Furthermore, the iterative fitting provides information on micro- and
macroturbulence and $v\sin i$.
By fitting the extended wings of the H$\alpha$ and H$\beta$ Balmer lines, we
determine the $T_{\rm eff}$ to be 5200 K and 5100 K respectively. Using
instead the Na i doublet at $\lambda$5890Å, we find a $T_{\rm eff}$ of 5280 K.
The lower value derived from the H$\beta$ line wings is explained by the many
metal lines contributing to the profile. We tried to use the Mg i b triplet to
evaluate $\log g$ but as for the VWA analysis we found that it is difficult to
assign the continuum level, so instead we used the wide Ca i lines. From the
SME analysis we find the $\log g$ to be 4.43 from Mg i and 4.49 from Ca i. Our
evaluation of the metallicity gives ${\rm[M/H]}=+0.13$ and $v_{\rm
micro}=0.80$ km s-1.
The uncertainties using SME, as found by Valenti & Fischer (2005), and based
on a large sample (more than 1000 stars) are 44 K in $T_{\rm eff}$, 0.06 dex
in $\log g$ and 0.03 dex in [M/H], which we adopt for our SME analysis of
CoRoT-7. In a few cases, Valenti & Fischer (2005) found offsets of up to 0.3
dex for $\log g$. When we compare the results for CoRoT-7 for different lines
and methods used to constrain $\log g$, we find a scatter of 0.06 dex. This is
consistent with the results of Valenti & Fischer (2005) and we assign this as
the $1$-$\sigma$ uncertainty.
In summary, the parameters determined with SME for the H1 and H1-107 spectra
of CoRoT-7 give fully consistent results with the more extensive analysis with
VWA. Our results from the SME analysis are given in Table 2.
## 5 Absolute magnitude from the Wilson-Bappu effect
The width of the emission peaks seen in the core of the Ca ii H & K lines
(3934.8 and 3969.7 Å) in late-type stars are directly correlated to the value
of $\log g$, and thus to the mass and radius of the star. This implies that
the width can be calibrated in terms of the absolute luminosity (Gray 2008).
The calibration of the absolute magnitude is of the form: $M_{V}=a\,\log
W_{0}+b$, where $W_{0}$ is the width at the zero-level of the emission
component, and where also the constants $a$ and $b$ need to be properly
calibrated. This is usually done using data from clusters, and we have used
the recent calibration of Pace et al. (2003) who found $a=-18.0$ and $b=33.2$,
with a quoted uncertainty of 0.6 mag on $M_{V}$.
In Fig. 7, we show the Ca ii H & K lines of CoRoT-7. The emission components
with self-reversal in the line cores are clearly seen. By measuring the width
of both the H- and the K-line, following the method of Pace et al. (2003), we
find an absolute magnitude of $M_{V}=5.4\pm 0.6$. Given the spectroscopic
effective temperature, the location in the Hertzsprung-Russell diagram
indicates that CoRoT-7 is a main sequence star with spectral type in the range
is G8V – K0V. That the star is not evolved is in good agreement with the $\log
g$ determination.
Table 5: Parameters of CoRoT-7. Parameter | Value | Unit | Method
---|---|---|---
$T_{\rm eff}$ | $5250$ | $\pm 60$ | K | Spectroscopy
$\log g$ | $4.47$ | $\pm 0.05$ | | Spectroscopy
[Fe/H] | $+0.12$ | $\pm 0.06$ | | Spectroscopy
$L/M$ | $0.62$ | $\pm 0.08$ | ${\rm L}_{\odot}/{\rm M}_{\odot}$ | Spectr.: $L/M\propto T_{\rm eff}^{4}/g$
$M$ | $0.91$ | $\pm 0.03$ | M⊙ | Isochrone/tracks
$R$ | $0.82$ | $\pm 0.04$ | R⊙ | Isochrone/tracks
$L$ | $0.49$ | $\pm 0.07$ | L⊙ | Isochrone/tracks
$\log g$ | $4.57$ | $\pm 0.05$ | | Isochrone/tracks
_Notes:_ The mass, luminosity and radius are determined
from comparison with evolution models and rely on the
age limit of $A<2.3$ Gyr from Léger, Rouan, Schneider et al. (2009).
## 6 Evolutionary status
We will now evaluate the atmospheric parameters determined above for CoRoT-7
and compare with evolutionary models to constrain the mass, radius and
luminosity.
### 6.1 Final atmospheric parameters of CoRoT-7
There is generally good agreement for the determination of $T_{\rm eff}$ using
VWA and SME. With the VWA method we only used Fe i-ii lines while with SME we
also used the Balmer lines to constrain $T_{\rm eff}$. As mentioned, the
quoted uncertainties in Table 2 only include the intrinsic error of the
method, i.e. by varying the model parameters. However, the temperature and
pressure profile of the atmospheric model may not fully represent the actual
star. From the analysis of the Sun and $\alpha$ Cen B, we found good agreement
for their $T_{\rm eff}$ and $\log g$ determined from model-independent methods
(see Sect. 3.4). Thus, there appears to be no large systematic errors. Bruntt
et al. (2010) analysed a larger sample of stars, comparing the spectroscopic
$T_{\rm eff}$s with those from fundamental methods (as done for $\alpha$ Cen B
here) and found a systematic offset in $T_{\rm eff}$ of $-40\pm 20$ K. We have
included this offset to get the final value $T_{\rm eff}=5250\pm 60$ K. We
used several pressure sensitive spectral features to constrain $\log g$ and
the mean value we adopt is $\log g=4.47\pm 0.05$. For $T_{\rm eff}$ and $\log
g$ we have added systematic errors on 50 K and 0.05 dex, based on the
discussion by Bruntt et al. (2010). The mean metallicity is found to be $[{\rm
M/H}]=+0.12\pm 0.06$ where we have increased the uncertainty due to the
inclusion of systematic errors on $T_{\rm eff}$ and $\log g$. These are our
final estimates for the parameters of CoRoT-7 and they are summarised in Table
5.
Our new results for the fundamental parameters are in good agreement with
Léger, Rouan, Schneider et al. (2009). They found $T_{\rm eff}=5275\pm 75$ K
as a mean value of different groups using different spectroscopic analyses of
the UVES spectrum. They also used a calibration using 2MASS infrared
photometry, taking into account interstellar reddening, yielding $5300\pm 70$
K. They find $\log g=4.50\pm 0.10$ using the Fe i-ii equilibrium criterion and
the Mg i b and Na i D lines, which is also in good agreement with our value.
Léger, Rouan, Schneider et al. (2009) found a slightly lower metallicity,
$[{\rm M/H}]=+0.03\pm 0.06$ (our revised value for the same spectrum is $[{\rm
M/H}]=+0.11\pm 0.06$). In that analysis several strong lines were included,
while in this study we only used Fe i lines with ${\rm EW}<90$ mÅ. For other
elements (and Fe ii) we included lines with ${\rm EW}<140$ mÅ. This choice was
made because the strong lines start to be saturated and are therefore less
sensitive to changes in the atmospheric parameters. For comparison 250 Fe i
and 18 Fe ii lines were used by Léger, Rouan, Schneider et al. (2009) while we
used only 143 and 16, respectively. In our analysis we used Fe lines in the
wavelength range 4880–6865 Å, while Léger, Rouan, Schneider et al. (2009)
included several lines in the blue region (4515–6865 Å). We note that the
current version of VWA does not take into account molecular lines, which start
to become a problem for such a cool star, especially at short wavelengths.
Figure 8: BASTI isochrones with different ages and metallicities are shown,
and filled circles and boxes mark selected mass points. The determined $L/M$
ratios for CoRoT-7, $\alpha$ Cen B, and the Sun are plotted as open symbols.
Figure 9: Four ASTEC evolution tracks are shown for different mass and
metallicity, e.g. a track for $1.00\,{\rm M}_{\odot}$ and ${\rm[Fe/H]}=-0.02$
is shown near the Sun. The determined $L/M$ ratios for CoRoT-7, $\alpha$ Cen
B, and the Sun are plotted as open symbols. Dashed lines are used for ages
higher than the adopted limits on the age, i.e. 4.6 Gyr for the Sun, 2.3 Gyr
for CoRoT-7 and 6.5 Gyr for $\alpha$ Cen B, while the maximum possible age is
14 Gyr.
### 6.2 Stellar mass, luminosity and radius
In some cases the modelling of the transit light curve can be used to obtain
the mean density of the star. However, as pointed out by Léger, Rouan,
Schneider et al. (2009), the shallow eclipse combined with stellar activity
modulating the light curve seriously hampers such analyses. From the
spectroscopic value of $\log g$ we have an estimate of $g=GM/R^{2}$.
Multiplying this with the relation $L\propto R^{2}T_{\rm eff}^{4}$ we can
eliminate the radius, i.e. $L/M\propto T_{\rm eff}^{4}/g$. Thus, we determine
the luminosity-mass ratio: $(L/{\rm L}_{\odot})/(M/{\rm M}_{\odot})=0.62\pm
0.08$. The uncertainty is dominated by the uncertainty on the surface gravity.
In Figs. 8 and 9 we compare this estimate with isochrones from BASTI
(Pietrinferni et al. 2004) and evolution tracks from ASTEC (Christensen-
Dalsgaard 2008). These models do not include overshoot but this has no impact
on low-mass stars such as CoRoT-7. The mixing-length parameter for the ASTEC
grid was $\alpha_{\rm ML}=1.8$. The models express metallicity in terms of the
heavy element mass fraction, $Z$. To convert each $Z$ to spectroscopic values,
we adopted the solar value $Z_{\odot}=0.0156$ (Caffau et al. 2009) with an
assumed uncertainty of $0.002$. This corresponds to an increase in the
uncertainty of [Fe/H] by 0.05 dex.
In Fig. 8 we show two sets of isochrones with metallicity ${\rm[Fe/H]}=+0.10$
and $+0.28$ for ages of 2 and 7 Gyr, with several mass points indicated in the
range 0.8 to 1.1 ${\rm M}_{\odot}$. The lower metallicity is close to that of
CoRoT-7 and the higher metallicity represents $\alpha$ Cen B. The uncertainty
on $L/M$ for CoRoT-7 is relatively large, so we cannot constrain the mass
without further constraints. Fortunately, Léger, Rouan, Schneider et al.
(2009) estimated the age of CoRoT-7 from the rotation period and the activity
index of the Ca H & K lines: 1.2–2.3 Gyr. Adopting this age limit, we can
estimate the mass and radius from the isochrones: $M/{\rm M}_{\odot}=0.89\pm
0.03$ and $R/{\rm R}_{\odot}=0.80\pm 0.04$.
In Fig. 9 we show four selected ASTEC evolution tracks which represent the
Sun, CoRoT-7 (two tracks), and $\alpha$ Cen B. The dashed part of each track
is for ages above these adopted limits: 4.6 Gyr for the Sun ($1.00$ M⊙ track),
2.3 Gyr for CoRoT-7 (0.92 and 0.86 M⊙), and 6.5 Gyr for $\alpha$ Cen B (0.94
M⊙; see Miglio & Montalbán 2005 for discussion of the age of $\alpha$ Cen
A$+$B). Furthermore, the tracks all end at 14 Gyr. It is seen that the Sun is
quite well represented, although the $L/M$ ratio is quite high at 4.6 Gyr, but
this is explained by the available track having slightly too low metallicity.
The 0.94 M⊙ track for $\alpha$ Cen B agrees with the $L/M$ ratio within the
1-$\sigma$ limit. For CoRoT-7 the 0.86 M⊙ track does not reach the determined
$T_{\rm eff}$ and $L/M$ ratio in 2.3 Gyr. However, for the 0.92 M⊙ track there
is agreement with the $T_{\rm eff}$ and $L/M$ ratio. From similar tracks we
determine these limits on the mass and radius of CoRoT-7: $M/{\rm
M}_{\odot}=0.92\pm 0.03$ and $R/{\rm R}_{\odot}=0.83\pm 0.04$. This is in good
agreement with the result from the BASTI tracks. As our final result, we adopt
the mean radius and mass determined from the two sets of models: $M/{\rm
M}_{\odot}=0.91\pm 0.03$ and $R/{\rm R}_{\odot}=0.82\pm 0.04$.
Léger, Rouan, Schneider et al. (2009) determined slightly different values for
the mass and radius. They used the STAREVOL evolution tracks (Palacios,
private comm.) with slightly different stellar atmospheric values. Their
values are $M/{\rm M}_{\odot}=0.93\pm 0.03$ and $R/{\rm R}_{\odot}=0.87\pm
0.04$ (i.e. $\log g=4.53\pm 0.04$), which agree quite well with our revised
results given in Table 5.
From comparison with the BASTI and ASTEC models the determined $L/M$ ratio of
CoRoT-7 seems to be too large, although the uncertainty is large. In order to
determine the luminosity we therefore adjust the ratio by $-1\,\sigma$,
$(L/{\rm L}_{\odot})/(M/{\rm M}_{\odot})=0.54\pm 0.08$, and multiply by the
inferred mass to get $L/{\rm L}_{\odot}=0.49\pm 0.07$. The determined mass and
radius from the isochrones correspond to a surface gravity $\log g=4.57\pm
0.04$, which is slightly higher (1.6 $\sigma$) than the spectroscopic value of
$4.47\pm 0.05$.
To validate that the BASTI and ASTEC models can be used for CoRoT-7 we also
plot the Sun and $\alpha$ Cen B in Figs. 8 and 9. For $\alpha$ Cen B the
uncertainty is much smaller than for CoRoT-7: $(L/{\rm L}_{\odot})/(M/{\rm
M}_{\odot})=0.50\pm 0.02$. Miglio & Montalbán (2005) determined an age of
about 6.5 Gyr for the $\alpha$ Cen A$+$B system and with our metallicity of
$+0.3$ dex, there is good agreement with both sets of models. From comparison
with the BASTI and ASTEC models we get the mass $0.90\pm 0.03$ M⊙, which
agrees well with the dynamical mass of $0.934\pm 0.006$ M⊙. The radius is
$0.84\pm 0.04$ R⊙, where the interferometric result is $R=0.864\pm 0.005\,{\rm
R}_{\odot}$. Combining the mass and radius from the comparison with the
isochrones we get $\log g=4.54\pm 0.04$, which is in good agreement with the
spectroscopic value of $4.50\pm 0.03$.
## 7 Discussion and conclusion
We have presented a detailed spectroscopic analysis of the planet-hosting star
CoRoT-7. The analysis is based on HARPS spectra which have higher signal-to-
noise and better resolution than be UVES spectrum used to get a preliminary
result (Léger, Rouan, Schneider et al. 2009). We analysed both individual
spectra from different nights and co-added spectra and found excellent
agreement. Only for one of the single HARPS spectra did we find a systematic
error in $T_{\rm eff}$ and $\log g$, which is explained by the low S/N.
We described in detail the VWA tool which is used for determination of the
atmospheric parameters $T_{\rm eff}$ and $\log g$ using Fe i-ii lines and the
pressure sensitive Mg i b and Ca lines. We used the SME tool to analyse in
addition the Balmer and Na i lines. We find excellent agreement between the
different methods.
To evaluate the evolutionary status (age) and fundamental stellar parameters
(mass, radius) we compared the observed properties of CoRoT-7 with theoretical
isochrones. From the spectroscopic $T_{\rm eff}$ and $\log g$ we can estimate
the $L/M$ ratio. We compared this with isochrones but find that the
uncertainty is too large to constrain the evolutionary status. However, by
imposing constraints on the stellar age (1.2–2.3 Gyr from Léger, Rouan,
Schneider et al. 2009) we can constrain the mass and radius to $0.91\pm 0.03$
M⊙ and $0.82\pm 0.04$ R⊙. This is a only slight revision of the original value
from Léger, Rouan, Schneider et al. (2009) who used a lower metallicity. The
relatively large uncertainty of 7% on the stellar radius directly impacts the
accuracy of the determine radius and density of the transiting planet,
CoRoT-7b.
We have used the new stellar parameters to fit the transit light curve
reported by Léger, Rouan, Schneider et al. (2009). We used the formalism of
Giménez (2006) with fixed limb-darkening coefficients, and we explored the
parameter space which is consistent with the stellar parameters and their
associated uncertainty. The constraints to the fit include the orbital
inclination ($81.45\pm 1.10^{\circ}$), the phase of transit ingress
$\theta_{1}(0.02785\pm 0.00005$), and the ratio of planet-to-star radius
($0.0176\pm 0.0003$). We refer to Sect. 9 in Léger, Rouan, Schneider et al.
(2009) for a description of the methodology of the fitting procedure. With the
new stellar parameters we determine the radius of the planet to be slightly
smaller with radius $1.58\pm 0.10$ R⊕ (Léger, Rouan, Schneider et al. 2009
found $1.68\pm 0.09$ R⊕). The slightly smaller radius is mainly due to our
revision of the stellar radius.
The new stellar mass and the updated inclination were used, together with the
published values of the ephemeris (Léger, Rouan, Schneider et al. 2009) and
radial velocity semiamplitude (Queloz et al. 2009) to estimate the mass of the
planet CoRoT-7b as $5.2\pm 0.8\,{\rm M}_{\oplus}$. Combined with the radius of
the planet this results in a density of $7.2\pm 1.8{\rm\,g\,cm}^{-3}$. which
is consistent, but slightly more dense than the reported value of $5.6\pm 1.3$
g cm-3 in the previous work.
We also analysed spectra of the Sun and $\alpha$ Cen B, also observed with the
HARPS spectrograph. For these stars the fundamental parameters are known with
very good accuracy and they can therefore be used to validate the methods we
use for the much fainter star CoRoT-7. We compared the spectroscopically
determined $T_{\rm eff}$ and $\log g$ with the values from fundamental methods
for $\alpha$ Cen B, i.e. using the binary dynamical mass and the
interferometric determination of the radius. There is excellent agreement
within 1-$\sigma$, indicating that the adopted uncertainties are realistic.
This gives us some confidence that we can use theoretical evolution models to
constrain the radii and masses of stars, but requires that limits can be put
on the stellar age.
The exoplanet host star CoRoT-7 is a slowly rotating, metal rich, type G9V
star. The star is relatively faint and its fundamental parameters can only be
determined through indirect methods. The expected future discoveries of
similar planet systems with CoRoT and _Kepler_ will also be limited by our
ability to characterise the host stars. In the case of _Kepler_ we have the
additional advantage that for the brightest stars the solar-like pulsations
can be used to constrain the stellar radius (Christensen-Dalsgaard et al.
2010). This analysis also relies on evolution models but will be able to
constrain the stellar radius to about 2% (Stello et al. 2009; Basu et al.
2010). For most of the _Kepler_ targets astrometric parallaxes will be
available, while for CoRoT-7 we must wait for the _GAIA_ mission.
###### Acknowledgements.
We are thankful to Nikolai Piskunov (Uppsala Astronomical Observatory) for
making SME available to us and for answering numerous questions. We are
grateful for the availability of the VALD database for the atomic parameters
used in this work. Based on observations made with ESO Telescopes at the La
Silla and Paranal Observatories under programme IDs 081.C-0413(C), 082.C-0120,
082.C-0308(A), 282.C-5036(A), and 60.A-9036(A).
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|
arxiv-papers
| 2010-05-18T14:32:52 |
2024-09-04T02:49:10.474763
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. Bruntt, M. Deleuil, M. Fridlund, R. Alonso, F. Bouchy, A. Hatzes,\n M. Mayor, C. Moutou, D. Queloz",
"submitter": "Hans Bruntt",
"url": "https://arxiv.org/abs/1005.3208"
}
|
1005.3247
|
# Influence of Coulomb correlations on the quantum well intersubband
absorption at low temperatures
Thi Uyen-Khanh Dang uyen@itp.physik.tu-berlin.de Carsten Weber Institut für
Theoretische Physik, Nichtlineare Optik und Quantenelektronik, Technische
Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany Marten Richter
Institut für Theoretische Physik, Nichtlineare Optik und Quantenelektronik,
Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
Department of Chemistry, University of California Irvine, Irvine, California
92697, USA Andreas Knorr Institut für Theoretische Physik, Nichtlineare
Optik und Quantenelektronik, Technische Universität Berlin, Hardenbergstr. 36,
10623 Berlin, Germany
###### Abstract
We present a theory for the intersubband absorption including electronic
ground-state correlations in a doped GaAs/Al0.35Ga0.65As quantum well system.
Focusing on the influence of the Coulomb interaction among the carriers at low
temperatures, we find that the ground-state correlations lead to an increased
renormalization and spectral broadening of the absorption spectrum. At $T$ = 1
K, its full width at half maximum is increased by up to a factor 3. The
inclusion of electron-phonon scattering strongly reduces the relative impact
of the electronic correlations.
###### pacs:
73.21.Fg,78.67.De
As a source of Terahertz and infrared radiation, semiconductor quantum well
intersubband (ISB) transitions have been widely investigated experimentally.
Elsaesser and Woerner (1999); Williams (2007) In particular, an understanding
of the spectral linewidth is of crucial importance for the design and control
of heterostructureHelm et al. (1989) and laser devices.Chow et al. (1997)
Although the theoretical description of ISB dynamics has been the subject of
many studies, Iotti and Rossi (2001); Butscher et al. (2004); Pereira et al.
(2004); Li and Ning (2004); Waldmüller et al. (2004); Zheng et al. (2004);
Shih et al. (2005); Kira and Koch (2006); Waldmueller et al. (2006); Butscher
and Knorr (2006); Pasenow et al. (2008); Weber et al. (2009); Vogel et al.
(2009) current models for the ISB optics become insufficient at low
temperatures, yielding an intrinsic absorption linewidth which is much smaller
than experimentally observed.Kaindl (2000); Waldmüller et al. (2004); Shih et
al. (2005) For an equilibrium Fermi distribution of the electrons in the
ground-state, Pauli blocking prevents electron-electron scattering at low
temperatures, while the electron-phonon contribution is too weak to explain
the experimental findings. To approach the experimental findings in the low
temperature regime, some theoretical descriptions focus on disorder
corrections such as impuritiesBanit et al. (2005) or interface roughness.Li
and Ning (2004)
In this article, we demonstrate that even in the absence of disorder,
correlations in the electronic ground-state of the doped quantum well yield a
line broadening at low temperatures not observed before: Coulomb correlations
lead to modified distribution functions which partially reduce the strong
Pauli blocking in the low temperature regime by opening up new electron-
electron scattering channels.Takada and Yasuhara (1991); Daniel and Vosko
(1960) This leads to additional dephasing which manifests itself in an
enhanced broadening of the absorption spectrum. Even if the additional
inclusion of electron-phonon scattering strongly reduces the influence of the
correlations, a thorough investigation of ground-state correlations clearly
improves our physical understanding of the many-body effects in high quality
quantum well samples.
Our approach is as follows: (I) after summarizing the overall dynamics of a
two-dimensional electron gas in an ISB system, (II) we discuss the origin and
(III) aspects of ground-state correlations and finally show their impact on
the ISB absorption spectrum.
(I) ISB dynamics. For our investigations, we consider an n-doped
$\text{GaAs}/\text{Al}_{0.35}\text{Ga}_{0.65}\text{As}$ quantum well system
where only the kinetics of the two lowest conduction subbands are of
relevance. 111For the calculations, we used the following parameters: static
dielectric constant $\varepsilon_{0}$ = 12.9, high-frequency dielectric
constant $\varepsilon_{\text{bg}}$ = 10.9, longitudinal optical phonon energy
$\hbar\omega_{\rm LO}$ = 36 meV, subband gap $\varepsilon_{\rm gap}$ = 210.66
meV, effective masses $m_{1}^{*}=0.078m_{0}$, $m_{2}^{*}=0.131m_{0}$. This is
a good approximation since we are focusing on low temperatures and subband
gaps which are typically quite large ($k_{\text{B}}T\lesssim$ 9 meV $\ll$ 100
meV $\lesssim\varepsilon_{\text{gap}}$ ). To approximate the more realistic
finite potential of the quantum well, we use an effective well width for an
infinite potential well.Waldmüller et al. (2004) Non-parabolicity effects are
included in the form of different effective subband masses.Ekenberg (1989) A
sketch of the considered quantum well in-plane band structure is given in Fig.
1.
Figure 1: (Color online) Sketch of the two energetically lowest conduction
subbands of an n-doped quantum well. The arrows symbolize the different
Coulomb many-body interactions among the electrons. Optical excitation creates
an electron ($a^{\dagger}_{2,\vec{k},s}$) in band 2 and annihilates an
electron ($a_{1,\vec{k},s}$) in band 1.
The Hamiltonian of the investigated system consists of the in-plane kinetics
of the confined electrons ($H_{0,\text{el}}$), the Coulomb interaction
($H_{\text{C}}$), and the semiclassical coupling to an external light field
($H_{\text{em}}$):
$\displaystyle H_{0,\text{el}}$
$\displaystyle=\sum_{i,\vec{k}_{i},s_{i}}\varepsilon_{i,\vec{k}_{i}}a^{\dagger}_{i,{\vec{k}_{i}},s_{i}}a^{\phantom{\dagger}}_{i,{\vec{k}_{i}},s_{i}},$
(1) $\displaystyle H_{\text{C}}$
$\displaystyle=\frac{1}{2}\sum_{\\{ijlm\\}}V^{|\vec{k}{i}-\vec{k}_{l}|}_{\\{{ijlm}\\}}\,a^{\dagger}_{\\{i\\}}a^{\dagger}_{\\{j\\}}a_{\\{m\\}}a_{\\{l\\}},$
(2) $\displaystyle H_{\text{em}}$
$\displaystyle=\sum_{i,j,\vec{k}_{i},s_{i}}\hbar\Omega_{\rm
em}(t)a^{\dagger}_{i,{\vec{k}_{i}},s_{i}}a^{\phantom{\dagger}}_{j,{\vec{k}_{i}},s_{i}}.$
(3)
Here, $a^{\dagger}_{i,\vec{k}_{i},s_{i}}$
($a^{\phantom{\dagger}}_{i,\vec{k}_{i},s_{i}}$) denotes the creation
(annihilation) operator for an electron in subband $i$ with an in-plane wave
vector ${\vec{k}_{i}}$, spin $s_{i}$, and energy
$\varepsilon_{i,\vec{k}_{i}}=\varepsilon_{i}+(\hbar^{2}{\vec{k}^{2}_{i}}/2m_{i})$
(cf. Fig. 1). We introduce the compound index
$\\{i\\}=\\{i,\vec{k}_{i},s_{i}\\}$ to simplify the notation. Furthermore,
$V^{|\vec{k}{i}-\vec{k}_{l}|}_{\\{{ijlm}\\}}=V^{|\vec{k}{i}-\vec{k}_{l}|}_{{ijlm}}\delta_{\vec{k}_{i}+\vec{k}_{j},\vec{k}_{l}+\vec{k}_{m}}\delta_{s_{i},s_{l}}\delta_{s_{j},s_{m}}$
describes the Coulomb-induced transitions of electrons in the states
$\\{l,m\\}$ to the states $\\{i,j\\}$, where
$V^{|\vec{k}{i}-\vec{k}_{l}|}_{{ijlm}}$ is the Coulomb matrix
element.Waldmüller et al. (2004) The Rabi frequency $\Omega_{\rm em}(t)={\bf
d}_{12}\cdot{\bf E}(t)/\hbar$ (dipole moment ${\bf d}_{12}$) describes the
interaction between the external field ${\bf E}(t)$ and the electronic system.
The absorption coefficient $\alpha(\omega)$ of the ISB system is calculated
via the complex susceptibility $\chi(\omega)$ as
$\alpha(\omega)\propto\omega\text{Im}\left[\chi(\omega)\right]$,
$\chi(\omega)=P(\omega)/\epsilon_{0}E(\omega)$ with the macroscopic
polarization $P(\omega)$ determined by the microscopic ISB polarizations
$\rho_{ij,\vec{k},s}=\langle
a^{\dagger}_{i,{\vec{k}},s}a^{\phantom{\dagger}}_{j,{\vec{k}},s}\rangle$
$(i\neq j)$ [cf. Eq. (4)].Schäfer and Wegener (2002)
The calculations of the ISB dynamics for the polarizations
$\rho_{ij,\vec{k},s}$ are carried out within a density-matrix approach using a
correlation expansion.Lindberg and Koch (1988); Rossi and Kuhn (2002)
Evaluating the system up to second order, the dynamical equations for the
polarizations and the electronic populations $f_{i,{\vec{k}},s}=\langle
a^{\dagger}_{i,{\vec{k}},s}a^{\phantom{\dagger}}_{i,{\vec{k}},s}\rangle$ as
well as the second-order correlations $\sigma^{\text{c}}_{\\{ijlm\\}}=\langle
a^{\dagger}_{\\{i\\}}a^{\dagger}_{\\{j\\}}a^{\phantom{\dagger}}_{\\{l\\}}a^{\phantom{\dagger}}_{\\{m\\}}\rangle-(\langle
a^{\dagger}_{\\{i\\}}a^{\phantom{\dagger}}_{\\{m\\}}\rangle\langle
a^{\dagger}_{\\{j\\}}a^{\phantom{\dagger}}_{\\{l\\}}\rangle-\langle
a^{\dagger}_{\\{i\\}}a^{\phantom{\dagger}}_{\\{l\\}}\rangle\langle
a^{\dagger}_{\\{j\\}}a^{\phantom{\dagger}}_{\\{m\\}}\rangle)$ are coupled by
the electron-electron interaction. In the following, we will neglect the spin
index since the quantities of interest are spin independent for our system.
Evaluating the dynamical equations for the electronic polarizations between
the two lowest subbands $\rho_{12,\vec{k}}$ with the Hamiltonian given above
yields
$\displaystyle\frac{\text{d}}{\text{dt}}\rho_{12,{\vec{k}}}$
$\displaystyle=\frac{i}{\hbar}\langle[H_{0,\text{el}}+H_{\text{em}}+H_{\text{C}},a^{\dagger}_{1,{\vec{k}}}a^{\phantom{\dagger}}_{2,{\vec{k}}}]\rangle$
$\displaystyle=-\frac{i}{\hbar}(\tilde{\mathcal{E}}_{2,\vec{k}}-\tilde{\mathcal{E}}_{1,\vec{k}})\rho_{12,\vec{k}}-i\tilde{\Omega}(t)(f_{2,\vec{k}}-f_{1,\vec{k}})$
$\displaystyle\quad-\Gamma\\{\rho_{ij,\vec{k}},f_{i,\vec{k}}\\}.$ (4)
The energy $\tilde{\mathcal{E}}_{i,\vec{k}}$ combines the subband energy
$\varepsilon_{i,\vec{k}}$ and the energetic renormalization due to the Coulomb
exchange contribution. The renormalized Rabi frequency $\tilde{\Omega}(t)$
contains the Rabi frequency of the external light field $\Omega_{\rm em}(t)$
and a renormalization due to the Coulomb exciton and depolarization
contributions.Li and Ning (2004); Waldmüller et al. (2004) These
renormalizations result from a Hartree-Fock approximation, which is first
order in the Coulomb potential.Chuang et al. (1992); Nikonov et al. (1999) The
second-order correlation contribution yields the dephasing functional
$\Gamma\\{\rho_{ij,\vec{k}},f_{i,\vec{k}}\\}$ caused by Boltzmann scattering
between the electrons and includes both diagonal and nondiagonal Coulomb
scattering contributions which lead to a broadening of the absorption
spectrum. 222While we include all diagonal terms (proportional to
$\rho_{ij,\vec{k}}$) in the kinetic scattering contributions, we only consider
the nondiagonal terms in Ref. Waldmüller et al., 2004 which are proportional
to $\rho_{ij,\vec{k}-\vec{q}}$ since these are the dominant terms
counteracting the broadening of the diagonal terms. Depending on the
nonparabolicity of the bandstructure, a strong cancellation between diagonal
and nondiagonal terms can occur.Li and Ning (2004) For further details
concerning the Hartree-Fock and kinetic scattering contributions, see Ref.
Waldmüller et al., 2004.
Additional higher-order correlations are included via a phenomenological
damping $\gamma$ of the second-order correlation functions,Schilp et al.
(1994) which softens the strict energy conservation in
$\Gamma\\{\rho_{ij,\vec{k}},f_{i,\vec{k}}\\}$ typically used in Markovian
approaches. In particular, $\gamma$ enters
$\Gamma\\{\rho_{ij,\vec{k}},f_{i,\vec{k}}\\}$ and the correlation correction
of the ground-state distribution $\delta f_{{\vec{k}}}$ (discussed in the next
section) in a way that a strong cancellation of the influence of $\gamma$ on
the spectral broadening occurs.333The deviation $\delta f_{{\vec{k}}}$
increases for decreasing $\gamma$ while the linewidth of the absorption
spectrum neglecting correlation contributions, determined by
$\Gamma\\{\rho_{ij,\vec{k}},f_{i,\vec{k}}\\}$, is reduced for decreasing
$\gamma$. Thus, the broadening effect of $\delta f_{{\vec{k}}}$ on the
absorption spectrum is always in a reasonable proportion to the reduction of
the linewidth so that the influence of $\gamma$ on the spectral broadening is
strongly canceled. For the purpose of this paper, it is thus justified to
focus on a fixed value of $\hbar\gamma$ which is chosen as 5 meV here.
Due to Pauli-blocking in the dephasing functionals
$\Gamma\\{\rho_{ij,\vec{k}},f_{i,\vec{k}}\\}$, the broadening of the spectrum
becomes very narrow for low temperatures due to the sharp Fermi edge. This
result, which is not in agreement with experimental findings, leads us to the
assumption that some fundamental suppositions in the theory must be corrected.
Here, we address additional ground-state correlations neglected so far:
(II) Ground state correlations. In most theoretical descriptions of
$\tilde{\mathcal{E}}_{i,\vec{k}}$, $\tilde{\Omega}$, and $\Gamma$, the
electronic populations $f_{i,\vec{k}}$ in equilibrium are taken to be Fermi
distributions $f^{(0)}_{i,{\vec{k}}}$ of the non-interacting electron
gas.Butscher et al. (2004); Li and Ning (2004); Waldmüller et al. (2006) This
yields good agreement with the experiment in the high temperature
regime.Kaindl et al. (1998); Li and Ning (2004); Waldmüller et al. (2004) At
low temperatures, the electron-electron ground-state correlations are expected
to play an important role for the following reason: While for high
temperatures the kinetic energy of the electrons clearly dominates over the
Coulomb repulsion, the latter becomes more important for low temperatures
where the kinetic energy and the Coulomb repulsion may be of a similar
magnitude.
To extract the correlation of the ground-state beyond the usual second-order
Born-Markov approximation, we consider a deviation $\delta
f_{i,{\vec{k}}}=f_{i,{\vec{k}}}-f^{(0)}_{i,{\vec{k}}}$ from the equilibrium
Fermi distribution $f^{(0)}_{i,{\vec{k}}}$ (cf. Refs. Takada and Yasuhara,
1991; Daniel and Vosko, 1960 for a treatment of ground-state correlations in
metals) and include first-order memory effects. The evolution of $\delta
f_{i,{\vec{k}}}$ is described by the kinetics $H_{\text{0,el}}$ of the non-
interacting electron gas as well as the Coulomb coupling $H_{\text{C}}$:
$\displaystyle-i\hbar\frac{\text{d}}{\text{dt}}\delta
f_{i,{\vec{k}}}=-i\hbar\frac{\text{d}}{\text{dt}}f_{i,{\vec{k}}}=\langle[H_{0,\text{el}}+H_{\text{C}},a^{\dagger}_{i,{\vec{k}}}a^{{\phantom{\dagger}}}_{i,{\vec{k}}}]\rangle.$
(5)
We evolve Eq. (5) up to second order in the Coulomb coupling similar to the
derivation of the ISB polarization. Restricting the correlation effects to a
single subband (ground state), we neglect the subband index $i$ in the
following. In this case, the first-order correlation contributions vanish, and
in second order, we obtain linear differential equations for
$\sigma^{\text{c}}_{\\{ijlm\\}}$ with a time dependent inhomogeneity $Q(t)$:
$\displaystyle-i\hbar\frac{\text{d}}{\text{dt}}\delta f_{{\vec{k}}}$
$\displaystyle=\sum_{\vec{k}^{\prime},\vec{q}}\left[\sigma^{\text{c}}_{1}({\vec{k}},{\vec{q}},{\vec{k}^{\prime}})-\sigma^{\text{c}}_{2}({\vec{k}},{\vec{q}},{\vec{k}^{\prime}})\right]\tilde{V}^{|{\vec{q}}|},$
(6) $\displaystyle-i\hbar\frac{\text{d}}{\text{dt}}\sigma^{\text{c}}_{1/2}$
$\displaystyle=(\pm\Delta\varepsilon+i\hbar\gamma)\sigma^{\text{c}}_{1/2}\pm\tilde{W}^{|{\vec{k},\vec{q},\vec{k}^{\prime}}|}Q(t),$
(7)
with the abbreviations
$\tilde{W}^{|{\vec{k},\vec{q},\vec{k}^{\prime}}|}=2\tilde{V}^{|{\vec{q}}|}-\tilde{V}^{|{\vec{k}^{\prime}-\vec{q}-\vec{k}}|}$
and
$\Delta\varepsilon=\varepsilon_{\vec{k}-\vec{q}}+\varepsilon_{\vec{k^{\prime}}+\vec{q}}-\varepsilon_{\vec{k^{\prime}}}-\varepsilon_{\vec{k}}$,
where $\tilde{V}$ denotes the screened Coulomb matrix element of the lower
subband due to the modification of the potential by the presence of the other
electrons.Lee and Galbraith (1999) Integrating Eq. (7), and assuming the
memory of $Q(t)$ to be small, we can expand the inhomogeneity $Q(t^{\prime})$
in a perturbation series around the local time $t$, yielding the following
expression for $\sigma^{\text{c}}_{1/2}$:
$\displaystyle\sigma^{\text{c}}_{1/2}=\mp\frac{i}{\hbar}\int_{0}^{\infty}\mathrm{d}s\,e^{(\pm\frac{i}{\hbar}\Delta\varepsilon-\gamma)s}\tilde{W}^{|{\vec{k},\vec{q},\vec{k}^{\prime}}|}\left[Q(t)+s\
\dot{Q}(t)\right].$ (8)
The zeroth-order term $\propto Q(t)$ yields the typical Boltzmann scattering
contributions which vanish for the Fermi distribution functions. 444While this
term vanishes exactly for $\gamma=0$, it yields a small but finite value for
finite values of $\gamma$ which is neglected here. The second term includes
memory effects in first order (containing the temporal derivative of the
source) and leads to
$\displaystyle\sigma^{\text{c}}_{1/2}=\mp\frac{i}{\hbar}\tilde{W}^{|{\vec{k},\vec{q},\vec{k}^{\prime}}|}\dot{Q}(t)\left[\frac{1}{(\pm\frac{i}{\hbar}\Delta\varepsilon-\gamma)^{2}}\right],$
(9)
which is substituted in Eq. (6).
In a first order iteration, we take the electron occupations occuring in the
inhomogeneity $Q(t)$ to be Fermi distributions. This leads to the final
expression of the deviation $\delta f_{{\vec{k}}}$:
$\displaystyle\delta f_{\vec{k}}$
$\displaystyle=2\sum_{\vec{k}^{\prime},\vec{q}}\frac{\Delta\varepsilon^{2}-\hbar^{2}\gamma^{2}}{(\Delta\varepsilon^{2}+\hbar^{2}\gamma^{2})^{2}}\left(\tilde{V}^{|{\vec{k}}^{\prime}-{\vec{q}}-{\vec{k}}|}-2\tilde{V}^{|{\vec{q}}|}\right)\tilde{V}^{|{\vec{q}}|}$
$\displaystyle\hskip
28.45274pt\times\left[f^{0}_{\vec{k}}f^{0}_{\vec{k^{\prime}}}f^{-}_{\vec{k^{\prime}}+\vec{q}}f^{-}_{\vec{k}-\vec{q}}-f^{0}_{\vec{k^{\prime}}+\vec{q}}f^{0}_{\vec{k}-\vec{q}}f^{-}_{{\vec{k}^{\prime}}}f^{-}_{{\vec{k}}}\right],$
(10)
with the abbreviation $f^{-}_{\vec{k}}=1-f^{0}_{\vec{k}}$. Equation (10)
illustrates the interplay of electrons within one subband to renormalize the
equilibrium Fermi function: Two electrons with wave vectors
$\vec{k^{\prime}}+\vec{q},\vec{k}-\vec{q}$ are annihilated, while two
electrons with $\vec{k},\vec{k^{\prime}}$ are created, respectively. Again,
the phenomenological damping $\gamma$ represents higher order
correlations.Schilp et al. (1994)
Our result for the correlated ground-state distribution function $f_{\vec{k}}$
is plotted in Fig. 2 for three different temperatures: $T$ = 1 K (solid
lines), 50 K (dashed lines), and 100 K (dotted lines).
Figure 2: (Color online) Electron distribution function of the electronic
ground-state including (dark lines) and neglecting (light lines) Coulomb
correlations for a 5 nm quantum well with an electronic doping density of
$n_{\text{dop}}=6.0\times 10^{11}$ cm-2 and various temperatures. Inset:
Corresponding deviation $\delta f_{{\vec{k}}}=f_{\vec{k}}-f^{(0)}_{\vec{k}}$
from the Fermi functions.
Comparing $f_{\vec{k}}$ (dark lines) with the equilibrium Fermi distribution
$f^{(0)}_{\vec{k}}$ (light lines), we observe a slight decrease (increase) of
the electron distribution for wave vectors below (above) the Fermi edge,
already known from electron gases in metals.Takada and Yasuhara (1991); Daniel
and Vosko (1960) This is especially pronounced for $T$ = 1 K, where one now
finds available states below and a finite population above the Fermi edge.
Looking at the deviation from the Fermi function $\delta f_{{\vec{k}}}$
(plotted in the inset of Fig. 2), one can see the sharp edge around the Fermi
energy and a renormalization up to $3\%$ at a temperature of $T=1$ K. The
observed features soften for higher temperatures but remain on the same order
of magnitude.
Even though the total deviation from the ideal Fermi function does not change
strongly with temperature, we expect the influence of the correlations on
scattering processes to decrease for rising temperatures: (i) The relative
importance of the allowed scattering processes decreases with an increasing
softening of the Fermi edge (see Fig. 2) and, at the same time, (ii) the
deviation from the ideal Fermi function $\delta f_{{\vec{k}}}$ decreases close
to the Fermi edge. The scattering processes which go with the Coulomb coupling
$\tilde{V}^{q}\sim 1/|\vec{q}|$ are responsible for the heigth of $\delta
f_{{\vec{k}}}$. Since for larger temperatures the electron scattering is less
common around the Fermi edge, $\delta f_{{\vec{k}}}$ also decreases.
Even if the calculated $\delta f_{\vec{k}}$ is quantitatively quite small, it
opens up a new physical scenario for the low-temperature regime: $\delta
f_{\vec{k}}\neq 0$ leads to an occupation above and to available states below
the Fermi edge, in particular for temperatures near 0 K, and allows scattering
which is otherwise prohibited due to Pauli blocking.
Besides the dependence on the temperature, the observed correlation effects
also depend on the quantum well width, where a slightly stronger deviation is
found for smaller well widths (not shown). This is due to the stronger Coulomb
coupling between the electrons for decreasing well widths. The doping density
also influences the magnitude of $\delta f_{\vec{k}}$, where an optimal value
of the density leads to a maximal deviation (cf. also the absorption spectra
in Fig. 3). The decrease of the deviation for smaller densities can be
explained by the short-range nature of the electronic correlations: for a
larger mean-free path (Wigner-Seitz radius) $r_{s}$ between the
electrons,Ziman (1992) the probability of a momentum transfer decreases. On
the other hand, we expect a decreasing deviation at a certain point for large
densities since the the kinetic energy, which goes with $1/{r_{s}}^{2}$,
overweigths the Coulomb interaction, going with $1/r_{s}$ between the
electrons.Mahan (2000)
(III) Absorption spectra. Next, to discuss experimental observables, the
influence of the ground-state correlations on the linear quantum well
intersubband absorption spectrum is studied. Figure 3 shows the spectrum
calculated from the dynamics of the polarization given by Eq. (4) for
different doping densities at a temperature of $T$ = 1 K. Here, the influence
of the electronic ground-state correlations is clearly visible.
Figure 3: (Color online) ISB absorption spectrum of a 5 nm quantum well at a
temperature $T$ = 1 K and different doping densities $n_{\text{dop}}$ (in
$\text{cm}^{-2}$) including (dark lines) and neglecting (light lines) the
electronic ground-state correlations.
We find a strong broadening of the absorption line shape including ground-
state correlations (dark lines) compared to the spectrum neglecting the
electronic correlations (light lines). The full width at half maximum in the
former is up to three times larger than in the latter case. In addition, the
calculation including the correlated ground state shows a small energetic
renormalization.
Figure 4: (Color online) ISB absorption spectrum of a 5 nm quantum well with
$n_{\text{dop}}=6.0\times 10^{11}~{}\text{cm}^{-2}$ including (dark lines) and
neglecting (light lines) the electronic ground-state correlations for
different temperatures. Inset: Enlarged view of the absorption line shapes.
This substantiates our prior claim that even small ground-state correlations
have a strong influence at low temperatures: The absorption linewidth
increases due to the generation of available scattering states, reflecting a
partial cancellation of the strong Pauli blocking by the electronic
correlations. For the parameters used here, the difference between the
absorption spectrum including and neglecting ground-state correlations is
maximal for a doping density of $n_{\text{dop}}=6.0\times
10^{11}\text{cm}^{-2}$. The shift to lower energies for increasing densities
is mainly due to the energetic renormalization resulting from the Hartree-Fock
contributions,Li and Ning (2004); Waldmüller et al. (2004) leading to a
smaller effective band gap between the lower and upper subband. Since we want
to focus on the impact of the ground-state correlations on the absorption
spectrum, we will restrict the following investigations to a 5 nm quantum well
using a doping density of $n_{\text{dop}}=6.0\times 10^{11}\text{cm}^{-2}$.
In Fig. 4, we show the absorption spectra for different temperatures.
Comparing the results including (dark lines) and neglecting (light lines)
ground-state correlations, we find that for temperatures higher than 50 K, the
correlation effects are of no significant relevance as discussed earlier in
the temperature dependence of $\delta f_{{\vec{k}}}$ (cf. the inset of Fig.
4). Instead, nonparabolicity effects gain importance, leading to an asymmetric
line shape especially for smaller well widths due to the occupation of higher
energetic states.Waldmüller et al. (2004) For 50 K and below, the absorption
spectrum shows a significant broadening leading to a strongly reduced peak
absorption. Furthermore, the calculation with the correlated ground-state
shows a small energetic renormalization.
Although we find that the ground-state correlations lead to a significant
broadening of the spectrum at low temperatures, the linewidth is still
strongly underestimated compared to experimental findings.Kaindl (2000); Shih
et al. (2005) In order to present a full and consistent description of ISB
absorption including all important scattering processes which contribute to
the absorption line shape, electron-phonon interaction must also be taken into
account.Li and Ning (2004); Waldmüller et al. (2004) This is assumed to be the
major broadening mechanism for temperatures higher than $T$ =100 K.Butscher et
al. (2004) For low temperatures, spontaneous phonon emission yields a
temperature-independent contribution to the linewidth.
Figure 5: (Color online) ISB absorption spectrum of a 5 nm quantum well
including (dark lines) and neglecting (light lines) the electronic ground-
state correlations, with microscopically calculated Markovian electron-phonon
scattering rates for a doping density $n_{\text{dop}}=6.0\times
10^{11}\text{cm}^{-2}$ for different temperatures.
We therefore extract temperature-dependent Markovian electron-phonon
scattering rates from earlier microscopic calculations and include them as an
additional dephasing contribution $\gamma_{\text{phon}}$ in the
calculations.Butscher et al. (2004) The resulting calculated ISB absorption
spectra are found in Fig. 5. We find that the influence of the ground-state
correlations is strongly masked by the phonon-induced dephasing. Still, a
deviation between the absorption spectra neglecting ground-state correlations
(light solid line) and including ground-state correlations (dark solid line)
can be found for temperatures lower than 50 K, where the the spectra
neglecting ground-state correlations show less broadening than the ones
including ground-state correlations. For $T$ = 100 K, the difference is hardly
visible.
In conclusion, we have presented a microscopic theory for the description of
ISB quantum well absorption including electron-electron contributions and
electronic ground-state correlations. We showed that by including ground-state
correlations, the absorption linewidth for temperatures $T<$ 50 K shows
significant broadening, where the full width at half maximum is increased by
up to a factor 3 at $T$ = 1 K. The additional inclusion of electron-phonon
scattering masks the impact of the ground-state correlations on the spectral
width. In a next step, a more consistent treatment of both the electron-phonon
and electron-electron dephasing, including full non-Markovian effects,Butscher
and Knorr (2006) should be carried out to allow deeper insights into the
influence of electronic ground-state correlations.
###### Acknowledgements.
We acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG)
through the project KN 427/4-1 and the Alexander von Humboldt Foundation
through the Feodor-Lynen program.
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|
arxiv-papers
| 2010-05-18T16:37:12 |
2024-09-04T02:49:10.484541
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Thi Uyen-Khanh Dang, Carsten Weber, Marten Richter, Andreas Knorr",
"submitter": "Thi Uyen-Khanh Dang",
"url": "https://arxiv.org/abs/1005.3247"
}
|
1005.3324
|
# An LP with Integrality Gap $1+\epsilon$ for Multidimensional Knapsack
David Pritchard111École Polytechnique Fédérale de Lausanne and partially
supported by an NSERC post-doctoral fellowship.
###### Abstract
In this note we study packing or covering integer programs with at most $k$
constraints, which are also known as _$k$ -dimensional knapsack problems_. For
integer $k>0$ and real $\epsilon>0$, we observe there is a polynomial-sized LP
for the $k$-dimensional knapsack problem with integrality gap at most
$1+\epsilon$. The variables may be unbounded or have arbitrary upper bounds.
In the (classical) packing case, we can also remove the dependence of the LP
on the cost-function, yielding a polyhedral approximation of the integer hull.
This generalizes a recent result of Bienstock [3] on the classical knapsack
problem.
## 1 Introduction
The classical _knapsack problem_ is the following: given a collection of items
each with a value and a weight, and given a weight limit, find a subset of
items whose total weight is at most the weight limit, and whose value is
maximized. If $n$ denotes the number of items, this can be formulated as the
integer program $\\{\max\sum_{i=1}^{n}x_{i}v_{i}\mid
x\in\\{0,1\\}^{n},\sum_{i=1}^{n}x_{i}w_{i}\leq\ell\\}$ where $n$ denotes the
number of items, $v_{i}$ denotes the value of item $i$, $w_{i}$ denotes the
weight of item $i$, and $\ell$ denotes the weight limit.
In the more general _$k$ -dimensional knapsack_ (or $k$-constrained knapsack)
problem, there are $k$ different kinds of “weight” and a limit for each kind.
An example for $k=3$ would be a robber who is separately constrained by the
total mass, volume, and noisiness of the items he is choosing to steal. An
orthogonal generalization is that the robber could take multiple copies of
each item $i$, up to some prescribed limit of $d_{i}$ available copies. We
therefore model the $k$-dimensional knapsack problem as
$\\{\max cx\mid x\in\mathbb{Z}^{n},0\leq x\leq d,Ax\leq b\\}$ (1)
where $A$ is a $k$-by-$n$ matrix, $b$ is a vector of length $k$, and $d$ is a
vector of length $n$, all non-negative and integral. Two special cases are
common: if $d=\mathbf{1}$ we call it the _$0\textrm{-}1$ knapsack problem_; if
$d=+\infty$, we call it the _unbounded knapsack problem_.
Another natural generalization is the _$k$ -dimensional knapsack-cover
problem_,
$\\{\min cx\mid x\in\mathbb{Z}^{n},0\leq x\leq d,Ax\geq b\\}$
which has analogous unbounded and 0-1 special cases. We sometimes call this
version the _covering version_ and likewise (1) is the _packing version_.
On the positive side, for any fixed $k$, all above variants admit a simple
pseudo-polynomial-time dynamic programming solution. Chandra et al. [9] gave
the first PTAS (polynomial-time approximation scheme) for $k$-dimensional
knapsack in 1976, and later an LP-based scheme was given by Frieze and Clarke
[13]. See the book by Kellerer et al. [18, §9.4.2] for a more comprehensive
literature review. The case $k=1$ also admits a fully polynomial-time
approximation scheme (FPTAS), but for $k\geq 2$ there is no FPTAS unless
$\mathsf{P}$=$\mathsf{NP}$. This was originally shown for 0-1 $k$-dimensional
knapsack by Gens & Levner [15] and Korte & Schrader [20] (see also [18]) and
subsequently for arbitrary $d$ by Magazine & Chern [24].
Our main result is the following:
###### Theorem 1.
Let $k$ and $\epsilon$ be fixed. Given a $k$-dimensional knapsack (resp.
knapsack-cover) instance $\mathcal{K}$, there is a polynomial-sized extended
LP relaxation $\mathcal{L}$ of $\mathcal{P}$ with
$\mathrm{OPT}(\mathcal{P})\geq(1-\epsilon)\mathrm{OPT}(\mathcal{L})$ (resp.
with $\mathrm{OPT}(\mathcal{P})\leq(1+\epsilon)\mathrm{OPT}(\mathcal{L})$).
Here “polynomial-sized extended LP relaxation” means the following. First,
$\mathcal{P}$ has $n$ variables. Then $\mathcal{L}$ must have those $n$
variables plus a polynomial number of other ones. The projection
$\mathcal{L}^{\prime}$ of $\mathcal{L}$ onto the first $n$ variables must
contain the same integral solutions as $\mathcal{P}$. Finally, $\mathcal{L}$
and $\mathcal{P}$ must have the same objective function, i.e. the objective
function should ignore the extended variables.
In the proof, we will see that the LP can be constructed in polynomial time,
and that a near-optimal integral solution can be obtained from an optimal
extreme point fractional solution just by rounding down (resp. up). The number
of variables in the LP is $n^{O(k/\epsilon)}$ and the number of constraints is
$kn^{O(k/\epsilon)}$. The _integrality gap_ of an IP is the worst-case ratio
between the fractional and integral optimum and therefore Theorem 1 can be
equivalent stated as saying that $\mathcal{P}$ has integrality gap at most
$1+\epsilon$.
Our result and the techniques we use are a generalization of a recent result
of Bienstock [3], which dealt with the packing version for $k=1$. The key
observation we contribute is that his “filtering” approach was also
traditionally used to get a PTAS for multi-dimensional knapsack; in
_filtering_ we exhaustively guess the $\gamma$ max-cost items in the knapsack
for some constant $\gamma$.
The construction of $\mathcal{L}$ in Theorem 1 turns out to depend on the cost
function $c$. A more interesting and challenging problem is to find an
$\mathcal{L}$ which is independent of the cost-function, since this gives a
_polyhedral approximation_ $\mathcal{L}^{\prime}$ of $\mathcal{P}$ e.g. in the
packing case, it implies
$\mathcal{L}^{\prime}\supset\mathcal{P}\supset(1-\epsilon)\mathcal{L}^{\prime}$.
Bienstock’s result [3] actually gives an LP which does not depend on the item
cost/profits $c$. We will show (in Section 4) that in the packing case, our
approach can be similarly revised:
###### Theorem 2.
Let $k$ and $\epsilon$ be fixed. Given a $k$-dimensional knapsack instance
$\mathcal{K}$, there is a polynomial-sized extended LP relaxation
$\mathcal{L}$ of $\mathcal{P}$ with
$\mathrm{OPT}(\mathcal{P})\geq(1-\epsilon)\mathrm{OPT}(\mathcal{L})$, such
that $\mathcal{L}$ does not depend on $c$.
This comes as the cost of an increase in size to $kn^{O(k^{2}/\epsilon)}$. For
the covering case performing the same (a polynomial-sized extended LP
relaxation independent of $c$ with integrality gap $\leq 1+\epsilon$) is an
interesting open problem; we elaborate at the end.
### 1.1 Related Work
Knapsack (whether packing or covering) has an FPTAS by dynamic programming,
and it is well-known that dynamic programs of such a form can be solved as a
shortest-path problem, which has an LP formulation. Nonetheless, there is no
evident way to combine these steps to get an LP for knapsack with integrality
gap $1+\epsilon$. The problem (say, for packing, which is simpler) is that
last step in the FPTAS is not merely to return the last entry of the DP table,
but rather it finds the maximum scaled profit such that the minimum volume to
obtain it fits inside the knapsack (and then recovers the actual solution).
The naive fix is adding this volume constraint to the LP but it makes the LP
non-integral and then it is not clear how to proceed.
Bienstock & McClosky [4] extend the work of Bienstock [3] to covering problems
and other settings, and also give an LP of size
$n^{2}(1/\epsilon)^{\frac{1}{\epsilon}\log\frac{1}{\epsilon}}$ with
integrality gap $1+\epsilon$ for 1-dimensional, 0-1 covering knapsack.222They
use a disjunctive program; in essence, the LP guesses the most costly item in
the knapsack, then for
$i=1,\dotsc,O(\frac{1}{\epsilon}\log\frac{1}{\epsilon})$ it guesses the number
of items whose costs are $(1+\frac{1}{\epsilon})^{-(i,i+1]}$ times that cost,
with all guesses $>\frac{1}{\epsilon}$ deemed equivalent. In particular the LP
depends on the cost function. We remark that the method does not readily
extend to $k$-dimensional knapsack. There is some current work [5] on
obtaining primal-dual algorithms (that is, not needing the ellipsoid method or
interior-point subroutines) for knapsack-type covering problems with good
approximation ratio and [4] reports that the methods of [5] extend to a
combinatorial LP-based approximation scheme for 1-dimensional covering
knapsack.
Answering an open question of Bienstock [4] about the efficacy of automatic
relaxations for the knapsack problem, Karlin et al. [17] recently found that
the “Laserre hierarchy” of semidefinite programming relaxations, when applied
to the 1-dimensional 0-1 packing knapsack problem, gives an SDP with
integrality gap $1+\epsilon$ after $O(1/\epsilon^{2})$ rounds.
Knapsack problems have a couple of interesting basic properties. The first
contrasts with Lenstra’s result [22] that for any fixed $k$, integer programs
with $k$ constraints can be solved in polynomial time; in comparison, if we
have nonnegativity constraints for every variable plus _one other constraint_
, we get the unbounded (1-dimensional) knapsack problem, which is
$\mathsf{NP}$-hard [23]. Second, recall that for any optimization problem
whose objective is integral, and whose optimal value is polynomial in the
input size, any FPTAS can be used to get a pseudopolynomial-time algorithm. In
contrast, 0-1 2-dimensional knapsack shows the converse is false: it has a
pseudopolynomial-time algorithm, but getting an FPTAS is $\mathsf{NP}$-hard
even when each profit $c_{i}$ is 1, e.g. see [18, Thm. 9.4.1].
There is a line of work on maximizing constrained submodular functions. For
non-monotone submodular maximization subject to $k$ linear packing
constraints, the state of the art is by Lee et al. [21] who give a
$(5+\epsilon)$-approximation algorithm. For monotone submodular maximization
the state of the art is by Chekuri & Vondrák [10] who give a
$(e/(e-1)+\epsilon)$-approximation subject to $k$ knapsack constraints and a
matroid constraint. We note it is $\mathsf{NP}$-hard to obtain any factor
better than $e/(e-1)$ for monotone submodular maximization over a matroid
[12], so in this setting knapsack constraints only affect the best ratio by
$\epsilon$, just like in our setting of LP-relative approximation.
### 1.2 Overview
First, we review rounding and filtering. Rounding is a standard approach to
turn an optimal fractional solution into a nearly-optimal integral one, and
here we lose up to $k$ times the maximum per-item profit. Filtering works well
with rounding because it reduces the maximum per-item profit; the power of
these ideas is already enough to get an LP-based approximation scheme [13],
but it uses a separate LP for each “guess” made in filtering. Therefore, like
Bienstock [4], we use disjunctive programming [2] to combine all the separate
LPs into a single one. The approach has some similarity to the knapsack-cover
inequalities of Carr et al. [6].
## 2 Rounding and Filtering
We now explain the approach. A knapsack instance (1) is determined by the
parameters $(A,b,c,d)$. The naïve LP relaxation of the knapsack problem is
$\\{\max cx\mid x\in\mathbb{R}^{n},0\leq x\leq d,Ax\leq b\\}.$
$\mathcal{K}(A,b,c,d)$
In the following, _fractional_ means non-integral. The following lemma is
standard.
###### Lemma 3.
Let $x^{*}$ be an extreme point solution to the linear program
($\mathcal{K}(A,b,c,d)$). Then $x^{*}$ is fractional in at most $k$
coordinates.
###### Proof.
It follows from elementary LP theory that $x^{*}\in\mathbb{R}^{n}$ satisfies
$n$ (linearly independent) constraints with equality. There are $k$
constraints of the form $A_{j}x\leq b_{j}$; all other constraints are of the
form $x_{i}\geq 0$ or $x_{i}\leq d_{i}$, so at least $n-k$ of them hold with
equality. Clearly $x_{i}\geq 0$ and $x_{i}\leq d_{i}$ cannot both hold with
equality for the same $i$, so $x^{*}_{i}\in\\{0,d_{i}\\}$ for at least $n-k$
distinct $i$, which gives the result. ∎
Therefore, we obtain the following primitive guarantee on a rounding strategy.
Let $\lfloor\cdot\rfloor$ applied to a vector mean component-wise floor and
let $c_{\max}:=\max_{i}c_{i}$.
###### Corollary 4.
Let $x^{*}$ be an extreme point solution to the linear program
($\mathcal{K}(A,b,c,d)$). Then $c\lfloor x^{*}\rfloor\geq cx^{*}-kc_{\max}$.
Now the idea is to take $x^{*}$ to be an optimal fractional solution, and use
filtering (exhaustive guessing) to turn the additive guarantee into a
multiplicative factor of $1+\epsilon$. Let $\gamma$ denote a parameter, which
represents the size of a multi-set we will guess. For a non-negative vector
$z$ let the notation $\lVert z\rVert_{1}$ mean $\sum_{i}z_{i}$. A _guess_ is
an integral vector $g$ with $0\leq g\leq d,Ag\leq b$ and $\lVert
g\rVert_{1}\leq\gamma$. It is easy to see the number of possible guesses is
bounded by $(n+1)^{\gamma}$, and that for any constant $\gamma$ we can iterate
through all guesses in polynomial time.
From now on we assume without loss of generality (by reordering items if
necessary) that $c_{1}\leq c_{2}\leq\dotsb\leq c_{n}$. For a guess $g$ with
$\lVert g\rVert_{1}=\gamma$ we now define the _residual knapsack problem_ for
$g$. The residual problem models how to optimally select the remaining objects
_under the restriction_ that the $\gamma$ most profitable333To simplify the
description, even if $c_{i+1}=c_{i}$ we think of item $i+1$ as more profitable
than item $i$. items chosen (counting multiplicity) are $g$. Let $\mu(g)$
denote $\min\\{i\mid g_{i}>0\\}$. Define $d^{g}$ to be the first $\mu(g)$
coordinates of $d-g$ followed by $n-\mu(g)$ zeroes, and $b^{g}=b-Ag$. The
_residual knapsack problem_ for $g$ is $(A,b^{g},c,d^{g})$. The residual
problem for $g$ does not permit taking items with index more than $\mu(g)$ and
so its $c_{\max}$ value may be thought of as $c_{\mu(g)}$ or less, which is at
most $c\cdot g/\lVert g\rVert_{1}=c\cdot g/\gamma$.
If a guess $g$ has $\lVert g\rVert_{1}<\gamma$, define $b^{g}$ and $d^{g}$ to
be all-zero. Then Corollary 4 gives the following.
###### Corollary 5.
Let $x_{\mathrm{OPT}}$ be an optimal integral knapsack solution for
$(A,b,c,d)$. Let $g$ be the $\gamma$ most profitable items in
$x_{\mathrm{OPT}}$ (or all, if there are less than $\gamma$). Let $x^{*}$ be
an optimal extreme point solution to $\mathcal{K}(A,b^{g},c,d^{g})$. Then
$g+\lfloor x^{*}\rfloor$ is a feasible knapsack solution for $(A,b,c,d)$ with
value at least $1-k/\gamma$ times optimal.
###### Proof.
We use $\mathrm{OPT}$ to denote $c\cdot x_{\mathrm{OPT}}$. Note that
$x_{\mathrm{OPT}}-g$ is feasible for the residual problem for $g$. Therefore
$c\cdot x^{*}\geq\mathrm{OPT}-c\cdot g$. Moreover $c_{\max}$ in the residual
problem for $g$ is not more than $c\cdot
g/\gamma\leq\frac{\mathrm{OPT}}{\gamma}$, so Corollary 4 shows that
$c\cdot\lfloor x^{*}\rfloor\geq c\cdot
x^{*}-k\frac{\mathrm{OPT}}{\gamma}\geq\mathrm{OPT}-c\cdot
g-k\frac{\mathrm{OPT}}{\gamma}$
and consequently $\lfloor x^{*}\rfloor+g$ is a solution with value at least
$\mathrm{OPT}(1-\frac{k}{\gamma})$, as needed. ∎
By taking $\gamma=k/\epsilon$ and solving $\mathcal{K}(A,b^{g},c,d^{g})$ for
all possible $g$ we get the previously known PTAS for $k$-dimensional
knapsack; we now refine the approach to get a single LP.
## 3 Disjunctive Programming
We now review some disjunctive programming tools [2]. The only result we need
is that it is possible to write a compact LP for the convex hull of the union
of several polytopes, provided that we we have compact LPs for each one.
Suppose we have polyhedra $P^{1}=\\{x\in\mathbb{R}^{n}\mid A^{1}x\leq
b^{1}\\}$ and $P^{2}=\\{x\in\mathbb{R}^{n}\mid A^{2}x\leq b^{2}\\}$. Both of
these sets are convex and it is therefore easy to see that the convex hull of
their union is the set
$\textrm{conv.hull}(P^{1}\cup P^{2})=\\{x\in\mathbb{R}^{n}\mid x=\lambda
x^{1}+(1-\lambda)x^{2},0\leq\lambda\leq 1,A^{1}x^{1}\leq b^{1},A^{1}x^{2}\leq
b^{2}\\}.$
However, this is not a _linear_ program, e.g. since we multiply the variable
$\lambda$ by the variables $x^{1}$. Nonetheless, it is not hard to see that
the following is a linear formulation of the same set:
$\textrm{conv.hull}(P^{1}\cup P^{2})=\\{x\in\mathbb{R}^{n}\mid
x=x^{1}+x^{2},0\leq\lambda\leq 1,A^{1}x^{1}\leq\lambda
b^{1},A^{1}x^{2}\leq(1-\lambda)b^{2}\\}.$
A similar construction gives the convex hull of the union of any number of
polyhedra; we now apply this to the knapsack setting.
The LP $\mathcal{K}(A,b^{g},c,d^{g})$ was constructed to mean the left-over
problem after making a guess $g$ of the $\gamma$ most profitable items; we
similarly shift this LP to get $\\{y=x+g\mid x\in\mathbb{R}^{n},0\leq x\leq
d^{g},Ax\leq b^{g}\\}$ which is the same set, after the guessed part is added
back in.
Let $\mathcal{G}$ denote the set of all possible guesses $g$. Then the convex
hull of the union of the shifted polyhedra is given by the feasible region of
the following polyhedron:
$\Bigl{\\{}y\mid
y=\sum_{g\in\mathcal{G}}y^{g};\sum_{g\in\mathcal{G}}\lambda^{g}=1;\lambda\geq\mathbf{0};\forall
g:y^{g}=x^{g}+\lambda^{g}g,\mathbf{0}\leq
x^{g}\leq\lambda^{g}d^{g},Ay^{g}\leq\lambda^{g}b^{g}\Bigr{\\}}.$
($\mathcal{L}$)
We attach objective $\max c\cdot y$ to ($\mathcal{L}$) to make it into an LP,
and use it to prove Theorem 1.
###### Proof of Theorem 1, packing version.
Let $y$ be an optimal extreme point solution for ($\mathcal{L}$). It is
straightforward to argue that any extreme point solution has
$\lambda^{g^{*}}=1$ for some particular $g^{*}$, and $\lambda^{g}=0$ for all
other $g$. Hence $y=x^{g^{*}}+g^{*}$ where $x^{g^{*}}$ is an optimal extreme
point solution to $\mathcal{K}(A,b^{g^{*}},c,d^{g^{*}})$. We now show that
$\lfloor y\rfloor$ is a $(1-\epsilon)$-approximately optimal solution, re-
using the previous arguments.
If $\lVert g^{*}\rVert_{1}<\gamma$, then $x^{g^{*}}=0$ so $y$ is integral,
hence $y$ is an optimal knapsack solution. Otherwise, if $\lVert
g^{*}\rVert_{1}=\gamma$, then Corollary 4 shows that
$c\cdot\lfloor y\rfloor=c\cdot\lfloor x^{g^{*}}\rfloor+c\cdot g^{*}\geq c\cdot
x^{g^{*}}-k\frac{c\cdot g^{*}}{\gamma}+c\cdot g^{*}=c\cdot y-k\frac{c\cdot
g^{*}}{\gamma}\geq(1-\epsilon)c\cdot y,$
which completes the proof. ∎
The corresponding result for the covering version is very similar. One
difference is that we round up instead of down. The other is that some guesses
become inadmissible. Let $g$ be an integral vector with $0\leq g\leq d,\lVert
g\rVert_{1}\leq\gamma$; we define $\mu(g),d^{g}$ as before and call $g$ a
_guess_ only if $A(g+d^{g})\geq b$, in which case we set $b^{g}$ to be the
component-wise maximum of $\mathbf{0}$ and $b-Ag$.
## 4 Removing the Dependence on $c$ for Packing Problems
In the LPs described above, for each guess $g$, we treated that guess as the
set of most _profitable_ items. In particular, $b^{g}$ and $d^{g}$ are defined
in a way that depends on $c$. We now show in the packing case, how to write a
somewhat larger LP, still with integrality gap $1+\epsilon$, which is defined
independently of $c$. This exactly follows the approach of Bienstock [3]; what
we will do is guess the _biggest_ items for each constraint, rather than the
most profitable items. The technique does not seem to have an easy analogue
for covering problems.
In detail, previously, we guessed the multiset $g$ of $\gamma$ most profitable
items in the solution. Instead, let us guess a $k$-tuple
$(g^{1},g^{2},\dotsc,g^{k})$ where for each $k$, $g^{i}$ is the set of
$\gamma$ items in the solution which have largest coefficients with respect to
the $i$th constraint (breaking ties in each constraint in any consistent way).
What we need is that any extreme feasible solution with at most $k$ fractional
values can be rounded to an integral feasible solution at a relative cost
factor of at most $\epsilon$. Let the original extreme point LP solution be
$x$. We round each fractional value up to the closest integer, which causes
the solution to become an infeasible one, call it $y=\lceil x\rceil$. Then, to
retain feasibility, we go through each of the $k$ constraints, pick the
$c$-smallest set of $k$ items from $y$ whose deletion causes the constraint to
again become satisfied; and we delete the union of these sets from $y$,
obtaining $z$. Each set has $c$-cost at most $\frac{k}{\gamma}c(y)$ since for
each constraint $i$, any $k$ elements from $g^{i}$ form an eligible set for
deletion, and $y\subset g^{i}$ consists of at least $\gamma$ items. Thus
$c(z)\geq c(y)-k\frac{k}{\gamma}c(y)=(1-k^{2}/\gamma)c(x)$. Taking
$\gamma=k^{2}/\epsilon$ (compared to the previous $k/\epsilon$), we get the
desired result.
## 5 Discussion
We believe that the main result is a nice theoretical illustration of
techniques (filtering, rounding, disjunctive programming). However, it remains
to be seen if it could be given useful applications. The disjunctive
programming trick is definitely senseless sometimes: if we want to write an
LP-based computer program to $(1+\epsilon)$-approximately solve
multidimensional knapsack instances, it is more efficient to consider the LP
corresponding to each guess separately (as in [13]) rather than solve the
gigantic LP obtained by merging them together. Sometimes an LP-relative [19]
(or Lagrangian-preserving [11, 1]) approximation algorithm can be used as a
subroutine in ways that a non-LP-relative one could not. However, at least in
[11, 19, 1], the analysis relied on LP-relative or Lagrangian-preserving
analysis of the naïve LP, and an arbitrary LP would not have fared as well,
and the LP we build here seems not to be useful in this way.
Finding a compact formulation for $k$-dimensional covering knapsack with small
integrality gap and such that the LP does _not_ depend on the objective
function is an interesting open problem. For example, we are not aware of any
polynomial-sized extended LP for 1-dimensional covering knapsack with constant
integrality gap, in sharp contrast to the packing case. A partial result for
$k$-dimensional covering knapsack is the knapsack-cover LP [6] (see also [25,
8, 7] for applications); it has integrality gap at most $2k$, and while it is
not polynomial size, it can be $(1+\epsilon)$-approximately separated [6] and
hence $(1+\epsilon)$-approximately optimized [14, 16] in polynomial time.
From a theoretical perspective, it also seems challenging to find an LP for
2-dimensional (packing) knapsack where the size of the LP is a function of
$1/\epsilon$ times a polynomial in $n$, as was done in [4] for the
1-dimensional version.
### Acknowledgments
We thank Laura Sanità for helpful discussions on this topic.
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* [24] M. J. Magazine and M.-S. Chern. A note on approximation schemes for multidimensional knapsack problems. Math. of Oper. Research, 9(2):244–247, 1984.
* [25] D. Pritchard and D. Chakrabarty. Approximability of sparse integer programs. Algorithmica, 2010. In press. Preliminary versions at arXiv:0904.0859 and in Proc. 17th ESA, pages 83–94, 2009.
|
arxiv-papers
| 2010-05-18T20:44:52 |
2024-09-04T02:49:10.493049
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "David Pritchard",
"submitter": "David Pritchard",
"url": "https://arxiv.org/abs/1005.3324"
}
|
1005.3383
|
# The homotopical dimension of random 2-complexes
Daniel C. Cohen, Michael Farber and Thomas Kappeler Partially supported by
Louisiana Board of Regents grant NSF(2010)-PFUND-171.Partially supported by a
grant from the EPSRC.Partially supported by the Swiss National Science
Foundation.
###### Abstract
Stochastic algebraic topology aims at studying random or partly known spaces
which typically arise in applications as configuration spaces of large
systems. In this paper we study the Linial–Meshulam model of random two-
dimensional complexes. We prove that if the probability parameter $p$
satisfies $p\ll n^{-1-\epsilon}$, where $\epsilon>0$ is arbitrary and
independent of $n$, then a random 2-complex $Y$ is homotopically one
dimensional with probability tending to $1$ as $n\to\infty$. More precisely,
we show that under this assumption on $p$, the complex $Y$ can be collapsed to
a graph in finitely many steps. It is known that the homotopical dimension of
$Y$ is equal to $2$ for $p>3n^{-1}$.
## 1 Introduction
Since its inception in 1959 by Erdös and Rényi [ER60], the theory of random
graphs has developed into a rapidly growing and widely applicable branch of
discrete mathematics, bringing together ideas from graph theory,
combinatorics, and probability theory. In one model, a random graph is a
subgraph $\Gamma$ of a complete graph on $n$ vertices such that every edge of
the complete graph is included in $\Gamma$ with probability $p$, independently
of the other edges. One is interested in probabilistic features of $\Gamma$
and their dependence on $p$ when $n$ is large. Here $0<p<1$ is a probability
parameter which in general may depend on $n$. The theory of random graphs
[AS00, Bol08, JŁR00] offers many spectacular results and predictions, which
play an essential role in various engineering and computer science
applications. Random graphs also serve within mathematics as accessible models
for other, more complex random structures.
Higher dimensional analogs of the aforementioned Erdős–Rényi model were
recently suggested and studied by Linial–Meshulam in [LM06], and
Meshulam–Wallach in [MW09]. In these models, one generates a random
$d$-dimensional simplicial complex $Y$ by considering the full $d$-dimensional
skeleton of the simplex $\Delta_{n}$ on vertices $\\{1,\dots,n\\}$ and
retaining $d$-dimensional faces independently with probability $p$. Note that
in this construction $Y$ contains the $(d-1)$-dimensional skeleton of
$\Delta_{n}$. The work of Linial–Meshulam and Meshulam–Wallach provides
threshold functions for the vanishing of the $(d-1)$-st homology groups of
random complexes with coefficients in a finite abelian group. Threshold
functions for the vanishing of the $d$-th homology groups were subsequently
studied by Kozlov [Koz09].
In this paper, we focus on 2-dimensional random complexes. The corresponding
probability space $G(\Delta_{n}^{(2)},p)$ of the Linial–Meshulam model is
defined as follows. Let $\Delta_{n}$ denote the $(n-1)$-dimensional simplex
with vertices $\\{1,2,\dots,n\\}$. Then $G(\Delta_{n}^{(2)},p)$ denotes the
set of all 2-dimensional subcomplexes
$\Delta_{n}^{(1)}\subset Y\subset\Delta_{n}^{(2)},$
containing the one-dimensional skeleton $\Delta_{n}^{(1)}$. The probability
function $\mathbb{P}:G(\Delta_{n}^{(2)},p)\to{\mathbf{R}}$ is given by the
formula
$\mathbb{P}(Y)=p^{f(Y)}(1-p)^{{n\choose 3}-f(Y)},\quad Y\in
G(\Delta_{n}^{(2)},p),$
where $f(Y)$ denotes the number of faces in $Y$. In other words, each of the
2-dimensional simplexes of $\Delta_{n}^{(2)}$ is included in a random
2-complex $Y$ with probability $p$, independently of the other 2-simplexes. As
in the case of random graphs, $0<p<1$ is a probability parameter which may
depend on $n$. When $n$ grows, the model $G(\Delta_{n}^{(2)},p)$ includes all
finite $2$-dimensional complexes containing the full 1-skeleton
$\Delta_{n}^{(1)}$; however, the likelihood of various topological phenomena
is dependent on the value of $p$. The theory of deterministic 2-complexes
itself is a rich and active field of current research with many challenging
open questions, see [HMS93].
The fundamental group of a random 2-complex $Y\in G(\Delta_{n}^{(2)},p)$ was
investigated by Babson, Hoffman, and Kahle [BHK08]. They showed that for $p\gg
n^{-1/2}\cdot(3\log n)^{1/2}$, the group $\pi_{1}(Y)$ vanishes asymptotically
almost surely (i.e., the probability that $\pi_{1}(Y)$ is trivial tends to $1$
as $n\to\infty$). For $p\ll n^{-1/2-\epsilon}$, these authors use notions of
negative curvature due to Gromov to study the nontriviality and hyperbolicity
of $\pi_{1}(Y)$.
In this paper, we show that for $p\ll n^{-1-\epsilon}$ a random 2-complex $Y$
is homotopically 1-dimensional, a.a.s.111We use the abbreviation a.a.s. for
the phrase “asymptotically almost surely”. More precisely, we show that $Y$
can be collapsed to a graph in finitely many steps. This implies that $Y$ has
a free fundamental group and vanishing 2-dimensional homology. Note that the
vanishing of 2-dimensional homology in this range of $p$ also follows from a
result of Kozlov [Koz09]. In [CFK10], it is shown that for $p>3/n$, the
homology group $H_{2}(Y;{\mathbf{Z}})$ is nontrivial with probability tending
to $1$; see also [Koz09]. Thus, for $p>3/n$, the random 2-complex $Y$ is
homotopically two-dimensional a.a.s.
Our main result is as follows:
###### Theorem 1.
(a) If for some $k\geq 1$ the probability parameter $p$ satisfies222Recall
that the symbol $a_{n}\ll b_{n}$ means that $a_{n}>0$ and $a_{n}/b_{n}\to 0$
as $n\to\infty$.
$p\ll n^{-1-\frac{2}{k+1}},$
then a random 2-complex $Y\in G(\Delta_{n}^{(2)},p)$ is collapsible to a graph
in at most $k$ steps, asymptotically almost surely (a.a.s). (b) If for some
$k\geq 1$ the probability parameter $p$ satisfies
$p\gg n^{-1-\frac{1}{3\cdot 2^{k-1}-1}},$
then $Y$ is not collapsible to a graph in $k$ or fewer steps, a.a.s.
Loosely speaking, Theorem 1 combines with previously known results to suggest
that a random 2-complex with vanishing 2-dimensional homology is homotopically
one-dimensional.
Theorem 1 implies:
###### Corollary 2.
If for some $k\geq 1$ the probability parameter $p$ satisfies
$p\ll n^{-1-\frac{2}{k+1}}$
then the fundamental group $\pi_{1}(Y)$ of a random 2-complex $Y\in
G(\Delta_{n}^{(2)},p)$ is free and $H_{2}(Y;{\mathbf{Z}})=0$, a.a.s.
The proof of Theorem 1 is given at the very end of the paper. A key role is
played by Theorem 13, which states that there exists a finite list of
forbidden 2-complexes $\mathcal{L}_{k,r}$ with $k\geq 0$ and $r\geq 2$, such
that an arbitrary 2-complex of degree at most $r$ (see below) is collapsible
to a graph in $k$ steps if and only if it does not contain any of the
2-complexes from $\mathcal{L}_{k,r}$. This allows us to reduce the
collapsiblity problem to the containment problem for random complexes which
was studied in [CFK10].
### Acknowledgments
This research was implemented during visits of M. Farber to FIM ETH Zürich and
Louisiana State University, and a visit of D. Cohen to the University of
Zürich. Portions of this work were carried out during the Spring of 2010, when
the first two authors participated in the Mathematisches Forschungsinstitut
Oberwolfach Research in Pairs program. We thank the FIM ETH, LSU, the
University of Zürich, and the MFO for their support and hospitality, and for
providing productive mathematical environments.
## 2 Collapsibility of a 2-complex to a graph
### 2.1 Basic definitions
Let $Y$ be a finite 2-dimensional simplicial complex. An edge of $Y$ is called
free if it is included in exactly one 2-simplex.
The boundary $\partial Y$ is defined as the union of free edges. We say that a
2-complex $Y$ is closed if $\partial Y=\emptyset$.
A $2$-complex $Y$ is called pure if every maximal simplex is 2-dimensional. By
the pure part of a 2-complex we mean the maximal pure subcomplex, i.e. the
union of all 2-simplexes.
Let $Y$ be a simplicial 2-complex and let $\sigma$ and $\tau$ be two
2-simplexes of $Y$. We say that $\sigma$ and $\tau$ are adjacent if they
intersect in an edge. The distance between $\sigma$ and $\tau$,
$d_{Y}(\sigma,\tau)$, is the minimal integer $k$ such that there exists a
sequence of 2-simplexes $\sigma=\sigma_{0},\sigma_{1},\dots,\sigma_{k}=\tau$
with the property that $\sigma_{i}$ is adjacent to $\sigma_{i+1}$ for every
$0\leq i<k$. (If no such sequence exists then $d_{Y}(\sigma,\tau)=\infty$.)
The diameter ${\rm{diam}}(Y)$ is defined as the maximal value of
$d_{Y}(\sigma,\tau)$ taken over pairs of 2-simplexes of $Y$.
A simplicial 2-complex is strongly connected if it has a finite diameter.
A simplicial 2-complex has degree $\leq r$ if every edge is incident to at
most $r$ 2-simplexes.
A pseudo-surface is a finite, pure, strongly connected 2-dimensional
simplicial complex of degree at most $2$ (i.e., every edge is included in at
most two 2-simplexes).
More generally, for an integer $r>0$, an $r$-pseudo-surface is a finite, pure,
strongly connected 2-dimensional simplicial complex of degree at most $r$.
### 2.2 Simplicial collapse
Let $Y$ be a 2-complex. A 2-simplex of $Y$ is called free if at least one of
its edges is free. Let $\sigma_{1},\dots,\sigma_{k}$ be all free 2-simplexes
in $Y$, and let $e_{1},\dots,e_{k}$ be free edges with
$e_{i}\subset\sigma_{i}$. We say that the complex
$Y^{\prime}={Y-\cup_{i=1}^{k}{\rm{int}}(\sigma_{i})-\cup_{i=1}^{k}{\rm{int}}(e_{i})}$
is obtained from $Y$ by collapsing all free 2-simplexes. Clearly
$Y^{\prime}\subset Y$ is a deformation retract. The operation $Y\searrow
Y^{\prime}$ is called a simplicial collapse. Note that $Y^{\prime}$ is not
uniquely determined if one of the free simplexes of $Y$ has two free edges;
however the pure part of $Y^{\prime}$ (i.e. the union of 2-simplexes of
$Y^{\prime}$) is uniquely determined.
This process can be iterated $Y^{\prime}\searrow Y^{\prime\prime}$,
$Y^{\prime\prime}\searrow Y^{\prime\prime\prime}$, etc. We denote $Y=Y^{(0)}$,
$Y^{\prime}=Y^{(1)}$, $Y^{\prime\prime}=Y^{(2)}$ etc. The sequence of
subcomplexes $Y^{(0)}\supset Y^{(1)}\supset Y^{(2)}\supset\dots$ is decreasing
and there are two possibilities: either (a) for some $k$, the complex
$Y^{(k)}$ is one-dimensional (a graph), or (b) for some $k$, the complex
$Y^{(k)}$ is 2-dimensional and closed, i.e., $\partial Y^{(k)}=\emptyset$.
###### Definition 3.
We say that $Y$ is collapsible to a graph in at most $k$ steps if $Y^{(k)}$ is
a graph. We say that $Y$ is collapsible to a graph in $k$ steps if $Y^{(k)}$
is a graph and $\dim Y^{(k-1)}=2$.
Observe that if $Y$ is collapsible to a graph in at most $k$ steps then any
simplicial subcomplex $S\subset Y$ is also collapsible to a graph in at most
$k$ steps. At each step one removes the free triangles in $Y^{(i)}$ which
belong to $S$.
Let $Y$ be a 2-complex, and consider the sequence of collapses
$Y^{(0)}\searrow Y^{(1)}\searrow Y^{(2)}\searrow\dots\searrow
Y^{(k)}\searrow\dots.$
For a 2-simplex $\sigma\in Y$ define
$D_{Y}(\sigma)=\sup\\{i;\,\sigma\subset
Y^{(i)}\\}\,\,\in\\{0,1,\dots,\infty\\}.$
A 2-simplex $\sigma$ is free if and only if $D_{Y}(\sigma)=0$.
A 2-complex $Y$ is collapsible to a graph in at most $k+1$ steps if and only
if $D_{Y}(\sigma)\leq k$ for any $2$-simplex $\sigma$. If after performing
several collapses $Y^{(0)}\searrow Y^{(1)}\searrow Y^{(2)}\searrow\dots$ we
obtain a subcomplex $Y^{(r)}\subset Y$ with empty boundary $\partial
Y^{(r)}=\emptyset$, then $Y^{(r)}=Y^{(r+1)}=Y^{(r+2)}=\dots$ and
$D_{Y}(\sigma)=\infty$ for any simplex $\sigma$ in $Y^{(r)}$.
###### Lemma 4.
Let $\sigma$ be a 2-simplex with $D_{Y}(\sigma)=k$ where $0<k<\infty$. Then
one of the edges $e$ of $\sigma$ has the following property: for any 2-simplex
$\sigma^{\prime}$ of $Y$ which is incident to $e$ and distinct from $\sigma$
one has $D_{Y}(\sigma^{\prime})<k$ and there exists a 2-simplex
$\sigma^{\prime}$ incident to $e$ and distinct from $\sigma$ such that
$D_{Y}(\sigma^{\prime})=k-1$.
###### Proof.
Since $D_{Y}(\sigma)=k$, we know that after $k$ collapses an edge $e$ of
$\sigma$ becomes free. All other simplexes $\sigma^{\prime}$ of $Y$ incident
to $e$ must have been eliminated in previous steps, i.e., they satisfy
$D_{Y}(\sigma^{\prime})<k$. At least one of these simplexes $\sigma^{\prime}$
must have been eliminated in step $k-1$ since otherwise $\sigma$ would have
become free earlier. ∎
###### Lemma 5.
If $Z\subset Y$ is a subcomplex and $\sigma\subset Z$ is a $2$-simplex, then
$D_{Z}(\sigma)\leq D_{Y}(\sigma).$
###### Proof.
If a 2-simplex belongs to $Z$ and is not free in $Z$ then it is not free in
$Y$. This implies that $Z^{\prime}\subset Y^{\prime}$ and therefore
$Z^{(i)}\subset Y^{(i)}$ for any $i\geq 1$. Thus, the maximal $i$ such that
$\sigma$ is contained in $Z^{(i)}$ is less than or equal to the maximal $i$
such that $\sigma$ is contained in $Y$, which implies the statement of the
Lemma. ∎
### 2.3 $\sigma$-accessible boundary
###### Definition 6.
Let $Y$ be a 2-complex and let $\sigma,\tau$ be two 2-simplexes of $Y$ with
$D_{Y}(\tau)=0$ and $D_{Y}(\sigma)=k\geq 1$. A collapsing path from $\tau$ to
$\sigma$ is a sequence of 2-simplexes
$\tau=\sigma_{0},\sigma_{1},\dots,\sigma_{k-1},\sigma_{k}=\sigma$ such that
$D_{Y}(\sigma_{i})=i$ and each pair $\sigma_{i}$ and $\sigma_{i+1}$ has a
common edge, where $i=0,\dots,k-1$.
In a collapsing path, the initial simplex $\sigma_{0}=\tau$ is a free simplex,
and hence at least one of its edges belongs to the boundary $\partial Y$.
###### Definition 7.
Given a 2-simplex $\sigma$, we denote by $A_{Y}(\sigma)\subset\partial Y$ the
union of the edges in $\sigma_{0}\cap\partial Y$ which can appear in a
collapsing path $\sigma_{0},\sigma_{1},\dots,\sigma_{k}$ ending at $\sigma$.
We call $A_{Y}(\sigma)$ the $\sigma$-accessible part of the boundary.
In Definition 7, clearly $k=D_{Y}(\sigma)$. Note that
$A_{Y}(\sigma)\not=\emptyset$ if and only if $D_{Y}(\sigma)<\infty$.
###### Definition 8.
Let $\sigma$ be a 2-simplex of $Y$ with $D_{Y}(\sigma)\geq 1$. For an edge $e$
of $\sigma$ define
$A_{Y}(\sigma,e)\subset A_{Y}(\sigma)$
as the set of all edges $e^{\prime}$ of the boundary $\partial Y$ with the
property that there exists a collapsing path
$\sigma_{0},\sigma_{1},\dots,\sigma_{k}=\sigma$ such that $e^{\prime}$ is an
edge of $\sigma_{0}$ and $e=\sigma_{k-1}\cap\sigma_{k}$.
If $e_{1},e_{2},e_{3}$ are the edges of $\sigma$ then
$A_{Y}(\sigma)=\cup_{i=1}^{3}A_{Y}(\sigma,e_{i})$ and the sets
$A_{Y}(\sigma,e_{i})$ need not be mutually disjoint.
###### Lemma 9.
Let $\sigma$ and $\sigma^{\prime}$ be adjacent 2-simplexes of $Z$ with
$D_{Z}(\sigma)=D_{Z}(\sigma^{\prime})+1.$
Assume that any collapsing path in $Z$ ending at $\sigma$ passes through the
edge $e=\sigma\cap\sigma^{\prime}$. If $Z$ is embedded as a subcomplex
$Z\subset Y$ and
$D_{Z}(\sigma^{\prime})<D_{Y}(\sigma^{\prime}),$
then
$D_{Z}(\sigma)<D_{Y}(\sigma).$
###### Proof.
Let $k=D_{Z}(\sigma^{\prime})=D_{Z}(\sigma)-1$. We must show that
$D_{Y}(\sigma)\geq k+2.$ First we claim that the edge $e$ may become free only
after at least $k+2$ collapses in $Y$. Assume it is free in $Y$ after $k+1$
collapses. By assumption, $D_{Y}(\sigma^{\prime})\geq k+1.$ Hence the edge $e$
can only be free after $k+1$ collapses in $Y$ if $\sigma$ has been removed
already before, i.e., $D_{Y}(\sigma)\leq k.$ On the other hand, by Lemma 5,
$D_{Y}(\sigma)\geq D_{Z}(\sigma)=k+1$ which leads to a contradiction.
By assumption, the two edges of $\sigma$ different from $e$ are not free in
$Z^{(k+1)}$ and hence they are not free in $Y^{(k+1)}$. Thus
$D_{Y}(\sigma)\geq k+2$ as claimed. ∎
Note that the assumption of Lemma 9 that any collapsing path in $Z$ ending at
$\sigma$ passes through the edge $e$ is equivalent to
$A_{Z}(\sigma,e^{\prime})=\emptyset$ for the two remaining edges
$e^{\prime}\not=e$ of $\sigma$.
###### Lemma 10.
Let $Z\subset Y$ be a subcomplex. If $D_{Z}(\sigma)=D_{Y}(\sigma)$ for a
2-simplex $\sigma$ of $Z$ then there is an edge $e$ of $\sigma$ such that
$\emptyset\not=A_{Z}({\sigma,e})\subset A_{Y}({\sigma,e})\subset\partial Y.$
###### Proof.
Without loss of generality, we may assume that $Y$ is obtained from $Z$ by
attaching a single $2$-simplex.
The proof is by induction on $k=D_{Y}(\sigma)=D_{Z}(\sigma)$.
In the case $k=0$, there is an edge $e$ of $\sigma$ that is free in both $Z$
and $Y$. In particular, $e\subset\partial Y$.
We include the case $k=1$. Recall that $Z^{\prime}=Z^{(1)}$ denotes the result
of the first collapse of $Z$, $Z\searrow Z^{\prime}$. Since
$D_{Z}(\sigma)=D_{Y}(\sigma)=1$, there is an edge $e$ of $\sigma$ that is free
in $Y^{\prime}$ and hence in $Z^{\prime}$. Then every collapsing path
$\tau,\sigma$ in $Z$ with $e=\tau\cap\sigma$ is also a collapsing path in $Y$.
Hence $A_{Z}({\sigma,e})\subset A_{Y}({\sigma,e})$.
For the general case, assume that $D_{Y}(\sigma)=D_{Z}(\sigma)=k$. After $k$
collapses
$Z\searrow Z^{(1)}\searrow\dots\searrow Z^{(k)},\quad Y\searrow
Y^{(1)}\searrow\dots\searrow Y^{(k)},$
the $2$-simplex $\sigma$ is exposed in both $Z^{(k)}$ and $Y^{(k)}$. Thus,
$\sigma$ has a free edge $e$ in $Y^{(k)}$ (and hence in $Z^{(k)}$ as well).
Writing $Z^{\prime}=Z^{(1)}$ and $Y^{\prime}=Y^{(1)}$, by induction, we have
$\emptyset\not=A_{Z^{\prime}}({\sigma,e})\subset A_{Y^{\prime}}({\sigma,e})$
so that any collapsing path $\sigma_{1},\dots,\sigma_{k}$ from
$\sigma_{1}=\sigma^{\prime}\subset A_{Z^{\prime}}({\sigma,e})$ to
$\sigma_{k}=\sigma$ in $Z^{\prime}$ is also a collapsing path in $Y^{\prime}$.
Note in particular that every edge of $\sigma^{\prime}$ that is free in
$Z^{\prime}$ is also free in $Y^{\prime}$. Consequently, for every free
triangle $\tau$ in $Z$ which meets $\sigma^{\prime}$ in an edge free in
$Z^{\prime}$, the collapsing path
$\tau=\sigma_{0},\sigma_{1},\dots,\sigma_{k}$ in $Z$ is a collapsing path in
$Y$. The result follows. ∎
###### Corollary 11.
Let $Z\subset Y$ be 2-complexes such that for a 2-simplex $\sigma$ of $Z$ none
of the edges $e\in A_{Z}(\sigma)\subset\partial Z$ is free in $Y$. Then
$D_{Z}(\sigma)+1\leq D_{Y}(\sigma).$
###### Proof.
For a contradiction, assume that $D_{Y}(\sigma)\leq D_{Z}(\sigma)$. Then
$D_{Y}(\sigma)=D_{Z}(\sigma)$ by Lemma 5. We may now apply Lemma 10 which
claims that there is an edge $e$ of $\sigma$ for which
$\emptyset\not=A_{Z}(\sigma,e)\subset A_{Y}(\sigma,e)\subset\partial Y$. This
contradicts our assumption that no edge in $A_{Z}(\sigma)$ lies on the
boundary $\partial Y$. ∎
### 2.4 The list of forbidden $r$-pseudo-surfaces $\mathcal{L}_{k,r}$
For a pair of integers $k=0,1,\dots,$ and $r=2,3,\dots$ we denote by
$\mathcal{L}_{k,r}$ the set of all isomorphism types of $r$-pseudo-surfaces
$S$ with the following properties:
1. (a)
Each $S\in\mathcal{L}_{k,r}$ has a specified 2-simplex $\sigma_{\ast}$ (called
the center).
2. (b)
If $\partial S\not=\emptyset$ then $D_{S}(\sigma_{\ast})=k$.
3. (c)
$d_{S}(\sigma_{\ast},\sigma)\leq k$ for any 2-simplex $\sigma$.
Figure 1: Surfaces $\mathcal{L}_{1,2}$.
Note that $\mathcal{L}_{0,r}=\\{S\\}$ consists of a single complex
$S=\sigma_{\ast}$ (the triangle).
The set $\mathcal{L}_{1,2}$ consists of the three surfaces shown in Figure 1.
Each of the surfaces a, b, c is a union of 4 triangles. The surface c is a
tetrahedron, b is a tetrahedron with one face open, and a is a fully flattened
tetrahedron.
It is clear that $\mathcal{L}_{k,r}$ is finite and
$\mathcal{L}_{k,r}\subset\mathcal{L}_{k,r+1}$.
###### Example 12.
Consider the following important family of surfaces
$S_{k}\in\mathcal{L}_{k,2}$ where $k=0,1,2,\dots$. The first surface $S_{0}$
is defined as a single triangle $S_{0}=\sigma_{\ast}$. The next surface
$S_{1}$ is the shown in Figure 1 a. Surfaces $S_{2}$ and $S_{3}$ are shown in
Figure 2. In general, the surface $S_{k}$ is obtained from $S_{k-1}$ by adding
a triangle to every edge of the boundary $\partial S_{k-1}$. It is clear that
for the central triangle $\sigma_{\ast}$ of $S_{k}$, one has
$D_{S_{k}}(\sigma_{\ast})=k$. Thus $S_{k}$ is not collapsible to a graph in
$k$ steps, but is collapsible in $k+1$ steps.
Figure 2: Surfaces $S_{k}\in\mathcal{L}_{k,2}$.
The following Theorem plays a key role in this paper:
###### Theorem 13.
A 2-complex $Y$ of degree at most $r\geq 2$ is not collapsible to a graph in
$k$ steps, where $k=0,1,2,\dots$, if and only if there is a surface
$S\in\mathcal{L}_{k,r}$ which admits a simplicial embedding $S\to Y$.
In the proof, we will use the following statement:
###### Lemma 14.
Let $Y$ be a finite 2-dimensional simplicial complex of degree at most $r$ and
let $\sigma$ be a 2-simplex in $Y$ with $D_{Y}(\sigma)=k$, where
$k=0,1,2,\dots$. Then there exists a surface $S\in\mathcal{L}_{k,r}$ and a
simplicial embedding $S\to Y$ such that the central simplex $\sigma_{\ast}$ of
$S$ is mapped onto $\sigma$.
###### Proof of Lemma 14.
We will use induction on $k=D_{Y}(\sigma)$. For $k=0$, the statement is
obvious. Assume that it is true for all cases with $D_{Y}(\sigma)<k$, and
consider the situation when $D_{Y}(\sigma)=k>0$. If $Y\searrow Y^{\prime}$ is
the first collapse, then $\sigma\subset Y^{\prime}$ and clearly
$D_{Y^{\prime}}(\sigma)=k-1$
and $Y^{\prime}$ has degree at most $r$. By the inductive hypothesis, there
exists $S^{\prime}\in\mathcal{L}_{k-1,r}$ and a simplicial embedding
$S^{\prime}\to Y^{\prime}$, mapping the central simplex of $S^{\prime}$ onto
$\sigma$.
For each edge $e$ lying in $A_{S^{\prime}}(\sigma)$ choose a 2-simplex
$\sigma_{e}\subset Y$ as follows. If $e\subset\partial Y^{\prime}$, let
$\sigma_{e}$ be any free triangle in $Y$ containing $e$. If
$e\not\subset\partial Y^{\prime}$, let $\sigma_{e}$ be any triangle in
$Y^{\prime}$ containing $e$ which is not in $S^{\prime}$; such $\sigma_{e}$
exists since $e\not\subset\partial Y^{\prime}$.
Next we define a subcomplex $S\subset Y$ as the union
$S=S^{\prime}\cup\bigcup_{e}\sigma_{e}\,\subset Y,$
where $e$ runs over the edges in $A_{S^{\prime}}(\sigma)$. Note that $S$ is
finite, pure, and strongly connected since $S^{\prime}$ is an $r$-pseudo-
surface. Moreover, the degree of $S$ is at most $r$ since it is a subcomplex
of $Y$. One has $D_{S}(\sigma)\geq k$ by Corollary 11. More precisely, we
obtain that $D_{S}(\sigma)=k$ by Lemma 5. Finally we observe that obviously
$d_{S}(\sigma,\sigma^{\prime})\leq k$ for any 2-simplex $\sigma^{\prime}$ of
$S$. Thus, $S\in\mathcal{L}_{k,r}$. ∎
###### Proof of Theorem 13.
Consider the sequence of successive collapses $Y\searrow Y^{(1)}\searrow
Y^{(2)}\searrow Y^{(3)}\searrow\dots$. We assume that $Y$ is not collapsible
to a graph in $k$ steps, which implies that there are two possibilities:
either (a) $Y^{(i)}\not=Y^{(i+1)}$ for any $i<k$; or (b) for some $i<k$, one
has $\partial Y^{(i)}=\emptyset$.
In case (a), the complex $Y$ contains a 2-simplex with $D_{Y}(\sigma)=k$ and
Lemma 14 gives us an embedding of an $r$-pseudo-surface
$S\in\mathcal{L}_{k,r}$ into $Y$.
In case (b), we have $\partial Y^{(i)}=\emptyset$ for some $i<k$. Fix a
2-simplex $\sigma_{\ast}\in Y^{(i)}$ and consider distances
$d_{Y^{(i)}}(\sigma_{\ast},\sigma)$ to various 2-simplexes $\sigma$ of
$Y^{(i)}$. If all these distances are less than or equal to $k$, then
$Y^{(i)}$ belongs to $\mathcal{L}_{k,r}$ and we are done. If there are
simplexes $\sigma$ such that $d_{Y^{(i)}}(\sigma_{\ast},\sigma)>k$, then
consider the subcomplex $Z\subset Y^{(i)}$ defined as the union of all
$\sigma$ with $d_{Y^{(i)}}(\sigma_{\ast},\sigma)\leq k$.
Clearly $Z$ is not collapsible to a graph in $k$ steps. Therefore, in the
sequence of collapses $Z\searrow Z^{(1)}\searrow Z^{(2)}\searrow
Z^{(3)}\searrow\dots$, we again have either case (a) or (b) as above. In case
(a), we apply Lemma 14; and in case (b), we obtain a subcomplex $S\subset Z$
with $\partial S=\emptyset$ such that $d(\sigma_{\ast},\sigma)\leq k$ for any
$\sigma\subset S$. We have $S\in\mathcal{L}_{k,r}$ in either case, completing
the proof. ∎
## 3 Collapsibility of a random 2-complex
### 3.1 The degree sequence
Recall that the degree of an edge $e$ in a 2-complex is defined as the number
of 2-simplexes which contain $e$. The degree of an edge in a random 2-complex
$Y\in G(\Delta_{n}^{(2)},p)$ is an integer in the set $\\{0,1,\dots,n-2\\}$.
Let $X_{k}:G(\Delta_{n}^{(2)},p)\to{\mathbf{Z}}$ be the random variable
counting the number of edges of degree $k$ in a random $2$-complex, where
$k=0,1,2,\dots,n-2$. A straightforward calculation reveals that
${\mathbb{E}}(X_{k})={n\choose 2}{{n-2}\choose k}p^{k}(1-p)^{n-2-k}.$
The expectation of the number of edges of degree at least $r$ in a random
$2$-complex is
$\sum_{k=r}^{n-2}{\mathbb{E}}(X_{k})\leq
n^{2}\sum_{k=r}^{n-2}(pn)^{k}\leq\frac{n^{2}(pn)^{r}}{1-pn}.$ (1)
###### Corollary 15.
The probability that a random 2-complex $Y\in G(\Delta_{n}^{(2)},p)$ has an
edge of degree at least $r$ is less than or equal to
$\frac{n^{2+r}p^{r}}{1-pn}.$
Thus, if
$p\ll n^{-1-\frac{2}{r}},$
then a random 2-complex $Y\in G(\Delta_{n}^{(2)},p)$ has no edges of degree
$r$ or greater, a.a.s.
###### Proof.
This follows from inequality (1) by applying the first moment method, see, for
instance, [JŁR00]. ∎
### 3.2 The invariant $\tilde{\mu}(S)$.
Following [BHK08] and [CFK10], for a $2$-complex $S$ with $v=v(S)$ vertices
and $f=f(S)>0$ faces one defines
$\mu(S)=\frac{v}{f}\in\mathbb{Q},$
and
$\tilde{\mu}(S)=\min_{S^{\prime}\subset S}\mu(S^{\prime}),$
where $S^{\prime}$ runs over all subcomplexes of $S$ or, equivalently, over
all pure subcomplexes $S^{\prime}\subset S$. Note the following monotonicity
property of $\tilde{\mu}$:
$\displaystyle\mbox{if}\quad S\subset
T,\quad\mbox{then}\quad\tilde{\mu}(S)\geq\tilde{\mu}(T).$ (2)
The invariant $\tilde{\mu}$ controls embeddability of finite 2-complexes into
random 2-complexes as illustrated by the following result.
###### Theorem 16 ([CFK10]).
Let $S$ be a finite simplicial complex.
1. (a)
If $p\ll n^{-\tilde{\mu}(S)}$, the probability that $S$ admits a simplicial
embedding into a random 2-complex $Y\subset G(\Delta_{n}^{(2)},p)$ tends to
zero as $n\to\infty$;
2. (b)
If $p\gg n^{-\tilde{\mu}(S)}$, the probability that $S$ admits a simplicial
embedding into a random 2-complex $Y\subset G(\Delta_{n}^{(2)},p)$ tends to
one as $n\to\infty$.
###### Definition 17.
A 2-complex $S$ is called balanced if $\tilde{\mu}(S)=\mu(S)$, or,
equivalently, $\mu(S^{\prime})\geq\mu(S)$ for any subcomplex
$S^{\prime}\subset S$.
Any triangulated surface is balanced, see [CFK10].
###### Example 18.
Suppose that a 2-complex $S$ has a free triangle with two free edges, and that
the result $S^{\prime}$ of removing this triangle satisfies
$\mu(S^{\prime})<1$. Then $\mu(S)>\mu(S^{\prime})$ and $S$ is unbalanced.
Indeed, if $\mu(S^{\prime})=v/f$, where $v=v(S^{\prime})$ and
$f=f(S^{\prime})$, then $v<f$ and we have $\mu(S)=(v+1)/(f+1)>v/f$. In this
way one produces many unbalance 2-complexes, including 2-disks.
Next, we examine the $\tilde{\mu}$ invariants of 2-complexes
$S\in\mathcal{L}_{k,r}$.
###### Lemma 19.
Let $S$ be a closed 2-complex, i.e., $\partial S=\emptyset$. Then
$\tilde{\mu}(S)\leq 1$.
###### Proof.
Without loss of generality, we may assume that $S$ is connected, since
otherwise we can apply the following arguments to a connected component of $S$
and use the monotonicity property (2). Moreover, we may assume that $S$ is
pure, since otherwise we may deal with the maximal pure subcomplex of $S$
instead of $S$.
Suppose first that $H_{2}(S;{\mathbf{Z}}_{2})=0$. Then by the Euler–Poincaré
theorem, $\chi(S)\leq 1$, and we have
$v-e+f=\chi(S)\leq 1,\quad\mbox{and}\quad 3f\geq 2e,$
where $v,e,f$ denote the numbers of vertices, edges and faces in $S$. In the
latter inequality we used the assumptions that $S$ is pure and closed. These
inequalities imply
$v-f/2\leq\chi(S)\leq 1,\quad\mbox{and}\quad\mu(S)\leq 1/2+1/f.$
Since $f\geq 4$ we obtain that $\tilde{\mu}(S)\leq\mu(S)\leq 3/4<1.$
Assume now that $H_{2}(S;{\mathbf{Z}}_{2})\not=0$. We will show that there is
a subcomplex $S^{\prime}\subset S$ which is also closed, $\partial
S^{\prime}=\emptyset$, and satisfies
$H_{2}(S^{\prime};{\mathbf{Z}}_{2})={\mathbf{Z}}_{2}$. Indeed, consider a
nonzero two-dimensional cycle $c=\sum_{i\in I}\sigma_{i}$ with
${\mathbf{Z}}_{2}$ coefficients, where the $\sigma_{i}$ are distinct
2-simplexes of $S$. Let $I^{\prime}\subseteq I$ be the minimal subset of the
indexing set $I$ for which $c^{\prime}=\sum_{i\in I^{\prime}}\sigma_{i}$ is
still a cycle, and let $S^{\prime}=\bigcup_{i\in I^{\prime}}\sigma_{i}$ be the
corresponding subcomplex of $S$. Then clearly
$H_{2}(S^{\prime};{\mathbf{Z}}_{2})={\mathbf{Z}}_{2}$ and $S^{\prime}$ is
closed and pure.
By the Euler–Poincaré theorem, $\chi(S^{\prime})\leq 2$, and we have
$v^{\prime}-e^{\prime}+f^{\prime}=\chi(S^{\prime})\leq 2,\quad\mbox{and}\quad
3f^{\prime}\geq 2e^{\prime},$
where $v^{\prime},e^{\prime},f^{\prime}$ denote the numbers of vertices, edges
and faces in $S^{\prime}$. This gives
$v^{\prime}-f^{\prime}/2\leq\chi(S^{\prime})\leq 2,$
and
$\displaystyle\mu(S^{\prime})\leq\frac{1}{2}+\frac{2}{f^{\prime}}.$ (3)
Since $f^{\prime}\geq 4$, the last inequality gives $\mu(S^{\prime})\leq 1$.
Finally, we have $\tilde{\mu}(S)\leq\mu(S^{\prime})\leq 1$. ∎
###### Lemma 20.
If $S\in\mathcal{L}_{k,r}$ for some $k\geq 0$, $r\geq 2$ then one has
$\displaystyle\tilde{\mu}(S)\leq 1+\frac{2}{k+1}.$ (4)
###### Proof.
If $S$ is closed the result follows from Lemma 19. Assume now that $\partial
S\not=\emptyset$. Let $\sigma_{\ast}$ be the central simplex of $S$ and let
$\sigma_{0},\sigma_{1},\dots,\sigma_{k}=\sigma_{\ast}$ be a collapsing path
leading to $\sigma_{\ast}$. Here $D_{S}(\sigma_{i})=i$ and
$\sigma_{i}\cap\sigma_{i+1}$ is an edge, see Definition 7. Then the union
$S^{\prime}=\cup_{i=0}^{k}\sigma_{i}$ is a subcomplex having exactly $k+1$
faces and at most $k+3$ vertices. Thus,
$\mu(S^{\prime})\leq\frac{k+3}{k+1}=1+\frac{2}{k+1},$
establishing (4). ∎
### 3.3 The threshold for $k$-collapsibility.
###### Definition 21.
Let $\tilde{\mu}_{k,r}$ denote the largest possible value of the invariant
$\tilde{\mu}(S)$ for $S$ a forbidden $r$-pseudo-surface,
$\tilde{\mu}_{k,r}\,=\,\max_{S\in\mathcal{L}_{k,r}}\tilde{\mu}(S)\,\in\mathbb{Q}.$
For instance, examining the surfaces shown in Figure 1 reveals that
$\tilde{\mu}_{1,2}=3/2$.
###### Theorem 22.
Consider a random 2-complex $Y\in G(\Delta_{n}^{(2)},p)$.
1. (a)
If for some $r\geq 2$ and $k\geq 1$, one has
$p\ll n^{-1-\frac{2}{r+1}}\quad\mbox{and}\quad p\ll n^{-\tilde{\mu}_{k,r}},$
then $Y$ is collapsible to a graph in at most $k$ steps, a.a.s.
2. (b)
If for some $r\geq 2$ and $k\geq 1$, one has $p\gg n^{-\tilde{\mu}_{k,r}}$,
then $Y$ is not collapsible to a graph in $k$ or fewer steps, a.a.s.
###### Proof.
By Corollary 15, if $p\ll n^{-1-\frac{2}{r+1}}$, then a random 2-complex $Y\in
G(\Delta_{n}^{(2)},p)$ has degree at most $r$, a.a.s. Next, we apply Theorem
13 and examine the embeddability of complexes $S\in\mathcal{L}_{k,r}$ into
$Y$. By Theorem 16 (a), if $p\ll n^{-\tilde{\mu}(S)}$, then $S$ does not embed
into $Y$, a.a.s. Since $\tilde{\mu}_{k,r}\geq\tilde{\mu}(S)$, we see that the
assumption $p\ll n^{-\tilde{\mu}_{k,r}}$ implies that no
$S\in\mathcal{L}_{k,r}$ can be embedded into $Y$, a.a.s. Thus, by Theorem 13,
we see that $Y$ is collapsible to a graph in $k$ or fewer steps. This proves
part (a).
To prove part (b), we apply Theorem 16 (b) to conclude that if $p\gg
n^{-\tilde{\mu}_{k,r}}$, then there exists $S\in\mathcal{L}_{k,r}$ which is
embeddable into $Y$, a.a.s. This implies that $Y$ is not collapsible to a
graph in at most $k$ steps, a.a.s. ∎
###### Example 23.
Consider the surface $S_{k}\in\mathcal{L}_{k,2}$ introduced in Example 12.
Note that $S_{k}\in\mathcal{L}_{k,r}$ for any $r\geq 2$. The numbers of
vertices $v_{k}$ and faces $f_{k}$ of $S_{k}$ satisfy the recurrence relations
$\displaystyle v_{k}=2\cdot v_{k-1}\quad\mbox{and}\quad
f_{k}=v_{k-1}+f_{k-1}.$ (5)
Indeed, viewing $S_{k-1}$ as a subcomplex of $S_{k}$, we see that all vertices
of $S_{k-1}$ lie on the boundary, and each edge of the boundary of $S_{k-1}$
adds a vertex to $S_{k}$. This explains the first equation. For the second,
note that the number of new triangles in $S_{k}$ is equal to the number of
edges on $\partial S_{k-1}$.
Since $v_{0}=3$ and $f_{0}=1$, solving the recurrence relations (5) yields
$v_{k}=3\cdot 2^{k}\quad\mbox{and}\quad f_{k}=3\cdot 2^{k}-2.$
Consequently,
$\mu(S_{k})=1+\frac{1}{3\cdot 2^{k-1}-1}.$
###### Lemma 24.
The surface $S_{k}$ is balanced, and hence
$\tilde{\mu}(S_{k})=\mu(S_{k})=1+\frac{1}{3\cdot 2^{k-1}-1}.$
###### Proof.
Let $S$ be a pure subcomplex of $S_{k}$ with $v=v(S)$ vertices and $f=f(S)$
faces. Write $v=v_{k}-m$ and $f=f_{k}-n$, where $v_{k}$ and $f_{k}$ are as
above and $m$ and $n$ are the number of vertices and faces which are in
$S_{k}$, but not in $S$. We claim that $m=v_{k}-v\leq f_{k}-f=n$. This
assertion is established by induction.
The case $k=0$ is trivial. So assume inductively that for any $i<k$ and
$S^{\prime}\subset S_{i}$ a pure subcomplex, we have
$v(S_{i})-v(S^{\prime})\leq f(S_{i})-f(S^{\prime})$.
For a pure subcomplex $S\subset S_{k}$ as above, let $S^{\prime}$ be the pure
part of $S\cap S_{k-1}$. Then, $m=m^{\prime}+m^{\prime\prime}$ and
$n=n^{\prime}+n^{\prime\prime}$, where $v(S^{\prime})=v_{k-1}-m^{\prime}$,
$f(S^{\prime})=f_{k-1}-n^{\prime}$, $m^{\prime\prime}$ is the number of
vertices in $S_{k}\smallsetminus S_{k-1}$ which are not in $S$, and
$n^{\prime\prime}$ is the number of faces in $S_{k}\smallsetminus S_{k-1}$
which are not in $S$.
We have $m^{\prime}\leq n^{\prime}$ by induction. Observe that the vertices of
$S_{k}\smallsetminus S_{k-1}$ are in one-to-one correspondence with the faces
of $S_{k}\smallsetminus S_{k-1}$. If such a vertex is not in $S$, then the
corresponding face cannot be in $S$ either. Consequently,
$m^{\prime\prime}=n^{\prime\prime}$, and $m=m^{\prime}+m^{\prime\prime}\leq
n^{\prime}+n^{\prime\prime}=n$, completing the proof of the claim.
It follows immediately that $\mu(S)\geq\mu(S_{k})=\mu_{k}$. Indeed,
$\frac{v}{f}-\frac{v_{k}}{f_{k}}=\frac{v_{k}-m}{f_{k}-n}-\frac{v_{k}}{f_{k}}=\frac{nv_{k}-mf_{k}}{f_{k}(f_{k}-n)}=\frac{\mu_{k}n-m}{f_{k}-n}\geq\frac{n-m}{f_{k}-n}\geq
0.$
Thus, $S_{k}$ is balanced. ∎
From Lemmas 20 and 24 we obtain:
###### Corollary 25.
For any $r\geq 2$ and $k\geq 0$, one has the following inequalities:
$1+\frac{1}{3\cdot
2^{k-1}-1}\,\leq\,\tilde{\mu}_{k,r}\,\leq\,1+\frac{2}{k+1}.$
Note that the obtained upper and lower bounds for $\tilde{\mu}_{k,r}$ are
independent of $r$.
We believe that $\tilde{\mu}_{k,r}=1+1/(3\cdot 2^{k-1}-1)$.
###### Proof of Theorem 1.
The main theorem is now an immediate consequence of Theorem 22 and Corollary
25:
(a) Assume that $p\ll n^{-1-2/(k+1)}$ for some $k\geq 1$. According to
Corollary 25, $\tilde{\mu}_{k,r}\leq 1+2/(k+1).$ Choosing $r=\mbox{max}(2,k)$,
it then follows from Theorem 22 (a) that $Y\in G(\Delta_{n}^{(2)},p)$ is
collapsible to a graph in at most $k$ steps, a.a.s.
(b) Assume that $p\gg n^{-1-1/(3\cdot 2^{k-1}-1)}$ for some $k\geq 1$. Then by
Theorem 16 and Lemma 24 the surface $S_{k}$ (see Example 12) embeds into $Y$,
a.a.s. Since $S_{k}$ cannot be collapsed to a graph in $k$ or fewer steps we
obtain that $Y$ is not collapsible to a graph in $k$ or fewer steps. ∎
## References
* [AS00] N. Alon, J. Spencer, The Probabilistic Method, Third edition, Wiley-Intersci. Ser. Discrete Math. Optim., John Wiley & Sons, Inc., Hoboken, NJ, 2008. MR2437651
* [BHK08] E. Babson, C. Hoffman, M. Kahle, The fundamental group of random $2$-complexes, preprint 2008. arXiv:0711.2704
* [Bol08] B. Bollobás, Random Graphs, Second edition, Cambridge University Press, 2008. Cambridge Stud. Adv. Math., 73, Cambridge, 2001. MR1864966
* [CFK10] A. Costa, M. Farber, T. Kappeler, Topology of random 2-complexes, preprint 2010.
* [ER60] P. Erdős, A. Rényi, On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci. 5 (1960), 17–61. MR0125031
* [HMS93] C. Hog-Angeloni, W. Metzler, A. Sieradski, Two-dimensional homotopy and combinatorial group theory, London Math. Soc. Lecture Note Ser., 197, Cambridge University Press, Cambridge, 1993. MR1279174
* [JŁR00] S. Janson, T. Łuczak, A. Ruciński, Random graphs, Wiley-Intersci. Ser. Discrete Math. Optim., Wiley-Interscience, New York, 2000. MR1782847
* [Koz09] D. Kozlov, The threshold function for vanishing of the top homology group of random $d$-complexes, preprint 2009.
arXiv:0904.1652.
* [LM06] N. Linial, R. Meshulam, Homological connectivity of random $2$-complexes, Combinatorica 26 (2006), 475–487. MR2260850
* [MW09] R. Meshulam, N. Wallach, Homological connectivity of random $k$-complexes, Random Structures & Algorithms 34 (2009), 408–417. MR2504405
Daniel C. Cohen
Department of Mathematics
Louisiana State University
Baton Rouge, LA 70803 USA
cohen@math.lsu.edu
www.math.lsu.edu/$\sim$cohen
Michael Farber
Department of Mathematical Sciences
Durham University
Durham, DH1 3LE, UK
Michael.farber@durham.ac.uk
http://maths.dur.ac.uk/$\sim$dma0mf/
Thomas Kappeler
Mathematical Insitutte
University of Zurich
Winterthurerstrasse 190, CH-8057
Zurich, Switzerland
thomas.kappeler@math.uzh.ch
|
arxiv-papers
| 2010-05-19T08:54:37 |
2024-09-04T02:49:10.502591
|
{
"license": "Public Domain",
"authors": "Daniel C. Cohen, Michael Farber and Thomas Kappeler",
"submitter": "Michael Farber",
"url": "https://arxiv.org/abs/1005.3383"
}
|
1005.3540
|
# The extraordinary mid-infrared spectral properties of FeLoBAL Quasars
D. Farrah11affiliation: Astronomy Centre, University of Sussex, Brighton, UK
T. Urrutia22affiliation: Spitzer Science Center, California Institute of
Technology, Pasadena, CA 91125, USA M. Lacy33affiliation: National Radio
Astronomy Observatory, Charlottesville, Virginia, USA V.
Lebouteiller44affiliation: Department of Astronomy, Cornell University,
Ithaca, NY, USA H. W. W. Spoon44affiliation: Department of Astronomy, Cornell
University, Ithaca, NY, USA J. Bernard-Salas44affiliation: Department of
Astronomy, Cornell University, Ithaca, NY, USA N. Connolly55affiliation:
Physics Department, Hamilton College, Clinton, NY 13323, USA J.
Afonso66affiliation: Observatório Astronómico de Lisboa, Faculdade de
Ciências, Universidade de Lisboa, Tapada da Ajuda, 1349-018 Lisbon, Portugal
77affiliation: Centro de Astronomia e Astrofísica da Universidade de Lisboa,
Lisbon, Portugal B. Connolly88affiliation: Department of Physics and
Astronomy, University of Pennsylvania, Philadelphia, PA 19104-6396, USA J.
Houck44affiliation: Department of Astronomy, Cornell University, Ithaca, NY,
USA
###### Abstract
We present mid-infrared spectra of six FeLoBAL QSOs at $1<z<1.8$, taken with
the Spitzer space telescope. The spectra span a range of shapes, from hot dust
dominated AGN with silicate emission at 9.7$\mu$m, to moderately obscured
starbursts with strong Polycyclic Aromatic Hydrocarbon (PAH) emission. The
spectrum of one object, SDSS 1214-0001, shows the most prominent PAHs yet seen
in any QSO at any redshift, implying that the starburst dominates the mid-IR
emission with an associated star formation rate of order 2700 M⊙ yr-1. With
the caveats that our sample is small and not robustly selected, we combine our
mid-IR spectral diagnostics with previous observations to propose that FeLoBAL
QSOs are at least largely comprised of systems in which (a) a merger driven
starburst is ending, (b) a luminous AGN is in the last stages of burning
through its surrounding dust, and (c) which we may be viewing over a
restricted line of sight range.
###### Subject headings:
galaxies: active – quasars: absorption lines – infrared: galaxies – galaxies:
evolution
## 1\. Introduction
It is now well-established that there exist intimate links between the growth
of supermassive black holes and the production of stars in galaxies. Indirect
evidence for such links comes from, for example, the tight relationship
between central black hole and stellar bulge masses (e.g. Magorrian et al.
1998; Gebhardt et al. 2000; Gadotti & Kauffmann 2009), and the coeval stellar
populations in many massive elliptical galaxies (Dunlop et al., 1996; Ellis et
al., 1997; Rakos et al., 2007), suggesting their stars formed within less than
a Gyr of each other. Direct evidence comes from the existence of galaxies that
simultaneously harbor both high rates of star formation (of order tens to
thousands M⊙yr-1) and rapid accretion onto a central black hole (Sanders &
Mirabel, 1996; Genzel et al., 1998; Farrah et al., 2003; Lonsdale et al.,
2006), all enveloped in large quantities of dust, making them extremely
luminous in the infrared. The importance of these links for understanding the
assembly history of galaxies over at least a substantial fraction of the age
of the Universe is demonstrated by the strong evolution of the luminosity
function of IR-luminous galaxies with redshift (Saunders et al., 1990; Le
Floc’h et al., 2005), and the existence of a profusion of IR-luminous sources
in the high-redshift Universe (e.g. Barger et al. 1998; Eales et al. 1999;
Coppin et al. 2006; Austermann et al. 2009).
An insightful way to study these links is to identify active galaxies at ‘key’
points, where they are either rapidly building up stellar or central black
hole mass, or shifting from one evolutionary phase to the next. One such point
may be the ”youthful” QSOs; if the starburst precedes or is coeval with the
QSO111The (optical) QSO phase is not necessarily the period in which the SMBH
gains the bulk of its mass; indeed, there is evidence that the major periods
of BH growth are coeval with or precede the starburst, see e.g. Martínez-
Sansigre et al. (2005); Merloni et al. (2010); Treister et al. (2010), then
young QSOs should be those in which the starburst is drawing to a close, and a
QSO is starting to emerge from its surrounding dust. Such objects would
provide an excellent laboratory for testing feedback mechanisms between the
starburst and AGN (e.g. Silk & Rees 1998; Ciotti & Ostriker 2007; Lagos et al.
2008; Moe et al. 2009; Ceverino & Klypin 2009).
Finding the youthful QSO population is a challenging and ongoing problem (e.g.
Sanders et al. 1988; Canalizo & Stockton 2001; Lacy 2006; Coppin et al. 2008;
Georgakakis et al. 2009; Lípari et al. 2009). Significant attention has
focused on the Broad Absorption Line (BAL) QSOs (Lynds, 1967; Weymann et al.,
1991; Brotherton et al., 1998; Schmidt & Hines, 1999; Arav et al., 2001; Green
et al., 2001; Hall et al., 2002; Reichard et al., 2003; Priddey et al., 2007;
Gibson et al., 2009), whose properties222BAL QSOs come in 3 subtypes. High
Ionization BAL QSOs (HiBALs) show absorption in CIV $\lambda$1549, NV
$\lambda$1240, SiIV $\lambda$1394 and Ly$\alpha$, and comprise about 10% of
all QSOs (Trump et al., 2006). Low Ionization BAL QSOs (LoBALs) additionally
show absorption in MgII $\lambda$2799 and other lower ionization species, and
comprise $\sim 1.3$% of all QSOs. Finally, FeLoBAL QSOs, in addition to
showing all the absorption lines seen in LoBALs, also show weak iron
absorption features (Hazard et al., 1987; Becker et al., 1997). They account
for around $0.33\%$ of all QSOs, though the exact fraction is unclear. have
been explained either as arising from being observed at a particular
orientation, or because they are young objects still partially surrounded by
dust. The properties of HiBALs are now thought to be primarily an orientation
effect (Surdej & Hutsemekers, 1987; Murray et al., 1995; Schmidt & Hines,
1999; Gallagher et al., 2007; Doi et al., 2009), while the LoBALs remain
controversial, with evidence favoring both orientation and evolution (e.g.
Voit et al. 1993; Ogle et al. 1999; Gallagher et al. 2007; Ghosh & Punsly
2007; Montenegro-Montes et al. 2008; Urrutia et al. 2009; Zhang et al. 2010).
For FeLoBALs there is also controversy, but the evidence favoring the
evolution scenario is (arguably) stronger than for the LoBALs. Based on rest-
frame UV and optical spectra, Hall et al (2002) suggest that FeLoBALs are
young objects still enveloped in dust. Similar conclusions are reached by
Gregg et al (2002), who also postulate that FeLoBALs may be associated with
mergers. Further evidence comes from the discovery of iron absorption lines in
the UV spectra of two low-redshift ULIRGs which harbor obscured AGN (Farrah et
al., 2005). Finally, mid-IR photometry observations (Farrah et al. 2007a,
hereafter F07) found that many FeLoBAL QSOs were IR-luminous, and may harbor
high rates of star formation.
Excellent distinguishing evidence of the ‘youth vs. orientation’ debate for
FeLoBAL QSOs would be observations that show directly that high rates of star
formation accompany the AGN. Given their reddened nature, a good place to look
for such evidence is in the md-infrared, in which the spectral shapes of
reddened AGN differ markedly from those of star-forming regions. The
outstanding capabilities of the Infrared Spectrograph (IRS, Houck et al. 2004)
on-board the Spitzer space telescope (Werner et al., 2004) provided a huge
step forward in available mid-IR spectroscopic capabilities, with the
opportunity to shed light on the FeLoBAL phenomenon. In this paper, we use
Spitzer-IRS observations of six FeLoBAL QSOs to examine the idea that they are
young objects. We assume a spatially flat cosmology, with $H_{0}=70$ km s-1
Mpc-1, $\Omega=1$, and $\Omega_{\Lambda}=0.7$.
## 2\. Sample Selection
We selected our sample in early 2006, with the requirements that candidates be
confirmed as FeLoBAL QSOs via rest-frame UV spectroscopy, and lie in a
redshift range where we can observe useful diagnostics with the IRS.
Accordingly, we chose six objects (Table 1) at random from F07. The F07 sample
is drawn randomly from the FeLoBAL QSO population known at the time, lie in
the redshift range $1.0<z<1.8$, thus placing important mid-IR spectral
features in the IRS bandpass, and have photometry observations at 24$\mu$m,
70$\mu$m and 160$\mu$m, which provide good constraints on IR luminosities.
A downside of this selection though is that our sample is heterogeneous. When
we selected our sample, the few known FeLoBAL QSOs had been found in several
different surveys. Since then, larger, homogeneous samples of FeLoBAL QSOs
have been published (Trump et al., 2006; Scaringi et al., 2009), but these
samples were not available to us. Accordingly, some level of bias is
inevitable in our sample, though it is difficult to quantify what effect this
may have. We therefore simply list the origins of our sample; three objects
were found via the Sloan Digital Sky Survey (Hall et al., 2002; Adelman-
McCarthy et al., 2008), one (ISO 0056-2738) was discovered serendipitously
from followup of distant clusters (Duc et al., 2002), and two (SDSS 1427+2709
& SDSS 1556+3517) were discovered during spectroscopic followup of quasars
from the FIRST survey (Becker et al., 1997; Najita et al., 2000). SDSS
1556+3517 is radio-loud, while the other five are radio quiet.
Furthermore, at least three of our sample appear to have atypical333We include
the qualifier as the rest-frame UV absorption line properties of FeLoBAL QSOs
have not been studied exhaustively iron absorption features (Hall et al.,
2002). SDSS 1154+0300 has troughs where the absorption remains significant at
velocities comparable to the spacing between absorption features (so
$\gtrsim$12,000 km s-1), causing them to overlap each other. SDSS 2215-0045
and possibly SDSS 1214-0001 have stronger FeIII absorption than FeII
absorption, suggesting that the gas in which the BALs occur is dense, hot, and
moderately highly ionized. We therefore do not correlate mid-IR spectral
properties with those of the iron absorption features.
## 3\. Methods
We observed with the IRS using the first order of the short-low module, and
both orders of the long-low module, giving observed-frame wavelength coverage
of 7.5$\mu$m to 35$\mu$m. As the 24$\mu$m fluxes of our sample span a range of
values, we used different observation times for each object so as to to give a
signal-to-noise of at least 15 in the continuum at observed-frame 24$\mu$m.
Observations were performed in staring mode, using ‘high’ accuracy peak up
observations performed with the blue array from a nearby Two Micron All-Sky
Survey (2MASS, Skrutskie et al. 2006) star.
The data were processed through the Spitzer Science Center’s pipeline software
(version 18.7), which performs standard tasks such as ramp fitting and dark
current subtraction, and produces Basic Calibrated Data (BCD) frames. Starting
with these frames, we produced reduced spectra using the SMART v8.0 software
package, following the methods described in Lebouteiller et al. (2010), which
we summarize here. Individual frames were cleaned of ‘bad’ pixels using the
IRSCLEAN task. The first and last five pixels, corresponding to regions of
reduced sensitivity on the detector, were then removed. The individual frames
at each nod position were then median combined with equal weighting on each
resolution element. Sky background was removed from each image by subtracting
the image for the same order taken with the other nod position (i.e. ‘nod-nod’
sky subtraction). One-dimensional spectra were then extracted using ‘optimal’
extraction with default parameters, and defringed using the internal SMART
algorithm. We found that, in all cases, the sources were point-like, with a
FWHM that was never wider than the PSF. We also checked that the sources were
centered in the IRS slit by extracting spectra using ‘simple’ (i.e. not PSF
weighted) extraction. We found that the resulting spectra were of slightly
lower signal-to-noise but consistent with the optimally extracted spectra,
confirming that any slit offset is insignificant.
This procedure results in separate spectra for each nod and for each order.
The spectra for each nod were inspected; features present in only one nod were
treated as artifacts and removed. The two nod positions were then combined. As
the two nod positions have slightly offset wavelength grids, we combined the
nods by first interleaving them, and then interpolating the fully sampled
spectrum onto a reference wavelength grid. The nod-combined spectra in the
three orders were then merged to give the final spectrum for each object.
Overall, we obtained excellent continuum matches between different orders.
Only in one case (SDSS 1556+3517) was there a significant order mismatch,
between the SL1 and LL2 orders. For this object, we scaled the SL1 spectrum by
a factor of 1.10 to match the blue end of the LL2 continuum
There remain however several uncertainties over error propagation in the IRS
reduction process. For example, defringing is still not completely understood,
so some residuals likely remain that are not included in the ‘formal’ errors.
Another example is the order mismatch between SL1 and LL2 for SDSS 1556+3517 -
a PSF weighted ’optimal’ extraction should in principle not produce this
effect, and while the scaling is small, we do not understand why it is
necessary. Overall therefore, we regard the resulting errors on each
resolution element to be somewhat smaller than they should be, though the
degree of this underestimate is likely insignificant.
## 4\. Results
The IRS spectra are presented in Figure 1. Spectral measurements are presented
in Table 2. We found one further object serendipitously in the slit of ISO
0056-2738, which appears to be a PAH dominated system at $z\simeq 1.42$, but
do not consider it further in this paper.
### 4.1. Spectral Features
The spectra show a variety of spectral features. Four objects show one or more
broad emission features at 6.2$\mu$m, 7.7$\mu$m, 11.2$\mu$m and 12.7$\mu$m,
attributed to bending and stretching modes in Polycyclic Aromatic Hydrocarbons
(PAHs, the 12.7$\mu$m feature also contains a contribution from
[NeII]$\lambda$12.81). The redshifts determined from the PAHs were in all
cases consistent with the optical emission line redshifts, rather than the
redshifts at which the UV absorption features peak, placing the source of the
PAH emission in the host galaxy rather than in the outflow. The PAH fluxes and
equivalent widths (EWs) were computed by integrating the flux after
subtracting off a spline interpolated local continuum. An example of the
spline fits is shown in Figure 2.
At least three objects contain a broad feature centered approximately at
9.7$\mu$m, seen in both emission and absorption, that arises from an Si-O
stretching mode in Silicate dust (Knacke & Thomson, 1973). We measured the
strengths of these features via:
$S_{sil}=ln\left(\frac{F_{obs}(9.7\mu m)}{F_{cont}(9.7\mu m)}\right)$ (1)
where $F_{obs}$ is the observed flux density at rest-frame 9.7$\mu$m, and
$F_{cont}$ is the flux at the same wavelength deduced from a spline fit to the
continuum on either side (Spoon et al., 2007; Levenson et al., 2007; Sirocky
et al., 2008).
We also searched for other features seen in IR-luminous systems, though these
are reliably measurable only with higher resolution data, so we do not present
flux measurements. Both SDSS 1214-0001 and SDSS 1427+2709 show strong
[NeIII]$\lambda$15.56, and weak but significant [NeV]$\lambda$14.32. SDSS
1214-0001 additionally has weak detections at the positions of
[ArII]$\lambda$6.99 and H2S(2)$\lambda$12.29. The other four objects show no
further features that we can identify. SDSS 1154+0300 has an apparent feature
at rest-frame $\sim 8\mu$m that we cannot find a reliable ID for, though as
the spectrum is low S/N it is possible this feature arises from the
juxtaposition of a declining PAH and a rising silicate feature. ISO 0056-2738
is too low S/N to infer the presence or otherwise of other features. We also
note that all six spectra likely contain residual structure due to fringing
that the defringing algorithm was unable to completely remove.
### 4.2. Qualitative Comparisons
#### 4.2.1 Objects 3 & 4
We start by comparing the two objects with prominent PAHs, SDSS 1214-0001 and
SDSS 1427+2709, to well studied low-redshift objects (Figure 3). Both objects
closely resemble PAH dominated ULIRGs. It is superficially interesting that
SDSS 1214-0001 is a good match to IRAS 15206+3342, a local ULIRG with iron
absorption features in its UV spectrum, though not of great significance given
that many local ULIRGs have similar mid-IR spectral shapes, see e.g. the
position of IRAS 15206+3342 in the ‘network’ plot of Farrah et al. (2009b).
SDSS 1427+2709 on the other hand is well matched by ULIRGs with weaker PAHs
and a more pronounced continuum, such as Mrk 231. Neither system resembles
heavily absorbed ULIRG spectra such as Arp 220.
Moving on to comparisons with larger samples, we are hampered as there does
not exist a comprehensive mid-IR spectroscopic survey of HiBALs or LoBALs to
compare to. Even comparing to the general AGN population though, the
peculiarity of these two objects, in particular SDSS 1214-0001, is thrown into
sharp relief. Their PAHs are extraordinarily prominent, far more so than for
any QSO, radio loud or radio quiet, that we are aware of (Haas et al., 2005;
Hao et al., 2005; Shi et al., 2006; Maiolino et al., 2007), including those
selected to be far-IR luminous (Lutz et al., 2008; Martínez-Sansigre et al.,
2008), as well as the X-ray luminous ‘type 2 QSO’ objects (Sturm et al.,
2006). There are however a few systems with prominent PAHs among the Narrow
Line Seyfert 1 (NLS1, objects with H$\beta$ FWHMs of $<2000$km s-1, Osterbrock
& Pogge 1985) population (Sani et al., 2010), and several examples of Sy2
ULIRGs with strong PAHs (Armus et al. 2007; Imanishi et al. 2007; Farrah et
al. 2007b, see also the type 2 object LH901A in Sturm et al. 2006).
#### 4.2.2 Objects 1, 2, 5 & 6
Turning to the other four sources; SDSS 2215-0045 is a good match to QSOs with
strong silicate emission and weak but detectable PAHs, of which several exist
(Hao et al., 2005). Interestingly, SDSS 2215-0045 is an excellent match to
PG1351+640, a IR-luminous, CO detected QSO with narrow BALs, FeII in emission
under H$\beta$, and slight morphological disturbance (Gelderman & Whittle,
1994; Falcke et al., 1995; Zheng et al., 2001; Evans et al., 2001).
Conversely, SDSS 1556+3517, with its weaker silicate emission feature and
negligible PAHs, is a good match both to some classical QSOs, and to NLS1
systems such as PG1211+143 (which has a high velocity outflow, Pounds & Page
2006; Shi et al. 2007) and IZw1 (which has optical and UV FeII emission and a
decaying starburst (Marziani et al., 1996; Schinnerer et al., 1998; Surace et
al., 1998)). SDSS 1154+0300 has a poorer quality IRS spectrum but is also
consistent with QSOs with weak silicate emission and negligible PAHs. ISO
0056-2738 is of too low signal-to-noise to draw any conclusions other than it
appears to be consistent with AGN generally.
#### 4.2.3 The sample as a whole
Finally, we compare the ensemble properties of our sample to other classes of
active galaxies. We start by looking for matches with IR-selected AGN samples.
One sample where we might expect overlap is with the X-ray detected LIRGs in
Brand et al. (2008b), but the two samples do not resemble each other. None of
the 16 Brand et al. 2008b objects have detectable PAHs; though 9 have some
silicate absorption, the rest are featureless power laws. Our two PAH
dominated sources resemble some sources in optically faint 70$\mu$m selected
samples (Brand et al., 2008a; Farrah et al., 2009a), but 70$\mu$m selected
samples have no sources with silicates in emission. We do better if we instead
compare to IRS observations of high-redshift 15$\mu$m selected samples
(Hernán-Caballero et al., 2009), likely because we’re selecting on hotter
dust; 15$\mu$m samples contain objects comparable to our PAH strong objects,
but have few sources with silicates in emission. The one optically selected
sample that appears reasonably matched to ours is the NLS1 sample of Sani et
al. (2010), which contains both PAH dominated objects, and a few objects with
what appears to be weak silicate emission.
### 4.3. Spectral Diagnostics
We move on to quantitative diagnostics. As our spectra are of relatively low
S/N and do not have coverage in all the IRS modules, we use simple diagnostics
that allow for easy comparisons with other samples.
We start with the PAH features. The PAH flux ratios of our sample, considered
either as functions of each other or of PAH luminosity (Figures 4 and 5) are
slightly offset from those of low-redshift ULIRGs, but lie within the
dispersion of high-redshift 70$\mu$m or 24/r selected samples. This is
straighforward to understand. Local ULIRGs are selected without a bias towards
AGN, and have lower IR luminosities, on average, than our sample. Conversely,
the Sajina et al. (2007) & Dasyra et al. (2009) samples are (arguably) biased
towards AGN, and have comparable total IR luminosities to our objects. It is
thus not surprising that the PAH flux ratios and luminosities of our sample
resemble the high redshift comparison objects. It is also interesting that the
two systems with the strongest PAH detections (SDSS 1214-0001 & SDSS
1427+2709) are also the strongest 160$\mu$m detections, consistent with the
idea that starbursts are associated with colder dust, but the small size of
our sample means this consistency could simply be coincidence, so we do not
comment on it further.
We estimate star formation rates from the PAH features using:
$SFR[M_{\odot}yr^{-1}]=1.18\times 10^{-41}L_{P}[ergs\ s^{-1}]$ (2)
where $L_{P}$ is the combined luminosity of the 6.2$\mu$m and 11.2$\mu$m PAH
features (Farrah et al., 2007b). The values are listed in Table 2444Estimating
the star formation rates using the formula in Houck et al. (2007) gives
comparable results. Overall, they are comparable to those derived for other
high redshift IR-luminous sources, including ‘bump’ selected starbursts
(Farrah et al., 2008), sub-mm selected starbusts (Pope et al., 2008; Menéndez-
Delmestre et al., 2009), and far-IR bright QSOs (Lutz et al., 2008). The star
formation rate in SDSS 1214-0001 is extraordinarily high; assuming continuous
star formation then this star formation rate is capable of manufacturing
$10^{11}$M⊙ of stars in less than 100Myr, and $\lesssim 50$Myr if we assume a
late stage exponentially decaying burst.
Next, we employ the ‘Fork’ diagnostic of Spoon et al. 2007 (Figure 6), which
employs both the 6.2$\mu$m PAH feature and the 9.7$\mu$m silicate feature.
Here our sample is not distributed in a similar way to the 70$\mu$m or 24/r
selected sources, but instead lies along the lower branch of the fork. Spoon
et al. 2007 postulate that sources move around the Fork diagram as their power
source evolves, starting on the upper branch and then moving diagonally or
vertically downwards as starburst/AGN activity clears the obscuration from the
nuclear regions. Therefore, solely from this diagnostic, we would classify our
sample as late-stage ULIRGs.
We move on to consider mid-IR continuum diagnostics. Given the limited
wavelength coverage, we use the rest-frame 6$\mu$m continuum luminosity as a
proxy for the bolometric AGN luminosity (Nardini et al., 2008; Watabe et al.,
2009). Following Nardini et al. 2008, we define:
$R=\frac{L_{6}}{L_{IR}}$ (3)
where $L_{6}$ is the 6$\mu$m luminosity as measured from the IRS spectra and
$L_{IR}$ is the 1-1000$\mu$m luminosity from F07. The fractional contribution
of an AGN to the 6$\mu$m luminosity, $\alpha_{6}$, is then:
$\alpha_{6}=\frac{1}{R_{S}-R_{A}}\left(\frac{R_{A}R_{S}}{R}-R_{A}\right)$ (4)
where $R_{S}$ and $R_{A}$ are the equivalents of $R$ for ‘pure’ starbursts and
AGN, with values of $(117\pm 8)\times 10^{-4}$ and $0.32\pm 0.1$ respectively
(Nardini et al., 2008). Therefore, the fractional contribution of the AGN to
the total IR luminosity, $\alpha_{bol}$, is:
$\alpha_{bol}=\frac{\alpha_{6}}{\alpha_{6}+\left(\frac{R_{A}}{R_{S}}\right)(1-\alpha_{6})}$
(5)
The resulting $\alpha_{bol}$ values are listed in Table 1. For comparison, we
also list the $\alpha_{bol}$ values obtained from the SEDs in F07. In four
cases the values are consistent, but for two (SDSS 1556+3517 & SDSS 2215-0045)
they are not. We think it likely that the SED derived values are more
reliable, as they are direct measurements from the (albeit sparsely sampled)
full SEDs, while the values computed using Equation 5 are calibrated using a
wide range of sources. It is interesting that these two objects are the only
two with detected silicate emission, but we do not know if this is the cause
of the discrepancy, or coincidence. Considering either method though, the
sources with strong PAHs have weak AGN contributions to the total IR
luminosity, and the range in $\alpha_{bol}$ values for the whole sample is
similar to that seen in local ULIRGs.
Finally, we fit the source with the most prominent PAHs, SDSS 1214-0001, with
PAHFIT (Smith et al., 2007). This is not an attempt to reconstruct star
formation parameters as PAHFIT is intended for systems where emission from
stars provides at least most of the flux across the mid-IR, but it does serve
as a test of how much of the mid-IR flux in SDSS 1214-0001 is attributable to
star formation. The result is shown in Figure 7. The fit is good, with
$\chi^{2}_{red}=1.4$. The fit is poor at $\lesssim 6\mu$m, with a steeply
rising contribution from ‘starlight’ with decreasing wavelength that is
probably an attempt to fit the AGN continuum. At $\lambda\gtrsim 6\mu$m
though, we obtain an excellent fit. The PAHs are well fitted by the Drude
profiles, and the continuum is predicted to mostly come from dust at 50k, with
small contributions from the 135k and 300k components.
## 5\. Discussion & Conclusions
A general caveat to all of what follows is our small and heterogeneous sample.
Our conclusions should thus be regarded as tentative.
The mid-IR spectra of our sample span a wide range of shapes. We see classical
QSO spectra with hot silicate dust, together with classical starburst spectra
with strong PAHs. The ‘starburst’ spectra have more prominent PAHs than those
seen in any other QSO so far observed, while the hot silicate dust sources do
not have a close match in any purely mid/far-IR selected sample that we are
aware of. It is possible that our heterogeneous selection leads to the
heterogeneity of the spectra, but if this were true then we might expect the
ISO selected object to have the strongest PAHs, which is not the case. Indeed,
the two spectra at the extreme ends of the range of spectral shapes are both
SDSS objects. The star formation rates of our sample span levels comparable to
those in the most luminous starbursts found in any survey at any wavelength,
to those seen in moderately IR-luminous AGN. The spectral diagnostics mark our
sample as late-stage ULIRGs with a wide range of fractional AGN contributions.
These results, combined with their optical classification as reddened QSOs
with a strong outflow, are in principle compatible with a continuum of ‘end-
of-ULIRG’ vs ‘peculiar QSO’ contributions to the FeLoBAL QSO population. To
frame the discussion, we describe three points on this continuum in detail:
1: FeLoBAL QSOs are a transition stage between starburst dominated ULIRGs and
QSOs. As our sample span the entirety of this transition, and because FeLoBALs
are observed to be rare555A further reason behind the rarity of FeLoBALs is
that they they are hard to detect; FeLoBAL QSOs are red, have few or no broad
emission lines and no UV excess, and are thus hard to find in standard QSO
searches (see e.g. Appenzeller et al. 2005). The iron absorption features
themselves are weak., we can set constraints on the timescale of this
transition depending on how intrinsically common the Fe absorption features
are. If Fe absorption is ubiquitous in such a transition, then the transition
must be short in comparison to the ULIRG or QSO lifetime. Taking the ULIRG and
QSO lifetimes to be $\sim$108 years (Tacconi et al., 2008), then the
transition must take $\lesssim$107 years (see also Gregg et al. 2002). If on
the other hand the iron absorption features are intrinsically rare, then the
transition can take longer.
2: FeLoBALs are comprised of comparable fractions of two galaxy populations,
one that is transitioning from a ULIRG to a QSO, and another that is an
unusual phase that QSOs go through, unconnected to mergers. Here the
heterogeneity of our sample arises from observing two unconnected populations.
3: FeLoBAL QSOs are entirely an ‘unusual QSO’ class, and have nothing to do
with mergers or otherwise ‘transitioning’ systems. Instead, the iron
absorption features mark QSOs with both an outflow and atypically large
amounts of iron in their ISM.
Formally, we cannot rule any of these scenarios out. The third though seems
unlikely. It is hard to see how such a heterogeneous set of mid-IR spectra
could arise from observing a peculiar class of QSO in which nothing more
interesting than an outflow with large quantities of iron is going on,
especially if we assume that the restrictive selection on rest-frame UV
properties means we are viewing them over a restricted range in line of sight.
Furthermore, most systems with star formation rates in the range seen in our
sample are ULIRGs, which (at low redshift, at least) are nearly all mergers.
Distinguishing between the first and second scenarios is however more
difficult. On one hand, the first scenario uses a single origin for which
there is independent evidence from studying ULIRGs. The second scenario thus
seems contrived, as it proposes two origins where one will do. On the other
hand, our sample is small, and it is tempting to place undue emphasis on the
properties of SDSS 1214-0001, as it is so peculiar. If this object were
removed then the ‘unusual QSO’ scenario would become more attractive. That
said, the counter-argument also holds; remove (say) SDSS 1556+3517, and the
‘end-of-ULIRG’ scenario becomes more attractive. We are mindful though that it
is only the properties of SDSS 1214-0001 (and to an extent SDSS 1427+2709)
that make us seriously consider the ‘end-of-ULIRG’ scenario.
Recent work that may shed light on this is the study of NLS1’s by Sani et al.
(2010). Their spectra show that at least some NLS1’s harbor intense star
formation and a weak mid-IR AGN continuum. NLS1’s also have strong optical Fe
emission lines, a large soft X-ray excess, and exhibit rapid, large amplitude
X-ray variability. Furthermore, their black hole masses appear to be smaller
than those in BLS1’s of comparable luminosity, though there is controversy
over whether NLS1’s lie below (Peterson et al., 2000; Grupe & Mathur, 2004) or
on (Botte et al., 2005; Komossa & Xu, 2007; Decarli et al., 2008) the
$M_{BH}-\sigma_{*}$ relation (Tremaine et al., 2002). On one level this hints
at interesting links between NLS1s and FeLoBAL QSOs; for example, FeLoBAL QSOs
may be analogues of NLS1s seen more pole-on than edge-on, and in which a
strong outflow has formed. Exploring this idea in detail requires
significantly larger samples, so we do not pursue it here. More generally
however, it implies that relative orientation may also play a role in the
FeLoBAL QSO phenomenon.
We think it likely that the simplest solution that is consistent with all the
observations, and the apparent rarity of FeLoBALs, is the correct one. We
therefore propose that FeLoBAL QSOs as a class are rapidly evolving, youthful
QSOs in which a starburst is coming to an end, and the AGN is in the last
stages of burning through its surrounding dust. We also propose, with more
reserve, that (1) we view FeLoBAL QSOs over a restricted line of sight range,
where we see them more pole-on than edge on, and (2) the outflow has in some
cases partially cleared the dust from around the AGN, leaving the IR emission
from the starburst to dominate.
There are three ways to test this conclusion. First, FeLoBAL QSO hosts should
be in the final stages of merging, so their host galaxies should show slight
signs of morphological disturbance. Second, larger mid-IR spectroscopic
surveys of FeLoBAL QSOs should show a similar range of spectral shapes to
ours. Third, far-IR photometric surveys of FeLoBALs should find both a
moderate enhancement of the fraction of far-IR bright sources compared to the
classical QSO population, corresponding to those FeLoBAL QSOs that still
harbor high star formation rates, and a correlation between far-IR luminosity
and PAH equivalent width. Moreover, it would be useful to perform radiative
transfer modelling of a BAL wind through a starburst to determine the
conditions under which iron absorption is observed in such systems, and so
provide insight into whether our conclusions are feasible.
We close with a caveat and a speculative comment. First, there is an important
parameter that we cannot address; the way in which iron absorption features in
FeLoBALs signpost starburst and AGN activity. For example, if the Fe
absorption only occurs when a starburst is ending and an AGN is blowing away
its surrounding dust, but can occur throughout such a transition, then our
first scenario is likely correct. As a contrived counter-example, if Fe
absorption can only occur during the early phases of such a transition, but
can occur randomly in reddened AGN in which there is no significant star
formation, then in a small sample such as ours we would conclude, incorrectly,
that all FeLoBAL QSOs were a ULIRG-to-QSO transition where the Fe absorption
is present throughout. Resolving this issue is beyond the scope of this paper,
so we note it as a caveat to our conclusions.
Second, Hall et al. (2002) note that two of our sample, SDSS 1214-0001 and
SDSS 2215-0045 have [FeIII]/[FeII] ratios greater than unity, implying the
BALs arise in unusually dense, hot gas. The two ratios however are different;
SDSS 2215-0045 has a [FeIII]/[FeII] ratio much greater than unity, while SDSS
1214-0001 has an [FeIII]/[FeII] ratio only slightly greater than unity. While
conclusions drawn on only two objects are not trustworthy, this difference in
ratio is consistent with our proposed evolutionary sequence. SDSS 1214-0001
has strong PAHs and silicates in absorption, while SDSS 2215-0045 has weak
PAHs and silicates in emission, implying that SDSS 1214-0001 is at an earlier
evolutionary phase than SDSS 2215-0045. Moreover, the enhanced [FeIII]/[FeII]
ratio in SDSS 2215-0045 compared to SDSS 1214-0001 could be interpreted as the
BAL outflow in SDSS 2215-0045 being more developed666This is consistent with
the ‘youth’ hypothesis that Hall et al. (2002) give for SDSS 2215-0045.
Further support for this idea would be an [FeIII]/[FeII] ratio greater than
unity in SDSS 1427+2709, but the only published spectrum we are aware of is in
Becker et al. (2000), in which the FeIII lines lie in a noisy region at the
extreme blue end of the bandpass. They do not however appear to be
significantly stronger than the FeII lines, so we simply note this as an
interesting avenue for future work.
We thank the referee for a helpful report. 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 was provided by NASA. This research has made extensive
use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet
Propulsion Laboratory, California Institute of Technology, under contract with
NASA. DF thanks STFC for support via an Advanced Fellowship. NC gratefully
acknowledges support from a Cottrell College Science Award from the Research
Corporation and from an NSF RUI grant.
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Table 1FeLoBAL QSO Sample
Object | RA (2000) | Dec | $z_{sys}$aaSystemic redshift from narrow optical emission lines | $z_{abs}$bbRedshift of peak absorption from the broad UV absorption lines | $f_{24}$ | $f_{70}$ | $f_{160}$ | $\alpha_{bol}|_{6\mu m}$ddFractional AGN luminosity, computed using the prescription in §4.3 (Nardini et al., 2008). | $\alpha_{bol}|_{sed}$eeFractional AGN luminosity, derived from the SED fits in F07. | $L_{IR}$
---|---|---|---|---|---|---|---|---|---|---
(1) ISO J005645.1-273816 | 00 56 45.2 | -27 38 15.6 | 1.78 | 1.75 | 1.6 | $<$7.0 | $<$50 | 0.09 - 0.40 | $>0.26$ | 12.5-13.0
(2) SDSS J115436.60+030006.3 | 11 54 36.6 | 03 00 06.4 | 1.46ccHall et al. (2002) | 1.36 | 7.6 | 17.9 | $<$50 | 0.35 - 0.60 | $>0.40$ | 12.9-13.1
(3) SDSS J121441.42-000137.8 | 12 14 41.4 | -00 01 37.9 | 1.05 | 0.99 | 4.7 | 38.3 | 78.9 | 0.07 - 0.15 | $<0.37$ | 12.7-12.9
(4) SDSS J142703.62+270940.3 | 14 27 03.6 | 27 09 40.3 | 1.17 | ? | 4.8 | 32.6 | 68.1 | 0.09 - 0.17 | $<0.61$ | 12.8-13.0
(5) SDSS J155633.78+351757.3 | 15 56 33.8 | 35 17 58.0 | 1.50 | 1.48 | 13.9 | 24.7 | $<$50 | 0.40 - 0.70 | $>0.75$ | 13.1-13.3
(6) SDSS J221511.93-004549.9 | 22 15 11.9 | -00 45 49.9 | 1.48 | 1.36 | 10.4 | 27.2 | $<$50 | 0.09 - 0.30 | $>0.36$ | 13.0-13.4
Note. — All flux densities are quoted in mJy. Errors are typically 10% at
24$\mu$m, 20% at 70$\mu$m, and 25% at 160$\mu$m. Luminosities are the
logarithm of the rest-frame 1-1000$\mu$m luminosity, in units of solar
luminosities (3.826$\times 10^{26}$ Watts), taken from F07. Limits and ranges
are 3$\sigma$.
Table 2Spectral Measurements
Object | PAH 6.2$\mu$m | PAH 7.7$\mu$m | PAH 11.2$\mu$m | $S_{sil}$ | SFRaaStar formation rate, determined from Equation 2. Error is solely that derived from the uncertainty in the fluxes.
---|---|---|---|---|---
| Flux | EW | Flux | EW | Flux | EW | | M⊙ yr-1
(1) ISO 0056-2738 | 6.1$\pm$8.9 | 0.06$\pm$0.09 | 6.0$\pm$8.0 | 0.08$\pm$0.12 | 4.1$\pm$3.9 | 0.08$\pm$0.08 | 0.25$\pm$0.11 | $2600\pm 2500$
(2) SDSS 1154+0300 | 5.2$\pm$5.7 | 0.01$\pm$0.01 | 15.1$\pm$9.2 | 0.04$\pm$0.02 | 0.3$\pm$4.6 | 0.01$\pm$0.02 | 0.15$\pm$0.08 | $900\pm 1100$
(3) SDSS 1214-0001 | 16.3$\pm$3.3 | 0.06$\pm$0.01 | 56.5$\pm$7.0 | 0.25$\pm$0.05 | 22.5$\pm$5.6 | 0.24$\pm$0.06 | -0.25$\pm$0.05 | $2700\pm 500$
(4) SDSS 1427+2709 | 7.3$\pm$2.6 | 0.03$\pm$0.01 | 25.0$\pm$6.4 | 0.10$\pm$0.03 | 2.4$\pm$2.1 | 0.02$\pm$0.01 | -0.10$\pm$0.05 | $900\pm 350$
(5) SDSS 1556+3517 | 1.0$\pm$3.0 | 0.01$\pm$0.01 | 11.2$\pm$6.6 | 0.02$\pm$0.01 | 3.0$\pm$3.9 | 0.01$\pm$0.02 | 0.22$\pm$0.04 | $700\pm 800$
(6) SDSS 2215-0045 | 10.5$\pm$5.3 | 0.03$\pm$0.01 | 28.1$\pm$6.8 | 0.09$\pm$0.02 | 7.7$\pm$3.7 | 0.02$\pm$0.01 | 0.62$\pm$0.10 | $2700\pm 1000$
Note. — PAH fluxes are in units of $10^{-22}$W cm-2 and equivalent widths in
$\mu$m.
Figure 1.— Spitzer-IRS low-resolution spectra of our sample, plotted in the
rest-frame using the optical emission line redshifts. The data are plotted in
black, with 1$\sigma$ errors in grey. Flux densities are in mJy and
wavelengths are in $\mu$m. The number in the top left of each panel is the ID
number in Table 1. The green solid (dashed) lines mark the 6.2$\mu$m,
7.7$\mu$m, 8.6$\mu$m, 11.2$\mu$m and 12.7$\mu$m PAH features assuming the
systemic (peak absorption, if available) redshift. The red lines perform the
same function for the [Ne II]$\lambda$12.81, [Ne V]$\lambda$14.32 and [Ne
III]$\lambda$15.56 lines. The yellow shaded region shows the approximate
extent of the 9.7$\mu$m silicate absorption feature, but see Bowey et al.
1998; Draine 2003; Nikutta et al. 2009.
Figure 2.— An example of the spline fits used to determine the shape of the
underlying continuum for the PAH fluxes, in this case for the 6.2$\mu$m PAH
feature in SDSS 1214-0001. The (rest-frame) spectrum is plotted in black. The
points used to define the continuum are shown in green, while the spline fit
and its residuals are shown in blue. The errors on the spectrum have been
omitted for clarity. The dashed line shows the adopted peak wavelength of the
PAH feature, while the dotted lines show the lower and upper limits of the
integration. This plot also demonstrates the excellent agreement between the
two nods; the ‘double’ blue error bars show separately the error bars for the
two nod positions, which have slightly offset wavelength grids.
Figure 3.— Comparison of our spectra to well studied low-redshift objects.
Objects showing silicate absorption in the left panel, and the others are in
the right panel. Comparison spectra are taken from Brandl et al. (2006); Armus
et al. (2007); Shi et al. (2007) and Imanishi et al. (2007). Left: Red - Mrk
231. Orange - composite of local $\simeq 10^{11}$L⊙ starbursts. Green - Arp
220. Blue - IRAS 15206+3342 (a local ULIRG with iron absorption features in
its UV spectrum). Right: Red - 3C 273. Orange - I Zw 1 (a low-z NLS1). Green -
PG 1211+143. Blue - PG 1351+640 (both far-IR luminous QSOs).
Figure 4.— PAH ratio diagnostic plots, divided by comparisons to (top row)
low-redshift and (bottom row) high redshift samples. Black: FeLoBALs. Green:
low-z ULIRGs with measurable 7.7$\mu$m PAHs (Spoon et al., 2007; Desai et al.,
2007). Red: low-z AGN (Weedman et al., 2005). Blue: low-z starbursts with
L${}_{IR}<10^{11}$L⊙ (Brandl et al., 2006). Purple: high-z $70\mu$m selected
(Brand et al., 2008a; Farrah et al., 2009a). Grey: high-z ‘bump’ selected
(Farrah et al., 2008). Orange: high-$24\mu$m/0.7$\mu$m selected Dasyra et al.
(2009). Cyan: high-z $24\mu$m/8$\mu$m and $24\mu$m/0.7$\mu$m selected Sajina
et al. (2007). Yellow: high-z sub-mm selected (Pope et al., 2008; Menéndez-
Delmestre et al., 2009).
Figure 5.— PAH luminosity diagnostic plots. Color coding is the same as in
Figure 4.
Figure 6.— ‘Fork’ diagnostic diagram (Spoon et al., 2007). The FeLoBALs are
plotted in black, while the other points are color-coded as in Figure 4. Plots
of the composite spectra for each region are in figure 2 of Spoon et al.
(2007). With respect to the comparison objects plotted in Figure 3; Mrk 231 is
class 1A, IRAS 15206+3342 is class 1B, Arp 220 is class 3B, the starburst
composite is class 1C, and the remaining objects are all class 1A. The ‘bump’
sources lie close together on this plot, so, for clarity, we plot their
average as a single point.
Figure 7.— PAHfit results for SDSS 1214-0001. Details of the model parameters
can be found in Smith et al. (2007). Vertical black lines: observed data.
Green: combined best-fit model. Red solid lines: thermal dust continuum
components. Magenta: starlight continuum. Black solid: combined
starlight+thermal dust continuum. Violet: fine structure+molecular features.
Blue: PAH features. Black dotted: extinction curve.
|
arxiv-papers
| 2010-05-19T20:00:07 |
2024-09-04T02:49:10.514719
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. Farrah (Sussex), T. Urrutia (Caltech), M. Lacy (NRAO), V.\n Lebouteiller (Cornell), H. W. W. Spoon (Cornell), J. Bernard-Salas (Cornell),\n N. Connolly (Hamilton College), J. Afonso (Lisbon), B. Connolly\n (Pennsylvania), J. Houck (Cornell)",
"submitter": "Duncan Farrah",
"url": "https://arxiv.org/abs/1005.3540"
}
|
1005.3640
|
# Algebraic and analytic properties of quasimetric spaces with
dilations††thanks: This research was partially supported by Federal Target
Grant ”Scientific and educational personnel of innovation Russia” for
2009-2013 (government contract No. P2224) and by the program “Leading
Scientific Schools” (project N. NSh-5682.2008.1).
Svetlana Selivanova, Sergei Vodopyanov
###### Abstract
We provide an axiomatic approach to the theory of local tangent cones of
regular sub-Riemannian manifolds and the differentiability of mappings between
such spaces. This axiomatic approach relies on a notion of a dilation
structure which is introduced in the general framework of quasimetric spaces.
Considering quasimetrics allows us to cover a general case including, in
particular, minimal smoothness assumptions on the vector fields defining the
sub-Riemannian structure. It is important to note that the theory existing for
metric spaces can not be directly extended to quasimetric spaces.
Key words: Dilations, local group, contractible group, Mal’tsev’s theorem,
tangent cone, Carnot-Carathéodory space, differentiability
MSC: Primary 22E05, 53C17; Secondary 20F17, 22D05, 54E50.
## 1 Introduction
We study algebraic and analytic properties of quasimetric spaces endowed with
dilations (roughly speaking, dilations are continuous one-parameter families
of contractive homeomorphisms given in a neighborhood of each point).
Our work is motivated by investigation of metric properties of Carnot-
Carathéodory spaces, also referred to as sub-Riemannian manifolds which model
nonholonomic processes and naturally arise in many applications (see e. g. [1,
2, 5, 11, 12, 25, 27, 18, 29, 32, 36, 39, 45, 49] and references therein).
Let us first recall the “classical” definition of a sub-Riemannian manifold.
Given a smooth connected manifold $\mathbb{M}$ of dimension $N$ and smooth
“horizontal” vector fields $X_{1},\ldots,X_{m}\in C^{\infty}$ on $\mathbb{M}$
(where $m\leq N$), it is assumed that these vector fields span, together with
their commutators, the tangent space to $\mathbb{M}$ at each point
(Hörmander’s condition [27]). By Rashevskiǐ-Chow’s Theorem, any two points of
$\mathbb{M}$ can be connected by a horizontal curve and, therefore, there
exists an intrinsic sub-Riemannian metric $d_{c}$ on $\mathbb{M}$ defined as
the infimum over lengths of all horizontal curves.
Recently discovered applications have lead to considering a more general
situation [28, 29, 46, 54, 55, 56] when
1) a maximal possible reduction of smoothness of the vector fields is made
(see also [4, 22, 35]);
2) instead the Hörmander’s condition, a weaker one of a “weighted” filtration
of $T\mathbb{M}$ (see Definition 10) is assumed (see also [17, 18, 22, 39,
49]).
Under these general assumptions, the intrinsic metric $d_{c}$ might not exist,
but a certain quasimetric (a distance function meeting a generalized triangle
inequality, see Definition 1) can be introduced (see [39] where various
quasimetrics induced by families of vector fields on $\mathbb{R}^{N}$ were
studied).
On the other hand, recent development of analysis on general metric spaces has
lead to the question of describing the most general approach to the metric
geometry of sub-Riemannian manifolds. Among possible approaches is considering
metric spaces with dilations [2, 6, 9, 18].
Motivated by these considerations, we extend the notion of a dilation
structure to quasimetric spaces and investigate local properties of the
obtained object.
In 1981 M. Gromov has defined [23, 24] the tangent cone to a metric space
$({\mathbb{X}},d)$ at a point $x\in{\mathbb{X}}$ as the limit of pointed
scaled metric spaces $({\mathbb{X}},x,\lambda\cdot d)$ (when
$\lambda\to\infty$) w. r. t. Gromov-Hausdorff distance. This notion
generalizes the concept of the tangent space to a manifold and is useful in
the general theory of metric spaces (see e. g. [3, 11, 13, 15, 43]), in
particular, Carnot-Carathéodory spaces [32, 34].
A straightforward generalization of Gromov’s theory would make no sense for
quasimetric spaces, see Remark 6. In [46, 47] a convergence theory for
quasimetric spaces with the following properties was developed:
1) it includes the Gromov-Hausdorff convergence for metric spaces as a
particular case;
2) the limit is unique up to isometry for boundedly compact quasimetric
spaces;
3) it allows to introduce the notion of the tangent cone in the same way as
for metric spaces.
In [47] the existence of the tangent cone (w. r. t. the introduced
convergence) to a quasimetric space with dilations is proved (see Definition
2, Axioms (A0) —(A3), and Theorem 2). This statement contains as a particular
case a similar result by M. Buliga for metric spaces, see for instance [6],
where an axiomatic approach to metric spaces with dilations is introduced. A
similar approach was informally sketched by A. Bellaiche [2].
The main results of the present paper are Theorems 4 and 7. Theorem 4 (cf.
[7]) asserts that an additional axiom (A4) (saying that the limit of a certain
combination of dilations exists) allows to describe the algebraic structure of
the tangent cone: it is a simply connected Lie group, the Lie algebra of which
is graded and nilpotent.
In particular, this result allows to define the differential of a mapping
acting between two quasimetric spaces with dilations in the same way as it is
done in [50] for Carnot-Carathéodory spaces. A brief comparison of this
approach with Margulis-Mostow’s concept of differentiability [32] is given
below in Remark 14.
Thus, Theorem 4 allows to establish algebraic and analytic properties of the
considered space from metric and topological assumptions only. In the present
paper we do not attempt to prove that axioms of a dilation structure recover
sub-Riemannian geometry when the underlying space is a manifold (or which
axioms should be added to prove this). But we prove that
1) regular sub-Riemannian manifolds are examples of quasimetric spaces with
dilations (Theorem 7);
2) the tangent cones to quasimetric spaces with dilations are the same
algebraic objects as for regular sub-Riemannian manifolds (Theorem 4),
which can be viewed as a first step in this direction.
In our opinion, the proof of Theorem 4 is interesting in its own right. The
main step is to apply a theorem on local and global topological groups due to
A. I. Mal’tsev [31], which helps to overcome difficulties concerned with
investigation of a local version of the Hilbert’s Fifth Problem [58, 19, 37],
see Remark 2. As an auxiliary assertion we prove a generalized triangle
inequality for local groups endowed with (quasi)metrics and dilations (see
Proposition 8, Assertion 3)), which is of independent interest and gives an
alternative proof of a similar fact for (global) homogeneous groups [18].
In Section 4, we describe regular Carnot-Carathéodory spaces as the main
example of quasimetric spaces with dilations. In this case Axiom (A3) is just
a local approximation theorem, and (A4) is a consequence of estimates on
divergence of integral lines of the initial vector fields and the
nilpotentized ones.
In this paper we extend the approach to the subject given in our short
communication [57].
We are grateful to Isaac Goldbring for a discussion on some algebraic aspects
of the subject under consideration (see Remark 9) and for the references [40,
20]. We thank also the anonymous referee for the careful reading of our paper,
interesting questions and references, as well as useful hints concerning the
presentation and exposition of our results.
## 2 Basic notions and preliminary results
###### Definition 1.
A quasimetric space $({\mathbb{X}},d_{\mathbb{X}})$ is a topological space
${\mathbb{X}}$ with a quasimetric $d_{\mathbb{X}}$. A quasimetric is a
mapping $d_{\mathbb{X}}:{\mathbb{X}}\times{\mathbb{X}}\to\mathbb{R}^{+}$ with
the following properties:
(1) $d_{\mathbb{X}}(u,v)\geq 0$; $d_{\mathbb{X}}(u,v)=0$ if and only if $u=v$
$($non-degeneracy$)$;
(2) $d_{\mathbb{X}}(u,v)\leq c_{\mathbb{X}}d_{\mathbb{X}}(v,u)$ where $1\leq
c_{\mathbb{X}}<\infty$ is a constant independent of $u,v\in{\mathbb{X}}$
(generalized symmetry property);
(3) $d_{\mathbb{X}}(u,v)\leq
Q_{\mathbb{X}}(d_{\mathbb{X}}(u,w)+d_{\mathbb{X}}(w,v))$ where $1\leq
Q_{\mathbb{X}}<\infty$ is a constant independent of $u,v,w\in{\mathbb{X}}$
$($generalized triangle inequality$)$;
(4) the function $d_{\mathbb{X}}(u,v)$ is upper semi-continuous on the first
argument.
If $c_{\mathbb{X}}=1$, $Q_{\mathbb{X}}=1$, then
$({\mathbb{X}},d_{\mathbb{X}})$ is a metric space.
###### Remark 1.
Note that some authors introduce the notion of a quasimetric space without
assuming neither this space be topological nor the quasimetric be continuous
in any sense. Within such framework, the quasimetric balls need not be open
(see e. g. [41, 14, 26]). However, due to a theorem by R. A. Macìas and C.
Segovia [30], any quasimetric $d$ is equivalent to some other quasimetric
$\tilde{d}$, the balls associated to which are open (such a quasimetric looks
like $\rho(x,y)^{\frac{1}{\beta}}$, where $0<\beta\leq 1$ and $\rho(x,y)$ is a
metric) and, hence, define a topology.
In the present paper we study tangent cone questions. It is important to note,
that having the tangent cone to a (quasi)metric space, one can say nothing
about the existence of the tangent cone to the space with an equivalent
(quasi)metric, thus we would like the balls defined by the initial quasimetric
be open. For this reason we add the upper-continuity condition (4) to the
Definition 1 of a quasimetric space (as it is done e. g. in [49] for the case
of $\mathbb{R}^{n}$). This condition guarantees that the balls
$B^{d_{\mathbb{X}}}(x,r)$ are open sets, and that convergence w. r. t. the
initial topology of $\mathbb{X}$ implies convergence w. r. t. the topology
defined by $d_{\mathbb{X}}$.
Actually, we can assume the initial topology on $\mathbb{X}$ coincide with the
topology induced by the equivalent quasimetric $\tilde{d}$. Then the
topologies induced by $d$ and convergence w. r. t. initial topology on
$\mathbb{X}$ are equivalent. Further we always assume, w. l. o. g., this to
hold.
We denote by $B^{d_{\mathbb{X}}}(x,r)=\\{y\in\mathbb{X}\mid
d_{\mathbb{X}}(y,x)<r\\}$ a ball centered at $x$ of radius $r$, w. r. t. the
(quasi)metric $d_{\mathbb{X}}$. The symbol $\bar{A}$ stands for the closure of
the set $A$. A (quasi)metric space ${\mathbb{X}}$ is said to be boundedly
compact if all closed bounded subsets of ${\mathbb{X}}$ are compact.
###### Definition 2.
Let $({\mathbb{X}},d)$ be a complete boundedly compact quasimetric space and
the quasimetric $d$ be continuous on both arguments. The quasimetric space
${\mathbb{X}}$ is endowed with a dilation structure, denoted as
$({\mathbb{X}},d,\delta)$, if the following axioms (A0) — (A3) hold.
(A0) For all $x\in{\mathbb{X}}$ and for $\varepsilon\in(0,1]$, in some
neighborhood $U(x)$ of $x$ there are homeomorphisms called dilations
$\delta_{\varepsilon}^{x}:U(x)\to V_{\varepsilon}(x)$ and
$\delta_{\varepsilon^{-1}}^{x}:W_{\varepsilon^{-1}}(x)\to U(x)$, where
$V_{\varepsilon}(x)\subseteq W_{\varepsilon^{-1}}(x)\subseteq U(x)$. The
family $\\{\delta^{x}_{\varepsilon}\\}_{\varepsilon\in(0,1]}$ is continuous on
$\varepsilon$ (w. r. t. the initial topology on $\mathbb{X}$, see Remark 1,
and the ordinary topology on $(0,1]$). It is assumed that there exists an
$R>0$ such that $\bar{B}^{d}(x,R)\subseteq U(x)$ for all $x\in{\mathbb{X}}$,
and for all $\varepsilon<1$ and $\tilde{r}>0$ with the property
$\bar{B}^{d}(x,\tilde{r})\subseteq U(x)$ we have the inclusion
$B^{d}(x,\tilde{r}\varepsilon)\subseteq\delta_{\varepsilon}^{x}B^{d}(x,\tilde{r})\subset
B^{d}(x,\tilde{r})$.
(A1) For all $x\in{\mathbb{X}}$, $y\in U(x)$, we have
$\delta_{\varepsilon}^{x}x=x,\ \delta_{1}^{x}=\text{id},\
\lim\limits_{\varepsilon\to 0}\delta_{\varepsilon}^{x}y=x$.
(A2) For all $x\in{\mathbb{X}}$ and $u\in U(x)$, we have
$\delta_{\varepsilon}^{x}\delta_{\mu}^{x}u=\delta_{\varepsilon\mu}^{x}u$
provided that both parts of this equality are defined.
(A3) For any $x\in{\mathbb{X}}$, uniformly on $u,v\in\bar{B}^{d}(x,R)$ there
exists the limit
$\lim\limits_{\varepsilon\to
0}\frac{1}{\varepsilon}d(\delta_{\varepsilon}^{x}u,\delta_{\varepsilon}^{x}v)=d^{x}(u,v).$
(2.1)
If the function $d^{x}:U(x)\times U(x)\to\mathbb{R}^{+}$ is such that
$d^{x}(u,v)=0$ implies $u=v$, then the dilation structure is called
nondegenerate.
If the convergence in $(A3)$ is uniform on $x$ in some compact set, then the
dilation structure is said to be uniform.
If the following axiom (A4) holds, then we say that ${\mathbb{X}}$ is endowed
with a strong dilation structure.
(A4) The limit of the value
$\Lambda_{\varepsilon}^{x}(u,v)=\delta_{\varepsilon^{-1}}^{\delta_{\varepsilon}^{x}u}\delta_{\varepsilon}^{x}v$
exists:
$\lim\limits_{\varepsilon\to
0}\Lambda_{\varepsilon}^{x}(u,v)=\Lambda^{x}(u,v)\in B^{d}(x,R),$ (2.2)
This limit is uniform on $x$ in some compact set and $u,v\in B^{d}(x,r)$ for
some $0<r\leq R$. See Proposition 4 regarding possible choices of $r$.
###### Remark 2.
These axioms of dilations are a slight modification and simplification of
those proposed in [6] for metric spaces. Essential for proving Theorem 4 is
that, in (A0), we require the continuity of dilations on the parameter
$\varepsilon$ which was missed in [6]. Note also that axioms (A1), (A2), (A4)
do not depend on the quasimetric. The condition $\lim\limits_{\varepsilon\to
0}\delta_{\varepsilon}^{x}y=x$ informally states that the topological space
${\mathbb{X}}$ is locally contractible.
###### Example 1.
In the case when ${\mathbb{X}}$ is a Riemannian manifold, dilations can be
introduced as homotheties induced by the Euclidean ones. See [6]–[10] for more
examples.
###### Remark 3.
For a general (quasi)metric space $(\mathbb{X},d)$, the closure of a ball need
not coincide with the corresponding closed ball, only the inclusion
$\bar{B}^{d}(x,r)\subseteq\\{y:d(y,x)\leq r\\}$ holds. But, in the case of a
(quasi)metric space endowed with a dilation structure, also the converse
inclusion is true. Indeed, let $z\in\\{y:d(y,x)\leq r\\}$ be such that
$d(z,x)=r$; let $z_{n}=\delta^{x}_{1-\varepsilon_{n}}z\in B^{d}(x,r)$, where
$\varepsilon_{n}\to 0$. Then $d(z_{n},z)\to 0$, according to (A0), (A1) and
Remark 1, hence $z\in\bar{B}^{d}(x,r)$.
###### Remark 4.
By virtue of (A3) and continuity of $d(u,v)$, the function $d^{x}(u,v)$ is
continuous on both arguments. Further, the functions $d^{x}$ and $d$ define
the same topology on $U(x)$ (the equivalence of convergences induced by
$d^{x}$ and $d$ can be verified straightforwardly, using uniformity on $u,v$
in (A3)) and, hence, $(U(x),d^{x})$ is boundedly compact.
Remark 4 and Axiom (A3) imply
###### Proposition 1.
If $({\mathbb{X}},d,\delta)$ is a nondegenerate dilation structure, then
$d^{x}$ is a quasimetric on $B^{d}(x,R)$ with the same constants
$c_{\mathbb{X}}$, $Q_{\mathbb{X}}$ $($see $(2)$, $(3)$ of Definition 1$)$ as
for the initial quasimetric $d$.
In the same way as for metric spaces [6], Axioms (A2), (A3) imply
###### Proposition 2.
The function $d^{x}$ from Axiom $(A3)$ meets the cone property
$d^{x}(u,v)=\frac{1}{\mu}d^{x}(\delta_{\mu}^{x}u,\delta_{\mu}^{x}v)$
for all $u,v\in B^{d}(x,R)$ and $\mu$ such that
$\delta_{\mu}^{x}u,\delta_{\mu}^{x}v\in B^{d}(x,R)$ $($in particular, for all
$\mu\leq 1$$)$.
###### Proposition 3.
If $({\mathbb{X}},d,\delta)$ is a strong dilation structure then the limits of
the expressions
$\Sigma_{\varepsilon}^{x}(u,v)=\delta_{\varepsilon^{-1}}^{x}\delta_{\varepsilon}^{\delta_{\varepsilon}^{x}u}v,\
\operatorname{inv}^{x}_{\varepsilon}(u)=\delta_{\varepsilon^{-1}}^{\delta_{\varepsilon}^{x}u}x$
exist:
$\lim\limits_{\varepsilon\to
0}\Sigma_{\varepsilon}^{x}(u,v)=\Sigma^{x}(u,v)\in B^{d}(x,R),\
\lim\limits_{\varepsilon\to
0}\operatorname{inv}^{x}_{\varepsilon}(u)=\operatorname{inv}^{x}(u)\in
B^{d}(x,R).$ (2.3)
These limits are uniform on $x$ in some compact set and $u,v\in
B^{d}(x,\hat{r})$.
Conversely, if the limits 2.3 exist and are uniform, then Axiom (A4) holds.
###### Proof.
The assertion about the second limit follows from the fact that
$\operatorname{inv}^{x}_{\varepsilon}(u)=\Lambda_{\varepsilon}^{x}(u,x)$. Easy
calculations show that
$\Sigma^{x}_{\varepsilon}(u,v)=\Lambda_{\varepsilon}^{\delta_{\varepsilon}^{x}u}(\text{inv}^{x}_{\varepsilon}u,v)$
from where, taking in account the uniformity of convergence in (A4), the
existence and uniformity of the first limit follows.
Moreover, it is easy to see that
$\Sigma_{\varepsilon}^{\delta_{\varepsilon}^{x}u}(\text{inv}^{x}_{\varepsilon}u,v)=\Lambda^{x}_{\varepsilon}(u,v),$
hence
$\Lambda^{x}(u,v)=\Sigma^{x}(\text{inv}^{x}u,v).$ (2.4)
Therefore, from the existence and uniformity of the limits 2.3, Axiom (A4)
follows. ∎
Further we assume, w. l. o. g., that $\hat{r}=r$ (otherwise, take the
intersection of the corresponding balls), i. e. functions $\Lambda^{x}$ and
$\Sigma^{x}$ are defined on the same subset of $U(x)\times U(x)$. The
following proposition can be viewed as an example of existence of one of the
combinations from Proposition 3 (cf. the arguments of Bellaiche [2], the last
section).
###### Proposition 4.
Let $({\mathbb{X}},d,\delta)$ be a uniform dilation structure. Then there are
$r,\varepsilon_{0}>0$ such that for all $\varepsilon\in(0,\varepsilon_{0}]$,
$u,v\in B^{d}(x,r)$ the combination
$\Sigma_{\varepsilon}^{x}(u,v)=\delta_{\varepsilon^{-1}}^{x}\delta_{\varepsilon}^{\delta_{\varepsilon}^{x}u}v\in
U(x)$ from Proposition 3 is defined.
###### Proof.
Let $x^{\prime}=\delta_{\varepsilon}^{x}u$,
$x^{\prime\prime}=\delta_{\varepsilon}^{x^{\prime}}v$. To show the existence
of the combination $\Sigma_{\varepsilon}^{x}(u,v)\in U(x)$ it suffices to
verify that $x^{\prime\prime}\in W_{\varepsilon^{-1}}(x)$. Let us prove that,
for suitable $u,v,\varepsilon$, it is true that $x^{\prime\prime}\in
B^{d}(x,R\varepsilon)\subseteq W_{\varepsilon^{-1}}(x)$. It follows from
Proposition 2 that
$d^{x}(x,x^{\prime})=d^{x}(x,\delta_{\varepsilon}^{x}u)=\varepsilon
d^{x}(x,u)$, $d^{x^{\prime}}(x^{\prime},x^{\prime\prime})=\varepsilon
d^{x^{\prime}}(x^{\prime},v)$. Due to (A3), for any $\delta>0$ there is an
$\varepsilon>0$ such that: if $d^{x}(p,q)=O(\varepsilon)$, then
$d^{x}(p,q)(1-\delta)\leq d(p,q)\leq d^{x}(p,q)(1+\delta)$. Let $p=x$,
$q=x^{\prime}$ and consider arbitrary $r,R^{x}>0$ such that
$B^{d}(x,r)\subseteq B^{d^{x}}(x,R^{x})\subseteq B^{d}(x,R)$ (such reals exist
according to Remark 4). For any $\delta>0$ there is an
$\varepsilon_{0}^{\prime}>0$ such that for $u\in B^{d}(x,r)$,
$\varepsilon\in(0,\varepsilon_{0}^{\prime}]$ we have
$d(x,x^{\prime})\leq\varepsilon R^{x}(1+\delta)$. Analogously, there is an
$\varepsilon_{0}^{\prime\prime}>0$ such that for $v\in B^{d}(x,r)$,
$\varepsilon\in(0,\varepsilon_{0}^{\prime\prime}]$ we have
$d(x^{\prime},x^{\prime\prime})\leq\varepsilon R^{x^{\prime}}(1+\delta)$. Due
to uniformity of the limit in (A3) we can assume, w. l. o. g., that
$R^{x}=R^{x^{\prime}}=\xi$. Let
$\varepsilon_{0}=\min\\{\varepsilon_{0}^{\prime},\varepsilon_{0}^{\prime\prime}\\}$.
The generalized triangle inequality implies $d(x,x^{\prime\prime})\leq
Q_{\mathbb{X}}\left(d(x,x^{\prime})+d(x^{\prime},x^{\prime\prime})\right)\leq
2Q_{\mathbb{X}}\varepsilon\xi(1+\delta)$. To satisfy the desired inequality
$d(x,x^{\prime\prime})\leq R\varepsilon$ it suffices to take an arbitrary
$\xi<\frac{R}{2Q_{\mathbb{X}}}$ such that $B^{d^{x}}(x,\xi)\subseteq
B^{d}(x,R)$. Then an arbitrary number $r$ satisfying $B^{d}(x,r)\subseteq
B^{d^{x}}(x,\xi)$ will be as desired. ∎
A pointed $($quasi$)$metric space is a pair $({\mathbb{X}},p)$ consisting of a
(quasi)metric space ${\mathbb{X}}$ and a point $p\in{\mathbb{X}}$. Whenever we
want to emphasize what kind of (quasi)metric is on ${\mathbb{X}}$, we shall
write the pointed space as a triple
$({\mathbb{\mathbb{X}}},p,d_{\mathbb{X}})$.
###### Definition 3 ([46, 47]).
A sequence $({\mathbb{X}}_{n},p_{n},d_{{\mathbb{X}}_{n}})$ of pointed
quasimetric spaces converges to the pointed space
$({\mathbb{X}},p,d_{\mathbb{X}})$, if there exists a sequence of reals
$\delta_{n}\to 0$ such that for each $r>0$ there exist mappings
$f_{n,r}:B^{d_{{\mathbb{X}}_{n}}}(p_{n},r+\delta_{n})\to{\mathbb{X}},\
g_{n,r}:B^{d_{{\mathbb{X}}}}(p,r+2\delta_{n})\to{\mathbb{X}}_{n}$ such that
1) $f_{n,r}(p_{n})=p,\ g_{n,r}(p)=p_{n}$;
2) $\operatorname{dis}(f_{n,r})<\delta_{n},\
\operatorname{dis}(g_{n,r})<\delta_{n};$
3) $\sup\limits_{x\in
B^{d_{{\mathbb{X}}_{n}}}(p_{n},r+\delta_{n})}d_{{\mathbb{X}}_{n}}(x,g_{n,r}(f_{n,r}(x)))<\delta_{n}$.
Here
$\operatorname{dis}(f)=\sup\limits_{u,v\in{\mathbb{X}}}|d_{\mathbb{Y}}(f(u),f(v))-d_{\mathbb{X}}(u,v)|$
is the distortion of a mapping
$f:({\mathbb{X}},d_{\mathbb{X}})\to({\mathbb{Y}},d_{\mathbb{Y}})$ which
characterizes the difference of $f$ from an isometry.
###### Theorem 1 ([47]).
1\. Reduced to the case of metric spaces, the convergence of Definition 3 is
equivalent to the Gromov-Hausdorff one;
2) Let $(X,p),\ (Y,q)$ be two complete pointed quasimetric spaces, each
obtained as a limit of the same sequence $(X_{n},p_{n})$ such that the
constants $\\{Q_{X_{n}}\\}$ are uniformly bounded: $|Q_{X_{n}}|\leq C$ for all
$n\in\mathbb{N}$. If $X$ is boundedly compact, then $X$ and $Y$ are isometric.
###### Remark 5.
Note that a straightforward generalization of Gromov’s theory to the case of
quasimetric spaces is, for various reasons, impossible. For example, the
Gromov-Hausdorff distance between two bounded quasimetric spaces is equal to
zero [21] and, thus, makes no sense in this context. Besides that, in [25, 2]
convergence is first defined for compact spaces; convergence of boundedly
compact spaces is defined as convergence of all (compact) balls. For
quasimetric spaces, this approach would not yield uniqueness of the limit up
to isometry.
###### Definition 4.
Let ${\mathbb{\mathbb{X}}}$ be a boundedly compact (quasi)metric space, $p\in
X$. If the limit of pointed spaces
$\lim\limits_{\lambda\to\infty}(\lambda{\mathbb{X}},p)=(T_{p}{\mathbb{X}},e)$
exists (in the sense of Definition 3), then $T_{p}{\mathbb{X}}$ is called the
tangent cone to ${\mathbb{X}}$ at $p$. Here
$\lambda{\mathbb{X}}=({\mathbb{X}},\lambda\cdot d_{\mathbb{X}})$; the symbol
$\lim\limits_{\lambda\to\infty}(\lambda{\mathbb{X}},p)$ means that, for any
sequence $\lambda_{n}\to\infty$, there exists
$\lim\limits_{\lambda_{n}\to\infty}(\lambda_{n}{\mathbb{X}},p)$ which is
independent of the choice of sequence $\lambda_{n}\to\infty$ as $n\to\infty$.
Any neighborhood $U(e)\subseteq T_{p}{\mathbb{X}}$ of the basepoint element
$e\in T_{p}{\mathbb{X}}$ is said to be a local tangent cone to ${\mathbb{X}}$
at $p$.
###### Remark 6.
Theorem 1 implies that, for complete boundedly compact quasimetric spaces, the
tangent cone is unique up to isometry, i. e. one should treat the tangent cone
from Definition 4 as a class of pointed quasimetric spaces isometric to each
other. Note also that the tangent cone is completely defined by any
(arbitrarily small) neighborhood of the point. More precisely, if $U$ is a
neighborhood of the point $p\in{\mathbb{X}}$ then the tangent cones of $U$ and
${\mathbb{X}}$ at $p$ are isometric. Moreover, the quasimetric space
$(T_{p}{\mathbb{X}},e)$ is a cone in the sense that it is invariant under
scalings, i. e. $(T_{p}{\mathbb{X}},e)$ is isometric to $(\lambda
T_{p}{\mathbb{X}},e)$ for all $\lambda>0$.
###### Theorem 2 ([47]).
Let $({\mathbb{X}},d,\delta)$ be a nondegenerate dilation structure. Then
$(U(x),x,d^{x})$ is a local tangent cone to ${\mathbb{X}}$ at $x$.
Note that on the neighborhood $U(x)\subseteq\mathbb{X}$ we have two
(quasi)metric structures $d$ and $d^{x}$, thus it is natural to denote the
local tangent cone to $\mathbb{X}$ at $x$ as $(U(x),d^{x})$, not introducing
any other underlying set for the tangent cone.
One of the main goals of the present paper is to describe the algebraic
properties of the (local) tangent cone in the case when
$({\mathbb{X}},d,\delta)$ is a strong uniform nondegenerate dilation
structure. Having only axioms (A0) — (A3) we can say nothing substantial about
this.
## 3 Algebraic properties of the tangent cone
###### Definition 5 ([44, 20]).
A local group is a tuple $({\mathcal{G}},e,i,p)$ where ${\mathcal{G}}$ is a
Hausdorff topological space with a fixed identity element $e\in{\mathcal{G}}$
and continuous functions $i:\Upsilon\to{\mathcal{G}}$ (the inverse element
function), and $p:\Omega\to{\mathcal{G}}$ (the product function) given on some
subsets $\Upsilon\subseteq{\mathcal{G}}$,
$\Omega\subseteq{\mathcal{G}}\times{\mathcal{G}}$ such that $e\in\Upsilon$,
$\\{e\\}\times{\mathcal{G}}\subseteq\Omega$,
${\mathcal{G}}\times\\{e\\}\subseteq\Omega$, and for all
$x,y,z\in{\mathcal{G}}$ the following properties hold:
1) $p(e,x)=p(x,e)=x$;
2) if $x\in\Upsilon$, then $(x,i(x))\in\Omega$, $(i(x),x)\in\Omega$ and
$p(x,i(x))=p(i(x),x)=e$;
3) if $(x,y),(y,z)\in\Omega$ and $(p(x,y),z),(x,p(y,z))\in\Omega$, then
$p(p(x,y),z)=p(x,p(y,z))$.
Assertions close to the next three propositions can be found in [6], but in
our consideration, some details are different. We include the proofs for the
reader’s convenience.
###### Proposition 5.
Let $({\mathbb{X}},d,\delta)$ be a strong dilation structure. Then the
function introduced in Axiom $(A4)$ yields a product and an inverse element
functions in a neighborhood of the given point. Precisely,
${\mathcal{G}}^{x}=(U(x),x,\operatorname{inv}^{x},\Sigma^{x})$ $($where
$\operatorname{inv}^{x},\Sigma^{x}$ are from Proposition 3$)$ is a local
group. For the inverse element, the following property holds:
$\operatorname{inv}^{x}(\operatorname{inv}^{x}(u))=u$.
###### Proof.
Let $u,v,w\in B^{d}(x,r)$, $\varepsilon\leq\varepsilon_{0}$, where $r$ is from
Axiom $(A4)$, and $\varepsilon_{0}$ is such that
$\Sigma_{\varepsilon}^{x}(u,v)$ is defined for all
$\varepsilon\leq\varepsilon_{0}$, for example, as in Proposition 4. By direct
calculation and using the uniformity of the limit in (A4) one can verify the
following relations:
$\Sigma^{x}_{\varepsilon}(x,u)=u;\
\Sigma^{x}_{\varepsilon}(u,\delta^{x}_{\varepsilon}u)=u;$
if both parts of the following equality are defined, then
$\Sigma^{x}_{\varepsilon}(u,\Sigma^{\delta^{x}_{\varepsilon}}_{\varepsilon}(v,w))=\Sigma^{x}_{\varepsilon}(\Sigma^{x}_{\varepsilon}(u,v),w);$
$\Sigma^{x}(u,\text{inv}_{\varepsilon}^{x}(u))=x;\
\Sigma^{\delta^{x}_{\varepsilon}u}(\text{inv}_{\varepsilon}^{x}(u),u)=\delta^{x}_{\varepsilon}u;$
$\text{inv}_{\varepsilon}^{\delta^{x}_{\varepsilon}u}\text{inv}^{x}_{\varepsilon}u=x.$
Passing to the limit when $\varepsilon\to 0$, we obtain that $\Sigma^{x}(u,v)$
is the product function w. r. t. the identity element $x$ and inverse function
$\text{inv}^{x}(u)$ such that
$\operatorname{inv}^{x}(\operatorname{inv}^{x}(u))=u$. The domains of the
product and inverse functions are some areas $\Omega\supseteq B^{d}(x,r)\times
B^{d}(x,r)$, $\Upsilon\supseteq B^{d}(x,r)$ where $r$ is from (A4). The
continuity of functions $\Sigma^{x}(u,v)$ and $\operatorname{inv}^{x}u$ is
obvious from (A0), (A4) and Proposition 3. ∎
###### Proposition 6.
The following identities
$\delta_{\mu}^{x}\Sigma^{x}(u,v)=\Sigma^{x}(\delta_{\mu}^{x}(u),\delta_{\mu}^{x}(v)),\
\operatorname{inv}^{x}(\delta_{\mu}^{x}u)=\delta_{\mu}^{x}\operatorname{inv}^{x}u$
hold provided both parts of the equality are defined $($in particular, when
$\Sigma^{x}(u,v)$ exists and $\mu\leq 1$$)$.
###### Proof.
For the function
$\Lambda^{x}=\lim\limits_{\varepsilon\to
0}\Lambda_{\varepsilon}^{x}(u,v)=\lim\limits_{\varepsilon\to
0}\delta_{\varepsilon^{-1}}^{\delta_{\varepsilon}^{x}u}\delta_{\varepsilon}^{x}v$
from Axiom (A4), direct calculations show that
$\Lambda^{x}_{\varepsilon}(\delta_{\mu}^{x}u,\delta_{\mu}^{x}v)=\delta_{\mu}^{\delta_{\varepsilon\mu}^{x}u}\Lambda^{x}_{\varepsilon\mu}(u,v),$
hence
$\delta_{\mu}^{x}\Lambda^{x}(u,v)=\Lambda^{x}(\delta^{x}_{\mu}u,\delta_{\mu}^{x}v),$
(3.1)
provided both parts of the last equality are defined. From here the second
equality of the proposition is obvious, since
$\operatorname{inv}^{x}(u)=\Lambda^{x}(u,x)$.
The first equality of the proposition follows from (3.1), (2.4) and from the
second equality. ∎
###### Proposition 7.
Let $({\mathbb{X}},d,\delta)$ be a strong nondegenerate uniform dilation
structure. Then for all $u\in B^{d}(x,r)$ the function $\Sigma^{x}(u,\cdot)$
$($see Proposition 3$)$ is a $d^{x}$-isometry on $B^{d}(x,r)$.
###### Proof.
Using Proposition 2 and uniformity in Axiom $(A3)$, we get
$\lim\limits_{\varepsilon\to 0}\frac{1}{\varepsilon}\mid
d(\delta_{\varepsilon}^{x}v,\delta_{\varepsilon}^{x}w)-d^{\delta_{\varepsilon}^{x}u}(\delta_{\varepsilon}^{x}v,\delta_{\varepsilon}^{x}w)\mid=\lim\limits_{\varepsilon\to
0}\mid\frac{1}{\varepsilon}d(\delta_{\varepsilon}^{x}v,\delta_{\varepsilon}^{x}w)-d^{\delta_{\varepsilon}^{x}u}(\delta_{\varepsilon^{-1}}^{\delta_{\varepsilon}^{x}u}\delta_{\varepsilon}^{x}v,\delta_{\varepsilon^{-1}}^{\delta_{\varepsilon}^{x}u}\delta_{\varepsilon}^{x}w)\mid=$
$=\mid d^{x}(v,w)-d^{x}(\Lambda^{x}(u,v),\Lambda^{x}(u,w))\mid=0,$
where $\Lambda^{x}$ is from Axiom (A4). Further, we have
$d^{x}(v,w)=d^{x}(\Lambda^{x}(u,\Sigma^{x}(u,v)),\Lambda^{x}(u,\Sigma^{x}(u,w)))=d^{x}(\Sigma^{x}(u,v),\Sigma^{x}(u,w)).$
From here the assertion follows. ∎
It is interesting to compare the following proposition with the definition and
properties of homogeneous norm on a homogeneous Lie group [18].
###### Proposition 8.
Let $({\mathbb{X}},d,\delta)$ be a strong nondegenerate dilation structure.
Then the function $|\cdot|:B^{d}(x,R)\to\mathbb{R}$, defined as
$|u|=d^{x}(x,u)$, meets the following properties:
$1)$ homogeneity$:$ if $u\in B^{d}(x,R)$ and $\delta_{r}^{x}u\in B^{d}(x,R)$
is defined then $|\delta_{r}^{x}u|=r|u|$;
$2)$ non-degeneracy: $u=x$ if and only if $|u|=0.$
$3)$ generalized triangle inequality$:$ if for $u,v\in B^{d}(x,R)$ the value
$\Sigma^{x}(u,v)\in B^{d}(x,R)$ is defined then the following inequality
holds:
$|\Sigma^{x}(u,v)|\leq c\left(|u|+|v|\right),$ (3.2)
where $1\leq c<\infty$ and $c=c(x)$ does not depend on $u,v$.
###### Proof.
The first property directly follows from the conical property; the second one
is equivalent to the assumption of non-degeneracy of the dilation structure.
Let us show 3). Due to continuity of the product function
$(u,v)\mapsto\Sigma^{x}(u,v)$ there exists $0<\tau\leq R$ such that
$\bar{B}^{d^{x}}(x,\tau)\subseteq B^{d}(x,r)$ and for all
$u,v\in\bar{B}^{d^{x}}(x,\tau)$ we have $\Sigma^{x}(u,v)\in B^{d}(x,R)\cap
B^{d^{x}}(x,R)$. W. l. o. g. assume $|v|\leq|u|$ and consider first the case
when $|u|\leq\tau$ (then $\varepsilon=\varepsilon(u)=\tau^{-1}|u|\leq 1$).
Let us show that the elements $\delta^{x}_{\tau|u|^{-1}}u$,
$\delta^{x}_{\tau|u|^{-1}}v$ exist and belong to $B^{d}(x,R)$.
Indeed, it is sufficient to verify that $u\in W_{\varepsilon^{-1}}(x)$. Since
$\varepsilon\tau=|u|$, we have $u\in\bar{B}^{d^{x}}(x,\tau\varepsilon)$ (see
Remark 3). According to the choice of $\tau$ the following inclusions hold
$\bar{B}^{d^{x}}(x,\tau)\subseteq B^{d}(x,r)\subseteq B^{d}(x,R)$, therefore,
due to axiom (A0) and Proposition 2, it is true that
$u\in\bar{B}^{d^{x}}(x,\tau\varepsilon)=\delta_{\varepsilon}^{x}\bar{B}^{d^{x}}(x,\tau)\subseteq\delta_{\varepsilon}^{x}B^{d}(x,R)\subseteq
V_{\varepsilon}(x)\subseteq W_{\varepsilon^{-1}}(x)$. Note that it can not
happen that $\delta_{\varepsilon^{-1}}^{x}u\in U(x)\setminus B^{d}(x,R)$,
because
$\delta_{\varepsilon^{-1}}^{x}B^{d}(x,R\varepsilon)\subseteq\delta_{\varepsilon^{-1}}^{x}\delta_{\varepsilon}^{x}B^{d}(x,R)=B^{d}(x,R)$.
Thus, due to 1),
$|\delta^{x}_{\tau|u|^{-1}}u|=d^{x}(x,\delta^{x}_{\tau|u|^{-1}}u)=\tau,\
|\delta^{x}_{\tau|u|^{-1}}v|\leq\tau$. Hence, by choice of $\tau$, the value
$\Sigma^{x}(\delta^{x}_{\tau|u|^{-1}}u,\delta^{x}_{\tau|u|^{-1}}v)\in
B^{d}(x,R)\cap B^{d^{x}}(x,R)$ is defined. Thus, from Proposition 6, we can
derive
$\Sigma^{x}(u,v)=\delta^{x}_{\tau^{-1}|u|}\Sigma^{x}(\delta^{x}_{\tau|u|^{-1}}u,\delta^{x}_{\tau|u|^{-1}}v).$
It follows immediately that
$|\Sigma^{x}(u,v)|=|\delta^{x}_{\tau^{-1}|u|}(\Sigma^{x}(\delta^{x}_{\tau|u|^{-1}}u,\delta^{x}_{\tau|u|^{-1}}v))|\\\
=\tau^{-1}|u||\Sigma^{x}(\delta^{x}_{\tau|u|^{-1}}u,\delta^{x}_{\tau|u|^{-1}}v)|\leq
c|u|\leq c(|u|+|v|),$
where $c=\tau^{-1}R$.
Let now be $|u|>\tau$ and $\Sigma^{x}(u,v)\in B^{d}(x,R)$ be defined. Choose
$0<\mu<1$ such that $\delta_{\mu}^{x}u,\delta_{\mu}^{x}u\in B^{d^{x}}(x,\tau)$
(such $\mu$ exists due to continuity of dilations). Then
$\mu|\Sigma^{x}(u,v)|=|\delta_{\mu}^{x}\Sigma^{x}(u,v)|=|\Sigma^{x}(\delta_{\mu}^{x}u,\delta_{\mu}^{x}v)|\leq
c(|\delta_{\mu}^{x}u|+|\delta_{\mu}^{x}v|)=c\mu(|u|+|v|).$
It follows (3.2). ∎
###### Definition 6.
The function $|\cdot|$, introduced in Proposition 8, is said to be the
homogeneous norm on the local group ${\mathcal{G}}^{x}$.
###### Definition 7 ([31]).
It is said that for a local group ${\mathcal{G}}$ the global associativity
property holds if there is a neighborhood of the identity
$V\subseteq{\mathcal{G}}$ such that for each $n$-tuple of elements
$a_{1},a_{2}\ldots,a_{n}\in V$ whenever there exist two different ways of
introducing parentheses in this $n$-tuple, so that all intermediate products
are defined, the resulting products are equal.
###### Theorem 3 (Mal’tsev [31]).
A local topological group ${\mathcal{G}}$ is locally isomorphic to a some
topological group $G$ if and only if the global associativity property in
${\mathcal{G}}$ holds.
###### Remark 7.
Unlike in the case of global groups, the verification of the global
associativity property for local groups is a nontrivial task. This
verification can not be done by a trivial induction as for global groups since
it would require to assume the existence of all intermediate products which
is, in general, not true for local groups. See comments in [40, 20] where
there are some references to papers with mistakes caused by misunderstandings
of this fact. In the local group ${\mathcal{G}}^{x}$ under our consideration
it is easy to provide examples for $n=4$ such that $u_{i}\in B^{d}(x,R)$ and
combinations $u=\Sigma^{x}(\Sigma^{x}(u_{1},\Sigma^{x}(u_{2},u_{3})),u_{4})$
and $u^{\prime}=\Sigma^{x}(u_{1},\Sigma^{x}(u_{2},\Sigma^{x}(u_{3},u_{4})))$
exist while the combination
$\Sigma^{x}(\Sigma^{x}(u_{1},u_{2}),\Sigma^{x}(u_{3},u_{4})))$ is not defined.
More examples can be found in [31, 40].
###### Proposition 9.
For the local group ${\mathcal{G}}^{x}$, the global associativity property
holds.
###### Proof.
Let $u_{1},u_{2},\ldots,u_{n}\in B^{d}(x,R)$, and $u,u^{\prime}$ be elements
obtained from the $n$-tuple $(u_{1},u_{2},\ldots,u_{n})$ by introducing
parentheses such that the products exist. We need to show that $u=u^{\prime}$.
Let $\tau$ be such as in the proof of Proposition 8, $R_{x}=\inf\\{\xi\mid
B^{d}(x,R)\subseteq B^{d^{x}}(x,\xi)\\}$, $c_{n}=nc^{n-1}$ where $c$ is from
(3.2). Let $s_{n}=\frac{\tau}{c_{n-1}R_{x}}$ and
$\tilde{u_{i}}=\delta_{s_{n}}^{x}u_{i}$. By induction on $n$ and using (3.2)
it is easy to show that all possible products of length not bigger than $n$ of
the elements $\tilde{u_{i}}$ are defined. Thus it can be trivially shown (as
for global groups) that
$\delta_{s_{n}}^{x}(u)=\delta_{s_{n}}^{x}(u^{\prime})$. Applying to both sides
of the last equality the homeomorphism $\delta^{x}_{s_{n}^{-1}}$ (which is, in
particular, an injective mapping), we get $u=u^{\prime}$ and finish the proof.
∎
###### Definition 8 ([48], Proposition 5.4).
A topological group $G$ is contractible if there is an automorphism $\tau:G\to
G$ such that $\lim\limits_{n\to\infty}\tau^{n}g=e$ for all $g\in G$.
###### Definition 9.
A topological space is locally compact if any of its points has a neighborhood
the closure of which is compact. A local group is locally compact if there is
a neighborhood of its identity element the closure of which is compact.
The proof of Theorem 4 relies on the following statement, see Remark 2 for
comments.
###### Proposition 10 ([48], Corollary 2.4).
For a connected locally compact group $G$ the following assertions are
equivalent:
$(1)$ $G$ is contractible;
$(2)$ $G$ is a simply connected Lie group the Lie algebra $V$ of which is
nilpotent and graded, i. e. there is a decomposition
$V=\bigoplus\limits_{s>0}V_{s}$ such that $[V_{s},V_{t}]\subseteq V_{s+t}$ for
all $s,t>0.$ In particular, $V$ is nilpotent.
###### Theorem 4.
Let $({\mathbb{X}},d,\delta)$ be a strong nondegenerate dilation structure.
Then
$1)$ For any $x\in{\mathbb{X}}$, the local group ${\mathcal{G}}^{x}$ is
locally isomorphic to a connected simply connected Lie group $G^{x}$ the Lie
algebra of which is nilpotent and graded;
$2)$ If the dilation structure is, in addition, uniform, then the Lie group
$G^{x}$ is the tangent cone $($in the sense of Definition 4$)$ to
${\mathbb{X}}$ at $x$, i. e., left translations on $G^{x}$ are isometries w.
r. t. quasimetric $\tilde{d}^{x}$ on $G^{x}$ which arises from $d^{x}$ in a
natural way. The local group ${\mathcal{G}}^{x}$ is a local tangent cone.
###### Proof.
Since ${\mathbb{X}}$ is boundedly compact, ${\mathcal{G}}^{x}$ is a locally
compact local group. Due to existence on ${\mathcal{G}}^{x}$ of a one-
parameter family of dilations this local group is linearly connected (indeed,
any two points $u,v\in U(x)$ can be connected by the continuous curve
$\\{\delta^{x}_{\varepsilon}(u)\\}_{1\geq\varepsilon\geq
0}\circ\\{\delta^{x}_{\varepsilon}(v)\\}_{0\leq\varepsilon\leq 1}$), hence
${\mathcal{G}}^{x}$ is connected.
According to Proposition 9, the global associativity property in
${\mathcal{G}}^{x}$ holds. Hence, by Theorem 3, ${\mathcal{G}}^{x}$ is locally
isomorphic to some topological group $G^{x}$. Let us use the construction of
this group given in the proof of Theorem 3 in [31] and in more details in
[16]: $G^{x}$ is obtained as the group of equivalence classes of words
arranged from elements of the initial local group ${\mathcal{G}}^{x}$.
Namely, let ${\mathcal{G}}^{x}_{(n)}=\\{(u_{1},\ldots,u_{n})\mid
u_{i}\in{\mathcal{G}}^{x}\\}$ be the set of words of length $n$, and
$\tilde{G}^{x}=\bigcup\limits_{n\in\mathbb{N}}{\mathcal{G}}^{x}_{(n)}$. On
$\tilde{G}^{x}$ the following two operations can be introduced. The
contraction is defined as
$(u_{1},\ldots,u_{n})\in{\mathcal{G}}^{x}_{(n)}\mapsto(u_{1},\ldots,u_{i-1},\Sigma^{x}(u_{i},u_{i+1}),u_{i+2},\ldots,u_{n})\in{\mathcal{G}}^{x}_{(n-1)},$
if $\Sigma^{x}(u_{i},u_{i+1})$ exists. The expansion is defined as
$(u_{1},\ldots,u_{n})\in{\mathcal{G}}^{x}_{(n)}\mapsto(u_{1},\ldots,u_{i-1},v,w,u_{i+1},\ldots,u_{n})\in{\mathcal{G}}^{x}_{(n+1)},$
if $u_{i}=\Sigma^{x}(v,w)$. Two words $(u_{1},\ldots,u_{n})$ and
$(v_{1},\ldots,v_{m})$ are called equivalent (which is denoted as
$(u_{1},\ldots,u_{n})\sim(v_{1},\ldots,v_{m})$) if they can be obtained one
from another by a finite sequence of contractions and expansions. Finally, let
$G^{x}=\tilde{G}^{x}/\sim$. The product and inverse functions and the neutral
element on $G^{x}$ are defined respectively as
$[(u_{1},\ldots,u_{n})]\cdot[(v_{1},\ldots,v_{m})]=[(u_{1},\ldots,u_{n},v_{1},\ldots,v_{m})],$
$[(u_{1},\ldots,u_{n})]^{-1}=[(\text{inv}^{x}u_{n},\ldots,\text{inv}^{x}u_{1})],\
e_{G^{x}}=[(e_{{\mathcal{G}}^{x}})].$
It is easy to verify that the function $\varphi:{\mathcal{G}}^{x}\to G^{x}$
which maps the element $g$ to the equivalence class $[(g)]$, is a local
isomorphism.
The topology on $G^{x}$ is defined as follows: if ${\mathcal{B}}$ is the basis
of topology of ${\mathcal{G}}^{x}$, then $B=\\{\varphi(U)\mid
U\in{\mathcal{B}}\\}$ is the base of topology of $G^{x}$. The verification of
axioms of a topological basis can be done straightforwardly.
For an arbitrary $s<1$ define a contractive automorphism on $G^{x}$ as
$\tau([(u_{1},\ldots,u_{n})])=[(\delta^{x}_{s}(u_{1}),\ldots,\delta^{x}_{s}(u_{n}))].$
Due to the linear connectedness of the group $G^{x}$ (because of the obvious
relation $[(e,e,$ $\ldots,e)]$$=[(e)]$ and the fact that the local group
${\mathcal{G}}^{x}$ is linearly connected), by Proposition 10 we get the first
assertion of the theorem.
Now let $s_{mn}=s_{\max\\{m,n\\}}$ (in notation of the proof of Proposition 9)
and define on $G^{x}$ a quasimetric as
$\tilde{d}^{x}([(u_{1},\ldots,u_{n})],[(v_{1},\ldots,v_{m})])\\\
=\frac{1}{s_{mn}}d^{x}(\Sigma^{x}(\delta_{s_{mn}}^{x}u_{1},\ldots,\delta_{s_{mn}}^{x}u_{n}),\Sigma^{x}(\delta_{s_{mn}}^{x}v_{1},\ldots,\delta_{s_{mn}}^{x}v_{m})).$
Note that Propositions 2, 6 imply the generalized triangle inequality for
$\tilde{d}^{x}$ with the constant $Q_{\mathbb{X}}$ and that $\varphi$ is an
isometry. Taking into account Theorem 2 and Proposition 7 we obtain the second
assertion. ∎
###### Remark 8.
Let us give a brief overview of the proof of Proposition 10, for showing that
it can not be straightforwardly applied to the case of local groups. The
crucial part of this proof is to show that a connected locally compact
contractible group is a Lie group. This proof heavily relies on several main
theorems from the book of Montgomery and Zippin [37], where the solution of H5
is given. The proofs of those theorems are long and complicated, and, as noted
in [37, p. 119], “Most of the Lemmas can be also proved by essentially the
same arguments for the case of a locally compact connected local group but we
shall not complicate the statements and proofs of the Lemmas by inserting the
necessary qualifications.” This last statement shows, that proving the
theorems (based on the mentioned lemmas) that we would need, for the case of
local groups, is, at least, nontrivial (and not done by anybody, as far as we
know). It would require a careful study of large parts of the book [37].
Overcoming this difficulty we apply Mal’tsev’s theorem 3 to reduce the
consideration to the case of (global) groups, for which Proposition 10 can be
applied.
###### Remark 9.
There is an another look at the proof of Proposition 9. It actually can be
proved without the triangle inequality (3.2) and any (quasi)metric structure,
by means of the following simple topological fact ([44, Chapter 3, Section 23,
E], see also [20]): in any local group there is a decreasing sequence of
neighborhoods $\\{{\mathcal{U}}_{n}\\}_{n\in\mathbb{N}}$ such that, for all
elements $u_{1},\ldots u_{n}\in{\mathcal{U}}_{n}$, their products are defined
with any combinations of parentheses. Using this fact, an analog of Theorem 4,
for locally compact topological spaces with dilations, can be proved (for this
purpose, axioms of Definition 2 should be modified in a natural way).
Globalizability of locally compact locally connected contractible local groups
was proved in [16], independently of our paper. The result of [16] can be
viewed as a generalization of the first assertion of Theorem 4.
On the other hand, using the (quasi)metric structure allows to make the proof
of global associativity more constructive in comparison with the purely
topological one.
## 4 Example: Carnot-Carathéodory spaces
###### Definition 10 ([2, 25, 28, 39, 29, 52, 53]).
Fix a connected Riemannian $C^{\infty}$-manifold $\mathbb{M}$ of dimension
$N$. The manifold $\mathbb{M}$ is called a regular Carnot-Carathéodory space
if in the tangent bundle $T\mathbb{M}$ there is a filtration
$H\mathbb{M}=H_{1}\mathbb{M}\subseteq\ldots\subseteq
H_{i}\mathbb{M}\subseteq\ldots\subseteq H_{M}\mathbb{M}=T\mathbb{M}$
of subbundles of the tangent bundle $T\mathbb{M}$, such that, for each point
$p\in\mathbb{M}$, there exists a neighborhood $U\subset\mathbb{M}$ with a
collection of $C^{1,\alpha}$ (where $\alpha\in(0,1]$) vector fields
$X_{1},\dots,X_{N}$ on $U$ enjoying the following three properties. For each
$v\in U$ we have
$(1)$ $X_{1}(v),\dots,X_{N}(v)$ constitutes a basis of $T_{v}\mathbb{M}$;
$(2)$ $H_{i}(v)=\operatorname{span}\\{X_{1}(v),\dots,X_{\dim H_{i}}(v)\\}$ is
a subspace of $T_{v}\mathbb{M}$ of dimension $\dim H_{i}$, $i=1,\ldots,M$,
where $H_{1}(v)=H_{v}\mathbb{M}$;
$(3)$
$[X_{i},X_{j}](v)=\sum\limits_{\operatorname{deg}X_{k}\leq\operatorname{deg}X_{i}+\operatorname{deg}X_{j}}c_{ijk}(v)X_{k}(v)$
(4.1)
where the degree $\deg X_{k}$ equals $\min\\{m\mid X_{k}\in H_{m}\\}$;
The number $M$ is called the depth of the Carnot-Carathéodory space.
###### Remark 10.
According to [29], all statements below are also valid for the case when
$X_{i}\in C^{1}$ and $M=2$.
###### Definition 11.
For any point $g\in\mathbb{M}$, define the mapping
$\theta_{g}(v_{1},\ldots,v_{N})=\exp\biggl{(}\sum\limits_{i=1}^{N}v_{i}X_{i}\biggr{)}(g).$
(4.2)
It is known that $\theta_{g}$ is a $C^{1}$-diffeomorphism of the Euclidean
ball $B_{E}(0,r)\subseteq\mathbb{R}^{N}$ to $\mathbb{M}$, where $0\leq
r<r_{g}$ for some (small enough) $r_{g}$. The collection
$\\{v_{i}\\}_{i=1}^{N}$ is called the normal coordinates or the coordinates of
the $1^{\text{st}}$ kind $($with respect to $u\in\mathbb{M})$ of the point
$v\in U_{g}=\theta_{g}(B_{E}(0,r_{g}))$. Further we will consider a compactly
embedded neighborhood ${\mathcal{U}}\subseteq\mathbb{M}$ such that
${\mathcal{U}}\subseteq\bigcap\limits_{g\in{\mathcal{U}}}U_{g}$.
###### Definition 12.
By means of coordinates (4.2), introduce on ${\mathcal{U}}$ the following
quasimetric $d_{\infty}$. For $u,v\in{\mathcal{U}}$ such that
$v=\exp\Bigl{(}\sum\limits_{i=1}^{N}v_{i}X_{i}\Bigr{)}(u)$ let
$d_{\infty}(u,v)=\max\limits_{i}\\{|v_{i}|^{\frac{1}{\deg X_{i}}}\\}.$
The properties (1), (2) of Definition 1 for the function $d_{\infty}$ and its
continuity on both arguments obviously follow from properties of the
exponential mapping. The generalized triangle inequality is proved in [28,
29]. We denote the balls w. r. t. $d_{\infty}$ as
$\operatorname{Box}(u,r)=\\{v\in{\mathcal{U}}\mid d_{\infty}(v,u)<r\\}.$
###### Definition 13.
Define in ${\mathcal{U}}$ the action of the dilation group
$\Delta^{g}_{\varepsilon}$ as follows: it maps an element
$x=\exp\Bigl{(}\sum\limits_{i=1}^{N}x_{i}X_{i}\Bigr{)}(g)\in{\mathcal{U}}$ to
the element
$\Delta^{g}_{\varepsilon}x=\exp\Bigl{(}\sum\limits_{i=1}^{N}x_{i}\varepsilon^{\deg
X_{i}}X_{i}\Bigr{)}(g)\in{\mathcal{U}}$
in the case when the right-hand part of the last expression makes sense.
###### Proposition 11 ([29]).
The coefficients
$\bar{c}_{ijk}=\begin{cases}c_{ijk}(g)\text{ of \eqref{tcomm} },&\text{if
}\operatorname{deg}X_{i}+\operatorname{deg}X_{j}=\operatorname{deg}X_{k}\\\
0,&\text{in other cases}\end{cases}$
define a graded nilpotent Lie algebra.
This Lie algebra can be represented by vector fields
$\\{(\widehat{X}_{i}^{g}\\}_{i=1}^{N}\in C^{\alpha}$ on ${\mathcal{U}}$ such
that
$[\widehat{X}_{i}^{g},\widehat{X}_{j}^{g}]=\sum\limits_{\operatorname{deg}X_{k}=\operatorname{deg}X_{i}+\operatorname{deg}X_{j}}c_{ijk}(g)\widehat{X}_{k}^{g}$
(4.3)
and $\widehat{X}_{i}^{g}(g)=X_{i}(g)$.
###### Definition 14.
To the Lie algebra $\\{\widehat{X}_{i}^{g}\\}_{i=1}^{N}$ there corresponds the
Lie group ${\mathcal{G}}^{g}=({\mathcal{U}},g,^{-1},*)$ at $g$. The product
function $*$ is defined as follows: if
$x=\exp\Bigl{(}\sum\limits_{i=1}^{N}x_{i}\widehat{X}^{g}_{i}\Bigr{)}(g)$,
$y=\exp\Bigl{(}\sum\limits_{i=1}^{N}y_{i}\widehat{X}^{g}_{i}\Bigr{)}(g)$, then
$x*y=\exp\Bigl{(}\sum\limits_{i=1}^{N}y_{i}\widehat{X}^{g}_{i}\Bigr{)}\circ\exp\Bigl{(}\sum\limits_{i=1}^{N}x_{i}\widehat{X}^{g}_{i}\Bigr{)}(g)=\exp\Bigl{(}\sum\limits_{i=1}^{N}z_{i}\widehat{X}^{g}_{i}\Bigr{)}(g)$,
where $z_{i}$ are computed via Campbell-Hausdorff formula. The inverse element
to $x=\exp\Bigl{(}\sum\limits_{i=1}^{N}x_{i}\widehat{X}^{g}_{i}\Bigr{)}(g)$ is
defined as
$x^{-1}=\exp\Bigl{(}\sum\limits_{i=1}^{N}(-x_{i})\widehat{X}^{g}_{i}\Bigr{)}(g)$.
###### Remark 11.
In the “classical” sub-Riemannian setting (see Introduction), the local Lie
group from Definition 14 is locally isomorphic to a Carnot group, i.e., a
connected simply connected Lie group the Lie algebra $V$ of which can be
decomposed into a direct sum $V=V_{1}\oplus\ldots\oplus V_{M}$ such that
$[V_{1},V_{i}]=V_{i+1},\ i=1,\ldots M-1$, $[V_{1},V_{M}]=\\{0\\}$. In the case
under our consideration, for the Lie algebra of the local group
${\mathcal{G}}^{g}$ only the inclusion $[V_{1},V_{i}]\subseteq V_{i+1}$ is
true. The converse inclusion will hold if we require an additional condition
[28, 29] in Definition 10: the quotient mapping
$[\,\cdot,\cdot\,]_{0}:H_{1}\times H_{j}/H_{j-1}\mapsto H_{j+1}/H_{j}$ induced
by Lie brackets is an epimorphism for all $1\leq j<M$. Under this additional
assumption, an analog of the Rashevskii-Chow theorem can be proved.
Strictly speaking, the group operation is defined on a neighborhood defined by
vector fields $\\{\widehat{X}^{g}_{i}\\}$, but, w. l. o. g., we can assume
that this neighborhood coincides with ${\mathcal{U}}$ [29, 53]. Note also that
the mapping $\theta_{g}$ is a local isometric isomorphism between the local
Lie group $({\mathcal{G}}^{g},*)$ and the Lie group $(\mathbb{R}^{N},*)$, and
$\theta_{g}(0)=g$. The group operation $*$ on $\mathbb{R}^{N}$ is introduced
by analogy with Definition 14, by means of $C^{\infty}$ vector fields
$\\{(\widehat{X}^{g}_{i})^{\prime}\\}$ on $\mathbb{R}^{N}$, such that
$\widehat{X}_{i}^{g}=(\theta_{g})_{*}(\widehat{X}_{i}^{g})^{\prime}$, where
$(\theta_{g})_{*}\langle{Y}\rangle$$(\theta_{g}(x))=D\theta_{g}(x)\langle{Y}(x)\rangle$,
${Y}\in T\mathbb{R}^{N}$ (see details in [28, 29, 53]). In what follows, we
will identify the neighborhood ${\mathcal{U}}$ with its image
$\theta_{g}^{-1}({\mathcal{U}})\subseteq\mathbb{R}^{N}$.
This identification allows, in particular, to define canonical coordinates of
the first kind, induced by the nilpotentized vector fields in a similar way as
11.
###### Definition 15.
For $u,v\in\mathbb{R}^{N}$ such that
$v=\exp\Bigl{(}\sum\limits_{i=1}^{N}v_{i}(\widehat{X}^{g}_{i})^{\prime}\Bigr{)}(u)$,
let $d_{\infty}^{g}(u,v)=\max\limits_{i}\\{|v_{i}|^{\frac{1}{\deg X_{i}}}\\}.$
It is known [18] that $d_{\infty}^{g}$ is a quasimetric. We denote the balls
w. r. t. this quasimetric as
$\operatorname{Box}^{g}(u,r)=\\{v\in\mathbb{R}^{N}\mid
d_{\infty}^{g}(v,u)<r\\}.$
###### Proposition 12 ([29, 50]).
If $r$ is such that $\operatorname{Box}(g,r)\subseteq{\mathcal{U}}$ then
$\operatorname{Box}(g,r)=\operatorname{Box}^{g}(g,r)$.
###### Definition 16.
The nilpotentized vector fields also define dilations on ${\mathcal{U}}$: the
element
$x=\exp\Bigl{(}\sum\limits_{i=1}^{N}x_{i}\widehat{X}^{g}_{i}\Bigr{)}(g)\in{\mathcal{U}}$
is mapped to the element
$\delta^{g}_{g,\varepsilon}x=\exp\Bigl{(}\sum\limits_{i=1}^{N}x_{i}\varepsilon^{\deg
X_{i}}\widehat{X}^{g}_{i}\Bigr{)}(g)\in{\mathcal{U}}$
in the case when the right-hand part of the last expression makes sense.
###### Proposition 13 ([29, 50]).
For all $\varepsilon>0$ and $u\in{\mathcal{U}}$, we have
$\Delta^{g}_{\varepsilon}u=\delta^{g}_{g,\varepsilon}u$, if both parts of this
equality are defined.
###### Proposition 14 ([18, 29, 53]).
The cone property for the quasimetric $d_{\infty}^{g}(u,v)$ holds:
$d_{\infty}^{g}(u,v)=\frac{1}{\varepsilon}d_{\infty}^{g}(\Delta^{g}_{\varepsilon}u,\Delta^{g}_{\varepsilon}v)$
for all possible $\varepsilon>0$.
###### Theorem 5 (Estimate on divergence of integral lines [28, 29]).
Consider points $u,v\in\mathcal{U}$ and
$w_{\varepsilon}=\exp\Bigl{(}\sum\limits_{i=1}^{N}w_{i}\varepsilon^{\operatorname{deg}X_{i}}X_{i}\Bigr{)}(v)\text{
and
}\widehat{w}_{\varepsilon}=\exp\Bigl{(}\sum\limits_{i=1}^{N}w_{i}\varepsilon^{\operatorname{deg}X_{i}}\widehat{X}^{u}_{i}\Bigr{)}(v).$
Then
$\max\\{d_{\infty}^{u}(w_{\varepsilon},\widehat{w}_{\varepsilon}),d_{\infty}^{u^{\prime}}(w_{\varepsilon},\widehat{w}_{\varepsilon})\\}=\varepsilon[\Theta(u,v,\alpha,M)]\rho(u,v)^{\frac{\alpha}{M}},$
(4.4)
where $\Theta$ is uniformly bounded on $u,v\in{\mathcal{U}}.$
###### Theorem 6 (Local approximation theorem [2, 21, 25, 28, 29, 52]).
If $u,v\in\operatorname{Box}(g,\varepsilon)$, then
$\left|d_{\infty}(u,v)-d_{\infty}^{g}(u,v)\right|=O(\varepsilon^{1+\frac{\alpha}{M}})$
uniformly on $g\in{\mathcal{U}},\ u,v\in\operatorname{Box}(g,\varepsilon)$.
###### Theorem 7.
Dilations from Definition 13 induce on the quasimetric space
$({\mathcal{U}},d_{\infty})$ a strong uniform nondegenerate dilation structure
with the conical quasimetric $($$d^{x}$ from Axiom $(A3)$$)$ $d^{g}_{\infty}$.
###### Proof.
Axioms (A0) — (A2) and non-degeneracy of Definition 2 obviously hold due to
properties of exponential mappings; (A3) and uniformity directly follow from
Theorem 6.
Axiom (A4) follows from group operation properties and Theorem 5. Indeed, let
$u=\exp\Bigl{(}\sum\limits_{i=1}^{N}u_{i}X_{i}\Bigr{)}(g),\ $
$v=\exp\Bigl{(}\sum\limits_{i=1}^{N}v_{i}X_{i}\Bigr{)}(g)\in{\mathcal{U}}$. We
need to show the existence and uniformity of the limits of
$\Sigma_{\varepsilon}^{g}(u,v)=\Delta_{\varepsilon^{-1}}^{g}\Delta_{\varepsilon}^{\Delta_{\varepsilon}^{g}u}v$
and
$\operatorname{inv}^{g}_{\varepsilon}(u)=\Delta_{\varepsilon^{-1}}^{\Delta_{\varepsilon}^{g}u}g$,
when $\varepsilon\to 0$ (see Proposition 3).
First we prove the existence of the limit on the local group (i. e. replacing
$\Delta^{g}_{\varepsilon}$ by $\delta^{g}_{g,\varepsilon}$) According to (A2),
$\lim\limits_{\varepsilon\to
0}\Delta_{\varepsilon}^{x}u=\lim\limits_{\varepsilon\to 0}u_{\varepsilon}=g$.
By means of (11) we can write
$v=\exp\Bigl{(}\sum\limits_{i=1}^{N}\tilde{v}^{\varepsilon}_{i}X_{i}\Bigr{)}(u_{\varepsilon}).$
Since the coordinates of the first kind are uniquely defined,
$\lim\limits_{\varepsilon\to 0}\tilde{v}^{\varepsilon}_{i}=v_{i},\
i=1,\ldots,N.$ (4.5)
Now let
$a=\delta^{u_{\varepsilon}}_{g,\varepsilon}v=\exp\Bigl{(}\sum\limits_{i=1}^{N}\tilde{v}^{\varepsilon}_{i}\varepsilon^{\operatorname{deg}X^{g}_{i}}\widehat{X}_{i}\Bigr{)}\circ\exp\Bigl{(}\sum\limits_{i=1}^{N}u_{i}\varepsilon^{\operatorname{deg}X_{i}}\widehat{X}^{g}_{i}\Bigr{)}(g).$
Then
$\Sigma_{\varepsilon}^{g}(u,v)=\delta_{g,\varepsilon^{-1}}^{g}a=\exp\Bigl{(}\sum\limits_{i=1}^{N}\tilde{v}^{\varepsilon}_{i}(\delta^{g}_{g,\varepsilon^{-1}})_{*}(\varepsilon^{\operatorname{deg}X_{i}}\widehat{X}^{g}_{i})\Bigr{)}\circ\exp\Bigl{(}\sum\limits_{i=1}^{N}u_{i}(\delta^{g}_{g,\varepsilon^{-1}})_{*}(\varepsilon^{\operatorname{deg}X_{i}}\widehat{X}^{g}_{i})\Bigr{)}(g).$
Using group homogeneity and (4.5), we get the existence of the uniform (on
$g$) limit
$\lim\limits_{\varepsilon\to
0}\Sigma_{\varepsilon}^{g}(u,v)=\exp\Bigl{(}\sum\limits_{i=1}^{N}v_{i}\widehat{X}^{g}_{i}\Bigr{)}\circ\exp\Bigl{(}\sum\limits_{i=1}^{N}u_{i}\widehat{X}^{g}_{i}\Bigr{)}(g).$
Now let us estimate the difference between the two combinations. From
Properties 13, 14 and Theorem 5 we infer
$d_{\infty}^{g}\left(\Delta_{\varepsilon^{-1}}^{g}\Delta_{\varepsilon}^{\Delta_{\varepsilon}^{g}u}v,\delta_{g,\varepsilon^{-1}}^{g}\delta_{g,\varepsilon}^{\delta_{g,\varepsilon}^{g}u}v\right)=d_{\infty}^{g}\left(\Delta_{\varepsilon^{-1}}^{g}\Delta_{\varepsilon}^{\Delta_{\varepsilon}^{g}u}v,\Delta_{\varepsilon^{-1}}^{g}\delta_{g,\varepsilon}^{\Delta_{\varepsilon}^{g}u}v\right)=$
$=\varepsilon^{-1}d_{\infty}^{g}\left(\Delta_{\varepsilon}^{u_{\varepsilon}}v,\delta_{g,\varepsilon}^{u_{\varepsilon}}v\right)=\varepsilon^{-1}\cdot
O\left(\varepsilon^{1+\frac{1}{\alpha}}\right)\to 0$
when $\varepsilon\to 0$, which implies the uniform convergence of
$\Sigma_{\varepsilon}^{g}(u,v)$.
Concerning the inverse element, we have
$u_{\varepsilon}=\exp\Bigl{(}\sum\limits_{i=1}^{N}u_{i}\varepsilon^{\deg
X_{i}}X_{i}\Bigr{)}(g),\
g=\exp\Bigl{(}\sum\limits_{i=1}^{N}-u_{i}\varepsilon^{\deg
X_{i}}X_{i}\Bigr{)}(u_{\varepsilon}),$
hence
$\text{inv}^{g}(u,v)=\text{inv}_{\varepsilon}^{g}(u,v)=\exp\Bigl{(}\sum\limits_{i=1}^{N}-u_{i}X_{i}\Bigr{)}(g),$
which finishes the proof.
∎
###### Remark 12.
In contrast to the proof of a similar assertion in [8], we do not use, for
proving Theorem 7, the normal frames technique [2].
Nevertheless, our considerations include, as a particular case, the
“classical” sub-Riemannian setting, although in this setting the number of
nontrivial commutators of “horizontal” vector fields can be bigger then the
dimension $N$ of the manifold $\mathbb{M}$. Indeed, the nilpotent Lie
algebras, defined by different bases, are isomorphic to each other due to the
functorial property of the tangent cone [50, 29]. Analogs of the basic
Theorems 6, 5, needed for the proof of Theorem 7 for the intrinsic metric
$d_{c}$ are proved in [2, 29, 52].
###### Remark 13.
An analog of Theorem 7 can be proved for some other quasimetrics equivalent to
$d_{\infty}$, looking like e. g. in [2].
Note also that proofs in [28] do not use tools concerned with the Baker-
Campbell-Hausdorff formula.
## 5 Differentiability
Let $(\mathbb{X},d_{\mathbb{X}},\delta)$ and
$(\mathbb{Y},d_{\mathbb{Y}},\tilde{\delta})$ be two quasimetric spaces with
strong nondegenerate dilation structures. In this section we denote the local
group $\mathcal{G}^{x}$ at $x\in\mathbb{X}$ ($\mathcal{G}^{y}$ at
$y\in\mathbb{Y}$ ) by the symbol $\mathcal{G}^{x}\mathbb{X}$
($\mathcal{G}^{y}\mathbb{Y}$). Quasimetrics on them will be denoted by $d^{x}$
and $d^{y}$ respectively.
Recall that a $\delta$-homogeneous homomorphism of graded nilpotent groups
$\mathbb{G}$ and $\widetilde{\mathbb{G}}$ with one-parameter groups of
dilations $\delta$ and $\tilde{\delta}$ [18] respectively is a continuous
homomorphism $L:\mathbb{G}\to\widetilde{\mathbb{G}}$ of these groups such that
$L\circ\delta=\tilde{\delta}\circ L.$
The case of local graded nilpotent groups $\mathcal{G}$ and
$\widetilde{\mathcal{G}}$ with one-parameter groups of dilations $\delta$ and
$\tilde{\delta}$ respectively is different from this only in that the equality
$L\circ\delta(v)=\tilde{\delta}\circ L(v)$ holds only for $v\in{\mathcal{G}}$
and $t>0$ such that $\delta_{t}v\in{\mathcal{G}}$ and
$\tilde{\delta}_{t}L(v)\in\widetilde{\mathcal{G}}$.
###### Definition 17.
Given two quasimetric spaces $(\mathbb{X},d_{\mathbb{X}},\delta)$ and
$(\mathbb{Y},d_{\mathbb{Y}},\tilde{\delta})$ with strong uniform nondegenerate
dilation structures, and a set $E\subset\mathbb{X}$. A mapping
$f:E\to{\mathbb{Y}}$ is called $\delta$-differentiable at a point $g\in E$ if
there exists a $\delta$-homogeneous homomorphism
$L:\bigl{(}\mathcal{G}^{g}\mathbb{X},d^{g}\bigr{)}\to\bigl{(}\mathcal{G}^{f(g)}\mathbb{Y},d^{f(g)}\bigr{)}$
of the local nilpotent tangent cones such that
$d^{f(g)}(f(v),L(v))=o\bigl{(}d^{g}(g,v)\bigr{)}\quad\text{as
$E\cap{\mathcal{G}^{g}\mathbb{X}}\ni v\to g$}.$ (5.1)
A $\delta$-homogeneous homomorphism
$L:\bigl{(}{\mathcal{G}}^{g}\mathbb{X},d^{g}\bigr{)}\to\bigl{(}{\mathcal{G}}^{f(g)}\mathbb{Y},{d}^{f(g)}\bigr{)}$
satisfying condition (5.1), is called a $\delta$-differential of the mapping
$f:E\to{\mathbb{Y}}$ at $g\in E$ on $E$ and is denoted by $Df(g)$. It can be
proved like in [50, 51] that if ${E=\mathbb{X}}$ then the
$\delta$-differential is unique.
Moreover, it is easy to verify that a homomorphism
$L:\bigl{(}{\mathcal{G}}^{g}\mathbb{X},d^{g}\bigr{)}\to\bigl{(}{\mathcal{G}}^{f(g)}\mathbb{Y},{d}^{f(g)}\bigr{)}$
satisfying (5.1) commutes with the one-parameter dilation group:
$\tilde{\delta}^{f(g)}_{t}\circ L=L\circ\delta^{g}_{t},$ (5.2)
i.e., $L$ is $\delta$-homogeneous homomorphism.
In the case of Carnot groups, the introduced concept of differentiability
coincides with the concept of $P$-differentiability given by P. Pansu in [42].
The following assertion is similar to the corresponding statement of [51,
Proposition 2.3].
###### Proposition 15.
Definition 17 is equivalent to each of the following assertions:
$1)$
$d^{f(g)}\bigl{(}\tilde{\delta}^{f(g)}_{t^{-1}}f\bigl{(}\delta^{g}_{t}(v)\bigr{)},L(v)\bigr{)}=o(1)$
as $t\to 0$, where $o(\cdot)$ is uniform in the points $v$ of any compact part
of $\mathcal{G}^{g}\mathbb{X};$
$2)$ $d^{f(g)}(f(v),L(v))=o\bigl{(}d_{\mathbb{X}}(g,v)\bigr{)}$ as
$E\cap{\mathcal{G}^{g}\mathbb{X}}\ni v\to g;$
$3)$ $d_{\mathbb{Y}}(f(v),L(v))=o\bigl{(}d^{g}(g,v)\bigr{)}$ as
$E\cap{\mathcal{G}^{g}\mathbb{X}}\ni v\to g;$
$4)$ $d_{\mathbb{Y}}(f(v),L(v))=o\bigl{(}d_{\mathbb{X}}(g,v)\bigr{)}$ as
$E\cap{\mathcal{G}^{g}\mathbb{X}}\ni v\to g;$
$5)$
$d_{\mathbb{Y}}\bigl{(}f\bigl{(}\delta^{g}_{t}(v)\bigr{)},L\bigl{(}\delta^{g}_{t}v\bigr{)}\bigr{)}=o(t)$
as $t\to 0$, where $o(\cdot)$ is uniform in the points $v$ of any compact part
of $\mathcal{G}^{g}\mathbb{X}$.
###### Proof.
Consider a point $v$ of a compact part of $\mathcal{G}^{g}\mathbb{X}$ and a
sequence $\varepsilon_{i}\to 0$ as $i\to 0$ such that
$\delta^{g}_{\varepsilon_{i}}v\in E$ for all $i\in\mathbb{N}$. From (5.1) we
have
$d^{f(g)}\bigl{(}f\bigl{(}\tilde{\delta}^{g}_{\varepsilon_{i}}v\bigr{)},L\bigl{(}\tilde{\delta}^{g}_{\varepsilon_{i}}v\bigr{)}\bigr{)}=o\bigl{(}d^{g}\bigl{(}g,\delta^{g}_{\varepsilon_{i}}v\bigr{)}\bigr{)}=o({\varepsilon_{i}})$.
In view of (5.2), we infer
$d^{f(g)}\bigl{(}\tilde{\delta}^{f(g)}_{\varepsilon_{i}}\bigl{(}\tilde{\delta}^{f(g)}_{\varepsilon_{i}^{-1}}f\bigl{(}\delta^{g}_{\varepsilon_{i}}v\bigr{)}\bigr{)},\tilde{\delta}^{f(g)}_{\varepsilon_{i}}L(v)\bigr{)}=o(\varepsilon_{i})\quad\text{uniformly
in\leavevmode\nobreak\ $v$.}$
From here, applying the cone property of Proposition 2, we obtain just item 1.
Obviously, the arguments are reversible. Item 1 is equivalent to item 5 since
in view of (2.1) we have
$\bigl{|}d_{\mathbb{Y}}\bigl{(}\tilde{\delta}^{f(g)}_{\varepsilon_{i}}\bigl{(}\tilde{\delta}^{f(g)}_{\varepsilon_{i}^{-1}}f\bigl{(}\delta^{g}_{\varepsilon_{i}}v\bigr{)}\bigr{)},\tilde{\delta}^{f(g)}_{\varepsilon_{i}}L(v)\bigr{)}-d^{f(g)}\bigl{(}\tilde{\delta}^{f(g)}_{\varepsilon_{i}}\bigl{(}\tilde{\delta}^{f(g)}_{\varepsilon_{i}^{-1}}f\bigl{(}\delta^{g}_{\varepsilon_{i}}v\bigr{)}\bigr{)},\tilde{\delta}^{f(g)}_{\varepsilon_{i}}L(v)\bigr{)}\bigr{|}\\\
=\bigl{|}d_{\mathbb{Y}}\bigl{(}\tilde{\delta}^{f(g)}_{\varepsilon_{i}}\bigl{(}\tilde{\delta}^{f(g)}_{\varepsilon_{i}^{-1}}f\bigl{(}\delta^{g}_{\varepsilon_{i}}v\bigr{)}\bigr{)},\tilde{\delta}^{f(g)}_{\varepsilon_{i}}L(v)\bigr{)}-o(\varepsilon_{i})\bigr{|}=o(\varepsilon_{i})\quad\text{uniformly
in\leavevmode\nobreak\ $v$.}$ (5.3)
Item 5 implies item 3 and vice versa. By comparing the metrics:
$d^{g}(g,v)=O\bigl{(}d_{\mathbb{X}}(g,v)\bigr{)}$ and
$d_{\mathbb{X}}(g,v)=O\bigl{(}d^{g}(g,v)\bigr{)}$, we obtain the equivalence
of the items 3 and 4. The proof of an equivalence of the items 4 and 2 is
similar to (5.3). ∎
Let us generalize the chain rule of paper [51].
###### Theorem 8.
Suppose that $\mathbb{X},{\mathbb{Y}},\mathbb{Z}$ are three quasimetric spaces
with strong uniform nondegenerate dilation structures, $E$ is a set in
$\mathbb{X}$, and $f:E\to{\mathbb{Y}}$ is a mapping from $E$ into $\mathbb{Y}$
$\delta$-differentiable at a point $g\in E$. Suppose also that $F$ is a set in
$\mathbb{Y}$, $f(E)\subset Y$ and $\varphi:F\to\mathbb{Z}$ is a mapping from
$F$ into $\mathbb{Z}$ $\tilde{\delta}$-differentiable at
$p=f(g)\in{\mathbb{Y}}$. Then the composition $\varphi\circ
f:E\to{\mathbb{Z}}$ is $\delta$-differentiable at $g$ and
$D(\varphi\circ f)(g)=D\varphi(p)\circ Df(g).$
###### Proof.
By hypothesis, $d^{f(g)}(f(v),Df(g)(v))=o\bigl{(}d^{g}(g,v)\bigr{)}$ as $v\to
g$ and also $d^{\varphi(p)}(\varphi(w)$,
$D\varphi(p)(w))=o\bigl{(}d^{p}(p,w)\bigr{)}$ as $w\to p$. It follows that $f$
is continuous in $g\in E$ and $\varphi$ is continuous in $p\in F$. We now
infer
$d^{\varphi(p)}((\varphi\circ f)(v),(D\varphi(p)\circ Df(g))(v))\\\ \leq
Q_{\mathbb{Z}}\bigl{[}d^{\varphi(p)}(\varphi(f(v)),D\varphi(p)(f(v)))+d^{\varphi(p)}(D\varphi(p)(f(v)),D\varphi(p)(Df(g)(v)))\bigr{]}\\\
\leq
o\bigl{(}d^{p}(p,f(v))\bigr{)}+O\bigl{(}d^{p}\bigl{(}f(v),Df(g)(v)\bigr{)}\bigr{)}\\\
\leq
o\bigl{(}d^{g}(g,v)\bigr{)}+O\bigl{(}o\bigl{(}d^{g}(g,v)\bigr{)}\bigr{)}=o\bigl{(}d^{g}(g,v)\bigr{)}\quad\text{as
$v\to g$},$
since
$d^{p}\bigl{(}p,f(v)\bigr{)}\leq
Q_{\mathbb{Y}}\left[d^{p}\bigl{(}p,Df(g)(v)\bigr{)}+d^{p}\bigl{(}f(v),Df(g)(v)\bigr{)}\right]\\\
=O\bigl{(}d^{g}(g,v)\bigr{)}+o\bigl{(}d^{g}(g,v)\bigr{)}=O\bigl{(}d^{g}(g,v)\bigr{)}\quad\text{as
}v\to g.$
(The estimate $d^{p}\bigl{(}p,Df(g)(v)\bigr{)}=O\bigl{(}d^{g}(g,v)\bigr{)}$ as
$v\to g$ follows from the continuity of the homomorphism $Df(g)$ and (5.2).) ∎
###### Remark 14.
Note that the concept of differentiability for the quasiconformal mappings of
Carnot-Carathéodory manifolds was first suggested by Margulis and Mostow in
[32] and is essentially based on Mitchell’s paper [34]: A quasiconformal
mapping $\varphi:{\mathbf{M}}\to{\mathbf{N}}$ is differentiable at a point
$x_{0}$ in the sense of [32] if the family of mappings
$\varphi_{t}:({\mathbf{M}},td_{\mathbf{M}})\to({\mathbf{N}},td_{\mathbf{N}})$
induced by the mapping
$\varphi:({\mathbf{M}},d_{\mathbf{M}})\to({\mathbf{N}},d_{\mathbf{N}})$
converges to a horizontal homomorphism of the tangent cones at the points
$x_{0}\in{\mathbf{M}}$ and $\varphi(x_{0})\in{\mathbf{N}}$ as $t\to\infty$
uniformly on compact sets. Unfortunately, this definition is not well suitable
for studying the differentials. The problem is that the tangent cone is a
class of isometric spaces. Dealing with differentials, one would prefer to
know what happens in a fixed direction of a tangent space. In this context, in
applications of differentials it is important to know how a concrete
representative of the tangent cone is geometrically and analytically connected
with the given (quasi)metric space.
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|
arxiv-papers
| 2010-05-20T09:30:28 |
2024-09-04T02:49:10.526514
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Svetlana Selivanova, Sergey Vodopyanov",
"submitter": "Sergey Vodopyanov K.",
"url": "https://arxiv.org/abs/1005.3640"
}
|
1005.3739
|
Convex Bodies With Minimal Volume
Product in $\mathbb{R}^{2}$ — A New Proof
Lin Youjiang and Leng Gangsong
${}^{a}Department$ of Mathematics, Shanghai University, Shanghai 200444, P. R.
China
linyoujiang@shu.edu.cn, gleng@staff.shu.edu.cn
†† 2010 Mathematics Subject Classification. Primary: 52A10, 52A40. Key words
and phrases. Convex body, Duality, Mahler Conjecture, Polytopes. The authors
would like to acknowledge the support from the National Natural Science
Foundation of China (10971128), Shanghai Leading Academic Discipline Project
(S30104).
Abstract. In this paper, a new proof of the following result is given: The
product of the volumes of an origin symmetric convex bodies $K$ in
$\mathbb{R}^{2}$ and of its polar body is minimal if and only if $K$ is a
parallelogram.
## 1\. Introduction
A well-known problem in the theory of convex sets is to find a lower bound for
the product of volumes $\mathcal{P}(K)=V(K)V(K^{\ast})$, which is called the
volume-product of K , where $K$ is an $n$-dimensional origin symmetric convex
body and $k^{\ast}$ is the polar body of $K$ (see definition in Section 2). Is
it true that we always have
$\displaystyle\mathcal{P}(K)$ $\displaystyle\geq$
$\displaystyle\mathcal{P}(B_{\infty}^{n}),$ (1.1)
where $B_{\infty}^{n}=\\{x\in\mathbb{R}^{n}:~{}|x_{i}|\leq 1,~{}1\leq i\leq
n\\}$?
For some particular classes of convex symmetric bodies in $\mathbb{R}^{n}$, a
sharper estimate for the lower bound of $\mathcal{P}(K)$ has been obtained. If
$K$ is the unit ball of a normed $n$-dimensional space with a 1-unconditional
basis, J. Saint-Raymond [12] proved that $\mathcal{P}(K)\geq 4^{n}/n!$; the
equality case, obtained for $1-\infty$ spaces, is discussed in [6] and [11].
When $K$ is a zonoid it was proved in [2] and [10] that the same inequality
holds, with equality if and only if $K$ is an $n$-cube.
In [1], J. Bourgain and V. D. Milman proved that there exist some $c>0$ such
that for every $n$ and every convex body $K$ of $\mathbb{R}^{n}$,
$\mathcal{P}(K)\geq c^{n}\mathcal{P}(B_{2}^{n}).$
The best known constant $c=\frac{\pi}{4}$ is due to Kuperberg [3].
In [4], K. Mahler proved (1.2) when $n=2$. There are several other proofs of
the two-dimensional result, see for example the proof of M. Meyer, [7], but
the question is still open even in the three-dimensional case.
In this paper, we present a new proof about the problem when $n=2$, which is
different from the proof in [4] and [7]. Firstly, we prove that any origin
symmetric polygon satisfies the conjecture. Then, using the continuity of
$\mathcal{P}(K)$ with respect to the Hausdorff metric, we can easily prove
that the conjecture is also correct for any origin symmetric convex bodies in
$\mathbb{R}^{2}$. For the three-dimensional case, the conjecture maybe can be
solved by use of the same idea.
Finally, let us mention the problem of giving an upper bound to
$\mathcal{P}(K)$; it was proved by L. A. Santaló [13]:
$P(K)\leq\mathcal{P}(B_{2}^{n})$, where $B_{2}^{n}$ is the $n$-dimensional
Euclidean unit ball. In [5], [8] and [9], it was shown that the equality holds
only if $K$ is an ellipsoid.
## 2\. Notations and background materials
As usual, $S^{n-1}$ denotes the unit sphere, $B^{n}$ the unit ball centered at
the origin, $o$ the origin and $\|\cdot\|$ the norm in Euclidean $n$-space
$\mathbb{R}^{n}$. If $x$, $y\in\mathbb{R}^{n}$, then $\langle x,y\rangle$ is
the inner product of $x$ and $y$.
If $K$ is a set, $\partial K$ is its boundary, $int\;K$ is its interior, and
$conv~{}K$ denotes its convex hull. Let $\mathbb{R}^{n}\backslash K$ denote
the complement of $K$, i.e., $\mathbb{R}^{n}\backslash
K=\\{x\in\mathbb{R}^{n}:x\notin K\\}.$ If $K$ is a $n$-dimensional convex
subset of $\mathbb{R}^{n}$, then $V(k)$ is its volume $V_{n}(K)$.
Let $\mathcal{K}^{n}$ denote the set of convex bodies (compact, convex subsets
with non-empty interiors) in $\mathbb{R}^{n}$. Let $\mathcal{K}^{n}_{o}$
denote the subset of $\mathcal{K}^{n}$ that contains the origin in its
interior. Let $h(K,\cdot):S^{n-1}\rightarrow\mathbb{R}$, denote the support
function of $K\in\mathcal{K}^{n}_{o}$; i.e.,
$\displaystyle h(K,u)=\max\\{u\cdot x:~{}x\in K\\},u\in S^{n-1},$ (2.1)
and let $\rho(K,\cdot):S^{n-1}\rightarrow\mathbb{R}$, denote the radial
function of $K\in\mathcal{K}^{n}_{o}$; i.e.,
$\displaystyle\rho(K,u)=\max\\{\lambda\geq 0:~{}\lambda u\in K\\},u\in
S^{n-1}.$ (2.2)
A linear transformation (or affine transformation) of $\mathbb{R}^{n}$ is a
map $\phi$ from $\mathbb{R}^{n}$ to itself such that $\phi x~{}=~{}Ax$ (or
$\phi x~{}=~{}Ax+t$, respectively), where $A$ is an $n\times n$ matrix and
$t\in\mathbb{R}^{n}$. By definition, for any parallelograms centered at the
origin $ABCD$ and $A^{\prime}B^{\prime}C^{\prime}D^{\prime}$, there always is
an linear transformation $\mathcal{A}$ taking $ABCD$ to
$A^{\prime}B^{\prime}C^{\prime}D^{\prime}$.
Geometrically, an affine transformation in Euclidean space is one that
preserves:
(1). The collinearity relation between points; i.e., three points which lie on
a line continue to be collinear after the transformation.
(2) Ratios of distances along a line; i.e., for distinct collinear points
$P_{1}$, $P_{2}$, $P_{3}$, the ratio $|P_{2}-P_{1}|/|P_{3}-P_{2}|$ is
preserved.
If $K\in{K}^{n}_{o}$, we define the polar body of $K$, $K^{\ast}$, by
$K^{\ast}=\\{x\in\mathbb{R}^{n}:~{}x\cdot y\leq 1~{},\forall y\in K\\}.$
It is easy to verify that (see p.44 in [14])
$\displaystyle
h(K^{\ast},u)=\frac{1}{\rho(K,u)}~{}~{}~{}~{}~{}~{}and~{}~{}~{}~{}~{}~{}~{}\rho(K^{\ast},u)=\frac{1}{h(K,u)}$
(2.3)
If $P$ is a polygon, i.e., $P=conv\\{p_{1},\cdots,p_{m}\\}$, where $p_{i}$
$(i=1,\cdots,m)$ are vertices of polygon $P$. By the definition of polar body,
we have
$\displaystyle P^{\ast}$ $\displaystyle=$
$\displaystyle\\{x\in\mathbb{R}^{2}:x\cdot p_{1}\leq 1,\cdots,x\cdot p_{m}\leq
1\\}$ (2.4) $\displaystyle=$
$\displaystyle\bigcap_{i=1}^{m}\\{x\in\mathbb{R}^{2}:x\cdot p_{i}\leq 1\\},$
which implies that $P^{\ast}$ is the intersection of $m$ closed half-planes
with exterior normal vector $p_{i}$ and the distance of straight line
$\\{x\in\mathbb{R}^{2}:x\cdot p_{i}=1\\}$ from the origin is $1/\|p_{i}\|$.
Thus, if $P$ is an inscribed polygon in a unit circle, then $P^{\ast}$ is
polygon circumscribed around the unit circle. In the proof of Lemma 3.3, we
shall make use of these properties.
For $K$, $L\in\mathcal{K}^{n}$ the Hausdorff distance is defined by
$\displaystyle d(K,L)=\min\\{\lambda\geq 0:~{}K\subset L+\lambda
B^{n},~{}L\subset K+\lambda B^{n}\\},$ (2.5)
which can be conveniently defined by (see p.53 in [14])
$\displaystyle d(K,L)=\max_{u\in S^{n-1}}|h(K,u)-L(K,u)|,$ (2.6)
therefore, a sequence of convex bodies $K_{i}$ converges to $K$ if and only if
the sequence of support function $h(K_{i},\cdot)$ converges uniformly to
$h(K,\cdot)$.
In $\mathcal{K}^{n}_{o}$, the convergence of convex bodies is equivalent to
the uniform convergence of their radial functions. Because the conclusion will
be used in the proof of Lemma 3.5, we prove this conclusion (this proof is due
to Professor Zhang Gaoyong and we listened his lecture in Chongqing).
Let $K\in\mathcal{K}^{n}_{o}$. Define
$\displaystyle r_{1}=\max_{u\in S^{n-1}}\rho(K,u),$ (2.7) $\displaystyle
r_{0}=\min_{u\in S^{n-1}}\rho(K,u).$ (2.8)
It is easily seen that
$\displaystyle r_{1}=\max_{u\in S^{n-1}}h(K,u),$ (2.9) $\displaystyle
r_{0}=\min_{u\in S^{n-1}}h(K,u).$ (2.10)
Lemma 2.1. If $K\in\mathcal{K}^{n}_{o}$, then
$\displaystyle\rho(K+tB^{n},u)\leq\rho(K,u)+\frac{r_{1}}{r_{0}}t,$ (2.11)
$\displaystyle|u\cdot v(x)|\geq\frac{r_{0}}{r_{1}},$ (2.12)
where $x=u\rho(K,u)\in\partial K.$
Proof. For $x\in\partial K$, let $x^{\prime}$ be the point on
$\partial(K+tB^{n})$ and has the same direction as $x$. Let
$u=x/\|x\|=x^{\prime}/\|x^{\prime}\|$. Then
$\rho(K+tB^{n},u)-\rho(K,u)=\|x^{\prime}-x\|.$
Since $K$ and $K+tB^{n}$ are parallel, the projection length of $x^{\prime}-x$
onto the normal $v(x)$ is less than $t$,
$\|x^{\prime}-x\|\leq\frac{t}{|u\cdot v(x)|}.$
There is
$\displaystyle|u\cdot v(x)|$ $\displaystyle=$ $\displaystyle\frac{|x\cdot
v(x)|}{\|x\|}$ (2.13) $\displaystyle=$ $\displaystyle\frac{h(K,v(x))}{\|x\|}$
$\displaystyle\geq$ $\displaystyle\frac{r_{0}}{r_{1}}.$
The desired inequalities follow. $\Box$
Theorem 2.2. If a sequence of convex bodies $K_{i}\in\mathcal{K}^{n}_{0}$
converges to $K\in\mathcal{K}^{n}_{0}$ in the Hausdorff metric, then the
sequence of radial functions $\rho(K_{i},\cdot)$ converges to $\rho(K,\cdot)$
uniformly.
Proof. Assume that $d(K_{i},K)<\varepsilon$. Then $K_{i}\subset K+\varepsilon
B^{n}$, and $K\subset K_{i}+\varepsilon B^{n}$. By Lemma 2.1, (2.9) and (2.10)
$\rho(K_{i},\cdot)\leq\rho(K,\cdot)+\frac{r_{1}}{r_{0}}\varepsilon,$
$\rho(K,\cdot)\leq\rho(K_{i},\cdot)+\frac{r_{1}+\varepsilon}{r_{0}-\varepsilon}\varepsilon.$
When $\varepsilon<r_{0}/2$, we have
$|\rho(K_{i},\cdot)-\rho(K,\cdot)|\leq\frac{4r_{1}}{r_{0}}\varepsilon,$
therefore the sequence of radial functions $\rho(K_{i},\cdot)$ converges to
$\rho(K,\cdot)$ uniformly. $\Box$
## 3\. Main result and its proof
First, looking the following important theorem:
Theorem 3.1. For any origin symmetric convex body $K\subset\mathbb{R}^{n}$,
$\mathcal{P}(K)$ is linear invariant, that is, for every linear transformation
$A:~{}\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$, we have
$\mathcal{P}(AK)=\mathcal{P}(K)$.
Proof. For any $u\in S^{n-1}$, we have
$\rho((AK)^{\ast},u)=\frac{1}{h(AK,u)}=\frac{1}{h(K,A^{t}u)}=\rho(K^{\ast},A^{t}u)=\rho(A^{-t}K^{\ast},u).$
Hence, $(AK)^{\ast}=A^{-t}K^{\ast}$, therefore
$\mathcal{P}(AK)=V(AK)V((AK)^{\ast})=V(AK)V(A^{-t}K^{\ast})$
$=|A||A^{-t}|V(K)V(K^{\ast})=V(K)V(K^{\ast})=\mathcal{P}(K).$
$\Box$
Because any parallelogram can been linear transformed into a unit square,
therefore their volume product is same (this value is equal to 8). By the
theorem above, we consider linear transformation of origin symmetric polygon.
We obtain the following theorem, which is critical in our proof.
Theorem 3.2. In $\mathbb{R}^{2}$, for any origin symmetric polygon $P$, there
exists a linear transformation $\mathcal{A}:~{}P\rightarrow P^{\prime}$, where
$P^{\prime}$ satisfies that $P^{\prime}\subset B^{2}$ and there exist three
continuous vertices contained in $\partial B^{2}$.
Proof. Since $P$ is origin symmetric polygon, its number of sides is an even
and corresponding two sides are parallel. Let $A_{1},\cdots,A_{n},B_{1},\cdots
B_{n}$ denote all vertices of $P$. In order to prove this theorem, we need
three steps.
The first step, transforming parallelogram $A_{1}A_{2}B_{1}B_{2}$ into
rectangular $A_{1}^{\prime}A_{2}^{\prime}B_{1}^{\prime}B_{2}^{\prime}$
inscribed in $B^{2}$. Now $P$ is transformed into $P_{1}$ (see (2) or
$(2)^{\prime}$ in Figure 3.1.1 and 3.1.2).
The second step, transforming $P_{1}$ into $P_{2}$ (see (3) in Figure 3.1.2).
For polygon $P_{1}$, if there exist some vertices
$\\{A_{i}^{\prime}:~{}i\in
I\subset\\{3,\cdots,n\\}\\}\subset\mathbb{R}^{2}\backslash B^{2},$
there exists a linear transformation $\mathcal{A}_{1}:P_{1}\rightarrow P_{2}$,
which shortens segment $A_{1}^{\prime}A_{2}^{\prime}$ and
$B_{1}^{\prime}B_{2}^{\prime}$ into $A_{1}^{\prime\prime}A_{2}^{\prime\prime}$
and $B_{1}^{\prime\prime}B_{2}^{\prime\prime}$, simultaneously makes some
vertices $\\{A_{i}^{\prime\prime}:i\in I_{2}\subset\\{3,\cdots,n\\}\\}$ on
boundary of $B^{2}$ and $P_{2}\subset B^{2}$. If
$\\{A_{3}^{\prime},\cdots,A_{n}^{\prime},B_{3}^{\prime},\cdots,B_{n}^{\prime}\\}\subset
int~{}B^{2},$
then there exists a linear transformation
$\mathcal{A}_{1}^{\prime}:P_{1}\rightarrow P_{2}$, which lengthens segments
$A_{1}^{\prime}A_{2}^{\prime}$ and $B_{1}^{\prime}B_{2}^{\prime}$ into
$A_{1}^{\prime\prime}A_{2}^{\prime\prime}$ and
$B_{1}^{\prime\prime}B_{2}^{\prime\prime}$ respectively, simultaneously makes
some vertices $\\{A_{i}^{\prime\prime}:i\in I_{1}\subset\\{3,\cdots,n\\}\\}$
on boundary of $B^{2}$ and $P_{2}\subset B^{2}$. (see (3) in Figure 3.1.2).
The third step, transforming $P_{2}$ into $P_{3}$. If
$A_{1}^{\prime\prime},A_{2}^{\prime\prime},A_{i}^{\prime\prime}$ are three
continuous vertices contained in $\partial B^{2}$, then this theorem has been
proved; otherwise rotation transforming $P_{2}$ into $P_{3}^{\prime}$, which
satisfies that $A_{2}^{\prime\prime}A_{i}^{\prime\prime}$ parallels x-axis
(see (4) in Figure 3.2). Then we transform $P_{3}^{\prime}$ into $P_{3}$,
lengthening segments $A_{2}^{\prime\prime}B_{i}^{\prime\prime}$ and
$A_{i}^{\prime\prime}B_{2}^{\prime\prime}$ into $A_{2}^{(3)}B_{i}^{(3)}$ and
$A_{i}^{(3)}B_{2}^{(3)}$ respectively, simultaneously making some vertices
$\\{A_{j}^{(3)}:j\in I_{3}\subset\\{3,\cdots,i-1\\}\\}$ on boundary of $B^{2}$
and $P_{3}\subset B^{2}$ (Since it is easy to prove that vertices
$\\{A_{i+1}^{(3)},\cdots,A_{n}^{(3)},B_{1}^{(3)}\\}$ are in the internal of
$B^{2}$ ) (see (5) in Figure 3.2).
Repeating the third step finite times, we can get a polygon $P^{\prime}$, in
which there exist three continuous vertices contained in $\partial B^{2}$,
which completes the proof. $\Box$
By above theorem, we consider the volume-product of polygon with three
continuous vertices in $\partial B^{2}$.
Lemma 3.3. Suppose that $P^{\prime}\subset B^{2}$ is an origin symmetric
polygon and $A,C,B$ are three continuous vertices of $P^{\prime}$ contained in
$\partial B^{2}$, then
$\mathcal{P}(P^{\prime\prime})\leq\mathcal{P}(P^{\prime})$, where
$P^{\prime\prime}$ is a new polygon from $P^{\prime}$ by deleting vertices $C$
and $C^{\prime}$.
Proof. Suppose side $AB$ parallels X-axis (see Figure 3.3.), straight lines
$l$, $l_{1}$ and $l_{2}$ are tangent lines to the unit circle $B^{2}$ passing
through points $C$, $A$ and $B$ respectively.
Let $A=(-x_{0},y_{0})$, then $B=(x_{0},y_{0})$. Let $\theta$ denote $\angle
xOC$. It is clear that $\pi/2\leq\theta\leq\pi-\arctan(y_{0}/x_{0})$ when
point $C$ is in third quadrant. We have the following equations of straight
lines:
$l_{1}:~{}~{}y-y_{0}=\frac{x_{0}}{y_{0}}(x+x_{0}),$
$l_{2}:~{}~{}y-y_{0}=-\frac{x_{0}}{y_{0}}(x-x_{0}),$
$l:~{}~{}y-\sin\theta=-\frac{\cos\theta}{\sin\theta}(x-\cos\theta).$
Let point $N$ denote the intersection of $l$ and Y-axis and point $M$ denote
the intersection of $l_{1}$ and Y-axis. We can easily get $N(0,1/\sin\theta)$
and $M(0,1/y_{0})$. In order to obtain the abscissas of intersection of $l$
and $l_{1}$, $l_{2}$, we solve the following equation systems:
$\left\\{\begin{aligned}
y-\sin\theta&=-\frac{\cos\theta}{\sin\theta}(x-\cos\theta)\\\
y-y_{0}&=\frac{x_{0}}{y_{0}}(x+x_{0})\end{aligned}\right.$ (3.1)
and
$\left\\{\begin{aligned}
y-\sin\theta&=-\frac{\cos\theta}{\sin\theta}(x-\cos\theta)\\\
y-y_{0}&=-\frac{x_{0}}{y_{0}}(x-x_{0})\end{aligned}\right.$ (3.2)
We can get abscissas of points $H$ and $L$:
$x_{1}=\frac{y_{0}-\sin\theta}{y_{0}\cos\theta+x_{0}\sin\theta}$
and
$x_{2}=\frac{y_{0}-\sin\theta}{y_{0}\cos\theta-x_{0}\sin\theta}.$
Therefore we can obtain the area of $\triangle MHL$:
$S_{\triangle MHL}=\frac{x_{0}}{y_{0}}\cdot\frac{\sin\theta-
y_{0}}{\sin\theta+y_{0}}.$
Let $V=V(P^{\prime\prime})$ and $V^{0}=V({P^{\prime\prime}}^{\ast})$, where
$P^{\prime\prime}$ denotes the new polygon from $P^{\prime}$ by deleting
vertices $C$ and $C^{\prime}$, then $\mathcal{P}(P^{\prime})$ is a function
$f(\theta)$, where
$\displaystyle f(\theta)$ $\displaystyle=$
$\displaystyle\left(V+2x_{0}(\sin\theta-
y_{0})\right)\left(V^{0}-\frac{2x_{0}}{y_{0}}\cdot\frac{\sin\theta-
y_{0}}{\sin\theta+y_{0}}\right)$ (3.3)
and
$\frac{\pi}{2}\leq\theta\leq\pi-\arctan(\frac{y_{0}}{x_{0}}).$
We have
$\displaystyle f^{\prime}(\theta)$ $\displaystyle=$ $\displaystyle
2x_{0}\cos\theta\cdot\frac{(V^{0}y_{0}-2x_{0})(\sin\theta+y_{0})^{2}+2y_{0}(4x_{0}y_{0}-V)}{y_{0}(\sin\theta+y_{0})^{2}}.$
(3.4)
In (3.6), since $\cos\theta\leq 0$ and $y_{0}(\sin\theta+y_{0})^{2}\geq 0$, in
order to prove $f^{\prime}(\theta)\leq 0$, let $t=\sin\theta$, we just need to
prove $g(t)\geq 0$, where
$\displaystyle g(t)$ $\displaystyle=$
$\displaystyle(V^{0}y_{0}-2x_{0})(t+y_{0})^{2}+2y_{0}(4x_{0}y_{0}-V),~{}~{}~{}~{}t\in[y_{0},1].$
(3.5)
In order to prove $g(t)\geq 0$, we just need to prove that
$V^{0}y_{0}-2x_{0}>0$ and $g(y_{0})\geq 0$.
Because of $V({P^{\prime\prime}}^{\ast})\geq
V(conv\\{A,M,B,A^{\prime},M^{\prime},B^{\prime}\\})$,
$V^{0}\geq 4x_{0}y_{0}+2x_{0}(\frac{1}{y_{0}}-y_{0}),$
therefore,
$\displaystyle V^{0}y_{0}-2x_{0}$ $\displaystyle\geq$
$\displaystyle\left(4x_{0}y_{0}+2x_{0}(\frac{1}{y_{0}}-y_{0})\right)y_{0}-2x_{0}$
(3.6) $\displaystyle=$ $\displaystyle 2x_{0}y_{0}^{2}$ $\displaystyle>$
$\displaystyle 0,$
and therefore function $g(t)$ is a parabola opening upward. Thence, when
$t\in[y_{0},1]$, quadratic function $g(t)$ is increasing, thus we just need to
proof
$\displaystyle g(y_{0})$ $\displaystyle=$ $\displaystyle
2y_{0}(2V^{0}y_{0}^{2}-V)\geq 0.$ (3.7)
Let $\mathcal{D}$ denote the area of circular segment enclosed by arc
$\widehat{BA^{\prime}}$ and chord $\overline{BA^{\prime}}$, then
$\displaystyle V^{0}$ $\displaystyle\geq$ $\displaystyle
4x_{0}y_{0}+2x_{0}(\frac{1}{y_{0}}-y_{0})+2\mathcal{D}$ (3.8)
and
$\displaystyle V$ $\displaystyle\leq$ $\displaystyle
4x_{0}y_{0}+2\mathcal{D}.$ (3.9)
In order to prove (3.9), we just need to prove
$\displaystyle
2\left(4x_{0}y_{0}+2x_{0}(\frac{1}{y_{0}}-y_{0})+2\mathcal{D}\right)y_{0}^{2}$
$\displaystyle\geq$ $\displaystyle 4x_{0}y_{0}+2\mathcal{D},$ (3.10)
which equivalent to
$\displaystyle 2x_{0}y_{0}^{3}$ $\displaystyle\geq$
$\displaystyle\mathcal{D}(1-2y_{0}^{2}).$ (3.11)
And because
$\displaystyle\mathcal{D}$ $\displaystyle\leq$ $\displaystyle(1-x_{0})\cdot
2y_{0},$ (3.12)
hence, we just need to prove
$\displaystyle x_{0}y_{0}^{3}$ $\displaystyle\geq$ $\displaystyle
y_{0}(1-x_{0})(1-2y_{0}^{2}),$ (3.13)
which equivalent to
$\displaystyle x_{0}^{3}-2x_{0}^{2}+1$ $\displaystyle\geq$ $\displaystyle 0,$
(3.14)
which is clearly correct.
Summary, we get $f^{\prime}(\theta)\leq 0$ when
$\theta\in[\pi/2,\pi-\arctan(y_{0}/x_{0})]$, hence when
$\theta=\pi-\arctan(y_{0}/x_{0})$, which implies that point $C$ coincides with
point $A$, function $f(\theta)$ obtain minimal function value, therefore
$\mathcal{P}(P^{\prime\prime})\leq\mathcal{P}(P^{\prime})$. $\Box$
Making use of Lemma 3.3, we can obtain the following conclusion.
Theorem 3.4. If $P\subset\mathbb{R}^{2}$ is an origin symmetric polygon, then
$\mathcal{P}(P)\geq\mathcal{P}(S)$, where $S$ is square.
Proof. By Theorem 3.2, Lemma 3.3 and linear invariance of $\mathcal{P}(P)$, if
the number of sides of polygon $P$ is $2n$, there exists a polygon $P_{1}$
with $2(n-1)$ sides satisfying $\mathcal{P}(P_{1})\leq\mathcal{P}(P)$.
Repeating this process $n-2$ times, we can obtain a square $S$ satisfying
$\mathcal{P}(P)\geq\mathcal{P}(S)$. $\Box$
In order to obtain the main result in the paper, we first prove the following
lemma.
Lemma 3.5. The volume product $\mathcal{P}(K)$ is continuous under the
Hausdorff metric.
Proof. Let
$\lim_{i\rightarrow\infty}K_{i}=K.$
By Theorem 2.2, the sequence of radial function $\rho(K_{i},\cdot)$ converges
to $\rho(K,\cdot)$ uniformly, therefore the reciprocal of radial function
$1/\rho(K_{i},\cdot)$ converges to $1/\rho(K,\cdot)$ uniformly. Since
$\displaystyle d(K_{i}^{\ast},K^{\ast})$ $\displaystyle=$
$\displaystyle\max_{u\in S^{n-1}}|h(K^{\ast}_{i},u)-h(K^{\ast},u)|$ (3.15)
$\displaystyle=$ $\displaystyle\max_{u\in
S^{n-1}}\left|\frac{1}{\rho(K_{i},u)}-\frac{1}{\rho(K,u)}\right|,$
we have
$\displaystyle\lim_{i\rightarrow\infty}K^{\ast}_{i}$ $\displaystyle=$
$\displaystyle K^{\ast}.$ (3.16)
By continuity of the volume function $V(\cdot)$ under the Hausdorff metric, we
have
$\displaystyle\mathcal{P}(K)$ $\displaystyle=$ $\displaystyle V(K)V(K^{\ast})$
(3.17) $\displaystyle=$
$\displaystyle\lim_{i\rightarrow\infty}V(K_{i})\lim_{i\rightarrow\infty}V(K_{i}^{\ast})$
$\displaystyle=$
$\displaystyle\lim_{i\rightarrow\infty}V(K_{i})V(K_{i}^{\ast})$
$\displaystyle=$ $\displaystyle\lim_{i\rightarrow\infty}\mathcal{P}(K_{i}).$
$\Box$
Theorem 3.6. If $K\subset\mathbb{R}^{2}$ is an origin symmetric convex body
and $S\subset\mathbb{R}^{2}$ is a square, then
$\mathcal{P}(K)\geq\mathcal{P}(S)$.
Proof. For any origin symmetric convex body $K\subset\mathbb{R}^{2}$, there
exists a sequence of origin symmetric polytopes $\\{P_{i}\\}$ converging to
$K$ under the Hausdorff metric. By Theorem 3.4 and Lemma 3.5, we have
$\displaystyle\mathcal{P}(K)=\lim_{n\rightarrow\infty}\mathcal{P}(P_{i})\geq\mathcal{P}(S).$
(3.18)
$\Box$
## References
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|
arxiv-papers
| 2010-03-10T12:58:21 |
2024-09-04T02:49:10.539243
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Youjiang Lin",
"submitter": "Youjiang Lin",
"url": "https://arxiv.org/abs/1005.3739"
}
|
1005.3747
|
11institutetext: Lehrstuhl für Astronomie, University of Würzburg, Am Hubland,
D-97074 Würzburg
# Modelling the variability of 1ES1218+30.4
M. Weidinger 11 F. Spanier 11 fspanier@astro.uni-wuerzburg.de
(Received 22 February 2010 / Accepted 6 April 2010)
###### Abstract
Context. The blazar 1 ES 1218+30.4 has been previously detected by the VERITAS
and MAGIC telescopes in the very high energies. The new detection of VERITAS
from December 2008 to April 2009 proves that 1 ES 1218+30.4 is not static, but
shows short-time variability.
Aims. We show that the time variability may be explained in the context of a
self-consistent synchrotron-self Compton model, while the long time
observation do not necessarily require a time-resolved treatment.
Methods. The kinetic equations for electrons and photons in a plasma blob are
solved numerically including Fermi acceleration for electrons as well as
synchrotron radiation and Compton scattering.
Results. The light curve observed by VERITAS can be reproduced in our model by
assuming a changing level of electron injection compared to the constant state
of 1 ES 1218+30.4 . The multiwavelength behaviour during an outburst becomes
comprehensible by the model.
Conclusions. The long time measurements of VERITAS are still explainable via a
constant emission in the SSC context, but the short outbursts each require a
time-resolved treatment.
###### Key Words.:
galaxies: jets - relativistic processes - radiation mechanisms: non-thermal -
BL Lacertae objects: individual: 1 ES 1218+30.4 \- galaxies: active
††offprints: F. Spanier,
## 1 Introduction
Blazars are a special class of active galactic nuclei (AGN) exhibiting 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. The source of this emission is
presumably the relativistic jet emitted at a narrow angle to the line of sight
to the observer.
In high-peaked BL Lac objects (HBLs) the SED shows a double hump structure as
the most notable feature with the first hump in the UV- to X-ray regime and
the second hump in the gamma-ray regime. Indeed, a substantial fraction of the
known nearby HBLs have already been discovered with Cherenkov telescopes like
H.E.S.S., MAGIC or VERITAS. The origin of the first hump is mostly undisputed:
nonthermal, relativistic electrons in the jet are emitting synchrotron
radiation. The origin of the second hump is still controversially debated. Up
to now two kinds of models are discussed: leptonic (e.g. Maraschi et al.,
1992) and hadronic (e.g. Mannheim, 1993) ones, which are mostly applied for
other subclasses of blazars.
Another important feature of AGNs in general and HBLs in particular is their
strong variability. The dynamical timescale may range from minutes to years.
This requires complex models, which obviously have to include time dependence,
but this gives us also the chance to understand the mechanisms that drive
AGNs. We will apply a self-consistent leptonic model to new data observed for
the source 1 ES 1218+30.4 , because those are the ones favoured for HBLs.
The source HBL 1 ES 1218+30.4 has been discovered as a candidate BL Lac object
on the basis of its X-ray emission and has been identified with the X-ray
source 2A 1219+30.5 (Wilson et al., 1979; Ledden et al., 1981). For the first
time, 1 ES 1218+30.4 has been observed at VHE energies using the MAGIC
telescope in January 2005 (Albert et al., 2006) and later from VERITAS
(Acciari et al., 2009). Coverage of the optical/X-ray regime is provided by
BeppoSAX (Donato et al., 2005) and SWIFT (Tramacere et al., 2007),
unfortunately the data are not always simultaneous. During the observations
from December 2008 to April 2009 VERITAS also observed 1 ES 1218+30.4 showing
a time-variability (the VERITAS collaboration et al., 2010). The observations
from the MAGIC telescope have previously been modelled by Rüger et al. (2010).
the VERITAS collaboration et al. (2010) claim that their new observations
exhibiting variability challenge the previous models. We will show that a
timedependent model using a self-consistent treatment of electron acceleration
is able to model the new VERITAS data.
We present the kinetic equation, which we solve numerically, describing the
synchrotron-self Compton emission (Sect. 2). In Sect. 3 we apply our code to 1
ES 1218+30.4 , taking the VERITAS data into account and give a set of physical
parameters for the most acceptable fit. Finally, we discuss our results in the
light of particle acceleration theory and the multiwavelength features.
## 2 Model
Here we will give a brief description of the model used, for a complete
overview see (Weidinger et al., 2010; Weidinger & Spanier, 2010).
We start with the relativistic Vlasov equation (see e.g. Schlickeiser, 2002)
in the one dimensional diffusion approximation (e.g. Schlickeiser, 1984), here
the relativistic approximation $p\approx\gamma mc$ is used. This kinetic
equation will then be solved time-dependently in two spatially different
zones, the smaller acceleration zone and the radiation zone, which are assumed
to be spherical and homogeneous. Both contain isotropically distributed
electrons and a randomly oriented magnetic field as common for these models.
All calculations are made in the rest frame of the blob.
Electrons entering the acceleration zone (radius $R_{\text{acc}}$) from the
upstream of the jet are continuously accelerated through diffusive shock
acceleration. This extends the model of Kirk et al. (1998) with a stochastic
part. The energy gain due to the acceleration is balanced by radiative
(synchrotron) and escape losses, the latter scaling with $t_{\text{esc}}=\eta
R_{\text{acc}}/c$ with $\eta=10$ as an empirical factor reflecting the
diffusive nature of particle loss. Escaping electrons completely enter the
radiation zone (radius $R_{\text{rad}}$) downstream of the acceleration zone.
Here the electrons are suffering synchrotron losses as in the acceleration
zone and also inverse-Compton losses, but they do not undergo acceleration.
Pair production and other contributions do not alter the SED in typical SSC
conditions and are neglected (Chiang & Böttcher, 2002). The SED in the
observer’s frame is calculated by boosting the selfconsistently calculated
photons towards the observer’s frame and correcting for the redshift $z$:
$I_{\nu_{\text{obs}}}=\delta^{3}h\nu_{\text{obs}}/(4\pi)N_{\text{ph}}$ with
$\nu_{\text{obs}}=\delta/(1+z)\nu$. The acceleration zone will have no
contribution to $I_{\text{obs}}$ directly, due to the $R_{\text{i}}^{2}$
dependence of the observed flux at a distance $r$
($F_{\nu_{\text{obs}}}(r)=\pi I_{\nu_{\text{obs}}}R_{\text{rad}}^{2}r^{-2}$)
and the small size of the acceleration zone. The kinetic equation in the
acceleration zone is
$\displaystyle\frac{\partial n_{e}(\gamma,t)}{\partial t}=$
$\displaystyle\frac{\partial}{\partial\gamma}\left[(\beta_{s}\gamma^{2}-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\
\text{.}$ (1)
The injected electrons at $\gamma_{0}$, as the blob propagates through the
jet, are considered via
$Q_{\text{inj}}(\gamma,t):=Q_{0}\delta(\gamma-\gamma_{0})$. The synchrotron
losses are calculated using Eq. (2).
$\displaystyle P_{s}(\gamma)$
$\displaystyle=\frac{1}{6\pi}\frac{\sigma_{\text{T}}B^{2}}{mc}\gamma^{2}=\beta_{s}\gamma^{2}$
(2)
with the Thomson cross-section $\sigma_{\text{T}}$. The characteristic
timescale for the acceleration
$t_{\text{acc}}=\left(v_{s}^{2}/(4K_{||})+2v_{A}^{2}/(9K_{||})\right)^{-1}$ of
the system is found by comparing Eq. (2) with Schlickeiser (1984) with the
parallel spatial diffusion coefficient $K_{||}$ not depending on $\gamma$ when
using the hard sphere approximation. The characteristic timescale has an
additional factor ($\propto v_{A}^{2}$) arising from the Fermi-II processes
compared to shock acceleration by itself. The stochastic part of the
acceleration also gives rise to the second row in Eq. (2), while the first row
mainly depends on Fermi-I processes. This dependence of $t_{\text{acc}}$ is
important for the interpretation of the resulting electron spectra, e.g. of
their slopes (depending on $t_{\text{acc}}/t_{\text{esc}}$) or the maximum
energies (depending on $1/(t_{\text{acc}}\beta_{s})$), see Weidinger et al.
(2010) for details. For modelling SEDs and lightcurves it is primary important
to ensure sensible values for $t_{\text{acc}}$. Unlike in Drury et al. (1999),
the energy-dependence of the escape losses is also neglected because we do not
expect a pileup as suggested in Schlickeiser (1984) at typical SSC conditions.
$v_{s},v_{A}$ are the shock and Alfvén speed respectively. Hence $a$ in Eq.
(2) measures the efficiency of the shock acceleration compared to stochastic
processes. Setting $v_{A}=0$, i.e. $a\rightarrow\infty$, will result in a
shock-only model like Kirk et al. (1998).
This model takes account of a much more confined shock region. Fermi-I
acceleration will probably not occur over the whole blob when considering a
real blazar but rather at a small region in the blob’s front. Neglecting
acceleration simplifies the kinetic equation in the radiation zone to
$\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{.}$ (3)
$P_{\text{IC}}$ accounts for the inverse-Compton losses of the electrons
additionally occurring (beside the synchrotron losses) (e.g. Schlickeiser,
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}}}\leavevmode\nobreak\
\text{.}$ (4)
The photon energies are rewritten in terms of the electron rest mass,
$h\nu=\alpha mc^{2}$ for the scattered photons and $h\nu=\alpha_{1}mc^{2}$ for
the target photons respectively. Equation (4) is solved numerically using the
full Klein-Nishina cross-section for a single electron scattering off a photon
field (see e.g. Jones, 1968). Here $\alpha_{max}$ accounts for the kinematic
restrictions on IC scattering. In analogy to the acceleration zone the
catastrophic losses are considered via $t_{\text{esc,rad}}=\eta
R_{\text{rad}}/c$ with $\eta=10$. $t_{\text{esc,rad}}$ is the responding
timescale of the electron system, which is proportional to the variability
timescale in the observer’s frame (see e.g. Kerrick et al., 1995):
$\displaystyle
t_{\text{var}}\propto\frac{t_{\text{esc,rad}}}{\delta}\leavevmode\nobreak\ .$
(5)
To determine the time-dependent model SED of blazars the partial differential
equation for the differential photon number density has to be solved time-
dependently, which can be done numerically. The PDE (6) can be obtained from
the radiative transfer equation making use of the isotropy of the blob
$\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{,}$ (6)
where $R_{s}$ and $R_{c}$ are the production rates for synchrotron photon and
the inverse-Compton respectively. $R_{s}$ is calculated using the well known
Melrose approximation and the inverse-Compton production rate $R_{c}$ is
treated in the most exact way, i.e. using the full Klein-Nishina cross
section, see Weidinger et al. (2010). Below a critical energy the obtained
spectrum is self-absorbed due to synchrotron self-absorption, which is
described by $\alpha_{\nu}$ (Weidinger et al., 2010; Rüger et al., 2010). The
photon-loss rate is set to be the light-crossing time.
## 3 Results
Using the parameters summed up in Table 1 we were able to fit the emission of
1 ES 1218+30.4 as a steady state with our SSC model, see Fig. 1. We used all
the archival data from BeppoSAX, SWIFT in the X-ray band and the MAGIC 2006,
VERITAS 2009 as well as the new released VERITAS 2010 data in the VHE to model
the SED of 1 ES 1218+30.4 (Donato et al., 2005; Tramacere et al., 2007; Albert
et al., 2006; Acciari et al., 2009; the VERITAS collaboration et al., 2010).
The derived SED is absorbed in the VHE using the EBL model of Primack et al.
(2005) for the corresponding redshift of 1 ES 1218+30.4 .
The parameters of our SSC model are well winthin the standard SSC parameter
region with an equipartition parameter of $0.02$. Even though PIC and MHD
simulations suggest a higher magnetic field compared to particle energy (in
the range of 0.1), this is a common assumption in SSC models, but has to be
kept in mind with regard e.g. the stability of the blob. If one wishes to
enforce higher equipartition parameters one could to use the model of
Schlickeiser & Lerche (2007). In order to allow strong shocks to form
$v_{A}<v_{S}$ must be fulfilled, which is the case for $a=10$.
Due to relatively small deviation (within the error margins) between the MAGIC
2005/VERITAS 2008 and the averaged VHE data from the VERITAS 2009 campaign we
find a steady state the most plausible way to model the emission, i.e. the
small fluctuations (see the overall lightcurve in the VERITAS collaboration et
al. (2010)) are not contributing significantly to the averaged observed SEDs.
In the Fermi LAT energy regime our model yields a photon index of
$\alpha_{\text{Fer}}=-1.69$, whichagrees well with the Fermi measurement of
$-1.63\pm 0.12$ (Abdo et al., 2009).
Table 1: Model parameters for the low-state SED, basis to model the outburst
by varying $Q_{0}$.
$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$ | $\delta$
---|---|---|---|---|---|---
$6.25\cdot 10^{4}$ | $0.12$ | $6.0\cdot 10^{14}$ | $3.0\cdot 10^{15}$ | $1.11$ | $10$ | $44$
The lightcurve of the VERITAS collaboration et al. (2010) shows a relatively
strong outburst at $\approx$ MJD54861. Starting with the steady state emission
(solid line, Fig. 1; parameters: Table 1) we injected more electrons $Q_{0}$
into the emission region at low $\gamma_{0}\approx 3$. As the blob evolves in
time the emission in the model at higher energies rises and drops off again
when the injected electrons finally relax to the initial $Q_{0}$. This process
can be explained as density fluctuations along the jet axis and finally fits
the flare.
Figure 1: Model SED of 1 ES 1218+30.4 (black solid line) as derived using the
described model (see Sect. 2) and the parameters shown in Table 1. The VHE
parts of the model SEDs have been absorbed using the EBL model of Primack et
al. (2005). The BeppoSAX data are from Donato et al. (2005), SWIFT from
Tramacere et al. (2007), MAGIC from Albert et al. (2006), VERITAS 2009 from
Acciari et al. (2009) and the blue dots are the new VERITAS 2010 data from the
VERITAS collaboration et al. (2010). The dashed red curve shows the time
integrated SED over the strong outburst shown in Fig. 2, as measured by
VERITAS in 2009.
We found that nearly doubling the injected electron number density in a
$Q_{0}(t)=1+b(t/t_{\text{e,var}})^{3}$ way with a timescale
$t_{\text{e,var}}\approx 1.5$ days (as measured in the observer’s frame) and
then decreasing them to the initial $Q_{0}$ in an almost linear way on the
same timescale fits the strong outburst of 1 ES 1218+30.4 . The corresponding
lightcurve of the model as well as the observed one are summarized in Fig. 2
3.
Figure 2: Lightcurve of the photon flux above $200$ GeV as measured by the
VERITAS collaboration et al. (2010) (inset of their figure) in January 2009 to
February 2009 and our model (red solid line). The outburst was modelled by
injecting more electrons into the blob by varying $Q_{0}$ at a timescale of
$\approx 1.5$ days (see text for details).
Figure 3 shows a more detailed view of the lightcurve in the VHE (above $200$
GeV) as well as the corresponding lightcurves in the X-Ray (between $1.2$ keV
and $11$ keV) regime of BeppoSAX/SWIFT and the lower tail of the Fermi LAT
energy range (between $0.2$ GeV and $22$ GeV) as predicted by our model. The
latter two have been scaled down to the flux level of the VERITAS measurement,
see Fig. 3, because the real fluxes are higher than the VHE flux. The model
predicts the peak of the lightcurve in the Fermi regime to be $1.26$ hours
ahead of the VHE one, where the X-ray regime is delayed by $0.97$ hours for 1
ES 1218+30.4 . The delay of the X-ray band can be used to verify the model
when multiwavelength data of the flaring behaviour of 1 ES 1218+30.4 is
available, while the derivation of the $0.2$ GeV to $22$ GeV lightcurve is
beyond the resolution of Fermi for this source.
Figure 3: Detailed view of the high outburst shown already in Fig. 2 as well
as the behaviour of 1 ES 1218+30.4 in the lower Fermi LAT energy band and the
synchrotron regime, measurable by BeppoSAX/SWIFT during such a flare.
Additionally we plotted the time averaged SED (over the outburst from MJD54860
until MJD54864) into the SED of 1 ES 1218+30.4 , Fig. 1 (dashed red line). As
one can see only when separately considering the strongest outburst of 1 ES
1218+30.4 within the VERITAS campaign in 2009 the aberration from a presumed
steady state is significant. In contrast an average over the whole observation
of the VERITAS collaboration et al. (2010), which is low-state most of the
time, will result in a steady state as shown here or in Rüger et al. (2010).
For the IC photon index $\alpha$ ($\nu F_{\nu}\propto\nu^{\alpha+2}$) above
$200$ GeV we get $\alpha_{\text{VHE}}=-3.53$ for the low-state (solid curve in
Fig. 1), which slightly softens to $\alpha_{\text{VHE}}=-3.56$ when
considering the high-state as the time-average over the single outburst shown
in the lightcurve, Fig. 2 in the VERITAS range. Note that the photon index and
its behaviour during an outburst in this energy range is very sensitive to the
EBL absorption and thus to the EBL model used and shows a strong dependency on
the considered energy range. With our model we are able to compute the
spectral behaviour in the X-Ray energy range of the BeppoSAX/SWIFT satellites
(i.e. $1.2$ keV $<$ E $<$ $11$ keV). The model predicts the source to be
spectrally steady in this regime with a photon index
$\alpha_{\text{xray}}=-2.68$ for outbursts on timescales of days. Considering
shorter averaging timescales of the outburst of 1 ES 1218+30.4 , e.g. the
first or last two hours, two hours around the peak in the lightcurve, the
maximum derivation from $\alpha_{\text{xray}}$ $(\alpha_{\text{VHE}})$
predicted by the model is $\pm 0.05$ $(-0.07)$, which could not be measured
with current experiments and thus is considered as spectrally steady in this
case.
## 4 Discussion
Our results clearly show that the latest observations from the VERITAS
telescope for 1 ES 1218+30.4 still agree with a constant (steady state)
emission from a SSC model when averaged over a long observation period. This
is due to the relatively moderate variability of 1 ES 1218+30.4 compared to
the observation time.
The variability may be well explained in the context of the self-consistent
treatment of acceleration of electrons in the jet. We are aware that an
outburst of the timescale of roughly five days as measured from 1 ES 1218+30.4
does not necessarily require a shock in jet model, which scales down to a few
minutes depending on the SSC parameters (Weidinger & Spanier, 2010), but may
also be explained as e.g. different accretion states. Nevertheless the
fundamental statement remains the same: long time observation of slightly
variable blazars will result in a steady state emission, while an average over
a single outburst will, of course, result in a significantly different SED for
the source. We are not yet able to rule out different emission models or even
complex geometries of the emitting region. But we are able to model the
influence of short outbursts of a source on the SED and the lightcurves in the
different energy bands selfconsistently.
The VERITAS collaboration only shows an integrated spectrum for 1 ES 1218+30.4
, which is due to the low flux of the source and the photon index behaviour of
the combined high-states. This integrated spectrum does not show strong
variations with regard to the known low-state observed by MAGIC. Our model now
predicts a clear change in the spectrum, which is indicated by the dashed line
in Fig. 1, which shows the average over one outburst with a slight, currently
not detectable spectral softening in the VHE range, while the synchrotron peak
in the BeppoSAX/SWIFT regime remains spectrally constant. This situation
changes for shorter and/or stronger outbursts of an overall timescale of
hours, which will result in measurable spectral evolutions in all energy
regimes when considered with the presented model. Furthermore the time-
resolved SEDs during a flare are comprehensible with our model. Hence with
better time-resolved spectra or/and better multiwavelength coverage it should
be possible to prove this model, and if the model is indeed applicable it will
be a good tool to investigate the whole SED during an outburst without having
all energy regimes observationally covered.
Acknowledgments MW wants to thank the Elitenetzwerk Bayern and GK1147 for
their support. FS acknowledges support from the DFG through grant SP 1124/1.
## References
* Abdo et al. (2009) Abdo, A. A., Ackermann, M., Ajello, M., et al. 2009, Astrophys. J., 707, 1310
* Acciari et al. (2009) Acciari, V. A., Aliu, E., Arlen, T., et al. 2009, Astrophys. J., 695, 1370
* Albert et al. (2006) Albert, J., Aliu, E., Anderhub, H., et al. 2006, Astrophys. J., Lett., 642, L119
* Chiang & Böttcher (2002) Chiang, J. & Böttcher, M. 2002, Astrophys. J., 564, 92
* Donato et al. (2005) Donato, D., Sambruna, R. M., & Gliozzi, M. 2005, Astron. Astrophys., 433, 1163
* Drury et al. (1999) Drury, L. O., Duffy, P., Eichler, D., & Mastichiadis, A. 1999, Astron. Astrophys., 347, 370
* Jones (1968) Jones, F. C. 1968, Physical Review, 167, 1159
* Kerrick et al. (1995) Kerrick et al. 1995, Astrophys. J., Lett., 438, L59
* Kirk et al. (1998) Kirk, J. G., Rieger, F. M., & Mastichiadis, A. 1998, Astron. Astrophys., 333, 452
* Ledden et al. (1981) Ledden, J. E., Odell, S. L., Stein, W. A., & Wisniewski, W. Z. 1981, Astrophys. J., 243, 47
* Mannheim (1993) Mannheim, K. 1993, Astron. Astrophys., 269, 67
* Maraschi et al. (1992) Maraschi, L., Ghisellini, G., & Celotti, A. 1992, Astrophys. J., Lett., 397, L5
* Primack et al. (2005) Primack, J. R., Bullock, J. S., & Somerville, R. S. 2005, in American Institute of Physics Conference Series, Vol. 745, High Energy Gamma-Ray Astronomy, ed. F. A. Aharonian, H. J. Völk, & D. Horns, 23–33
* Rüger et al. (2010) Rüger, M., Spanier, F., & Mannheim, K. 2010, Mon. Not. R. Astron. Soc., 401, 973
* Schlickeiser (1984) Schlickeiser, R. 1984, Astron. Astrophys., 136, 227
* Schlickeiser (2002) Schlickeiser, R. 2002, Cosmic ray astrophysics (Astronomy and Astrophysics Library; Physics and Astronomy Online Library. Berlin: Springer. ISBN 3-540-66465-3, 2002, XV + 519 pp.)
* Schlickeiser & Lerche (2007) Schlickeiser, R. & Lerche, I. 2007, A&A, 476, 1
* the VERITAS collaboration et al. (2010) the VERITAS collaboration, Acciari, V. A., Aliu, E., et al. 2010, ASTROPHYSICAL JOURNAL LETTERS, 709, L163
* Tramacere et al. (2007) Tramacere, A., Giommi, P., Massaro, E., et al. 2007, Astron. Astrophys., 467, 501
* Weidinger et al. (2010) Weidinger, M., Rüger, M., & Spanier, F. 2010, Astrophysics and Space Sciences Transactions, 6, 1
* Weidinger & Spanier (2010) Weidinger, M. & Spanier, F. 2010, in Int. J. Mod. Phys. D, Vol. (subm.), HEPRO II conference proceedings, ed. G. Romero, F. Aharonian, & J. Paredes
* Wilson et al. (1979) Wilson, A. S., Ward, M. J., Axon, D. J., Elvis, M., & Meurs, E. J. A. 1979, Mon. Not. R. Astron. Soc., 187, 109
|
arxiv-papers
| 2010-05-20T16:03:15 |
2024-09-04T02:49:10.545046
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Matthias Weidinger and Felix Spanier",
"submitter": "Matthias Weidinger",
"url": "https://arxiv.org/abs/1005.3747"
}
|
1005.3807
|
# A model of the twisted $K$-theory related to bundles of finite order
A.V. Ershov ershov.andrei@gmail.com
###### Abstract.
In the present paper we propose a geometric model of the twisted $K$-theory
related to elements of finite order in
$H^{3}(X,\,\mathbb{Z})\times[X,\,\mathop{\rm BBSU}\nolimits_{\otimes}]$. For
this purpose we consider the monoid of endomorphisms of the direct limit of
matrix algebras which acts on the space of Fredholm operators, the
representing space of $K$-theory, in such a way that this action corresponds
to the multiplication of $K(X)$ by elements of finite order. Being well-
pointed and grouplike, this monoid has the classifying space which is the base
of the universal Dold fibration. This allows us to define the corresponding
twisted $K$-theory as the group of homotopy classes of sections of the
associated fibration of Fredholm operators.
Partially supported by the joint RFBR-DFG project (RFBR grant 07-01-91555 /
DFG project “K-Theory, $C^{*}$-algebras, and Index theory”)
## Introduction
The complex $K$-theory is a generalized cohomology theory represented by the
$\Omega$-spectrum $\\{K_{n}\\}_{n\geq 0}$, where
$K_{n}=\mathbb{Z}\times\mathop{\rm BU}\nolimits$ if $n$ is even and
$K_{n}={\rm U}$ if $n$ is odd. $K_{0}=\mathbb{Z}\times\mathop{\rm
BU}\nolimits$ is an $E_{\infty}$-ring space, and the corresponding space of
units $K_{\otimes}$ (which is an infinite loop space) is
$\mathbb{Z}/2\mathbb{Z}\times\mathop{\rm BU}\nolimits_{\otimes}$, where
$\mathop{\rm BU}\nolimits_{\otimes}$ denotes the space $\mathop{\rm
BU}\nolimits$ with the $H$-space structure induced by the tensor product of
virtual bundles of virtual dimension $1$. Twistings of the $K$-theory over a
compact space $X$ are classified by homotopy classes of maps $X\rightarrow{\rm
B}(\mathbb{Z}/2\mathbb{Z}\times\mathop{\rm
BU}\nolimits_{\otimes})\simeq\mathop{\rm
K}\nolimits(\mathbb{Z}/2\mathbb{Z},\,1)\times\mathop{\rm
BBU}\nolimits_{\otimes}$ (where $\rm B$ denotes the functor of classifying
space). The theorem that $\mathop{\rm BU}\nolimits_{\otimes}$ is an infinite
loop space was proved by G. Segal [18]. Moreover, the spectrum $\mathop{\rm
BU}\nolimits_{\otimes}$ can be decomposed as follows: $\mathop{\rm
BU}\nolimits_{\otimes}=\mathop{\rm
K}\nolimits(\mathbb{Z},\,2)\times\mathop{\rm BSU}\nolimits_{\otimes}$. This
implies that the twistings in $K$-theory can be classified by homotopy classes
of maps $X\rightarrow\mathop{\rm
K}\nolimits(\mathbb{Z}/2\mathbb{Z},\,1)\times\mathop{\rm
K}\nolimits(\mathbb{Z},\,3)\times\mathop{\rm BBSU}\nolimits_{\otimes}$. In
other words, for a compact space $X$ the twistings correspond to elements in
$H^{1}(X,\,\mathbb{Z}/2\mathbb{Z})\times
H^{3}(X,\,\mathbb{Z})\times[X,\,\mathop{\rm
BBSU}\nolimits_{\otimes}],\;[X,\,\mathop{\rm
BBSU}\nolimits_{\otimes}]=bsu^{1}_{\otimes}(X),$ where
$\\{bsu^{n}_{\otimes}\\}_{n}$ is the generalized cohomology theory
corresponding to the infinite loop space $\mathop{\rm
BSU}\nolimits_{\otimes}$.
Twisted K-theory (under the name “K-theory with local coefficients”) has its
origins in M. Karoubi’s PhD thesis [10] and in paper of P. Donovan and M.
Karoubi [6], where the case of a local coefficient system
$\alpha\in\mathbb{Z}/2\mathbb{Z}\times H^{1}(X,\,\mathbb{Z}/2\mathbb{Z})\times
H^{3}_{tors}(X,\,\mathbb{Z})$ was studied. The case of general (not
necessarily of finite order) twistings from $H^{3}(X,\,\mathbb{Z})$ was
considered by J. Rosenberg in [16]. A modern survey on this subject (including
historical remarks) is given in [11]. A very accessible introduction to the
subject is also given in [19].
The twisted $K$-theory corresponding to the twistings coming from
$H^{1}(X,\,\mathbb{Z}/2\mathbb{Z})\times H^{3}(X,\,\mathbb{Z})$ has been
intensively studied during the last decade, but not the general case (as far
as the author knows). It seems that the reason is that there is no known
appropriate geometric model for “nonabelian” twistings from $[X,\,\mathop{\rm
BBSU}\nolimits_{\otimes}]$.
In the present paper we make an attempt to give such a model for elements of
finite order in $H^{3}(X,\,\mathbb{Z})\times[X,\,\mathop{\rm
BBSU}\nolimits_{\otimes}]$.
For this purpose we consider the monoid of endomorphisms of the direct limit
of matrix algebras
$M_{kl^{\infty}}(\mathbb{C}):=\lim\limits_{\longrightarrow\atop{m}}M_{kl^{m}}(\mathbb{C})$
(the limit is taken over unital homomorphisms). More precisely, in the
infinite algebra $M_{kl^{\infty}}(\mathbb{C})$ we fix an increasing filtration
by unital subalgebras $A_{kl^{m}}\subset A_{kl^{m+1}}\subset\ldots,\quad
A_{kl^{m}}\cong M_{kl^{m}}(\mathbb{C})$ such that
$A_{kl^{m+1}}=M_{l}(A_{kl^{m}})$ and consider endomorphisms of
$M_{kl^{\infty}}(\mathbb{C})$ that induced by unital homomorphisms of the form
$h_{m,\,n}\colon A_{kl^{m}}\rightarrow A_{kl^{m+n}}$ (for some $m,\,n$), i.e.
that are of the form $M_{l^{\infty}}(h_{m,\,n}).$ Such endomorphisms form the
topological monoid $\mathop{\rm End}\nolimits(M_{kl^{\infty}}(\mathbb{C}))$
which is homotopy equivalent to the direct limit $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}:=\lim\limits_{\longrightarrow\atop{m,\,n}}\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$, where $\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}=\mathop{\rm
Hom}\nolimits_{alg}(M_{kl^{m}}(\mathbb{C}),\,M_{kl^{m+n}}(\mathbb{C}))$ is the
space of unital $*$-homomorphisms of matrix algebras, and the limit is not
contractible for pairs $\\{k,\,l\\}$ such that $(k,\,l)=1$. Note that
$\mathop{\rm Fr}\nolimits_{k,\,1}=\mathop{\rm PU}\nolimits(k),$ i.e. for
$m=n=0$ we return to the known case of abelian twistings of finite order
(which is described in the next section). Furthermore, the monoid $\mathop{\rm
End}\nolimits(M_{kl^{\infty}}(\mathbb{C}))$ naturally acts on the space of
Fredholm operators and this action induces the multiplication of $K(X)$ by
elements of order $k$. Moreover, the monoid $\mathop{\rm
End}\nolimits(M_{kl^{\infty}}(\mathbb{C}))$ is well-pointed and grouplike and
therefore it has the classifying space which is the base of the corresponding
universal principal fibration (in the sense of Dold, i.e. with the WCHP).
In fact, “usual” (abelian) twistings of order $k$ correspond to automorphisms
of $M_{kl^{\infty}}(\mathbb{C})$ (which form the group
$\lim\limits_{\longrightarrow\atop{m}}\mathop{\rm PU}\nolimits(kl^{m})$
because of $\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}=\mathop{\rm
PU}\nolimits(kl^{m})$ for $n=0$) while nonabelian ones correspond to general
endomorphisms. Note that these endomorphisms act on the localization of the
space of Fredholm operators over $l$ by homotopy auto-equivalences, i.e. they
are invertible in the sense of homotopy.
This paper is organized as follows.
In Section 1 we give a review of standard material about twisted $K$-theory
related to twistings from $H^{3}(X,\,\mathbb{Z})$. The definition is based on
the conjugation action of the projective unitary group $\mathop{\rm
PU}\nolimits({\mathcal{H}})$ of a separable Hilbert space ${\mathcal{H}}$ on
the space of Fredholm operators $\mathop{\rm Fred}\nolimits({\mathcal{H}})$,
the representing space of complex $K$-theory. This action induces the action
of the Picard group $Pic(X)$ on $K(X)$ by group automorphisms (Theorem 1). We
also consider the specialization of this construction to the case of twistings
of finite order in $H^{3}(X,\,\mathbb{Z})$ because precisely this particular
case we are going to generalize in what follows.
In Section 2 we study the spaces of unital $*$-homomorphisms of matrix
algebras $\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}$ which will play in the
subsequent consideration the same role as the groups $\mathop{\rm
PU}\nolimits(k)$ for twistings of finite order in $H^{3}(X,\,\mathbb{Z})$.
The key result of Section 3 is Theorem 17 which can be regarded as a
counterpart of Theorem 1. It states that in terms of the representing space
$\mathop{\rm Fred}\nolimits({\mathcal{H}})$ the multiplication of the
$K$-functor by (not necessarily line) bundles of finite order $k$ can be
represented by some maps $\gamma_{kl^{m},\,l^{n}}\colon\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\times\mathop{\rm
Fred}\nolimits({\mathcal{H}})\rightarrow\mathop{\rm
Fred}\nolimits({\mathcal{H}}).$
In order to organize the particular maps $\gamma_{kl^{m},\,l^{n}}$ for
different $m,\,n$ in a genuine action on $\mathop{\rm
Fred}\nolimits({\mathcal{H}})$ we should take the direct limit
$\lim\limits_{\longrightarrow\atop{m,\,n}}\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$. It turns out that this limit naturally is a
topological monoid, and we give its precise definition in Section 4\. In
Section 5 we investigate its action on $K$-theory.
Since $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}:=\lim\limits_{\longrightarrow\atop{m,\,n}}\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$ is a well-pointed grouplike topological monoid,
it has the classifying space $\mathop{\rm B}\nolimits\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ which is the base of the universal
principal $\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$-fibration.
This allows us to define the corresponding twisted $K$-theory as the set of
homotopy classes of sections of the associated fibration with fiber the space
of Fredholm operators. We do this in Section 6.
In Section 7 we sketch an approach via (a homotopy coherent version of) bundle
gerbes.
In Section 8 we define maps $\mathop{\rm B}\nolimits\mathop{\rm
Fr}\nolimits_{k^{t}l^{\infty},\,l^{\infty}}\times\mathop{\rm
B}\nolimits\mathop{\rm
Fr}\nolimits_{k^{u}l^{\infty},\,l^{\infty}}\rightarrow\mathop{\rm
B}\nolimits\mathop{\rm Fr}\nolimits_{k^{t+u}l^{\infty},\,l^{\infty}}$ and the
corresponding generalization of the (finite) Brauer group.
Sections 9 and 10 contains some results concerning homotopy types of
considered spaces, in particular, a calculation of homotopy groups of
$\mathop{\rm End}\nolimits(M_{kl^{\infty}}(\mathbb{C}))$ (which clarifies the
origin of the condition $(k,\,l)=1$).
Acknowledgments: I am grateful to Professor E.V. Troitsky for all-round
support and very helpful discussions. A number of related questions were
discussed with Professors A.S. Mishchenko, Thomas Schick and Georgy I.
Sharygin and I would like to express my gratitude to them.
## 1\. Twisted $K$-theory related to twistings from $H^{3}(X,\,\mathbb{Z})$
In order to establish the relation with the subsequent construction of more
general twistings, we begin with a review of the standard material about
twisted $K$-theory with twistings from $H^{3}(X,\,\mathbb{Z})$.
Let $X$ be a compact space, $Pic(X)$ its Picard group consisting of
isomorphism classes of line bundles with respect to the tensor product. The
Picard group is represented by the $H$-space $\mathop{\rm
BU}\nolimits(1)\simeq\mathbb{C}P^{\infty}\simeq\mathop{\rm
K}\nolimits(\mathbb{Z},\,2)$ whose multiplication is given by the tensor
product of line bundles or (in the appearance of the Eilenberg-MacLane space)
by the addition of two-dimensional integer cohomology classes. In particular,
the first Chern class $c_{1}$ defines the group isomorphism $c_{1}\colon
Pic(X)\stackrel{{\scriptstyle\cong}}{{\rightarrow}}H^{2}(X,\,\mathbb{Z}).$ The
group $Pic(X)$ is a subgroup of the multiplicative group of the ring $K(X)$
and therefore it acts on $K(X)$ by group automorphisms. This action is
functorial on $X$ and therefore it can be described in terms of classifying
spaces (see Theorem 1).
As a representing space for $K$-theory we take $\mathop{\rm
Fred}\nolimits({\mathcal{H}}),$ the space of Fredholm operators on the
separable Hilbert space ${\mathcal{H}}$. It is known [2] that for a compact
space $X$ the action of $Pic(X)$ on $K(X)$ is induced by the conjugation
action
$\gamma\colon\mathop{\rm PU}\nolimits({\mathcal{H}})\times\mathop{\rm
Fred}\nolimits({\mathcal{H}})\rightarrow\mathop{\rm
Fred}\nolimits({\mathcal{H}}),\;\gamma(g,\,T)=gTg^{-1}$
of the projective unitary group $\mathop{\rm PU}\nolimits({\mathcal{H}})$ of
the Hilbert space ${\mathcal{H}}$ on $\mathop{\rm
Fred}\nolimits({\mathcal{H}}).$ The precise statement is given by the
following theorem (recall that $\mathop{\rm
PU}\nolimits({\mathcal{H}})\simeq\mathbb{C}P^{\infty}\simeq\mathop{\rm
K}\nolimits(\mathbb{Z},\,2)$).
###### Theorem 1.
If $f_{\xi}\colon X\rightarrow\mathop{\rm Fred}\nolimits({\mathcal{H}})$ and
$\varphi_{\zeta}\colon X\rightarrow\mathop{\rm PU}\nolimits({\mathcal{H}})$
represent $\xi\in K(X)$ and $\zeta\in Pic(X)$ respectively, then the composite
map
(1) $X\stackrel{{\scriptstyle\mathop{\rm
diag}\nolimits}}{{\longrightarrow}}X\times
X\stackrel{{\scriptstyle\varphi_{\zeta}\times
f_{\xi}}}{{\longrightarrow}}\mathop{\rm
PU}\nolimits({\mathcal{H}})\times\mathop{\rm
Fred}\nolimits({\mathcal{H}})\stackrel{{\scriptstyle\gamma}}{{\rightarrow}}\mathop{\rm
Fred}\nolimits({\mathcal{H}})$
represents $\xi\otimes\zeta\in K(X)$.
Proof see in [2].$\quad\square$
It is essential for the theorem that the group $\mathop{\rm
PU}\nolimits({\mathcal{H}})$ has the homotopy type of the classifying space
for line bundles $\mathbb{C}P^{\infty}$ and from the other hand its
conjugation action on $\mathop{\rm Fred}\nolimits({\mathcal{H}})$ induces the
action of the Picard group on $K(X)$.
In order to define the corresponding version of $K$-theory consider
$\mathop{\rm Fred}\nolimits({\mathcal{H}})$-bundle $\widetilde{\mathop{\rm
Fred}\nolimits}({\mathcal{H}})\rightarrow\mathop{\rm
BPU}\nolimits({\mathcal{H}})$ associated (by means of the action $\gamma$)
with the universal principal $\mathop{\rm PU}\nolimits({\mathcal{H}})$-bundle
over the classifying space $\mathop{\rm
BPU}\nolimits({\mathcal{H}})\simeq\mathop{\rm K}\nolimits(\mathbb{Z},\,3)$ for
$\mathop{\rm PU}\nolimits({\mathcal{H}})$, i.e. the bundle
(2) $\textstyle{\mathop{\rm
Fred}\nolimits({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
EPU}\nolimits({\mathcal{H}}){\mathop{\times}\limits_{\mathop{\rm
PU}\nolimits({\mathcal{H}})}}\mathop{\rm
Fred}\nolimits({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
BPU}\nolimits({\mathcal{H}}).}$
Then for any map $f\colon X\rightarrow\mathop{\rm
BPU}\nolimits({\mathcal{H}})$ the corresponding twisted $K$-theory $K_{f}(X)$
is the set (in fact the group) of homotopy classes of sections
$[X,\,f^{*}\widetilde{\mathop{\rm Fred}\nolimits}({\mathcal{H}})]^{\prime}$
(here $[\ldots,\,\ldots]^{\prime}$ denotes the set of fibrewise homotopy
classes of sections). The group $K_{f}(X)$ depends up to isomorphism only on
the homotopy class $[f]$ of $f$, i.e. in fact on the corresponding cohomology
class in $H^{3}(X,\,\mathbb{Z})$ called the Dixmier-Douady class.
###### Remark 2.
Although the isomorphism class of the twisted $K$-theory group only depends on
the twisting class in $H^{3}(X,\,\mathbb{Z})$, it is important to note that
this isomorphism is not natural, but that instead one has a natural action of
$H^{2}(X,\,\mathbb{Z})$ on such isomorphisms111The author is grateful to
Thomas Schick who pointed me out to this important fact. [5].
In this paper we will consider twistings of finite order, in the abelian case
they are related to subgroups $\mathop{\rm PU}\nolimits(k)\subset\mathop{\rm
PU}\nolimits({\mathcal{H}}),\;k\in\mathbb{N}$.
###### Remark 3.
It is not true that for every $\alpha\in H^{3}(X,\,\mathbb{Z})$ such that
$k\alpha=0$ there exists a $\mathop{\rm PU}\nolimits(k)$-bundle with Dixmier-
Douady class $\alpha$: in general one has to consider all groups $\mathop{\rm
PU}\nolimits(k^{n}),\>n\in\mathbb{N}$. For example, for $k=2$ there is no
factorization $\mathop{\rm
K}\nolimits(\mathbb{Z}/2\mathbb{Z},\,2)\rightarrow\mathop{\rm
BPU}\nolimits(2)\rightarrow\mathop{\rm K}\nolimits(\mathbb{Z},\,3)$ of the
Bockstein map $\mathop{\rm
K}\nolimits(\mathbb{Z}/2\mathbb{Z},\,2)\rightarrow\mathop{\rm
K}\nolimits(\mathbb{Z},\,3)$ (otherwise applying the loop functor we obtain
the factorization from Remark 6 below which clearly does not exist [1]).
Let ${\mathcal{B}}({\mathcal{H}})$ be the algebra of bounded operators on the
separable Hilbert space ${\mathcal{H}}$,
$M_{k}({\mathcal{B}}({\mathcal{H}}))={\mathcal{B}}({\mathcal{H}}^{\oplus k})$
the matrix algebra over ${\mathcal{B}}({\mathcal{H}})$ (of course, it is
isomorphic to ${\mathcal{B}}({\mathcal{H}})$), $M_{k}(\mathbb{C})\rightarrow
M_{k}({\mathcal{B}}({\mathcal{H}}))$ the inclusion induced by the inclusion of
the unit
$\mathbb{C}\rightarrow{\mathcal{B}}({\mathcal{H}}),\;1\mapsto\mathop{\rm
Id}\nolimits.$ Thereby $\mathop{\rm U}\nolimits(k)$ is a subgroup of the
unitary group $\mathop{\rm U}\nolimits_{k}({\mathcal{H}})$ of the algebra
$M_{k}({\mathcal{B}}({\mathcal{H}}))$, and we have the injective homomorphism
(3) $i_{k}\colon\mathop{\rm PU}\nolimits(k)\hookrightarrow\mathop{\rm
PU}\nolimits_{k}({\mathcal{H}}),$
where $\mathop{\rm PU}\nolimits_{k}({\mathcal{H}})$ is the projective unitary
group on ${\mathcal{H}}^{\oplus k}$ (of course, $\mathop{\rm
PU}\nolimits_{k}({\mathcal{H}})\cong\mathop{\rm PU}\nolimits({\mathcal{H}})$).
The group $\mathop{\rm PU}\nolimits(k)$ is the base of the principal
$\mathop{\rm U}\nolimits(1)$-bundle
(4) $\mathop{\rm U}\nolimits(1)\rightarrow\mathop{\rm
U}\nolimits(k)\stackrel{{\scriptstyle\chi_{k}}}{{\rightarrow}}\mathop{\rm
PU}\nolimits(k).$
Let $\vartheta_{k,\,1}\rightarrow\mathop{\rm PU}\nolimits(k)$ be the complex
line bundle associated with (4) (we introduce the subscripts in
$\vartheta_{k,\,1}$ for unification with the subsequent notation).
Analogously, $\mathop{\rm PU}\nolimits({\mathcal{H}})$ is the base of the
universal principal $\mathop{\rm U}\nolimits(1)$-bundle
$\mathop{\rm U}\nolimits(1)\rightarrow\mathop{\rm
U}\nolimits({\mathcal{H}})\rightarrow\mathop{\rm PU}\nolimits({\mathcal{H}}).$
Let $[k]$ be the trivial $\mathbb{C}^{k}$-bundle over $X$.
###### Proposition 4.
If a line bundle $\zeta\rightarrow X$ satisfies the condition
(5) $[k]\otimes\zeta=\zeta^{\oplus k}\cong X\times\mathbb{C}^{k},$
then its classifying map $\varphi_{\zeta}\colon X\rightarrow\mathop{\rm
PU}\nolimits_{k}({\mathcal{H}})\cong\mathop{\rm PU}\nolimits({\mathcal{H}})$
can be lifted to a map $\widetilde{\varphi}_{\zeta}\colon
X\rightarrow\mathop{\rm PU}\nolimits(k)$ (see (3)) such that
$i_{k}\circ\widetilde{\varphi}_{\zeta}\simeq\varphi_{\zeta}$, and vice versa.
In particular, $\zeta\cong\widetilde{\varphi}_{\zeta}^{*}(\vartheta_{k,\,1}).$
Proof. Extend exact sequence (4) to the right to fibration
(6) $\mathop{\rm
PU}\nolimits(k)\stackrel{{\scriptstyle\psi_{k}}}{{\rightarrow}}\mathop{\rm
BU}\nolimits(1)\stackrel{{\scriptstyle\omega_{k}}}{{\rightarrow}}\mathop{\rm
BU}\nolimits(k).$
In particular, $\psi_{k}\colon\mathop{\rm
PU}\nolimits(k)\rightarrow\mathop{\rm
BU}\nolimits(1)\simeq\mathbb{C}P^{\infty}$ is a classifying map for
$\mathop{\rm U}\nolimits(1)$-bundle $\chi_{k}$ (4). It is easy to see that the
diagram
$\textstyle{\mathop{\rm
PU}\nolimits(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{k}}$$\scriptstyle{\psi_{k}}$$\textstyle{\mathop{\rm
BU}\nolimits(1)}$$\textstyle{\mathop{\rm
PU}\nolimits_{k}({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$
commutes, where the vertical arrow is a classifying map for the bundle
$\mathop{\rm U}\nolimits(1)\rightarrow\mathop{\rm
U}\nolimits({\mathcal{H}})\rightarrow\mathop{\rm PU}\nolimits({\mathcal{H}})$.
Let $\zeta\rightarrow X$ be a line bundle satisfying condition (5),
$\varphi^{\prime}_{\zeta}\colon X\rightarrow\mathop{\rm BU}\nolimits(1)$ its
classifying map. Since $\omega_{k}$ (see (6)) is induced by taking the direct
sum of a line bundle with itself $k$ times (followed by the extension of the
structural group to $\mathop{\rm U}\nolimits(k)$), we see that
$\omega_{k}\circ\varphi^{\prime}_{\zeta}\simeq*$. Now it is easy to see from
the exactness of (6) that $\varphi^{\prime}_{\zeta}\colon
X\rightarrow\mathop{\rm BU}\nolimits(1)$ has a lift
$\widetilde{\varphi}^{\prime}_{\zeta}\colon X\rightarrow\mathop{\rm
PU}\nolimits(k)$, and hence the same is true for
$\varphi_{\zeta}.\quad\square$
###### Remark 5.
Note that the choice of a lift $\widetilde{\varphi}_{\zeta}$ corresponds to
the choice of trivialization (5): two choices differ by a map
$X\rightarrow\mathop{\rm U}\nolimits(k)$. Thus, a lift is defined up to the
action of $[X,\,\mathop{\rm U}\nolimits(k)]$ on $[X,\,\mathop{\rm
PU}\nolimits(k)].$ The subgroup in $Pic(X)$ consisting of (classes of) line
bundles satisfying condition (5) is $\mathop{\rm
im}\nolimits\\{\psi_{k*}\colon[X,\,\mathop{\rm
PU}\nolimits(k)]\rightarrow[X,\,\mathbb{C}P^{\infty}]\\}$ (or the factor-group
$[X,\,\mathop{\rm PU}\nolimits(k)]/[X,\,\mathop{\rm U}\nolimits(k)]$:
$[X,\,\mathop{\rm U}\nolimits(k)]$ is a normal subgroup in $[X,\,\mathop{\rm
PU}\nolimits(k)]$ because it is the kernel of the group homomorphism
$i_{k*}\colon[X,\,\mathop{\rm PU}\nolimits(k)]\rightarrow[X,\,\mathop{\rm
PU}\nolimits_{k}({\mathcal{H}})]$, cf. (3)).
###### Remark 6.
We do not claim that every element $\zeta\in Pic(X),\>\zeta^{k}=1$ can be
represented by a map $X\rightarrow\mathop{\rm PU}\nolimits(k).$ For example,
for $k=2$ there is no factorization $\mathop{\rm
K}\nolimits(\mathbb{Z}/2\mathbb{Z},\,1)\simeq\mathbb{R}P^{\infty}\rightarrow\mathop{\rm
PU}\nolimits(2)\rightarrow\mathbb{C}P^{\infty}$ of the Bockstein map
$\mathop{\rm
K}\nolimits(\mathbb{Z}/2\mathbb{Z},\,1)\simeq\mathbb{R}P^{\infty}\rightarrow\mathbb{C}P^{\infty}\simeq\mathop{\rm
K}\nolimits(\mathbb{Z},\,2)$. In order to obtain all elements of order $k$ in
the sense of the group structure on $Pic(X)$ one has to consider all subgroups
$\mathop{\rm PU}\nolimits(k^{n}),\,n\in\mathbb{N}$ (cf. Remark (3)).
Let $\mathop{\rm Fred}\nolimits_{k}({\mathcal{H}})$ be the subspace of
Fredholm operators in $M_{k}({\mathcal{B}}({\mathcal{H}}))$. Clearly,
$\mathop{\rm Fred}\nolimits_{k}({\mathcal{H}})\cong\mathop{\rm
Fred}\nolimits({\mathcal{H}})$. Being a subgroup in $\mathop{\rm
PU}\nolimits_{k}({\mathcal{H}})$ (see (3)), the group $\mathop{\rm
PU}\nolimits(k)$ acts on $M_{k}({\mathcal{B}}({\mathcal{H}}))$. Let
(7) $\gamma_{k,\,1}\colon\mathop{\rm PU}\nolimits(k)\times\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})\rightarrow\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})$
be the restriction of this action on $\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})\subset M_{k}({\mathcal{B}}({\mathcal{H}}))$.
Then the diagram
$\textstyle{\mathop{\rm PU}\nolimits_{k}({\mathcal{H}})\times\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\qquad\gamma}$$\textstyle{\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})}$$\textstyle{\mathop{\rm
PU}\nolimits(k)\times\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{k}\times\mathop{\rm
id}\nolimits}$$\scriptstyle{\gamma_{k,\,1}}$
commutes and we have the following theorem which is a specialization of
Theorem 1.
###### Theorem 7.
Let $f_{\xi}\colon X\rightarrow\mathop{\rm Fred}\nolimits_{k}({\mathcal{H}})$
be a representing map for some element $\xi\in K(X).$ Let $\zeta$ be as in the
previous proposition. Then the composite map
$X\stackrel{{\scriptstyle\mathop{\rm diag}\nolimits}}{{\rightarrow}}X\times
X\stackrel{{\scriptstyle\widetilde{\varphi}_{\zeta}\times
f_{\xi}}}{{\longrightarrow}}\mathop{\rm PU}\nolimits(k)\times\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})\stackrel{{\scriptstyle\gamma_{k,\,1}}}{{\longrightarrow}}\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})$
represents the element $\xi\otimes\zeta\in K(X).$
###### Remark 8.
Note that the “subgroup” $\mathop{\rm U}\nolimits(k)\rightarrow\mathop{\rm
PU}\nolimits(k)$ acts homotopy trivially on $\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})$ (and hence trivially on $K(X)$), in
accordance with Remark 5. Indeed, if $\varphi_{\zeta}$ can be lifted to
$\mathop{\rm U}\nolimits(k)$, then $\zeta\cong[1]$ is a trivial line bundle
over $X$, from the other hand the action of $\mathop{\rm U}\nolimits(k)$ on
$\mathop{\rm Fred}\nolimits_{k}({\mathcal{H}})$ can be extended to the action
of the contractible group $\mathop{\rm
U}\nolimits_{k}({\mathcal{H}})\cong\mathop{\rm U}\nolimits({\mathcal{H}})$.
###### Remark 9.
It follows from the definition of the inclusion $i_{k}$ that the action
$\gamma_{k,\,1}$ is trivial on elements in $K(X)$ of the form $k\xi.$ Indeed,
a classifying map for $k\xi$ can be decomposed as $X\stackrel{{\scriptstyle
f_{\xi}}}{{\rightarrow}}\mathop{\rm
Fred}\nolimits({\mathcal{H}})\stackrel{{\scriptstyle\mathop{\rm
diag}\nolimits}}{{\rightarrow}}\mathop{\rm Fred}\nolimits_{k}({\mathcal{H}}).$
From the other hand, there is the relation $(1+(\zeta-1))\cdot
k\xi=k\xi+0=k\xi$ in the $K$-functor, or, equivalently,
$\zeta\otimes([k]\otimes\xi)=(\zeta\otimes[k])\otimes\xi=[k]\otimes\xi$ in
terms of bundles.
Note that since inclusion (3) is a group homomorphism, the group structure on
$\mathop{\rm PU}\nolimits(k)$ corresponds to the tensor product of line
bundles that are classified by this group.
Consider $\mathop{\rm Fred}\nolimits_{k}({\mathcal{H}})$-bundle
(8) $\begin{array}[]{c}\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
22.40698pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\\&\\\\}}}\ignorespaces{\hbox{\kern-22.40698pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
22.40698pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
46.40698pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
46.40698pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\mathop{\rm
EPU}\nolimits(k){\mathop{\times}\limits_{\mathop{\rm
PU}\nolimits(k)}}\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
97.63693pt\raise-6.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
97.63693pt\raise-30.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-3.0pt\raise-40.5pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
77.29314pt\raise-40.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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BPU}\nolimits(k)}$}}}}}}}\ignorespaces\ignorespaces}}}}}\end{array}$
associated with the universal principal $\mathop{\rm PU}\nolimits(k)$-bundle
$\mathop{\rm EPU}\nolimits(k)\rightarrow\mathop{\rm BPU}\nolimits(k)$ by means
of the action $\gamma_{k,\,1}$. This bundle is the pullback of (2) with
respect to $\mathop{\rm B}\nolimits i_{k}$. We will denote it by
$\widetilde{\mathop{\rm
Fred}\nolimits_{k}}({\mathcal{H}})\rightarrow\mathop{\rm BPU}\nolimits(k)$ for
short.
Now the version of the twisted $K$-theory related to the conjugation action of
$\mathop{\rm PU}\nolimits(k)$ on $\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})$, or, equivalently, to the action of the
group of (isomorphism classes of) line bundles classified by maps
$X\rightarrow\mathop{\rm PU}\nolimits(k)$ on $K(X)$, is defined as follows:
for a given map $f\colon X\rightarrow\mathop{\rm BPU}\nolimits(k)$ we define
$K_{f}(X)$ as the set $[X,\,f^{*}(\widetilde{\mathop{\rm
Fred}\nolimits_{k}}({\mathcal{H}}))]^{\prime}$ of homotopy classes of sections
of the induced bundle $f^{*}(\widetilde{\mathop{\rm
Fred}\nolimits_{k}}({\mathcal{H}}))\rightarrow X.$
Note that up to (noncanonical) isomorphism the twisted $K$-theory depends only
on the cohomology class $\beta=f^{*}(\alpha)\in H^{3}(X,\,\mathbb{Z}),$ where
$\alpha\in H^{3}(\mathop{\rm
BPU}\nolimits(k),\,\mathbb{Z})\cong\mathbb{Z}/k\mathbb{Z}$ is the generator,
therefore the more appropriate notation for it is $K_{\beta}(X)$.
Note that the considered constructions are well-behaved with respect to the
group homomorphisms
$\mathop{\rm PU}\nolimits(k^{n})\rightarrow\mathop{\rm
PU}\nolimits(k^{n+1}),\;T\mapsto T\otimes E_{k},$
i.e. we can take the corresponding direct limits.
There is another way to define the twisted $K$-theory. Namely, let
${\mathcal{K}}({\mathcal{H}})$ be the algebra of compact operators on the
separable Hilbert space ${\mathcal{H}}$. Recall that the group of
$*$-automorphisms of the algebra ${\mathcal{K}}({\mathcal{H}})$ is
$\mathop{\rm PU}\nolimits({\mathcal{H}})$. For a given $\mathop{\rm
PU}\nolimits({\mathcal{H}})$-cocycle on $X$ consider the corresponding
${\mathcal{K}}({\mathcal{H}})$-bundle $A\rightarrow X$ and define the
corresponding twisted $K$-theory as the algebraic $K$-theory of the Banach
algebra $\Gamma(A,\,X)$ of its continuous sections (we should only remember
that the algebra ${\mathcal{K}}({\mathcal{H}})$ is not unital). If the
Dixmier-Douady class of $A$ has finite order, then the
${\mathcal{K}}({\mathcal{H}})$-bundle $A\rightarrow X$ is of the form
$A_{k}\otimes{\mathcal{K}}({\mathcal{H}})$, where $A_{k}\rightarrow X$ is a
matrix algebra bundle with fiber $M_{k}(\mathbb{C})$ (for some $k$). In this
case the twisted $K$-theory can be defined as the $K$-theory of the algebra of
sections of $A_{k}\rightarrow X$.
The specific property of the finite-dimensional case is that algebras of
sections of nonisomorphic bundles $A_{k}\rightarrow X$ and
$A_{m}^{\prime}\rightarrow X$ can be Morita-equivalent, i.e. they can define
the same element in the Brauer group $Br(X)$ (note that if in addition
$(k,\,m)=1$, then $\Gamma(X,\,A_{k})$ is Morita-equivalent to $C(X)$). This
happens precisely when $A_{k}\otimes{\mathcal{K}}({\mathcal{H}})\cong
A_{m}^{\prime}\otimes{\mathcal{K}}({\mathcal{H}})$ as
${\mathcal{K}}({\mathcal{H}})$-bundles (let us notice the relation of this
fact to Remarks 5 and 8). In fact, there is the group isomorphism $Br(X)\cong
H^{3}(X,\,\mathbb{Z})$ defined by the assignment to an algebra bundle its
Dixmier-Douady class. The torsion subgroup in $Br(X)$, the so-called “finite
Brauer group”, corresponds to (finite dimensional) matrix algebra bundles.
For a fixed $\alpha\in H^{3}(X,\,\mathbb{Z}),\;\alpha\neq 0$ the twisted
$K$-theory $K_{\alpha}(X)$ is not a ring, only a $K(X)$-module. However there
are maps $K_{\alpha}(X)\otimes K_{\beta}(X)\rightarrow K_{\alpha+\beta}(X)$
which equip the direct sum $\mathop{\oplus}\limits_{\alpha\in
Br(X)}K_{\alpha}(X)$ with the structure of a graded ring.
## 2\. Spaces of unital homomorphisms of matrix algebras
In this section we study spaces of unital $*$-homomorphisms of matrix
algebras. They can be regarded as analogs of groups of $*$-automorphisms
$\mathop{\rm Aut}\nolimits(M_{k}(\mathbb{C}))\cong\mathop{\rm PU}\nolimits(k)$
in the subsequent constructions.
Fix a pair of positive integers $\\{k,\,l\\},\;(k,\,l)=1.$ Let $\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$ be the space of unital $*$-homomorphisms of
matrix algebras $\mathop{\rm
Hom}\nolimits_{alg}(M_{kl^{m}}(\mathbb{C}),\,M_{kl^{m+n}}(\mathbb{C}))$.
Recall that the group of $*$-automorphisms of the complex matrix algebra
$M_{n}(\mathbb{C})$ is the projective unitary group $\mathop{\rm
PU}\nolimits(n),$ therefore there are the left action of $\mathop{\rm
PU}\nolimits(kl^{m+n})$ and the right action of $\mathop{\rm
PU}\nolimits(kl^{m})$ on $\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}$.
Moreover, $\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}$ is a (left) homogeneous
space over $\mathop{\rm PU}\nolimits(kl^{m+n})$:
###### Proposition 10.
There is an isomorphism of homogeneous spaces
(9) $\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}\cong\mathop{\rm
PU}\nolimits(kl^{m+n})/(E_{kl^{m}}\otimes\mathop{\rm PU}\nolimits(l^{n})),$
where $E_{n}$ and the symbol “$\otimes$” denote the unit matrix and the
Kronecker product of matrices respectively.
Proof. It follows from Noether-Skolem’s theorem that the group $\mathop{\rm
PU}\nolimits(kl^{m+n})$ acts transitively on the set of unital
$*$-homomorphisms $M_{kl^{m}}(\mathbb{C})\rightarrow
M_{kl^{m+n}}(\mathbb{C})$. From the other hand, the stabilizer of such
homomorphism $M_{kl^{m}}(\mathbb{C})\rightarrow
M_{kl^{m+n}}(\mathbb{C}),\;T\mapsto T\otimes E_{l^{n}}$ is the subgroup
$E_{kl^{m}}\otimes\mathop{\rm PU}\nolimits(l^{n})\subset\mathop{\rm
PU}\nolimits(kl^{m+n}).\quad\square$
In particular, for $n=0$ we have $\mathop{\rm
Fr}\nolimits_{kl^{m},\,1}=\mathop{\rm PU}\nolimits(kl^{m})$.
###### Proposition 11.
A map $\varphi\colon X\rightarrow\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}$ is
the same thing as an embedding of trivial bundles $X\times
M_{kl^{m}}(\mathbb{C})\stackrel{{\scriptstyle\mu}}{{\hookrightarrow}}X\times
M_{kl^{m+n}}(\mathbb{C})$ whose restriction to a fiber is a unital
$*$-homomorphism of matrix algebras.
Proof. We have the bijection (in obvious notation) $\mathop{\rm
Mor}\nolimits(X,\,\mathop{\rm
Hom}\nolimits_{alg}(M_{kl^{m}}(\mathbb{C}),\,M_{kl^{m+n}}(\mathbb{C})))\cong\mathop{\rm
Mor}\nolimits(X\times
M_{kl^{m}}(\mathbb{C}),\,M_{kl^{m+n}}(\mathbb{C})),\;h(x)(T)\mapsto
h(x,\,T),\,x\in X,\,T\in M_{kl^{m}}(\mathbb{C})$. But for any map
$\lambda\colon X\times M_{kl^{m}}(\mathbb{C})\rightarrow
M_{kl^{m+n}}(\mathbb{C})$ there exists the unique map $\nu\colon X\times
M_{kl^{m}}(\mathbb{C})\rightarrow X\times
M_{kl^{m+n}}(\mathbb{C}),\;\nu(x,\,T)=(x,\,\lambda(x,\,T))$ which is the
identity on the first factor $X.\quad\square$
For an embedding $\mu$ as in the statement of Proposition 11 one can define
the subbundle
---
$\textstyle{B_{l^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subset\qquad}$$\textstyle{X\times
M_{kl^{m+n}}(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$
of centralizers for the image of $\mu$ which is an
$M_{l^{n}}(\mathbb{C})$-bundle such that $M_{kl^{m}}(\mathbb{C})\otimes
B_{l^{n}}=X\times M_{kl^{m+n}}(\mathbb{C}).$
In particular, applying the previous proposition to $\mathop{\rm
id}\nolimits\colon\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$ we obtain the canonical embedding
$\widetilde{\mu}\colon\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}\times
M_{kl^{m}}(\mathbb{C})\hookrightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\times
M_{kl^{m+n}}(\mathbb{C}),\>(h,\,T)\mapsto(h,\,h(T))$ and the corresponding
$M_{l^{n}}(\mathbb{C})$-bundle
${\mathcal{B}}_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$. Clearly, we have the canonical isomorphism
$M_{kl^{m}}(\mathbb{C})\otimes{\mathcal{B}}_{kl^{m},\,l^{n}}\cong\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\times M_{kl^{m+n}}(\mathbb{C})$ with the trivial
bundle, but let us notice that the bundle
${\mathcal{B}}_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$ itself is not trivial for $n>0$, as it follows
from the next proposition.
###### Proposition 12.
The $M_{l^{n}}(\mathbb{C})$-bundle
${\mathcal{B}}_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$ is associated with the principal $\mathop{\rm
PU}\nolimits(l^{n})$-bundle
(10) $\mathop{\rm PU}\nolimits(l^{n})\rightarrow\mathop{\rm
PU}\nolimits(kl^{m+n})\rightarrow\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}$
(see (9)).
Proof is trivial.$\quad\square$
Note that with respect to the above notation we have
$B_{l^{n}}=\varphi^{*}({\mathcal{B}}_{kl^{m},\,l^{n}}).$
There is the homeomorphism (cf. (9))
(11) $\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}\cong\mathop{\rm
U}\nolimits(kl^{m+n})/(E_{kl^{m}}\otimes\mathop{\rm U}\nolimits(l^{n})),$
therefore we have the principal $\mathop{\rm U}\nolimits(l^{n})$-bundle (cf.
(10))
(12) $\begin{array}[]{c}\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
13.90985pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\\&\crcr}}}\ignorespaces{\hbox{\kern-13.90985pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathop{\rm
U}\nolimits(l^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.90985pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
37.90985pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
37.90985pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\mathop{\rm
U}\nolimits(kl^{m+n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
59.21634pt\raise-5.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
59.21634pt\raise-30.8111pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-3.0pt\raise-40.64441pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
43.85753pt\raise-40.64441pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}}$}}}}}}}\ignorespaces\ignorespaces}}}}}\end{array}$
over $\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}$. Let
$\vartheta_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$ be the vector $\mathbb{C}^{l^{n}}$-bundle
associated with (12). For example, for $n=0$ we have the line bundle
$\vartheta_{kl^{m},\,1}\rightarrow\mathop{\rm PU}\nolimits(kl^{m})$ associated
with $\mathop{\rm U}\nolimits(1)\rightarrow\mathop{\rm
U}\nolimits(kl^{m})\rightarrow\mathop{\rm PU}\nolimits(kl^{m}).$ Note that
$\mathop{\rm
End}\nolimits(\vartheta_{kl^{m},\,l^{n}})={\mathcal{B}}_{kl^{m},\,l^{n}}.$
Let $X$ be a compact topological space. By $[n]$ denote the trivial vector
bundle with fiber $\mathbb{C}^{n}$. Note that there is the canonical
trivialization $[kl^{m}]\otimes\vartheta_{kl^{m},\,l^{n}}\cong[kl^{m+n}]$ of
the bundle $[kl^{m}]\otimes\vartheta_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$.
###### Proposition 13.
(cf. Proposition 4). For any vector $\mathbb{C}^{l^{n}}$-bundle
$\eta_{l^{n}}\rightarrow X$ such that
(13) $[kl^{m}]\otimes\eta_{l^{n}}\cong[kl^{m+n}]$
there is a map $\varphi=\varphi_{\eta_{l^{n}}}\colon X\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$ such that
$\varphi^{*}(\vartheta_{kl^{m},\,l^{n}})\cong\eta_{l^{n}},$ and vice versa.
Note that such $\varphi$ is not unique (even up to homotopy): it also depends
on the choice of trivialization (13).
Proof. Consider the fibration (cf. (12))
(14) $\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}\mathop{\rm
BU}\nolimits(l^{n})\stackrel{{\scriptstyle\beta}}{{\rightarrow}}\mathop{\rm
BU}\nolimits(kl^{m+n}),$
where $\alpha$ classifies $\vartheta_{kl^{m},\,l^{n}}$ as a
$\mathbb{C}^{l^{n}}$-bundle and $\beta$ is induced by the group homomorphism
$\mathop{\rm U}\nolimits(l^{n})\rightarrow\mathop{\rm
U}\nolimits(kl^{m+n}),\;T\mapsto E_{kl^{m}}\otimes T$ (the Kronecker product
of matrices), hence $\beta$ classifies $[kl^{m}]\otimes\xi_{l^{n}}^{univ}$ as
a $\mathbb{C}^{kl^{m+n}}$-bundle (here $\xi_{l^{n}}^{univ}$ is the universal
$\mathbb{C}^{l^{n}}$-bundle over $\mathop{\rm BU}\nolimits(l^{n})$).
Vector $\mathbb{C}^{l^{n}}$-bundle $\eta_{l^{n}}$ is represented by a map
$\varphi^{\prime}\colon X\rightarrow\mathop{\rm BU}\nolimits(l^{n})$, but
since its composition with $\beta$ is homotopy trivial (because of (13)), we
see that $\varphi^{\prime}$ has a lift $\varphi\colon X\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$ with the required property.$\quad\square$
Note that the previous proposition can be applied to all $\eta_{l^{n}}$ such
that $[k]\otimes\eta_{l^{n}}\cong[kl^{n}],$ i.e. those of order $k$. Such
bundles are classified (in the sense of Proposition 13) by maps
$X\rightarrow\mathop{\rm Fr}\nolimits_{k,\,l^{n}}$ (it is easy to see from
fibration (14) with $m=0$), and there are inclusions $\mathop{\rm
Fr}\nolimits_{k,\,l^{n}}\hookrightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$ (for example to a homomorphism $h\colon
M_{k}(\mathbb{C})\rightarrow M_{kl^{n}}(\mathbb{C})$ we can associate the
homomorphism $M_{l^{m}}(h)\colon M_{l^{m}}(M_{k}(\mathbb{C}))\rightarrow
M_{l^{m}}(M_{kl^{n}}(\mathbb{C}))$).
###### Remark 14.
We do not assert that every bundle $\eta_{l^{n}}\rightarrow X$ of order $k$ in
the sense of the group structure $K_{\otimes}$ can be classified by a map
$X\rightarrow\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}$ (cf. Remark 6). In
order to represent all elements of order $k$ for a compact $X$, one has to
consider all spaces $\mathop{\rm
Fr}\nolimits_{k^{r}l^{m},\,l^{n}},\;r,\,m,\,n\in\mathbb{N}$ (cf. Section 10).
The assignment
$\\{h\colon M_{kl^{m}}(\mathbb{C})\rightarrow
M_{kl^{m+n}}(\mathbb{C})\\}\mapsto\\{M_{l}(h)\colon
M_{l}(M_{kl^{m}}(\mathbb{C}))\rightarrow M_{l}(M_{kl^{m+n}}(\mathbb{C}))\\}$
defines the map $\iota_{m+1,\,n}\colon\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m+1},\,l^{n}}$ (recall that
$M_{m}(M_{n}(\mathbb{C}))=M_{mn}(\mathbb{C})$).
The assignment
$\\{h\colon M_{kl^{m}}(\mathbb{C})\rightarrow
M_{kl^{m+n}}(\mathbb{C})\\}\mapsto\\{M_{kl^{m}}(\mathbb{C})\stackrel{{\scriptstyle
h}}{{\rightarrow}}M_{kl^{m+n}}(\mathbb{C})\stackrel{{\scriptstyle
i}}{{\rightarrow}}M_{kl^{m+n+1}}(\mathbb{C})\\},$
where $i(T)=T\otimes E_{l}$, defines the map $\iota_{m,\,n+1}\colon\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n+1}}$.
###### Proposition 15.
$\iota^{*}_{m+1,\,n}(\vartheta_{kl^{m+1},\,l^{n}})=\vartheta_{kl^{m},\,l^{n}},\;\iota^{*}_{m,\,n+1}(\vartheta_{kl^{m},\,l^{n+1}})=\vartheta_{kl^{m},\,l^{n}}\otimes[l].$
Proof is trivial.$\quad\square$
In particular, for $\iota_{m,\,1}\colon\mathop{\rm
PU}\nolimits(kl^{m})\rightarrow\mathop{\rm Fr}\nolimits_{kl^{m},\,l}$ we have:
$\iota^{*}_{m,\,1}(\vartheta_{kl^{m},\,l})=\vartheta_{kl^{m},\,1}\otimes[l]$
(recall that $\mathop{\rm PU}\nolimits(kl^{m})=\mathop{\rm
Fr}\nolimits_{kl^{m},\,1}$).
## 3\. Relation to $K$-theory
Recall (see Section 1) that the group $\mathop{\rm PU}\nolimits(k)$ acts on
the representing space $\mathop{\rm Fred}\nolimits({\mathcal{H}})$ of
$K$-theory and this action induces the action of line bundles of order $k$ on
$K$-functor. In this section we will show that the spaces of unital
homomorphisms of matrix algebras allow us to describe the analogous “action”
of arbitrary (not necessarily line) bundles of finite order in terms of the
classifying space $\mathop{\rm Fred}\nolimits({\mathcal{H}})$.
Again, let ${\mathcal{B}}({\mathcal{H}})$ be the algebra of bounded operators
on the separable Hilbert space ${\mathcal{H}}$,
$M_{kl^{m}}({\mathcal{B}}({\mathcal{H}}))$ the matrix algebra over
${\mathcal{B}}({\mathcal{H}})$ (clearly, it is isomorphic to
${\mathcal{B}}({\mathcal{H}})$). One can think of
$M_{kl^{m}}({\mathcal{B}}({\mathcal{H}}))$ as the algebra of bounded operators
on ${\mathcal{H}}^{\oplus kl^{m}}$. Let $\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})$ be the subspace of Fredholm operators
in $M_{kl^{m}}({\mathcal{B}}({\mathcal{H}}))$. Of course, $\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})\cong\mathop{\rm
Fred}\nolimits({\mathcal{H}}).$
The evaluation map $\mathop{\rm
Hom}\nolimits_{alg}(M_{kl^{m}}(\mathbb{C}),\,M_{kl^{m+n}}(\mathbb{C}))\times
M_{kl^{m}}(\mathbb{C})\rightarrow M_{kl^{m+n}}(\mathbb{C})$, i.e.
(15) $ev_{kl^{m},\,l^{n}}\colon\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}\times
M_{kl^{m}}(\mathbb{C})\rightarrow M_{kl^{m+n}}(\mathbb{C}),\quad
ev_{kl^{m},\,l^{n}}(h,\,T)=h(T)$
induces the map (cf. (7))
(16) $\gamma_{kl^{m},\,l^{n}}\colon\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\times\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})\rightarrow\mathop{\rm
Fred}\nolimits_{kl^{m+n}}({\mathcal{H}}).$
###### Remark 16.
Note that map (15) can be decomposed as follows
(17) $\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}\times
M_{kl^{m}}(\mathbb{C})\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}{\mathop{\times}\limits_{\mathop{\rm
PU}\nolimits(kl^{m})}}M_{kl^{m}}(\mathbb{C})\rightarrow
M_{kl^{m+n}}(\mathbb{C}),$
where the last map is the projection
${\mathcal{A}}_{kl^{m},\,l^{n}}\rightarrow M_{kl^{m+n}}(\mathbb{C})$ of the
tautological $M_{kl^{m}}(\mathbb{C})$-bundle
${\mathcal{A}}_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Gr}\nolimits_{kl^{m},\,l^{n}}$ over the matrix Grassmannian $\mathop{\rm
Gr}\nolimits_{kl^{m},\,l^{n}}=\mathop{\rm PU}\nolimits(kl^{m+n})/(\mathop{\rm
PU}\nolimits(kl^{m})\otimes\mathop{\rm PU}\nolimits(l^{n}))$ which
parameterizes unital $kl^{m}$-subalgebras in the fixed $kl^{m+n}$-algebra
$M_{kl^{m+n}}(\mathbb{C})$ [7].
Let $\eta_{l^{n}}\rightarrow X$ be a vector $\mathbb{C}^{l^{n}}$-bundle over
$X$ satisfying (13), $\varphi=\varphi_{\eta_{l^{n}}}\colon
X\rightarrow\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}$ its classifying map (in
the sense of Proposition 13), $B_{l^{n}}=\mathop{\rm
End}\nolimits(\eta_{l^{n}})$.
###### Theorem 17.
(Cf. Theorem 7). Assume that $f_{\xi}\colon X\rightarrow\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})$ represents an element $\xi\in K(X)$.
Then the composition
$X\stackrel{{\scriptstyle\mathop{\rm
diag}\nolimits}}{{\longrightarrow}}X\times
X\stackrel{{\scriptstyle\varphi\times f_{\xi}}}{{\longrightarrow}}\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\times\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})\stackrel{{\scriptstyle\gamma_{kl^{m},\,l^{n}}}}{{\longrightarrow}}\mathop{\rm
Fred}\nolimits_{kl^{m+n}}({\mathcal{H}})$
represents the element $\xi\otimes\eta_{l^{n}}\in K(X).$
Proof. If $\xi$ is represented by a family of Fredholm operators
$F=\\{F_{x}\\}$ on the Hilbert space ${\mathcal{H}}^{\oplus kl^{m}}$, then
$\xi\otimes\eta_{l^{n}}$ is represented by the family of Fredholm operators
$\\{F_{x}\otimes 1_{B_{l^{n}}}\\}$ on the Hilbert bundle
${\mathcal{H}}^{\oplus kl^{m}}\otimes\eta_{l^{n}}$. It follows from
Proposition 11 that $\varphi$ defines a trivialization of the last bundle,
i.e. finally we obtain a family of Fredholm operators in the fixed space
$\mathop{\rm Fred}\nolimits_{kl^{m+n}}({\mathcal{H}}).\quad\square$
In particular, for $n=0$ ($\Rightarrow\,\mathop{\rm
Fr}\nolimits_{kl^{m},\,1}=\mathop{\rm PU}\nolimits(kl^{m})$) we have the
action of $\mathop{\rm PU}\nolimits(kl^{m})$ on $\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})$ (cf. (17)) which corresponds to the
tensor product $\xi\mapsto\xi\otimes\eta_{1}$ by the line bundle
$\eta_{1}=\varphi^{*}(\vartheta_{kl^{m},\,1})$ (see Theorem 7).
###### Remark 18.
The previous theorem can be regarded as a generalization of Theorem 7 which
corresponds to the special case $m=n=0,$ when the space of homomorphisms
$\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}$ is the group $\mathop{\rm
PU}\nolimits(k)$ (see Section 1).
Note that the following Theorem 20 can also be specialized to this case: as we
have already noticed, the group structure on $\mathop{\rm PU}\nolimits(k)$
corresponds to the tensor product of line bundles classified by this group.
###### Remark 19.
(Cf. Remark 9). Note that the “action” described in Theorem 17 is trivial on
elements of the form $kl^{m}\xi\in K(X)$ which are represented by the subspace
$\mathop{\rm Fred}\nolimits({\mathcal{H}})\stackrel{{\scriptstyle\mathop{\rm
diag}\nolimits}}{{\longrightarrow}}\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})$. Indeed, the center
$\mathbb{C}E_{kl^{m}}\subset M_{kl^{m}}(\mathbb{C})$ is fixed under map (15).
Note that the composition of homomorphisms of matrix algebras defines the map
$\kappa\colon\mathop{\rm
Hom}\nolimits_{alg}(M_{kl^{m+n}}(\mathbb{C}),\,M_{kl^{m+n+r}}(\mathbb{C}))\times\mathop{\rm
Hom}\nolimits_{alg}(M_{kl^{m}}(\mathbb{C}),\,M_{kl^{m+n}}(\mathbb{C}))$
$\rightarrow\mathop{\rm
Hom}\nolimits_{alg}(M_{kl^{m}}(\mathbb{C}),\,M_{kl^{m+n+r}}(\mathbb{C})),$
i.e.
(18) $\kappa\colon\mathop{\rm Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n+r}}.$
Clearly, the diagram
$\textstyle{\mathop{\rm Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\times\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\;\;\quad\mathop{\rm
id}\nolimits_{\mathop{\rm
Fr}\nolimits}\times\gamma}$$\scriptstyle{\kappa\times\mathop{\rm
id}\nolimits_{\mathop{\rm Fred}\nolimits}}$$\textstyle{\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Fred}\nolimits_{kl^{m+n}}({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n+r}}\times\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{\mathop{\rm
Fred}\nolimits_{kl^{m+n+r}}({\mathcal{H}})}$
is commutative.
Composition (18) corresponds to the composition $\mu_{2}\circ\mu_{1}$ of
embeddings $\mu_{1},\,\mu_{2}$ corresponding to maps $\varphi_{1}\colon
X\rightarrow\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}},\;\varphi_{2}\colon
X\rightarrow\mathop{\rm Fr}\nolimits_{kl^{m+n},\,l^{r}}$ (cf. Proposition 11).
Note that if $\mu_{1},\;\mu_{2}$ correspond to subbundles
$B_{l^{n}},\;B_{l^{r}}$ respectively, then $\mu_{2}\circ\mu_{1}$ corresponds
to the subbundle $B_{l^{n}}\otimes B_{l^{r}}$ in $X\times
M_{kl^{m+n+r}}(\mathbb{C})$.
Moreover, composition (18) corresponds to the tensor product
$\xi\otimes\eta_{l^{n}}\otimes\eta_{l^{r}}:$
###### Theorem 20.
Let $\varphi_{1}:=\varphi_{\eta_{l^{n}}}\colon X\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}},\;\varphi_{2}:=\varphi_{\eta_{l^{r}}}\colon
X\rightarrow\mathop{\rm Fr}\nolimits_{kl^{m+n},\,l^{r}}$ be classifying maps
for bundles $\eta_{l^{n}}\rightarrow X,\;\eta_{l^{r}}\rightarrow X$
respectively. Then the composition
$X\stackrel{{\scriptstyle\mathop{\rm diag}\nolimits}}{{\rightarrow}}X\times
X\times X\stackrel{{\scriptstyle\varphi_{2}\times\varphi_{1}\times
f_{\xi}}}{{\longrightarrow}}\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\times\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})\stackrel{{\scriptstyle\lambda}}{{\rightarrow}}\mathop{\rm
Fred}\nolimits_{kl^{m+n+r}}({\mathcal{H}}),$
where $\lambda=\gamma\circ(\mathop{\rm id}\nolimits_{\mathop{\rm
Fr}\nolimits}\times\gamma)=\gamma\circ(\kappa\times\mathop{\rm
id}\nolimits_{\mathop{\rm Fred}\nolimits})$ (see the above diagram) represents
$\xi\otimes\eta_{l^{n}}\otimes\eta_{l^{r}}\in K(X)$ (cf. Theorem 17).
Proof is trivial.$\quad\square$
In general for a given bundle $\eta_{l^{n}}$ there are lot of nonequivalent
trivializations (13) (i.e. there are lot of homotopy nonequivalent maps
$\varphi$ classifying $\eta_{l^{n}}$). However, different trivializations act
on $K$-functor trivially. The situation is similar to the one in the case of
finite Brauer group which is the quotient of the monoid of isomorphism classes
of (finite dimensional) matrix algebra bundles (with respect to the
“$\otimes$” operation) by the submonoid of “trivial” bundles of the form
$\mathop{\rm End}\nolimits(\xi)$. Recall (see Proposition 4) that a map
$X\rightarrow\mathop{\rm PU}\nolimits(k)$ is not just a line bundle
$\zeta\rightarrow X$ of order $k$ but also some choice of a trivialization
$[k]\otimes\zeta\cong X\times{\mathbb{C}}^{k}.$ The point is that the action
of $\mathop{\rm PU}\nolimits(k)$ on $\mathop{\rm
Fred}\nolimits({\mathcal{H}})$ factors through the action of $\mathop{\rm
PU}\nolimits({\mathcal{H}})$, and the action of $\mathop{\rm U}\nolimits(k)$
factors through the action of the contractible group $\mathop{\rm
U}\nolimits({\mathcal{H}})$ respectively, hence the necessity of the
factorization (cf. Remark 8).
## 4\. Topological monoid $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$
The space $\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}$ itself does not have any
natural algebraic operation, but there is composition (18) which relates such
spaces. Using these spaces we construct a topological monoid such that maps
(16) give rise to its action on the space of Fredholm operators. More
precisely, since maps (16) correspond to the multiplication of $K(X)$ by
$l^{n}$-dimensional bundles (for $n\in\mathbb{N}$), the monoid acts on the
localization of the space $\mathop{\rm Fred}\nolimits({\mathcal{H}})$ over
$l$. In fact, the theory does not depend (up to homotopy) on the choice of
$l,\;(k,\,l)=1$, cf. Proposition 36.
So, consider the direct limit of matrix algebras
$M_{kl^{\infty}}(\mathbb{C}):=\lim\limits_{\longrightarrow\atop{m}}M_{kl^{m}}(\mathbb{C})$
(the limit is taken over unital $*$-homomorphisms) and fix an increasing
filtration by unital $*$-subalgebras
(19) $A_{k}\subset A_{kl}\subset\ldots\subset A_{kl^{m}}\subset
A_{kl^{m+1}}\subset\ldots,\quad A_{kl^{m}}\cong M_{kl^{m}}(\mathbb{C})$
in it such that $A_{kl^{m+1}}=M_{l}(A_{kl^{m}})$ (the algebra of $l\times
l$-matrices with elements from $A_{kl^{m}}$) for all $m\geq 0$.
Consider the monoid $\mathop{\rm End}\nolimits(M_{kl^{\infty}}(\mathbb{C}))$
of endomorphisms of this direct limit. More precisely, an endomorphism
$h\in\mathop{\rm End}\nolimits(M_{kl^{\infty}}(\mathbb{C}))$ is induced by a
unital $*$-homomorphism of the form $h_{m,\,n}\colon A_{kl^{m}}\rightarrow
A_{kl^{m+n}}$ (for some $m,\,n$), i.e. has the form
$M_{l^{\infty}}(h_{m,\,n}),\;h_{m,\,n}\in\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}=\mathop{\rm
Hom}\nolimits_{alg}(A_{kl^{m}},\,A_{kl^{m+n}}).$ By
$M_{l^{\infty}}(h_{m,\,n})$ we denote the sequence of homomorphisms
(20) $M_{l^{r}}(h_{m,\,n})\colon A_{kl^{m+r}}=M_{l^{r}}(A_{kl^{m}})\rightarrow
A_{kl^{m+n+r}}=M_{l^{r}}(A_{kl^{m+n}}),\>r\in\mathbb{N}.$
In particular, for $n=0$ we have an automorphism
$M_{l^{\infty}}(h_{m,\,0})\in\mathop{\rm
Aut}\nolimits(M_{kl^{\infty}}(\mathbb{C}))$.
Note that the composition of such endomorphisms is well-defined. For example,
we define the composition $M_{l^{\infty}}(h_{2})\circ M_{l^{\infty}}(h_{1}),$
where $h_{1}\colon A_{kl}\rightarrow A_{kl^{2}}$ and $h_{2}\colon
A_{k}\rightarrow A_{kl}$ are displayed on the diagram
$\textstyle{\ldots}$$\textstyle{\ldots}$$\textstyle{\ldots}$$\textstyle{A_{kl^{4}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\textstyle{A_{kl^{4}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\textstyle{A_{kl^{4}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\textstyle{A_{kl^{3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{\qquad\qquad\qquad\qquad
M_{l^{2}}(h)}$$\textstyle{A_{kl^{3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{\qquad\qquad\qquad\qquad
M_{l^{3}}(h_{2})}$$\textstyle{A_{kl^{3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\textstyle{A_{kl^{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{\qquad\qquad\qquad\qquad
M_{l}(h_{1})}$$\textstyle{A_{kl^{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{\qquad\qquad\qquad\qquad
M_{l^{2}}(h_{2})}$$\textstyle{A_{kl^{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\textstyle{A_{kl}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{h_{1}}$$\textstyle{A_{kl}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{\qquad\qquad\qquad\qquad
M_{l}(h_{2})}$$\textstyle{A_{kl}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\textstyle{A_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\textstyle{A_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{h_{2}}$$\textstyle{A_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$
as $M_{l^{\infty}}(M_{l^{2}}(h_{2})\circ h_{1}).$ Clearly, the composition of
endomorphisms is associative and $M_{l^{\infty}}(\mathop{\rm
id}\nolimits_{A_{k}}),$ i.e. the sequience $\\{\mathop{\rm
id}\nolimits_{A_{k}},\,\mathop{\rm id}\nolimits_{A_{kl}},\,\mathop{\rm
id}\nolimits_{A_{kl^{2}}},\,\ldots\,\\}$ is its unit. This completes the
definition of the topological monoid $\mathop{\rm
End}\nolimits(M_{kl^{\infty}}(\mathbb{C}))$.
By assignment to a homomorphism $h_{m,\,n}\colon A_{kl^{m}}\rightarrow
A_{kl^{m+n}}$ the homomorphism $M_{l^{r}}(h_{m,\,n})\colon
A_{kl^{m+r}}\rightarrow A_{kl^{m+n+r}},\>r\in\mathbb{N}$ (cf. (20)) we define
the embedding $\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m+r},\,l^{n}}$. Furthermore, the composition of
$M_{l^{r}}(h_{m,\,n})$ with the homomorphism $A_{kl^{m+n+r}}\rightarrow
M_{l^{u}}(A_{kl^{m+n+r}})=A_{kl^{m+n+r+u}},\;T\mapsto M_{l^{u}}(T)$ defines
the embedding $\mathop{\rm
Fr}\nolimits_{kl^{m+r},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m+r},\,l^{n+u}}$. The composition of these two embeddings
defines the embedding $\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m+r},\,l^{n+u}}.$ Using these maps we define the direct
limit $\lim\limits_{\longrightarrow\atop{m,\,n}}\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$.
###### Proposition 21.
The monoid $\mathop{\rm End}\nolimits(M_{kl^{\infty}}(\mathbb{C}))$ is
isomorphic to the direct limit $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}:=\lim\limits_{\longrightarrow\atop{m,\,n}}\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$.
Proof. Note that for any pair $m,\,n\,\geq 0$ there is the obvious embedding
$\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}\hookrightarrow\mathop{\rm
End}\nolimits(M_{kl^{\infty}}(\mathbb{C}))$. Now the proposition follows from
the universal property of the direct limit.$\quad\square$
Because of the previous proposition we will denote the monoid $\mathop{\rm
End}\nolimits(M_{kl^{\infty}}(\mathbb{C}))$ also by $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ (a particular isomorphism
$\mathop{\rm End}\nolimits(M_{kl^{\infty}}(\mathbb{C}))\cong\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ is defined by the particular choice
of a filtration $\\{A_{kl^{m}}\\}_{m\in\mathbb{N}}$ in
$M_{kl^{\infty}}(\mathbb{C})$ as above).
Since $\mathop{\rm Fr}\nolimits_{kl^{m},\,1}=\mathop{\rm
PU}\nolimits(kl^{m})$, we see that the subgroup $\mathop{\rm
Aut}\nolimits(M_{kl^{\infty}}(\mathbb{C}))\subset\mathop{\rm
End}\nolimits(M_{kl^{\infty}}(\mathbb{C}))$ of the monoid $\mathop{\rm
End}\nolimits(M_{kl^{\infty}}(\mathbb{C}))=\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ consisting of $*$-automorphisms of
$M_{kl^{\infty}}(\mathbb{C})$ is $\mathop{\rm
PU}\nolimits(kl^{\infty}):=\lim\limits_{\longrightarrow\atop{m}}\mathop{\rm
PU}\nolimits(kl^{m}).$
Note that under the condition $(k,\,l)=1$ the monoid $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ is not contractible: its homotopy
groups are as follows: $\pi_{r}(\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}})=\mathbb{Z}/k\mathbb{Z}$ for $r$ odd
and $0$ for $r$ even (see Proposition 35). (It is easy to see that the monoid
$\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ is contractible if and
only if $p\mid k\,\Rightarrow\,p\mid l$ for any prime $p$). In particular,
$\pi_{0}(\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}})=0$ and hence the
monoid is grouplike. Besides, it is a CW-complex, therefore the embedding of
the unit is a cofibration and therefore it is well-pointed.
###### Remark 22.
In place of spaces (9) one can consider spaces $\widetilde{\mathop{\rm
Fr}\nolimits}_{kl^{m},\,l^{n}}:=\mathop{\rm
SU}\nolimits(kl^{m+n})/(E_{kl^{m}}\otimes\mathop{\rm SU}\nolimits(l^{n}))$
which are the universal coverings of the corresponding $\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$’s. The corresponding monoid
$\widetilde{\mathop{\rm Fr}\nolimits}_{kl^{\infty},\,l^{\infty}}$ is the
universal covering $\widetilde{\mathop{\rm
Fr}\nolimits}_{kl^{\infty},\,l^{\infty}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ (with fiber the group $\rho_{k}$ of
$k$’th roots of unity). This monoid gives the “$\mathop{\rm
SU}\nolimits$”-version of the subsequent constructions. In particular, its
action on the space of Fredholm operators (cf. the next section) corresponds
to the multiplication of $K(X)$ by $\mathop{\rm SU}\nolimits$-bundles of order
$k$.
The monoid $\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ has the
filtration $\mathop{\rm PU}\nolimits(kl^{\infty})=\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,1}\stackrel{{\scriptstyle\iota_{1}}}{{\hookrightarrow}}\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l}\stackrel{{\scriptstyle\iota_{2}}}{{\hookrightarrow}}\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{2}}\stackrel{{\scriptstyle\iota_{3}}}{{\hookrightarrow}}\ldots$.
Obviously that the multiplication in $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ induces maps
(21) $\mu_{n,\,s}\colon\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{n}}\times\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{s}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{n+s}}.$
Note that $\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{n}}$ in the base space
of the vector $\mathbb{C}^{l^{n}}$-bundle $\vartheta_{kl^{\infty},\,l^{n}}$
which restricts to $\vartheta_{kl^{m},\,l^{n}}$ under the inclusion
$\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}\subset\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{n}}$ (see Proposition 15). Furthermore,
(22)
$\mu_{n,\,s}^{*}(\vartheta_{kl^{\infty},\,l^{n+s}})=\vartheta_{kl^{\infty},\,l^{n}}\boxtimes\vartheta_{kl^{\infty},\,l^{s}}.$
We also have
$\iota_{n+1}^{*}(\vartheta_{kl^{\infty},\,l^{n+1}})=\vartheta_{kl^{\infty},\,l^{n}}\otimes[l]$
(see Proposition 15).
So we see that the multiplication in the monoid $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ corresponds to the tensor product of
bundles, like the product in projective groups corresponds to the tensor
product of appropriate line bundles. Thus the monoid $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ gives a model of a classifying
$H$-space for bundles of order $k$ with tensor product whose multiplication is
strictly associative and unital.
## 5\. An action of the monoid $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ on the space of Fredholm operators
$K(X)$ is a commutative ring, therefore its multiplicative group acts on
$K(X)$ by group automorphisms. Invertible elements in $K(X)$ are virtual
bundles of virtual dimension $\pm 1$ (which form the group with respect to the
tensor product), while the multiplicative group of the localization
$K(X)[\frac{1}{l}]$ over $l$ consists of virtual bundles of virtual dimension
$\pm l^{n},\,n\in\mathbb{Z}$ (for a compact $X$ if $n$ is a big enough
positive integer then a virtual bundle of virtual dimension $l^{n}$ can be
realized by a geometric bundle $\eta_{l^{n}}\rightarrow X$).
Because of the functoriality of the mentioned action it can be described in
terms of the representing space $\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})$ of the localized $K$-theory. In
this section we define the action of the monoid $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ on $\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})$ which induces the multiplication
of $K$-functor by bundles of dimensions $l^{n}$ of order $k$ and coincides
with maps from Theorem 17 on its finite subspaces $\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\subset\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$.
Let
$M_{kl^{\infty}}({\mathcal{B}}({\mathcal{H}})):=\lim\limits_{\longrightarrow\atop{m}}M_{kl^{m}}({\mathcal{B}}({\mathcal{H}}))$
(the limit is taken over unital $*$-homomorphisms of matrix algebras which
form filtration (19)), $\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})$ be the subspace of Fredholm
operators in $M_{kl^{\infty}}({\mathcal{B}}({\mathcal{H}}))$.
The tautological action of $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ on $M_{kl^{\infty}}(\mathbb{C})$
(recall that $\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}=\mathop{\rm
End}\nolimits(M_{kl^{\infty}}(\mathbb{C}))$) defines the action
(23) $\gamma_{kl^{\infty},\,l^{\infty}}\colon\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}\times\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})\rightarrow\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})$
of the monoid $\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ on the
space $\mathop{\rm Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})$ whose
restrictions to “finite” subspaces of the direct limits coincide with maps
(16).
###### Remark 23.
Consider the map
(24) $\gamma_{kl^{\infty},\,l^{n}}\colon\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{n}}\times\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})\rightarrow\mathop{\rm
Fred}\nolimits_{kl^{\infty+n}}({\mathcal{H}})$
which is the limit of (16) when $m\rightarrow\infty$. According to Theorem 17
it corresponds to the map
$\xi\mapsto\xi\otimes\varphi^{*}(\vartheta_{kl^{\infty},\,l^{n}}),\;\xi\in
K(X)[\frac{1}{l}]$ for $\varphi\colon X\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{n}}$ (see the end of the previous section).
Note that the space $\mathop{\rm Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})$
is the localization of $\mathop{\rm Fred}\nolimits({\mathcal{H}})$ over $l$
(in the sense that $l$ is invertible). It is not surprising because the action
of the monoid $\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ relates to
the tensor product by $l^{n}$-dimensional bundles (cf. Theorem 17). In
particular, $\mathop{\rm Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})$
represents $K$-theory localized over $l$, i.e. $[X,\,\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})]=K(X)[\frac{1}{l}]$.
Since $\pi_{0}(\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}})=0$ we see
that the monoid acts on $\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})$ by homotopy auto-equivalences
that are homotopic to the identity map.
Since the monoid $\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ is
grouplike, we see that the set of homotopy classes $[X,\,\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}]$ is a group. Then, using (23) we
obtain the representation $[X,\,\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}]\rightarrow\mathop{\rm
Aut}\nolimits(K(X)[\frac{1}{l}])$ which is functorial on $X$ (“$\mathop{\rm
Aut}\nolimits$” denotes group automorphisms).
The monoid $\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}=\mathop{\rm
End}\nolimits(M_{kl^{\infty}}(\mathbb{C}))$ contains the subgroup $\mathop{\rm
PU}\nolimits(kl^{\infty})=\mathop{\rm
Aut}\nolimits(M_{kl^{\infty}}(\mathbb{C}))$ which in turn contains the
“subgroup” $\mathop{\rm U}\nolimits(kl^{\infty})$ (corresponding to the direct
limit of the canonical epimorphisms $\mathop{\rm
U}\nolimits(kl^{m})\rightarrow\mathop{\rm PU}\nolimits(kl^{m})$). The action
of groups $\mathop{\rm U}\nolimits(kl^{m})$ on spaces $\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})$ is homotopy trivial, because it
factors through the action of the contractible group $\mathop{\rm
U}\nolimits({\mathcal{H}})$ (cf. Remark 8). Analogously, the action of groups
$\mathop{\rm PU}\nolimits(kl^{m})$ on $\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})$ factors through the action of
$\mathop{\rm PU}\nolimits({\mathcal{H}})$.
Consider the fibration
$\mathop{\rm U}\nolimits(kl^{m})\rightarrow\mathop{\rm
U}\nolimits(kl^{m+n})/(E_{kl^{m}}\otimes\mathop{\rm
U}\nolimits(l^{n}))\rightarrow\mathop{\rm U}\nolimits(kl^{m+n})/(\mathop{\rm
U}\nolimits(kl^{m})\otimes\mathop{\rm U}\nolimits(l^{n}))$
(cf. (11)) and take the direct limit as $m,\,n\to\infty$. Since
$\lim\limits_{\longrightarrow\atop{m,\,n}}\mathop{\rm
U}\nolimits(kl^{m+n})/(\mathop{\rm U}\nolimits(kl^{m})\otimes E_{l^{n}})$ is
contractible (see Lemma 37) and the group
$\lim\limits_{\longrightarrow\atop{n}}\mathop{\rm U}\nolimits(l^{n})$ acts
freely on it, we obtain the homotopy equivalence
$\lim\limits_{\longrightarrow\atop{m,\,n}}\mathop{\rm
U}\nolimits(kl^{m+n})/(\mathop{\rm U}\nolimits(kl^{m})\otimes\mathop{\rm
U}\nolimits(l^{n}))\simeq\mathop{\rm
BU}\nolimits(l^{\infty})=\lim\limits_{\longrightarrow\atop{n}}\mathop{\rm
BU}\nolimits(l^{n}).$ Now we see that the homotopy nontrivial part of the
action of $\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ on
$\mathop{\rm Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})$ corresponds to the
cokernel $\mathop{\rm U}\nolimits(kl^{\infty})\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ (more precisely, to the cokernel
$[X,\,\mathop{\rm U}\nolimits(kl^{\infty})]\rightarrow[X,\,\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}]$) or, equivalently, to the image of
the map $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}\rightarrow\mathop{\rm
BU}\nolimits(l^{\infty})$ (cf. (14)) which is a classifying map for the direct
limit of bundles $\vartheta_{kl^{m},\,l^{n}}$.
Note that the space $\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$
classifies bundles $\eta_{l^{n}}\rightarrow X$ with the following equivalence
relation:
(25)
$\eta_{l^{m}}\sim\eta_{l^{n}}\;\Leftrightarrow\;\eta_{l^{m}}\otimes[l^{t-m}]\cong\eta_{l^{n}}\otimes[l^{t-n}]\;\hbox{for
some}\;t\in\mathbb{N}.$
In other words, the induced action on the localized $K$-theory
$K(X)[\frac{1}{l}]$ is the action of the multiplicative group of equivalence
classes (25) of bundles of the form
$\eta_{l^{n}}=\varphi^{*}(\vartheta_{kl^{m},\,l^{n}}),\;\varphi\colon
X\rightarrow\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}\subset\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$, i.e. those whose classifying maps
can be lifted to $\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ (cf.
(14)), and the group structure is induced by the tensor product of such
bundles (cf. Theorem 20 and the end of Section 4). Thus, these automorphisms
have the form $\xi\mapsto\xi\otimes\varphi^{*}(\vartheta_{kl^{m},\,l^{n}})$,
cf. Theorems 17 and 20 (note that for a compact $X$ every map
$X\rightarrow\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ can be
factorized through $X\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\subset\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ for some $m,\,n\in\mathbb{N}$).
Thus, we obtain the following main theorem.
###### Theorem 24.
For any compact $X$ action (23) on the representing space $\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})$ of $K[\frac{1}{l}]$-theory
induces the action
$\xi\mapsto\xi\otimes\varphi^{*}(\vartheta_{kl^{m},\,l^{n}})$ on
$K(X)[\frac{1}{l}]$ of the multiplicative group of equivalence classes (25) of
bundles $\eta_{l^{n}}=\varphi^{*}(\vartheta_{kl^{m},\,l^{n}})\in
K(X)[\frac{1}{l}]$, where $\varphi\in[X,\,\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}].$
## 6\. A definition of the corresponding twisted $K$-theory
In order to define the twisted $K$-theory for more general twistings by
analogy with the definition of the twisted $K$-theory from Section 1 first we
should do is to construct the classifying space of the topological monoid
$\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$. Fortunately, a well-
pointed grouplike topological monoid has the classifying space given, for
example, by May’s geometric bar-construction [12], [17], pp. 210-214. Recall
that in our case $\pi_{0}(\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}})=0$, i.e. $\pi_{0}$ is a group and
hence our monoid is grouplike.
Thus, there exists the classifying space $\mathop{\rm B}\nolimits\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ and the universal principal
$\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$-quasifibration
$\mathop{\rm E}\nolimits\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}\rightarrow\mathop{\rm
B}\nolimits\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ (in
particular, the space $\mathop{\rm E}\nolimits\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ is aspherical and even contractible
because $\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ is a CW-
complex). Furthermore, there is the homotopy equivalence $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}\stackrel{{\scriptstyle\simeq}}{{\rightarrow}}\Omega\mathop{\rm
B}\nolimits\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ (and hence
$\pi_{r}(\mathop{\rm B}\nolimits\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}})=\mathbb{Z}/k\mathbb{Z}$ for $r>0$
even and $0$ for $r$ odd, cf. Proposition 35).
Applying the two-sided geometric bar-construction ([17], ibid.) to the action
(23) of $\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ on $\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}}):=\lim\limits_{\longrightarrow\atop{m}}\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})$ we construct the $\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})$-quasifibration
(26) $\widetilde{\mathop{\rm
Fred}\nolimits_{kl^{\infty}}}({\mathcal{H}})\rightarrow\mathop{\rm
B}\nolimits\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$
over $\mathop{\rm B}\nolimits\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$.
But quasifibration (26) is not appropriate for our purpose: we would like (by
analogy with the abelian case, cf. (2)) to define the twisted $K$-theory
corresponding to a map $f\colon X\rightarrow\mathop{\rm B}\nolimits\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ as the set of homotopy classes of
sections of the “induced quasifibration” $f^{*}(\widetilde{\mathop{\rm
Fred}\nolimits_{kl^{\infty}}}({\mathcal{H}}))\rightarrow X$, but the problem
is that the pull-back of a quasifibration is not a quasifibration in general.
Fortunately, there are constructions that provide locally homotopy trivial
fibrations instead of quasifibrations and therefore allow induced fibrations
and classification. One of such constructions is M. Fuch’s modified Dold-
Lashof construction [9], the other one [20] given by J. Wirth (note that the
homotopy type of the classifying space $\mathop{\rm B}\nolimits\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ does not depend on the choice of a
particular construction).
Applying one of these constructions we can assume that (26) is a fibration (in
the sense of Dold, i.e. with weak covering homotopy property). We propose this
fibration as a model for the twisted $K$-theory for twistings corresponding to
the action of bundles of order $k$ on $K(X)$ by the tensor product (cf.
Theorem 24). More precisely, for a map $f\colon X\rightarrow\mathop{\rm
B}\nolimits\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ we define the
corresponding twisted $K$-theory as the set $[X,\,f^{*}(\widetilde{\mathop{\rm
Fred}\nolimits_{kl^{\infty}}}({\mathcal{H}}))]^{\prime}$ of homotopy classes
of sections of the induced fibration $f^{*}(\widetilde{\mathop{\rm
Fred}\nolimits_{kl^{\infty}}}({\mathcal{H}}))\rightarrow X$.
In order to obtain the fibration with fiber the $\Omega$-spectrum
$\\{K_{n}\\}_{n\geq 0}$, we should verify that the homotopy equivalence
$\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})\rightarrow\Omega^{2}\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})$ is equivariant with respect to
action (23) of the monoid $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$. For this purpose we can use the
version of Bott periodicity for spaces of Fredholm operators given in [3].
Action (23) consists of the composition of inclusions $\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})\rightarrow\mathop{\rm
Fred}\nolimits_{kl^{m+n}}({\mathcal{H}})$ induced by inclusions of filtration
(19) and the conjugation action of $\mathop{\rm PU}\nolimits(kl^{m+n})$ on
$\mathop{\rm Fred}\nolimits_{kl^{m+n}}({\mathcal{H}})$. It is easy to see that
the homotopy equivalences defined in [3] are equivariant with respect to both
mentioned types of maps and therefore can be applied fiberwisely to fibration
(26).
## 7\. An approach by means of bundle gerbes
In this section we sketch an approach to twisted $K$-theory for “higher”
twistings by means of some generalization of bundle gerbes [13], [14]. For
this purpose we want to combine the idea of bundle gerbes and bundle gerbe
modules from [4] with the idea of homotopy transition cocycles from [20]
applying to our monoid $\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$
and using the observation that the multiplication (21) in the monoid
corresponds to the tensor product of bundles (22).
First, let us recall some facts about “abelian” bundle gerbes with Dixmier-
Douady class of finite order [4], [13], [14] in a form appropriate for our
purposes.
Recall that $\vartheta_{k,\,1}\rightarrow\mathop{\rm PU}\nolimits(k)$ is the
line bundle $\mathop{\rm U}\nolimits(k){\mathop{\times}\limits_{\mathop{\rm
U}\nolimits(1)}}\mathbb{C}$ associated with the principal bundle $\chi_{k}$
(see (4)). Let ${\mathcal{U}}=\\{U_{\alpha}\\}_{\alpha\in A}$ be an open cover
of a compact space $X$, $Y=Y_{\mathcal{U}}$ the disjoint union of all the
elements in the open cover, $\pi\colon Y\rightarrow X$ the corresponding
projection, $Y^{[2]}=Y\times_{\pi}Y$ the fibre product. For a given
$\mathop{\rm PU}\nolimits(k)$ 1-cocycle
$g=\\{g_{\alpha\beta}\\}_{\alpha,\,\beta\in A}$ one can associate a bundle
gerbe $L\rightarrow Y^{[2]}$ as follows:
(27)
$L_{\alpha\beta}:=g_{\alpha\beta}^{*}(\vartheta_{k,\,1}),\;g_{\alpha\beta}\colon
U_{\alpha\beta}\rightarrow\mathop{\rm PU}\nolimits(k)$
and the product
(28) $\theta_{\alpha\beta\gamma}\colon L_{\alpha\beta}\otimes
L_{\beta\gamma}\stackrel{{\scriptstyle\cong}}{{\rightarrow}}L_{\alpha\gamma}$
over $U_{\alpha}\cap U_{\beta}\cap U_{\gamma}$ is induced by the group
structure on $\mathop{\rm U}\nolimits(k)$ (because
$\mu^{*}(\vartheta_{k,\,1})=\vartheta_{k,\,1}\boxtimes\vartheta_{k,\,1}$ for
the group multiplication $\mu\colon\mathop{\rm
PU}\nolimits(k)\times\mathop{\rm PU}\nolimits(k)\rightarrow\mathop{\rm
PU}\nolimits(k)$). The bundle gerbe $(L,\,Y)$ is nontrivial (equivalently, its
Dixmier-Douady class $d(L,\,Y)\neq 0\in H^{3}(X,\,\mathbb{Z})$) iff there is
no lift of $g$ to a $\mathop{\rm U}\nolimits(k)$-cocycle $\widetilde{g}$, i.e.
there is no $\mathop{\rm U}\nolimits(k)$ 1-cocycle $\widetilde{g}$ such that
$\chi_{k}\circ\widetilde{g}=g.$ In other words, the nontriviality of $(L,\,Y)$
is an obstruction to the existence of such a lift.
Recall [14] that two bundle gerbes $(L,\,Y)$ and $(L^{\prime},\,Y^{\prime})$
are called stably isomorphic if there are trivial bundle gerbes $T_{1}$ and
$T_{2}$ such that
(29) $L\otimes T_{1}\cong L^{\prime}\otimes T_{2}$
(here “$\otimes$” denotes the product of bundle gerbes). Recall also that
$(L,\,Y)$ and $(L^{\prime},\,Y^{\prime})$ are stably isomorphic iff
$d(L,\,Y)=d(L^{\prime},\,Y^{\prime}).$ Any stably equivalence class of bundle
gerbes with Dixmier-Douady class of finite order in $H^{3}(X,\,\mathbb{Z})$
contains a representative of the above form (i.e. determined by a projective
cocycle $g$ for $\mathop{\rm PU}\nolimits(k)$ for some $k\in\mathbb{N}$). Note
also that the product of bundle gerbes $L\otimes L^{\prime}$ corresponds to
the “tensor product” of groups
$\tau\colon\mathop{\rm PU}\nolimits(k_{1})\times\mathop{\rm
PU}\nolimits(k_{2})\rightarrow\mathop{\rm
PU}\nolimits(k_{1})\otimes\mathop{\rm PU}\nolimits(k_{2})\subset\mathop{\rm
PU}\nolimits(k_{1}k_{2})$
in the sense that the compositions
(30) $(U_{\alpha}\cap U_{\beta})\cap(V_{\gamma}\cap
V_{\delta})\stackrel{{\scriptstyle\mathop{\rm
diag}\nolimits}}{{\rightarrow}}(U_{\alpha}\cap U_{\beta})\times(V_{\gamma}\cap
V_{\delta})\stackrel{{\scriptstyle g_{\alpha\beta}\times
g^{\prime}_{\gamma\delta}}}{{\longrightarrow}}\mathop{\rm
PU}\nolimits(k_{1})\times\mathop{\rm
PU}\nolimits(k_{2})\stackrel{{\scriptstyle\tau}}{{\rightarrow}}\mathop{\rm
PU}\nolimits(k_{1}k_{2})$
(for all $\alpha,\,\beta\in A;\;\gamma,\,\delta\in A^{\prime}$) form a
projective cocycle over $Y_{\pi}\times_{\pi^{\prime}}Y^{\prime}$ which
determines the product bundle gerbe (where
$Y_{\pi}\times_{\pi^{\prime}}Y^{\prime}$ is the fibre product).
Note that a projective cocycle $g$ with values in $\mathop{\rm
PU}\nolimits(k)$ not just determines a bundle gerbe but contains some
additional information. More precisely, it gives rise to a module over the
bundle gerbe $L=g^{*}(\vartheta_{k,\,1})$ (27). Its construction is based on
the following proposition.
###### Proposition 25.
A map $\varphi\colon X\rightarrow\mathop{\rm PU}\nolimits(k)$ is nothing but
an isomorphism
(31)
$\widehat{\varphi}\colon\varphi^{*}(\vartheta_{k,\,1})\otimes\mathbb{C}^{k}\rightarrow
X\times\mathbb{C}^{k}.$
Proof. By definition, the total space $\vartheta_{k,\,1}$ is the set of
equivalence classes $[g,\,l]$ of pairs $(g,\,l),\;(g,\,l)\sim(gu,\,u^{-1}l),$
where $g\in\mathop{\rm U}\nolimits(k),\,u\in\mathop{\rm
U}\nolimits(1),\,l\in\mathbb{C}.$ Then for $\varphi=\mathop{\rm
id}\nolimits,\,X=\mathop{\rm PU}\nolimits(k)$ isomorphism (31) is defined as
follows:
$[g,\,l]\otimes w\mapsto(\bar{g},\,g(l\otimes w)),$
where $w\in\mathbb{C}^{k},\,\bar{g}=\chi_{k}(g)\in\mathop{\rm
PU}\nolimits(k).\quad\square$
Applying this proposition to the projective cocycle $g=\\{g_{\alpha\beta}\\}$,
we obtain isomorphisms
$\widehat{g}_{\alpha\beta}\colon
L_{\alpha\beta}\otimes\mathbb{C}^{k}\stackrel{{\scriptstyle\cong}}{{\rightarrow}}U_{\alpha\beta}\times\mathbb{C}^{k}$
(recall that $L_{\alpha\beta}=g_{\alpha\beta}^{*}(\vartheta_{k,\,1})$). Let
$E_{\alpha}\rightarrow U_{\alpha}$ be trivial bundles
$U_{\alpha}\times\mathbb{C}^{k}.$ Thus we have isomorphisms
$\widehat{g}_{\alpha\beta}\colon L_{\alpha\beta}\otimes
E_{\beta}\stackrel{{\scriptstyle\cong}}{{\rightarrow}}E_{\alpha}$ over
$U_{\alpha}\cap U_{\beta}$. The cocycle condition
$g_{\alpha\beta}g_{\beta\gamma}=g_{\alpha\gamma}$ gives rise to the
“associativity” condition
$\textstyle{L_{\alpha\beta}\otimes L_{\beta\gamma}\otimes
E_{\gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\mathop{\rm
id}\nolimits\times\widehat{g}_{\beta\gamma}}$$\scriptstyle{\theta_{\alpha\beta\gamma}\times\mathop{\rm
id}\nolimits}$$\textstyle{L_{\alpha\beta}\otimes
E_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{g}_{\alpha\beta}}$$\textstyle{L_{\alpha\gamma}\otimes
E_{\gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{g}_{\alpha\gamma}}$$\textstyle{E_{\alpha}}$
over $U_{\alpha}\cap U_{\beta}\cap U_{\gamma}$.
Following [4], denote the set (in fact, the semi-group) of isomorphism classes
of bundle gerbe modules over $L=(L,\,Y)$ by ${\rm Mod}(L)$. The corresponding
Grothendieck group $K(L)$ is the twisted $K$-theory group $K_{d(L)}(X)$ [4].
For example, if $d(L)=0,$ we have an isomorphism ${\rm Mod}(L)\cong{\rm
Bun}(X)$ with the semi-group ${\rm Bun}(X)$ of all isomorphism classes of
vector bundles over $X$. Note that this isomorphism is not canonical, but
depends on the choice of a trivialization of $L$, i.e. on isomorphisms
$L_{\alpha\beta}\cong L_{\alpha}^{*}\otimes L_{\beta}$ for line bundles
$L_{\alpha}\rightarrow U_{\alpha}$. Hence even in case of trivial $L$ we can
not canonically identify $L$-modules and vector bundles over $X$.
How one can describe ${\rm Mod}(L)$? The above discussion shows that there is
a close relation between projective bundles and bundle gerbe modules. The
precise statement is that
(32) ${\rm Mod}(L)/{\rm Pic}(X)\cong{\rm Pro}(X,\,d(L)),$
the quotient set of ${\rm Mod}(L)$ by the (obvious) action of ${\rm Pic}(X)$
is ${\rm Pro}(X,\,d(L)),$ the set of all isomorphism classes of projective
bundles over $X$ with class $d(L)$ (see [4], Proposition 4.4.).
The outlined results concerning abelian bundle gerbes and their modules will
serve as a guideline for our generalization.
As above, let ${\mathcal{U}}=\\{U_{\alpha}\\}_{\alpha\in A}$ be an open cover
of a compact space $X$, $Y=Y_{\mathcal{U}}$ the disjoint union of all the
elements in the open cover, $\pi\colon Y\rightarrow X$ the corresponding
projection, $Y^{[2]}=Y\times_{\pi}Y$ the fibre product. In our generalization
the monoid $\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ will play the
same role as the projective group $\mathop{\rm PU}\nolimits(k)$ in the just
described abelian case. According to [20], the local description of a
fibration with a structural monoid can be given by a homotopy transition
cocycle $g$.
Let us introduce further notation for specific maps between spaces
$\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}$. By $\iota$ denote maps
(33) $\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n+1}},\;h\mapsto i\circ h,$
where $h\in\mathop{\rm
Hom}\nolimits_{alg}(A_{kl^{m}},\,A_{kl^{m+n}})=\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$ (see (19)) and $i\colon
A_{kl^{m+n}}\hookrightarrow M_{l}(A_{kl^{m+n}})=A_{kl^{m+n+1}}$ is the
inclusion in filtration (19). In the matrix form we have $i(a)=\mathop{\rm
diag}\nolimits(a),$ where $a\in A_{kl^{m+n}}$.
So, firstly, we have a collection of functions $g_{\alpha\beta}\colon
U_{\alpha}\cap U_{\beta}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$. For simplicity we assume that these
functions take values in the subspace $\mathop{\rm
Fr}\nolimits_{k,\,l}\subset\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}},$ i.e. in fact $g_{\alpha\beta}\colon
U_{\alpha}\cap U_{\beta}\rightarrow\mathop{\rm Fr}\nolimits_{k,\,l}$. Note
that below we make the analogous assumption for homotopies
$g_{\alpha\beta\gamma},$ etc.
Denote $U_{\alpha_{0}}\cap\ldots\cap U_{\alpha_{n}}$ by
$U_{\alpha_{0}\cdots\alpha_{n}}$ for short. So, for any ordered pair
$\\{\alpha,\,\beta\\}\in A^{2}$ we have a map $g_{\alpha\beta}\colon
U_{\alpha\beta}\rightarrow\mathop{\rm Fr}\nolimits_{k,\,l}.$ On triple
intersection $U_{\alpha\beta\gamma}$ we have the composition
$M_{l}(g_{\alpha\beta})\circ g_{\beta\gamma}\colon
U_{\alpha\beta\gamma}\rightarrow\mathop{\rm Fr}\nolimits_{k,\,l^{2}},$
where $M_{l}(g_{\alpha\beta})\colon U_{\alpha\beta}\rightarrow\mathop{\rm
Fr}\nolimits_{kl,\,l},\;M_{l}(g_{\alpha\beta})(x)=M_{l}(g_{\alpha\beta}(x)),\,x\in
U_{\alpha\beta}$ and “$\circ$” here is induced by the composition
$\mu\colon\mathop{\rm Fr}\nolimits_{kl,\,l}\times\mathop{\rm
Fr}\nolimits_{k,\,l}\rightarrow\mathop{\rm Fr}\nolimits_{k,\,l^{2}}$ of
homomorphisms, i.e. $M_{l}(g_{\alpha\beta})\circ
g_{\beta\gamma}=\mu(M_{l}(g_{\alpha\beta})\times g_{\beta\gamma})$. We also
have the composition $\iota\circ g_{\alpha\gamma}\colon
U_{\alpha\gamma}\rightarrow\mathop{\rm Fr}\nolimits_{k,\,l^{2}},$ where
$\iota\colon\mathop{\rm Fr}\nolimits_{k,\,l}\rightarrow\mathop{\rm
Fr}\nolimits_{k,\,l^{2}}$ as above.
Under our assumption there is a homotopy
$g_{\alpha\beta\gamma}\colon U_{\alpha\beta\gamma}\times
I\rightarrow\mathop{\rm
Fr}\nolimits_{k,\,l^{2}},\;g_{\alpha\beta\gamma}|_{U_{\alpha\beta\gamma}\times\\{0\\}}=M_{l}(g_{\alpha\beta})\circ
g_{\beta\gamma},\;g_{\alpha\beta\gamma}|_{U_{\alpha\beta\gamma}\times\\{1\\}}=\iota\circ
g_{\alpha\gamma}|_{U_{\alpha\beta\gamma}}.$
On 4-fold intersections $U_{\alpha\beta\gamma\delta}$ we have the diagram of
homotopies:
$\textstyle{M_{l^{2}}(g_{\alpha\beta})\circ M_{l}(g_{\beta\gamma})\circ
g_{\gamma\delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M_{l}(g_{\alpha\beta\gamma})\circ
g_{\gamma\delta}\qquad\qquad}$$\scriptstyle{M_{l^{2}}(g_{\alpha\beta})\circ
g_{\beta\gamma\delta}}$$\textstyle{M_{l}(\iota\circ g_{\alpha\gamma})\circ
g_{\gamma\delta}=M_{l}(\iota)\circ M_{l}(g_{\alpha\gamma})\circ
g_{\gamma\delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M_{l}(\iota)\circ
g_{\alpha\gamma\delta}}$$\textstyle{M_{l^{2}}(g_{\alpha\beta})\circ\iota\circ
g_{\beta\delta}=\iota\circ M_{l}(g_{\alpha\beta})\circ
g_{\beta\delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\iota\circ
g_{\alpha\beta\delta}}$$\textstyle{\iota\circ\iota\circ
g_{\alpha\delta}=M_{l}(\iota)\circ\iota\circ g_{\alpha\delta}.}$
The equality in the low left corner of the diagram follows from the equality
$M_{l}(h)\circ i=i\circ h$ (cf. (33)). Note that $M_{l}(\iota)\neq\iota$ but
$\iota\circ\iota=M_{l}(\iota)\circ\iota$ hence the equality in the low right
corner and therefore two compositions of homotopies depicted on the above
diagram are homotopies between maps
$M_{l^{2}}(g_{\alpha\beta})\circ M_{l}(g_{\beta\gamma})\circ
g_{\gamma\delta}\quad\hbox{and}\quad\iota\circ\iota\circ
g_{\alpha\delta}\colon U_{\alpha\beta\gamma\delta}\rightarrow\mathop{\rm
Fr}\nolimits_{k,\,l^{3}}.$
We assume that there is a homotopy
$g_{\alpha\beta\gamma\delta}\colon U_{\alpha\beta\gamma\delta}\times
I^{2}\rightarrow\mathop{\rm Fr}\nolimits_{k,\,l^{3}}$
such that
$g_{\alpha\beta\gamma\delta}|_{U_{\alpha\beta\gamma\delta}\times
I\times\\{0\\}}=M_{l}(g_{\alpha\beta\gamma})\circ
g_{\gamma\delta}|_{U_{\alpha\beta\gamma\delta}\times I},$
$g_{\alpha\beta\gamma\delta}|_{U_{\alpha\beta\gamma\delta}\times
I\times\\{1\\}}=\iota\circ
g_{\alpha\beta\delta}|_{U_{\alpha\beta\gamma\delta}\times I},$
$g_{\alpha\beta\gamma\delta}|_{U_{\alpha\beta\gamma\delta}\times\\{0\\}\times
I}=M_{l^{2}}(g_{\alpha\beta})\circ
g_{\beta\gamma\delta}|_{U_{\alpha\beta\gamma\delta}\times I},$
$g_{\alpha\beta\gamma\delta}|_{U_{\alpha\beta\gamma\delta}\times\\{1\\}\times
I}=M_{l}(\iota)\circ
g_{\alpha\gamma\delta}|_{U_{\alpha\beta\gamma\delta}\times I}.$
The general pattern now should be clear. We should consider a collection of
“higher” homotopies
$g_{\alpha_{0}\cdots\alpha_{n}}\colon U_{\alpha_{0}\cdots\alpha_{n}}\times
I^{n-1}\rightarrow\mathop{\rm Fr}\nolimits_{k,\,l^{n}}$
which are compatible with
$g_{\alpha_{0}\cdots\widehat{\alpha}_{k}\cdots\alpha_{n}}$ in the obvious way.
Now we are ready to define a homotopic analog of bundle gerbes. One can say
that homotopy bundle gerbes are in the same relation to bundle gerbes as
homotopy transition cocycles to usual transition cocycles (for projective
bundles). Recall (see Proposition 12) that there is a canonical
$M_{l^{n}}(\mathbb{C})$-bundle
${\mathcal{B}}_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}},\quad{\mathcal{B}}_{kl^{m},\,l^{n}}=\mathop{\rm
PU}\nolimits(kl^{m+n}){\mathop{\times}\limits_{\mathop{\rm
PU}\nolimits(l^{n})}}M_{l^{n}}(\mathbb{C}).$ Let
$B_{\alpha_{0}\cdots\alpha_{n}}\rightarrow
U_{\alpha_{0}\cdots\alpha_{n}}\times I^{n-1}$
be the pullback of ${\mathcal{B}}_{k,\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{k,\,l^{n}}$ via $g_{\alpha_{0}\cdots\alpha_{n}}\colon
U_{\alpha_{0}\cdots\alpha_{n}}\times I^{n-1}\rightarrow\mathop{\rm
Fr}\nolimits_{k,\,l^{n}},$ i.e.
$B_{\alpha_{0}\cdots\alpha_{n}}:=g_{\alpha_{0}\cdots\alpha_{n}}^{*}({\mathcal{B}}_{k,\,l^{n}}).$
So $B_{\alpha_{0}\cdots\alpha_{n}}$ is an $M_{l^{n}}(\mathbb{C})$-bundle over
$U_{\alpha_{0}\cdots\alpha_{n}}\times I^{n-1}$.
For example, we have $M_{l}(\mathbb{C})$-bundles $B_{\alpha\beta}\rightarrow
U_{\alpha\beta},\;B_{\alpha\beta}=g_{\alpha\beta}^{*}({\mathcal{B}}_{k,\,l})$
over double intersections $U_{\alpha}\cap U_{\beta}$ (cf. (27)). We also have
$M_{l^{2}}(\mathbb{C})$-bundles $B_{\alpha\beta\gamma}\rightarrow
U_{\alpha\beta\gamma}\times
I,\;B_{\alpha\beta\gamma}=g_{\alpha\beta\gamma}^{*}({\mathcal{B}}_{k,\,l^{2}})$
such that
$B_{\alpha\beta\gamma}|_{U_{\alpha\beta\gamma}\times\\{0\\}}=B_{\alpha\beta}\otimes
B_{\beta\gamma}|_{U_{\alpha\beta\gamma}}\;\hbox{(cf. (\ref{indmapps}) and
(\ref{boxtimes})) and}\quad
B_{\alpha\beta\gamma}|_{U_{\alpha\beta\gamma}\times\\{1\\}}=M_{l}(B_{\alpha\gamma})|_{U_{\alpha\beta\gamma}}.$
Further, we have $M_{l^{3}}(\mathbb{C})$-bundles
$B_{\alpha\beta\gamma\delta}\rightarrow U_{\alpha\beta\gamma\delta}\times
I^{2},\;B_{\alpha\beta\gamma\delta}=g_{\alpha\beta\gamma\delta}^{*}({\mathcal{B}}_{k,\,l^{3}})$
such that
$B_{\alpha\beta\gamma\delta}|_{U_{\alpha\beta\gamma\delta}\times
I\times\\{0\\}}=B_{\alpha\beta\gamma}\otimes B_{\gamma\delta},$
$B_{\alpha\beta\gamma\delta}|_{U_{\alpha\beta\gamma\delta}\times
I\times\\{1\\}}=M_{l}(B_{\alpha\beta\delta}),$
$B_{\alpha\beta\gamma\delta}|_{U_{\alpha\beta\gamma\delta}\times\\{0\\}\times
I}=B_{\alpha\beta}\otimes B_{\beta\gamma\delta},$
$B_{\alpha\beta\gamma\delta}|_{U_{\alpha\beta\gamma\delta}\times\\{1\\}\times
I}=M_{l}(B_{\alpha\gamma\delta})$
(cf. the above diagram), and so on.
We call such collection of bundles that are compatible to each other as
described above a homotopy bundle gerbe. In particular, we can regard
$B_{\alpha\beta\gamma}$ as an analog of bundle gerbe product from
$B_{\alpha\beta}\otimes B_{\beta\gamma}$ to $M_{l}(B_{\alpha\gamma})$ (cf.
(28)). Bundles $B_{\alpha\beta\gamma\delta}$ express (the first of infinite
number of) associativity conditions.
Using the product of monoids
$\mathop{\rm Fr}\nolimits_{k^{t}l^{\infty},\,l^{\infty}}\times\mathop{\rm
Fr}\nolimits_{k^{u}l^{\infty},\,l^{\infty}}\rightarrow\mathop{\rm
Fr}\nolimits_{k^{t+u}l^{\infty},\,l^{\infty}}$
(see (35) in the next section) for two homotopy bundle gerbes
$(B,\,Y),\;(B^{\prime},\,Y^{\prime})$ one can define their product $(B\otimes
B^{\prime},\,Y_{\pi}\times_{\pi^{\prime}}Y^{\prime})$ (cf. (30)). We call a
homotopy bundle gerbe $(T,\,Y)$ trivial if the corresponding homotopy
transition $\mathop{\rm Fr}\nolimits_{k^{t}l^{\infty},\,l^{\infty}}$-cocycle
can be lifted to the total space of the bundle $\mathop{\rm
PU}\nolimits(k^{t}l^{\infty})\rightarrow\mathop{\rm
Fr}\nolimits_{k^{t}l^{\infty},\,l^{\infty}}$ (which is the direct limit of
principal bundles $\mathop{\rm
PU}\nolimits(k^{t}l^{m+n})\rightarrow\mathop{\rm
Fr}\nolimits_{k^{t}l^{m},\,l^{n}}$ with fibers $\mathop{\rm
PU}\nolimits(l^{n})$). Now we can define the stable equivalence relation on
the set of homotopy bundle gerbes by analogy with (29).
It was shown in [4] that there is an “analysis-free” definition of twisted
$K$-theory by means of bundle gerbe modules. We have already seen above that
such modules can be constructed by projective cocycles. In our situation we
can assume that there is the similar relation to the appropriate notion of a
“homotopy bundle gerbe modules”. Rather than give a general definition we
consider a simple example of (a candidate for) such object below. We start
with the following observation (cf. Proposition 25).
###### Proposition 26.
A map $\varphi\colon X\rightarrow\mathop{\rm Fr}\nolimits_{k,\,l}$ is nothing
but an isomorphism
(34) $\widehat{\varphi}\colon B\otimes M_{k}(\mathbb{C})\cong X\times
M_{kl}(\mathbb{C}),$
where $B\stackrel{{\scriptstyle M_{l}(\mathbb{C})}}{{\longrightarrow}}X$ is
the pullback $\varphi^{*}({\mathcal{B}}_{k,\,l}).$
Proof. Recall that ${\mathcal{B}}_{k,\,l}=\mathop{\rm
PU}\nolimits(kl){\mathop{\times}\limits_{\mathop{\rm
PU}\nolimits(l)}}M_{l}(\mathbb{C}),$ i.e. elements of ${\mathcal{B}}_{k,\,l}$
are equivalence classes of pairs $(g,\,a),$ where
$(g,\,a)\sim(gu,\,u^{-1}a),\;g\in\mathop{\rm
PU}\nolimits(kl),\,u\in\mathop{\rm PU}\nolimits(l)=E_{k}\otimes\mathop{\rm
PU}\nolimits(l)\subset\mathop{\rm PU}\nolimits(kl),\,a\in M_{l}(\mathbb{C})$.
By $[g,\,a]\in{\mathcal{B}}_{k,\,l}$ we denote the corresponding equivalence
class. Then isomorphism (34) for $\varphi=\mathop{\rm
id}\nolimits,\,X=\mathop{\rm Fr}\nolimits_{k,\,l}$ is defined by
$[g,\,a]\otimes b\mapsto(\bar{g},\,g(a\otimes b)),$
where $b\in M_{k}(\mathbb{C})$ and $\bar{g}\in\mathop{\rm
Fr}\nolimits_{k,\,l}$ is the coset $\\{gu\mid u\in\mathop{\rm
PU}\nolimits(l)=E_{k}\otimes\mathop{\rm PU}\nolimits(l)\subset\mathop{\rm
PU}\nolimits(kl)\\}.\quad\square$
Note that a trivialization of $B$ is equivalent to a lift of $\varphi$ to
$X\rightarrow\mathop{\rm PU}\nolimits(kl)$ in the fibration $\mathop{\rm
PU}\nolimits(l)\rightarrow\mathop{\rm PU}\nolimits(kl)\rightarrow\mathop{\rm
Fr}\nolimits_{k,\,l}.$
Let ${\mathcal{U}}=\\{U_{\alpha}\\}_{\alpha\in A}$ be an open cover of a
compact space $X$. Suppose that there are trivial $M_{k}(\mathbb{C})$-bundles
$A_{\alpha}\rightarrow U_{\alpha}$ with given trivialization. Applying the
previous proposition, we see that the homotopy transition cocycle $g$ defines
isomorphisms
$\widehat{g}_{\alpha\beta}\colon B_{\alpha\beta}\otimes A_{\beta}\cong
M_{l}(A_{\alpha})$
(cf. the discussion after Proposition 25), where the trivialization
$M_{l}(A_{\alpha})\cong U_{\alpha}\times M_{kl}(\mathbb{C})$ is defined by the
trivialization of $A_{\alpha}$. Note that the map
$g_{\alpha\beta\gamma}\colon U_{\alpha\beta\gamma}\times
I\rightarrow\mathop{\rm
Fr}\nolimits_{k,\,l^{2}},\;g_{\alpha\beta\gamma}|_{U_{\alpha\beta\gamma}\times\\{0\\}}=M_{l}(g_{\alpha\beta})\circ
g_{\beta\gamma},\;g_{\alpha\beta\gamma}|_{U_{\alpha\beta\gamma}\times\\{1\\}}=\iota\circ
g_{\alpha\gamma}|_{U_{\alpha\beta\gamma}}.$
defines the map
$\widehat{g}_{\alpha\beta\gamma}\colon B_{\alpha\beta\gamma}\otimes
A_{\gamma}\rightarrow M_{l^{2}}(A_{\alpha})$
which is a homotopy (through isomorphisms) between the composition
$B_{\alpha\beta}\otimes B_{\beta\gamma}\otimes
A_{\gamma}\stackrel{{\scriptstyle
1\otimes\widehat{g}_{\beta\gamma}}}{{\longrightarrow}}B_{\alpha\beta}\otimes
M_{l}(A_{\beta})\cong M_{l}(B_{\alpha\beta}\otimes
A_{\beta})\stackrel{{\scriptstyle
M_{l}(\widehat{g}_{\alpha\beta})}}{{\longrightarrow}}M_{l^{2}}(A_{\alpha})$
and
$M_{l}(B_{\alpha\gamma})\otimes A_{\gamma}\stackrel{{\scriptstyle
M_{l}(\widehat{g}_{\alpha\gamma})}}{{\longrightarrow}}M_{l^{2}}(A_{\alpha}).$
On four-fold intersections $U_{\alpha\beta\gamma\delta}$ we have a homotopy
between homotopies, etc. This collection of data can be regarded as an analog
of a bundle gerbe module over the homotopy bundle gerbe
$B:=\\{B_{\alpha_{0}\cdots\alpha_{n}}\\}.$ One can define the notion of
isomorphism on such objects, form their direct sum with the diagonal “action”
of the bundle gerbe and therefore define the corresponding semi-group (whose
Grothendieck group is a candidate to the role of the corresponding twisted
$K$-theory localized over $l$), etc.
Let ${\rm AB}_{l}(X)$ be the group of equivalence classes of matrix algebra
bundles with fibers $M_{l^{n}}(\mathbb{C}),\;n\in\mathbb{N}$ (it is classified
by the $H$-space $\mathop{\rm BPU}\nolimits(l^{\infty})_{\otimes}$). It can be
regarded as a “noncommutative analog” of the Picard group ${\rm Pic}(X)$ and
it acts on the set of homotopy bundle gerbe modules. Then the counterpart of
(32) should be the following: ${\rm Mod}(B)/{\rm AB}_{l}(X)\cong{\rm
HTC}(X,\,d(B)),$ where ${\rm HTC}(X,\,d(B))$ is the set of equivalence classes
of homotopy transition cocycles corresponding to the stable equivalence class
of the homotopy bundle gerbe $B$.
###### Remark 27.
For the sake of clarity we have considered the “projective” version of
homotopy bundle gerbes and modules with matrix algebas as fibers. But in order
to define twisted $K$-theory one should consider “linear” version replacing
${\mathcal{B}}_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$ by vector bundles
$\vartheta_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$ (see (12)), etc.
###### Remark 28.
In fact, the assignment to the homotopy transition cocycle $g$ the stable
equivalence class of the corresponding homotopy bundle gerbe $B$ corresponds
to the projection in the fibration
$\mathop{\rm BU}\nolimits(kl^{\infty})\rightarrow\mathop{\rm
B}\nolimits\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}\rightarrow\mathop{\rm
B}\nolimits(\mathop{\rm BU}\nolimits(l^{\infty})_{\otimes}),$
i.e. $d(B)\in H^{3}(X,\,\mathbb{Z})\times bsu^{1}_{\otimes}[\frac{1}{l}],$ cf.
Remark 29 (and moreover, $d(B)$ has finite order).
## 8\. A generalization of the Brauer group
Note that the tensor product of matrix algebras induces the maps
$\mathop{\rm Fr}\nolimits_{k^{t}l^{m},\,l^{n}}\times\mathop{\rm
Fr}\nolimits_{k^{u}l^{r},\,l^{s}}\rightarrow\mathop{\rm
Fr}\nolimits_{k^{t+u}l^{m+r},\,l^{n+s}},\;(h_{1},\,h_{2})\mapsto h_{1}\otimes
h_{2}.$
Taking the direct limits as $m,\,n,\,r,\,s\to\infty$ we obtain the monoid
homomorphism
(35) $\mathop{\rm Fr}\nolimits_{k^{t}l^{\infty},\,l^{\infty}}\times\mathop{\rm
Fr}\nolimits_{k^{u}l^{\infty},\,l^{\infty}}\rightarrow\mathop{\rm
Fr}\nolimits_{k^{t+u}l^{\infty},\,l^{\infty}}$
and due to the functoriality of the classifying space constructions the
corresponding map of classifying spaces
(36) $\mathop{\rm B}\nolimits\mathop{\rm
Fr}\nolimits_{k^{t}l^{\infty},\,l^{\infty}}\times\mathop{\rm
B}\nolimits\mathop{\rm
Fr}\nolimits_{k^{u}l^{\infty},\,l^{\infty}}\rightarrow\mathop{\rm
B}\nolimits\mathop{\rm Fr}\nolimits_{k^{t+u}l^{\infty},\,l^{\infty}}.$
Note that homomorphisms (35) are defined by the tensor product of the direct
limits of matrix algebras
(37) $M_{k^{t}l^{\infty}}(\mathbb{C})\times
M_{k^{u}l^{\infty}}(\mathbb{C})\mapsto M_{k^{t}l^{\infty}}(\mathbb{C})\otimes
M_{k^{u}l^{\infty}}(\mathbb{C})\cong M_{k^{t+u}l^{\infty}}(\mathbb{C}).$
It is easy to see that maps (36) define the structure of an $H$-space on the
direct limit $\mathop{\rm B}\nolimits\mathop{\rm
Fr}\nolimits_{k^{\infty}l^{\infty},\,l^{\infty}}:=\lim\limits_{\longrightarrow\atop{t}}\mathop{\rm
B}\nolimits\mathop{\rm Fr}\nolimits_{k^{t}l^{\infty},\,l^{\infty}}$ (the
direct limit is induced by the monoid homomorphisms $\mathop{\rm
End}\nolimits(M_{k^{t}l^{\infty}}(\mathbb{C}))\rightarrow\mathop{\rm
End}\nolimits(M_{k^{t+1}l^{\infty}}(\mathbb{C}))$). Using Proposition 35 one
can compute the homotopy groups of the space $\mathop{\rm
B}\nolimits\mathop{\rm Fr}\nolimits_{k^{\infty}l^{\infty},\,l^{\infty}}$:
(38) $\pi_{r}(\mathop{\rm B}\nolimits\mathop{\rm
Fr}\nolimits_{k^{\infty}l^{\infty},\,l^{\infty}})=\lim\limits_{\longrightarrow\atop{t}}\mathbb{Z}/k^{t}\mathbb{Z}=\mathbb{Z}[\frac{1}{k}]/\mathbb{Z}\;\hbox{
for $r>0$ even and $0$ for $r$ odd.}$
As we have already mentioned, the monoids $\mathop{\rm
Fr}\nolimits_{k^{t}l^{\infty},\,l^{\infty}}$ play in our case the same role as
groups $\mathop{\rm PU}\nolimits(k^{t})$ in the “usual” twisted $K$-theory,
therefore the space $\mathop{\rm B}\nolimits\mathop{\rm
Fr}\nolimits_{k^{\infty}l^{\infty},\,l^{\infty}}$ can naturally be considered
as an analog of the $H$-space $\mathop{\rm
BPU}\nolimits(k^{\infty}):=\lim\limits_{\longrightarrow\atop{n}}\mathop{\rm
BPU}\nolimits(k^{n})$. We consider $\mathop{\rm BPU}\nolimits(k^{\infty})$ as
an $H$-space with respect to the product induced by maps $\mathop{\rm
BPU}\nolimits(k^{m})\times\mathop{\rm
BPU}\nolimits(k^{n})\rightarrow\mathop{\rm BPU}\nolimits(k^{m+n})$
corresponding to the tensor product of matrix algebras, while (36) are also
induced by tensor product (37).
Recall that the $k$-primary component $Br_{k}(X)$ of the “finite” Brauer group
is $\mathop{\rm coker}\nolimits\\{[X,\,\mathop{\rm
BU}\nolimits(k^{\infty})]\stackrel{{\scriptstyle\mathop{\rm
B}\nolimits\chi_{*}}}{{\longrightarrow}}[X,\,\mathop{\rm
BPU}\nolimits(k^{\infty})]\\},$ where $\chi\colon\mathop{\rm
U}\nolimits(k^{\infty})\rightarrow\mathop{\rm PU}\nolimits(k^{\infty})$ is
induced by the canonical group epimorphisms $\chi_{k^{m}}\colon\mathop{\rm
U}\nolimits(k^{m})\rightarrow\mathop{\rm PU}\nolimits(k^{m})$, see. (4).
Alternatively, it can be defined as $\mathop{\rm
im}\nolimits\\{[X,\,\mathop{\rm
BPU}\nolimits(k^{\infty})]\stackrel{{\scriptstyle\mathop{\rm
B}\nolimits\psi_{*}}}{{\longrightarrow}}[X,\,\mathop{\rm
K}\nolimits(\mathbb{Z},\,3)]\\}$ (cf. (6)), whence it is just
$H^{3}_{k-tors}(X,\,\mathbb{Z}).$ It can also be interpreted as the group of
obstructions for the lift (= the reduction of the structural group) of
$\mathop{\rm PU}\nolimits(k^{m})$-bundles to $\mathop{\rm
U}\nolimits(k^{m})$-bundles.
Note that there is the $H$-space homomorphism $\mathop{\rm
BU}\nolimits(k^{\infty}l^{\infty})\rightarrow\mathop{\rm
B}\nolimits\mathop{\rm Fr}\nolimits_{k^{\infty}l^{\infty},\,l^{\infty}}$
induced by the composition of homomorphisms $\mathop{\rm
U}\nolimits(k^{t}l^{\infty})\rightarrow\mathop{\rm
PU}\nolimits(k^{t}l^{\infty})$ with inclusions $\mathop{\rm
PU}\nolimits(k^{t}l^{\infty})\rightarrow\mathop{\rm
Fr}\nolimits_{k^{t}l^{\infty},\,l^{\infty}}$ of the subgroups of automorphisms
of $M_{k^{t}l^{\infty}}(\mathbb{C})$ to the monoids of endomorphisms. Thus it
is natural to define the $k$-primary component of the generalized Brauer group
as $GBr_{k}(X):=\mathop{\rm coker}\nolimits\\{[X,\,\mathop{\rm
BU}\nolimits(k^{\infty}l^{\infty})]\rightarrow[X,\,\mathop{\rm
B}\nolimits\mathop{\rm Fr}\nolimits_{k^{\infty}l^{\infty},\,l^{\infty}}]\\}.$
The new part of the generalized Brauer group comparing with the “classical”
one consists of those (classes of)
$M_{k^{t}l^{\infty}}(\mathbb{C})$-fibrations whose structural monoid
$\mathop{\rm End}\nolimits(M_{k^{t}l^{\infty}}(\mathbb{C}))$ can not be
reduced to the group $\mathop{\rm
Aut}\nolimits(M_{k^{t}l^{\infty}}(\mathbb{C}))\subset\mathop{\rm
End}\nolimits(M_{k^{t}l^{\infty}}(\mathbb{C})).$
As a justification of our definition let us note that the fibration induced
from $\widetilde{\mathop{\rm
Fred}\nolimits_{k^{t}l^{\infty}}}({\mathcal{H}})\rightarrow\mathop{\rm
B}\nolimits\mathop{\rm Fr}\nolimits_{k^{t}l^{\infty},\,l^{\infty}}$ (see (26))
by the map $\mathop{\rm BU}\nolimits(k^{t}l^{\infty})\rightarrow\mathop{\rm
B}\nolimits\mathop{\rm Fr}\nolimits_{k^{t}l^{\infty},\,l^{\infty}}$ is trivial
(cf. the discussion at the end of Section 5). It seems that like the
“classical” Brauer group, the generalized one parameterizes twisted
$K$-theories (cf. the end of Section 1). However in contrast with “classical”
it does not admit a simple cohomological description.
From the purely homotopy point of view the generalized Brauer group is the
extension of the “classical” one by 2-periodicity, as the homotopy groups (38)
show. While the unique obstruction (to reduction of the structural group from
$\mathop{\rm PU}\nolimits(k^{m})$ to $\mathop{\rm U}\nolimits(k^{m})$) in the
case of the “classical” Brauer group is the three-dimensional cohomology class
in $H^{3}_{tors}(X,\,\mathbb{Z})$, in case of $GBr_{k}$ there are obstructions
in all odd dimensions (cf. (38)). In this connection note that the homotopy
fiber of the map
(39) $\mathop{\rm BPU}\nolimits(k^{t}l^{\infty})\rightarrow\mathop{\rm
B}\nolimits\mathop{\rm Fr}\nolimits_{k^{t}l^{\infty},\,l^{\infty}}$
induced by inclusion of the subgroup $\mathop{\rm
PU}\nolimits(k^{t}l^{\infty})\hookrightarrow\mathop{\rm
Fr}\nolimits_{k^{t}l^{\infty},\,l^{\infty}},\;\mathop{\rm
PU}\nolimits(k^{t}l^{\infty})=\mathop{\rm
Aut}\nolimits(M_{k^{t}l^{\infty}}(\mathbb{C}))$ is the space $\mathop{\rm
Gr}\nolimits_{k^{t}l^{\infty},\,l^{\infty}}:=\lim\limits_{\longrightarrow\atop{m,\,n}}\mathop{\rm
Gr}\nolimits_{k^{t}l^{m},\,l^{n}},$ where $\mathop{\rm
Gr}\nolimits_{k^{t}l^{m},\,l^{n}}:=\mathop{\rm
PU}\nolimits(k^{t}l^{m+n})/(\mathop{\rm
PU}\nolimits(k^{t}l^{m})\otimes\mathop{\rm PU}\nolimits(l^{n}))$ is the so-
called “matrix Grassmannian” [7].
###### Remark 29.
The fibration
$\mathop{\rm Gr}\nolimits_{kl^{\infty},\,l^{\infty}}\rightarrow\mathop{\rm
BSU}\nolimits(kl^{\infty})\rightarrow\mathop{\rm
B}\nolimits\widetilde{\mathop{\rm Fr}\nolimits}_{kl^{\infty},\,l^{\infty}}$
relates to the part
$bsu^{0}_{\otimes}[\frac{1}{l}]\rightarrow
bsu^{0}_{\otimes}[\frac{1}{l}]\rightarrow
bsu^{0}_{\otimes}(\mathbb{Z}/k\mathbb{Z})$
of the exact sequence for the generalized cohomology theory
$\\{bsu^{n}_{\otimes}\\}_{n}$ (see the Introduction) corresponding to the
coefficient sequence
$0\rightarrow\mathbb{Z}[\frac{1}{l}]\stackrel{{\scriptstyle\cdot
k}}{{\rightarrow}}\mathbb{Z}[\frac{1}{l}]\rightarrow\mathbb{Z}/k\mathbb{Z}\rightarrow
0.$
In fact, our new twistings correspond to the coboundary map $\delta\colon
bsu^{0}_{\otimes}(\mathbb{Z}/k\mathbb{Z})\rightarrow
bsu^{1}_{\otimes}[\frac{1}{l}]$ (while “classical” ones of finite order $k$
correspond to the coboundary map $H^{2}(X,\,\mathbb{Z}/k\mathbb{Z})\rightarrow
H^{3}(X,\,\mathbb{Z})$).
###### Remark 30.
Note that $\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}$ is the total space of
the principal $\mathop{\rm PU}\nolimits(kl^{m})$-bundle $\mathop{\rm
PU}\nolimits(kl^{m})\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Gr}\nolimits_{kl^{m},\,l^{n}}$. There is the commutative diagram (cf. (18))
(40) $\begin{array}[]{c}\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
34.27698pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\\&\\\\}}}\ignorespaces{\hbox{\kern-33.61726pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
33.61726pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
58.93669pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-6.49998pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-30.49997pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
58.93669pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n+r}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
76.30624pt\raise-6.49998pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
76.30624pt\raise-30.49997pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-34.27698pt\raise-40.33328pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Gr}\nolimits_{kl^{m},\,l^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
34.27698pt\raise-40.33328pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
58.27698pt\raise-40.33328pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
58.27698pt\raise-40.33328pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathop{\rm
Gr}\nolimits_{kl^{m},\,l^{n+r}}}$}}}}}}}\ignorespaces\ignorespaces}}}}}\end{array}$
which defines the action of the monoid $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ on $\mathop{\rm
Gr}\nolimits_{kl^{\infty},\,l^{\infty}}$ and there is the equivalence
$\textstyle{\mathop{\rm E}\nolimits\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}{\mathop{\times}\limits_{\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}}}\mathop{\rm
Gr}\nolimits_{kl^{\infty},\,l^{\infty}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\simeq}$$\textstyle{\mathop{\rm
BPU}\nolimits(kl^{\infty})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
B}\nolimits\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}}$$\textstyle{\mathop{\rm
B}\nolimits\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}}$
of $\mathop{\rm Gr}\nolimits_{kl^{\infty},\,l^{\infty}}$-fibrations.
###### Remark 31.
In this remark we establish a relation to constructions from paper [8]. Let
$A_{kl^{m}}^{univ}\rightarrow\mathop{\rm BPU}\nolimits(kl^{m})$ be the
universal $M_{kl^{m}}(\mathbb{C})$-bundle. Applying the functor $\mathop{\rm
Hom}\nolimits_{alg}(\ldots,\,M_{kl^{m+n}}(\mathbb{C}))$ to it fiberwisely we
obtain the $\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}$-bundle
(41) $\begin{array}[]{c}\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
15.35881pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\\&\crcr}}}\ignorespaces{\hbox{\kern-15.35881pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
15.35881pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
39.77351pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
39.77351pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm
H}_{kl^{m},\,l^{n}}(A_{kl^{m}}^{univ})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
65.14021pt\raise-6.37776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
65.14021pt\raise-30.20387pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-3.0pt\raise-41.35387pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
39.35881pt\raise-41.35387pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathop{\rm
BPU}\nolimits(kl^{m}).}$}}}}}}}\ignorespaces\ignorespaces}}}}}\end{array}$
Its total space ${\rm H}_{kl^{m},\,l^{n}}(A_{kl^{m}}^{univ})$ is homotopy
equivalent to $\mathop{\rm Gr}\nolimits_{kl^{m},\,l^{n}}$ [8]. Moreover,
(homotopy classes of) lifts in (41) of a map $f\colon X\rightarrow\mathop{\rm
BPU}\nolimits(kl^{m})$ correspond to (homotopy classes of) bundle embeddings
(42) $\begin{array}[]{c}\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
18.86624pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\\\&&\\\\}}}\ignorespaces{\hbox{\kern-18.86624pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{f^{*}(A_{kl^{m}}^{univ})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
18.86626pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
81.93567pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
7.10059pt\raise-5.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
42.86624pt\raise-33.17902pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
47.40096pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 81.93567pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X\times
M_{kl^{m+n}}(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
105.48378pt\raise-5.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
57.93567pt\raise-34.42253pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-3.0pt\raise-39.00664pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
42.86624pt\raise-39.00664pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X}$}}}}}}}{\hbox{\kern
111.52333pt\raise-39.00664pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces\ignorespaces}}}}}\end{array}$
(note that not every map $f$ has such a lift, see [8]). Applying composition
map (18) to (41) fiberwisely, we obtain
$\textstyle{\mathop{\rm Fr}\nolimits_{kl^{m+n},\,l^{r}}\times{\rm
H}_{kl^{m},\,l^{n}}(A_{kl^{m}}^{univ})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda}$$\textstyle{{\rm
H}_{kl^{m},\,l^{n+r}}(A_{kl^{m}}^{univ})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
BPU}\nolimits(kl^{m})}$
which is equivalent (under ${\rm
H}_{kl^{m},\,l^{n}}(A_{kl^{m}}^{univ})\simeq\mathop{\rm
Gr}\nolimits_{kl^{m},\,l^{n}},\;{\rm
H}_{kl^{m},\,l^{n+r}}(A_{kl^{m}}^{univ})\simeq\mathop{\rm
Gr}\nolimits_{kl^{m},\,l^{n+r}}$) on total spaces to the bottom arrow in (40).
Given a map $\varphi\colon X\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}$ and a lift $\widetilde{f}$ of $f$ in (41) we
obtain some new bundle embedding $f^{*}(A_{kl^{m}}^{univ})\rightarrow X\times
M_{kl^{m+n+r}}(\mathbb{C})$ corresponding to the composition
$X\stackrel{{\scriptstyle diag}}{{\longrightarrow}}X\times
X\stackrel{{\scriptstyle\varphi\times\widetilde{f}}}{{\longrightarrow}}\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}\times{\rm
H}_{kl^{m},\,l^{n}}(A_{kl^{m}}^{univ})\stackrel{{\scriptstyle\lambda}}{{\rightarrow}}{\rm
H}_{kl^{m},\,l^{n+r}}(A_{kl^{m}}^{univ}).$
Such maps (after taking the direct limit) define the action of the monoid
$\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ on (classes of)
embeddings (42). This gives us an interpretation of the principal $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$-fibration induced from the universal
one by map (39) (with $t=1$).
Note that the existence of an embedding
$A_{k}\hookrightarrow X\times M_{kl}(\mathbb{C})$
for $A_{k}\stackrel{{\scriptstyle M_{k}(\mathbb{C})}}{{\longrightarrow}}X$
implies the triviality of the corresponding $\mathop{\rm
End}\nolimits(M_{kl^{\infty}}(\mathbb{C}))$-fibration ${\rm
H}_{kl^{\infty},\,l^{\infty}}(A_{k})\rightarrow X$.
In order to define a new cohomological obstruction consider the monoid
$\widetilde{\mathop{\rm Fr}\nolimits}_{k^{t}l^{\infty},\,l^{\infty}}$ from
Remark 22. An easy calculation shows that $H^{5}(\mathop{\rm
B}\nolimits\widetilde{\mathop{\rm
Fr}\nolimits}_{k^{t}l^{\infty},\,l^{\infty}},\,\mathbb{Z})\cong\mathbb{Z}/k^{t}\mathbb{Z}$
and this class is the first obstruction to the reduction of the structural
monoid to the group $\mathop{\rm SU}\nolimits(k^{t}l^{\infty}).$
Note that the important feature of the classical Brauer group is its relation
to the Morita-equivalence of $C^{*}$-algebras [15]. More precisely, there is
another (equivalent, see Definition 3.4 in [1]) definition of the “usual”
twisted $K$-theory as the $K$-theory of continuous-trace algebras of sections
of locally trivial algebra bundles with fibers
${\mathcal{K}}({\mathcal{H}})\subset{\mathcal{B}}({\mathcal{H}})$ (whose local
triviality follows from Fell’s condition) with the structural group
$\mathop{\rm PU}\nolimits({\mathcal{H}})$. In our case we have bundles of
algebras with fibers $M_{kl^{\infty}}(\mathbb{C})$ with the structural monoid
$\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ which are locally
homotopy trivial. It seems to be an interesting task to investigate the
relation of the generalized Brauer group to the Morita-equivalence of such
bundles.
## 9\. Appendix 1: Homotopy groups, etc.
###### Lemma 32.
The homotopy groups of the space $\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}$
up to dimension $\sim 2l^{n}$ are as follows: $\pi_{r}(\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}})=\mathbb{Z}/kl^{m}\mathbb{Z}$ for $r$ odd and
$0$ for $r$ even.
Proof follows from the homotopy sequence of principal fibration (12) together
with the Bott periodicity for unitary groups.$\quad\square$
Note that the Bott periodicity allows us to compute homotopy groups in the
previous Lemma only up to dimension $\sim 2l^{n}$. In what follows such
homotopy groups will be called “stable”.
Unital homomorphisms of matrix algebras induce maps $\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\hookrightarrow\mathop{\rm
Fr}\nolimits_{kl^{t},\,l^{u}}$ for all $t\geq m,\,u\geq n.$ We want to obtain
some information about the direct limit
$\lim\limits_{\longrightarrow\atop{m,\,n}}\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}.$
###### Lemma 33.
The maps $\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m+1},\,l^{n}}$ induce the injective homomorphisms of stable
homotopy groups.
Proof. Consider the morphism of homotopy sequences of principal fibrations
(12)
$\textstyle{\mathop{\rm
U}\nolimits(l^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
U}\nolimits(kl^{m+n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
U}\nolimits(l^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
U}\nolimits(kl^{m+n+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
Fr}\nolimits_{kl^{m+1},\,l^{n}}}$
which in stable odd dimensions gives the commutative diagram
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot
kl^{m}}$$\scriptstyle{=}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot
l}$$\textstyle{\mathbb{Z}/kl^{m}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot
kl^{m+1}}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}/kl^{m+1}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$
whence we get the injective homomorphisms
$\pi_{r}(\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}})\rightarrow\pi_{r}(\mathop{\rm
Fr}\nolimits_{kl^{m+1},\,l^{n}}),\;\mathbb{Z}/kl^{m}\mathbb{Z}\rightarrow\mathbb{Z}/kl^{m+1}\mathbb{Z},\;\alpha\,(\mathop{\rm
mod}\nolimits\,kl^{m})\mapsto l\alpha\,(\mathop{\rm mod}\nolimits\,kl^{m+1})$
in odd stable dimensions.$\quad\square$
###### Lemma 34.
The maps $\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n+1}}$ induce the following homomorphisms of stable
homotopy groups in odd dimensions:
$\pi_{r}(\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}})\rightarrow\pi_{r}(\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n+1}}),\;\mathbb{Z}/kl^{m}\mathbb{Z}\rightarrow\mathbb{Z}/kl^{m}\mathbb{Z},\;\alpha\,(\mathop{\rm
mod}\nolimits\,kl^{m})\mapsto l\alpha\,(\mathop{\rm mod}\nolimits\,kl^{m}).$
Hence such a homomorphism has the kernel $\cong\mathbb{Z}/k\mathbb{Z}$.
Proof. Again, consider the morphism of homotopy sequences of principal
fibrations (12)
$\textstyle{\mathop{\rm
U}\nolimits(l^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
U}\nolimits(kl^{m+n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
U}\nolimits(l^{n+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
U}\nolimits(kl^{m+n+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n+1}}}$
which in odd stable dimensions turns into the commutative diagram
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot
kl^{m}}$$\scriptstyle{\cdot
l}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot
l}$$\textstyle{\mathbb{Z}/kl^{m}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot
kl^{m}}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}/kl^{m}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
gives us homomorphisms $\pi_{r}(\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}})\rightarrow\pi_{r}(\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n+1}})$ as in the statement of the
lemma.$\quad\square$
###### Proposition 35.
The homotopy groups of the space
$\lim\limits_{\longrightarrow\atop{m,\,n}}\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$ are as follows: $\mathbb{Z}/k\mathbb{Z}$ in all
odd dimensions and $0$ in all even dimensions.
Proof follows from the previous lemmas. More precisely, we consider the direct
limit of cyclic groups with respect to the homomorphisms
$\textstyle{\ldots}$$\textstyle{\ldots}$$\textstyle{\mathbb{Z}/kl^{m+1}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot
l}$$\textstyle{\mathbb{Z}/kl^{m+1}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots}$$\textstyle{\mathbb{Z}/kl^{m}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot
l}$$\scriptstyle{\cdot
l}$$\textstyle{\mathbb{Z}/kl^{m}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot
l}$$\textstyle{\ldots,}$
where the horizontal arrows have nonzero kernels. Therefore the $l$-primary
component vanishes in the direct limit (recall that $(k,\,l)=1).\quad\square$
Note that the previous proposition shows the reason of the assumption
$(k,\,l)=1$. This guarantees the homotopy nontriviality of the space
$\lim\limits_{\longrightarrow\atop{m,\,n}}\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$.
###### Proposition 36.
The inclusion $\lim\limits_{\longrightarrow\atop{n}}\mathop{\rm
Fr}\nolimits_{k,\,l^{n}}\rightarrow\lim\limits_{\longrightarrow\atop{m,\,n}}\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$ is a homotopy equivalence. Moreover, the
homotopy type of $\lim\limits_{\longrightarrow\atop{n}}\mathop{\rm
Fr}\nolimits_{k,\,l^{n}}$ does not depend on the choice of $l$ such that
$(k,\,l)=1.$
Proof. Clearly, the considered spaces are CW-complexes, therefore it is
sufficient to prove their weak homotopy equivalence. It can be done in analogy
with the proofs of the previous lemmas.
More precisely, consider the diagram
$\textstyle{\mathop{\rm
Fr}\nolimits_{k,\,l^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
Fr}\nolimits_{k,\,l^{n+1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n+1}}}$
and the corresponding diagram of the homotopy sequences in odd stable
dimensions (cf. Lemma 32):
$\textstyle{\mathbb{Z}/k\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subset\qquad\qquad}$$\scriptstyle{\cdot
l}$$\textstyle{\mathbb{Z}/kl^{m}\mathbb{Z}\cong\mathbb{Z}/k\mathbb{Z}\oplus\mathbb{Z}/l^{m}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot
l}$$\textstyle{\mathbb{Z}/k\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subset\qquad\qquad\quad}$$\textstyle{\mathbb{Z}/kl^{m}\mathbb{Z}\cong\mathbb{Z}/k\mathbb{Z}\oplus\mathbb{Z}/l^{m}\mathbb{Z},}$
where the horizontal arrows are injective according to Lemma 33, and the
vertical ones are nilpotent on the $l$-primary component by Lemma 34.
To prove the second part first suppose that $(l,\,l^{\prime})=1$, then
$\mathop{\rm
Fr}\nolimits_{k,\,l^{\infty}}\stackrel{{\scriptstyle\simeq}}{{\rightarrow}}\mathop{\rm
Fr}\nolimits_{k,\,(ll^{\prime})^{\infty}}\stackrel{{\scriptstyle\simeq}}{{\leftarrow}}\mathop{\rm
Fr}\nolimits_{k,\,l^{\prime\infty}}$ are homotopy equivalences. In the case
$(l,\,l^{\prime})=d>1$ we take $l^{\prime\prime}$ such that
$(l,\,l^{\prime\prime})=1=(l^{\prime},\,l^{\prime\prime})$. Then $\mathop{\rm
Fr}\nolimits_{k,\,l^{\infty}}\stackrel{{\scriptstyle\simeq}}{{\rightarrow}}\mathop{\rm
Fr}\nolimits_{k,\,(ll^{\prime\prime})^{\infty}}\stackrel{{\scriptstyle\simeq}}{{\leftarrow}}\mathop{\rm
Fr}\nolimits_{k,\,l^{\prime\prime\infty}}\stackrel{{\scriptstyle\simeq}}{{\rightarrow}}\mathop{\rm
Fr}\nolimits_{k,\,(l^{\prime\prime}l^{\prime})^{\infty}}\stackrel{{\scriptstyle\simeq}}{{\leftarrow}}\mathop{\rm
Fr}\nolimits_{k,\,l^{\prime\infty}}.\quad\square$
###### Lemma 37.
The space $\lim\limits_{\longrightarrow\atop{m,\,n}}\mathop{\rm
U}\nolimits(kl^{m+n})/(\mathop{\rm U}\nolimits(kl^{m})\otimes E_{l^{n}})$ is
contractible.
Proof. Since this space is a CW-complex, it is sufficient to prove that it is
weakly homotopy equivalent to a point. But this is obvious because the only
nontrivial stable homotopy groups in odd dimensions map under
$\mathop{\rm U}\nolimits(kl^{2n})/(\mathop{\rm U}\nolimits(kl^{n})\otimes
E_{l^{n}})\rightarrow\mathop{\rm U}\nolimits(kl^{2n+2})/(\mathop{\rm
U}\nolimits(kl^{n+1})\otimes E_{l^{n+1}})$
as follows:
$\mathbb{Z}/l^{n}\mathbb{Z}\rightarrow\mathbb{Z}/l^{n+1}\mathbb{Z},\;\alpha\,(\mathop{\rm
mod}\nolimits\,l^{n})\mapsto l^{2}\alpha\,(\mathop{\rm
mod}\nolimits\,l^{n+1}).\quad\square$
## 10\. Appendix 2: $\mathop{\rm
Fr}\nolimits_{k^{\infty}l^{\infty},\,l^{\infty}}$ as a classifying space
Let us show that any bundle of order $k^{n}$ in $K_{\otimes}$ can be
represented by a map $X\rightarrow\mathop{\rm
Fr}\nolimits_{k^{\infty}l^{\infty},\,l^{\infty}},$ and vice versa.
Consider the fibration
(43) $\widetilde{\mathop{\rm
Fr}\nolimits}_{k^{m},\,l^{n}}\rightarrow\mathop{\rm
Gr}\nolimits_{k^{m},\,l^{n}}\stackrel{{\scriptstyle\beta_{m,\,n}}}{{\rightarrow}}\mathop{\rm
BSU}\nolimits(k^{m}),$
where $\widetilde{\mathop{\rm Fr}\nolimits}_{k^{m},\,l^{n}}:=\mathop{\rm
SU}\nolimits(k^{m}l^{n})/(E_{k^{m}}\otimes\mathop{\rm SU}\nolimits(l^{n}))$,
and the map $\beta_{m,\,n}$ is a classifying map for the tautological
$M_{k^{m}}(\mathbb{C})$-bundle over the matrix Grassmannian $\mathop{\rm
Gr}\nolimits_{k^{m},\,l^{n}}:=\mathop{\rm
SU}\nolimits(k^{m}l^{n})/(\mathop{\rm SU}\nolimits(k^{m})\otimes\mathop{\rm
SU}\nolimits(l^{n}))$ [7]. Now taking the limit in (43) as
$m,\,n\rightarrow\infty$ with respect to maps induced by the tensor product
and using the $H$-space isomorphism
$\lim\limits_{\longrightarrow\atop{m,\,n}}\mathop{\rm
Gr}\nolimits_{k^{m},\,l^{n}}\cong\mathop{\rm BSU}\nolimits_{\otimes}$ (where
the $H$-space structure on
$\lim\limits_{\longrightarrow\atop{m,\,n}}\mathop{\rm
Gr}\nolimits_{k^{m},\,l^{n}}$ is defined by the maps $\mathop{\rm
Gr}\nolimits_{k^{m},\,l^{n}}\times\mathop{\rm
Gr}\nolimits_{k^{t},\,l^{u}}\rightarrow\mathop{\rm
Gr}\nolimits_{k^{m+t},\,l^{n+u}}$ induced by the tensor product of matrix
algebras) [7] we see that $\widetilde{\mathop{\rm
Fr}\nolimits}_{k^{\infty},\,l^{\infty}}:=\lim\limits_{\longrightarrow\atop{m,\,n}}\widetilde{\mathop{\rm
Fr}\nolimits}_{k^{m},\,l^{n}}$ is the homotopy fiber of the localization map
$\lim\limits_{\longrightarrow\atop{m,\,n}}\beta_{m,\,n}\colon\mathop{\rm
BSU}\nolimits_{\otimes}\rightarrow\mathop{\rm
BSU}\nolimits_{\otimes}[\frac{1}{k}]$. In particular, for any $\mathop{\rm
SU}\nolimits$-bundle over $X$ of order $k^{n},\>n\in\mathbb{N}$ a classifying
map has a lift to $\widetilde{\mathop{\rm
Fr}\nolimits}_{k^{\infty},\,l^{\infty}}$.
The general case (recall that $\mathop{\rm
BU}\nolimits_{\otimes}\cong\mathop{\rm
K}\nolimits(\mathbb{Z},\,2)\times\mathop{\rm BSU}\nolimits_{\otimes}$)
corresponds to the fibration $\mathop{\rm
Fr}\nolimits_{k^{\infty},\,l^{\infty}}\rightarrow\mathop{\rm
BU}\nolimits_{\otimes}\rightarrow\mathop{\rm
BU}\nolimits_{\otimes}[\frac{1}{k}],$ and $\mathop{\rm
Fr}\nolimits_{k^{\infty}l^{\infty},\,l^{\infty}}$ itself is the fiber of the
fibration $\mathop{\rm
Fr}\nolimits_{k^{\infty}l^{\infty},\,l^{\infty}}\rightarrow\mathop{\rm
BU}\nolimits_{\otimes}[\frac{1}{l}]\rightarrow\mathop{\rm
BU}\nolimits_{\otimes}[\frac{1}{kl}]$ (cf. (14) and Proposition 36).
## References
* [1] M. Atiyah, G. Segal: Twisted K-theory // arXiv:math/0407054v2 [math.KT]
* [2] M. Atiyah, G. Segal: Twisted K-theory and cohomology // arXiv:math/0510674v1 [math.KT]
* [3] M.F. Atiyah, I.M. Singer: Index theory for skew-adjoint Fredholm operators. Publ. Math. I.H.E.S. Paris, 37 (1969), 5-26.
* [4] P. Bouwknegt, A. L. Carey, V. Mathai, M. K. Murray, D. Stevenson: Twisted K-theory and K-theory of bundle gerbes Commun.Math.Phys.228:17-49, 2002.
* [5] Ulrich Bunke, Thomas Schick: On the topology of T-duality . Rev. Math. Phys. 17 (2005), no. 1, 77–112.
* [6] P. Donovan, M. Karoubi: Graded Brauer groups and K-theory with local coefficients. Pub. Math. IHES N 38, p. 5-25 (1971).
* [7] A.V. Ershov: A generalization of the topological Brauer group Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology (2008), 2:407-444.
* [8] A.V. Ershov: Topological obstructions to embedding of a matrix algebra bundle into a trivial one // arXiv:0807.3544v13 [math.KT]
* [9] Martin Fuchs: A modified Dold-Lashof construction that does classify $H$-principal fibrations. Math. Ann., 192:328-340, 1971.
* [10] M. Karoubi: Alg‘ebres de Clifford et K-th eorie. Ann. Sci. Ecole Norm. Sup. (4), pp. 161-270 (1968).
* [11] M. Karoubi: Twisted K-theory, old and new. K-Theory and Noncommutative Geometry, EMS Series of Congress Reports (2008), arXiv:math/0701789v3 [math.KT],
* [12] J. Peter May: Classifying spaces and fibrations. Memoirs of the American Mathematical Society., issue 1, no 155, AMS, Providence, Rhode Island 1975.
* [13] Michael K. Murray: Bundle gerbes J.Lond.Math.Soc. 54 (1996) 403-416.
* [14] Michael K. Murray, Daniel Stevenson: Bundle gerbes: stable isomorphism and local theory J.Lond.Math.Soc. 62 (2000) 925-937.
* [15] I. Raeburn., D.P. Williams: Morita Equivalence and Continuous-Trace $C^{*}$-Algebras (Mathematical Surveys and Monographs).
* [16] J. Rosenberg: Continuous-trace algebras from the bundle theoretic point of view. J. Austral Math. Soc. Ser. A. 47(3): 368-381, 1989.
* [17] Yu.B. Rudyak: On Thom Spectra, Orientability and Cobordism. Springer Monogr. in Math., Springer (1998).
* [18] G.B. Segal: Categories and cohomology theories. Topology 13 (1974).
* [19] Claude Schochet: The Dixmier-Douady Invariant for Dummies. Notices of the AMS, Vol. 56, Issue 07 (2009), pp.809-816; Correction: Vol. 57, Issue 03, p.419
* [20] James Wirth and Jim Stasheff: Homotopy Transition Cocycles. Journal of Homotopy and Related Structures, Volume 1 (2006), No. 1, 273-283.
|
arxiv-papers
| 2010-05-20T19:53:52 |
2024-09-04T02:49:10.555098
|
{
"license": "Public Domain",
"authors": "A.V. Ershov",
"submitter": "Andrey Ershov V.",
"url": "https://arxiv.org/abs/1005.3807"
}
|
1005.3834
|
# What’s So Peculiar About the Cycle 23/24 Solar Minimum?
N. R. Sheeley, Jr.
###### Abstract
Traditionally, solar physicists become anxious around solar minimum, as they
await the high-latitude sunspot groups of the new cycle. Now, we are in an
extended sunspot minimum with conditions not seen in recent memory, and
interest in the sunspot cycle has increased again. In this paper, I will
describe some of the characteristics of the current solar minimum, including
its great depth, its extended duration, its weak polar magnetic fields, and
its small amount of open flux. Flux-transport simulations suggest that these
characteristics are a consequence of temporal variations of the Sun’s large-
scale meridional circulation.
Space Science Division, Naval Research Laboratory, Washington DC 20375-5352,
USA
## 1\. Introduction
When I was asked to give this talk, I wondered if this minimum was really
peculiar, or whether we were just feeling the anxiety that occurs toward the
end of every sunspot cycle as solar physicists await the first new-cycle
active regions.
We are all familiar with this anxiety. Flare researchers become anxious
because they have contracts to study solar flares. Energetic particle
researchers become anxious because they want more data. NASA managers become
anxious because their spacecraft missions were justified in terms of what they
would learn about solar activity. But most anxious of all are the scientists
who predicted the strength of the next sunspot cycle. So when you encounter
the forecasters, appreciate the stress they are under and be kind.
The level of anxiety increased during the 1976 minimum when Jack Eddy reminded
us that sunspots became particularly scarce during the 70-year interval
1645-1715 and that another interruption of the sunspot cycle might occur at
any time (Eddy 1976). His talks alarmed some people, and sent reporters to
solar observatories to find out if we were headed into another Maunder
Minimum. We did not enter another Maunder Minimum in 1976 and we have emerged
unscathed from two subsequent minima since that time. Let’s see how the
present sunspot minimum compares with some of these earlier ones.
## 2\. The Sunspot Number
Figure 1 shows the sunspot number during the interval 1895–2009. The monthly
means are plotted at the bottom of this panel with an arbitrary linear scale.
The natural logarithms of these monthly means are plotted at the top of the
panel to show the lower numbers in more detail. The logarithms are indicated
by diamonds (or squares placed at -3 when the monthly means vanished, as
happened for the most recent data point in August 2009).
Figure 1.: Monthly averaged sunspot number (bottom) and its natural logarithm
(top) plotted versus time in years, showing that the current minimum is
comparable to those that occurred during the first half of the 20th century.
Figure 2.: Same as previous figure, except extending back to 1745. Deep minima
were common prior to the space age.
This figure shows that deep minima were common during the first half of the
twentieth century, but that these deep minima suddenly disappeared after the
very strong sunspot cycle in 1958. Since that time the minima have been
becoming progressively deeper. The depth of the current cycle is now
comparable to the depths of the cycles prior to 1958, and as low as the very
deep minima in 1902 and 1913 if the August 2009 measurement is included.
Figure 2 provides a 250-year perspective, and shows that the series of deep
minima extended back for 10 sunspot cycles before being interrupted by two
shallow minima in 1833 and 1843. Even deeper minima preceded them during the
weak sunspot cycles in the Dalton minimum (1800-1830). Thus, the present
minimum is deeper than we have seen since the space age began, but not
unusually deep on a time scale of 250 years or even 100 years.
## 3\. The Duration of the Minimum
Figure 3 shows a time-latitude distribution of sunspot eruptions since 1875,
prepared by David Hathaway(http://solarscience.msfc.nasa.gov/images/bfly.gif).
I have added the vertical dashed lines at the end of each cycle to help
determine whether that cycle overlaps with the next one.
Figure 3.: The occurrence of sunspot groups as a function of time in years
(horizontal axis) and sine latitude (vertical axis). Vertical dashed lines are
placed at the end of each sunspot cycle. Since 1954, these lines have clipped
the high-latitude wings of the next cycle, indicating no separation between
consecutive cycles. Exceptions include the present cycle and the southern
hemisphere in 1966.
There is a clear, 1-year eruption-free interval at the end of cycle 23. Moving
backward in time, there are no comparable gaps between cycles until the minima
in 1913 and 1902. As indicated in Figure 1, These minima were especially deep,
and they bounded the very weak sunspot cycle that peaked in 1906. A shorter
eruption-free interval occurred in 1954, separating cycles 18 and 19.
Examining the northern and southern hemispheres separately, we find that the
southern hemisphere had an appreciable eruption-free interval in 1966,
corresponding to a one-year delay in the start of sunspot cycle 20 in the
southern hemisphere.
Figure 4.: Mount Wilson Observatory Ca II 3934 Å images, showing the enhanced
emission on the front (upper left) and back (lower left) sides of the Sun near
sunspot maximum in 1958, the lack of such emission at sunspot minimum in 1954,
and the north-south asymmetry at the start of the new cycle in 1966 (Sheeley
1967).
This delayed start of southern-hemisphere activity is one that I experienced,
but had forgotten until now. The Ca II 3934 Å spectroheliogram in the lower
right panel of Figure 4 shows the north-south asymmetry on June 5, 1966 when I
was on Kitt Peak, obtaining a time series of high-resolution K-line spectra of
a region in the northern hemisphere. During an exposure, the power to the
telescope drive suddenly failed and the solar image drifted westward across
the slit. This produced an integrated K-line spectrum of the northern
hemisphere. When the power was restored, I made a similar scan across the
southern hemisphere, this time intentionally. I thought that the two spectra
would be representative of conditions at sunspot maximum and minimum and
therefore reveal all at once whether we could detect the sunspot cycle of the
unresolved Sun from variations in its K-line emission, as Olin Wilson was
starting to do for other stars (Wilson 1978).
Figure 5.: Latitude-time displays of zonal flows obtained by subtracting the
solar rotation profile from Doppler observations at the Mount Wilson
Observatory (upperpanel) and from GONG/MDI global oscillation measurements
(lower panel). Identical blue tracks fit the equatorial progressions during
past sunspot cycles, but not the progression of cycle 24 (Howe et al. 2009).
The so-called torsional oscillations (Howard & Labonte 1980) provide further
evidence that the present sunspot minimum is different from the past three
minima. These features are alternating bands of prograde and retrograde
rotation obtained when the long-term solar rotation profile is subtracted from
the east-west component of the large-scale Doppler field (in the case of the
Mount Wilson Observatory (MWO) measurements) or the global oscillation data
(in the case of the Global Oscillation Network Group/Michelson Doppler
Interferometer (GONG/MDI) helioseismic observations). The lower branches of
these residual east-west flows migrate toward the equator alongside (but not
coincident with) the zones of sunspot eruption.
Figure 5 compares the MWO zonal flows observed during 1975–2009 (Ulrich &
Boyden 2005) with the GONG/MDI flows (Howe et al. 2009) during 1995–2009. In
this figure, a blue line was arbitrarily drawn along a boundary between the
prograde and retrograde flows in cycle 23, and then shifted uniformly to the
equatorial tracks in the other cycles. As one can see, these identical blue
tracks match all of the equatorward progressions except the one that began at
high latitude in 2003. The current migration has started more slowly than any
equatorward migration since the observations began at Mount Wilson in 1975. As
Howe et al. (2009) first noted, this slow migration is a precursor to the
delayed onset of sunspot cycle 24 and may provide another clue to the origin
of the delay.
## 4\. Weaker Polar Magnetic Fields
We have all heard that the polar field is weaker now than it has been for many
years. Figure 6 shows the Wilcox Solar Observatory (WSO) measurements since
1976. The polar field is about two-thirds as strong as it was during the
previous minimum. The axial component of the Sun’s dipole field shows a
corresponding decrease, suggesting that the open flux is also about two-thirds
of its previous value.
Figure 6.: Plots of the Sun’s polar magnetic fields and axial dipole derived
from observations at the Wilcox Solar Observatory during 1976–2009.
Figure 7.: The numbers of north and south polar faculae during their times of
greatest visibility (fall or spring), and the yearly sunspot number for the
full disk, multiplied by 0.3, and assigned the polarity of the following spots
in each hemisphere (dashed lines) (Sheeley 2008).
It is interesting to see how far back in time we can extend these polar field
measurements. Polar faculae are visible on white-light images obtained daily
at the Mount Wilson Observatory since 1906, and their numbers provide a
reliable indication of the polar magnetic field strengths. Figure 7 shows the
numbers of polar faculae counted during the favorable intervals of each year
(fall or spring) since 1906. These numbers have been assigned the polarities
of the corresponding polar magnetic fields (since the invention of the
magnetograph in 1952), or extrapolated smoothly through zero (in the
premagnetograph years).
This figure suggests that the polar fields are weaker now than they have been
in the last 100 years. Rapid changes in the early 1960s are probably due to
alternating bands of flux carried poleward by meridional flow. Aside from
these changes, this cycle of southern-hemisphere faculae was the second
smallest in this 100-yr record. We have already seen in Figures 3 and 4 that
the southern-hemisphere activity was delayed by about one year in 1966. Is it
a coincidence that the delayed onset of this activity and the delayed onset of
cycle 24 were both preceded by unusually weak polar fields? This provides a
motive for reexamining the white-light images during the extended minimum
around 1913 to see how weak the polar fields may have been during that time.
## 5\. Less Open Magnetic Flux
During the declining phase of the sunspot cycle, the eruption of flux creates
and maintains low-latitude coronal holes with accompanying warps of the
streamer belt. These coronal holes gradually die out at sunspot minimum, as
the old-cycle eruptions stop and the relatively strong polar fields grab the
dwindling remnants of open field lines at low latitude. During the present
minimum, the old-cycle eruptions stopped early in 2008, but the low-latitude
holes and the warped streamer belt persisted for at least another year. This
peculiarity is due to the relatively weak polar magnetic fields (Wang et al.
2009), as one can see from the experiment performed in Figure 8.
The left panels show the NSO photospheric field (top), the observed Fe XII 195
Å emission (second from top), the photospheric distribution of open flux
derived from a potential field extension of the observed field, and the
corresponding derived field at 2.5$R_{\odot}$. The map of open flux shows
colored areas at low latitude that correspond to dark coronal holes in the map
of Fe XII 195 Å emission. The right panels show the same maps with polar
fields that are twice as strong (12 Gauss compared to 6 Gauss). This change
caused the derived regions of open flux to disappear from low latitude and the
neutral line of the coronal field to flatten toward the equator. The same
experiment showed very little change nine rotations earlier when large active
regions were still present at low latitude.
The survival of a low-latitude coronal hole depends on its field strength
relative to the field strength of the polar hole of opposite polarity.
Consequently, the low-latitude holes last longer when the polar fields are
weak. Also, if the polar field strengths differ appreciably in the two
hemispheres, then the longer-lived low-latitude holes would have the polarity
of the stronger polar field. This allows us to predict that the surviving low-
latitude holes ought to have had negative polarity in 1966 when the weaker
south polar field had positive polarity.
Figure 8.: Carrington maps of photospheric field (top), Fe XII 195 Å
intensity, derived open flux, and source-surface field (bottom), showing that
the low-latitude coronal holes disappear and the source-surface neutral line
flattens when the polar field strength is increased from 6 G (left) to 12 G
(right).
Figure 9 compares the total open flux on the Sun, derived from potential field
extrapolations of the observed photospheric magnetic field, with the total
open flux, derived from in situ measurements of the radial component of the
interplanetary magnetic field. These overlapping curves show similar behaviors
with low values during each sunspot minimum, when nearly all of the flux
originates in the polar coronal holes, and high values near and after sunspot
maximum. These high values occur when flux erupts in longitudinal phase and
increases the strength of the equatorial dipole. The individual peaks decay
with a lifetime of about 1.5 years as meridional flow carries the flux to
midlatitudes where it is sheared by differential rotation and dissipated by
supergranular diffusion. As indicated by the arrow, the present amount of open
flux on the Sun and in the heliosphere is the lowest since observations began
in 1967.
Figure 9.: The total open flux on the Sun derived from photospheric
magnetograms (solid line) and derived from in situ measurements of the radial
magnetic field (dashed line). For comparison, the sunspot number is plotted
below (dashed-dot line). The current value (arrow) is the lowest since
observations began in 1967.
## 6\. The Effects of Meridional Circulation
Figure 10 shows a longitudinally averaged map of photospheric magnetic field
measurements obtained at the Mount Wilson Observatory (MWO) since 1967. From
the first 13 years of these observations, Howard & Labonte (1981) identified
‘episodic poleward surges’ of flux extending from the sunspot belts to the
poles of the Sun. They argued that these surges were evidence for a poleward
meridional flow because supergranular diffusion by itself (Leighton 1964)
would blur out the flux distribution and not show these concentrated streams.
Figure 10.: Longitudinally averaged photospheric field observed at the Mount
Wilson Observatory since 1967, showing surges of flux migrating poleward from
the sunspot belts during each sunspot cycle. The dotted curve marks a
positive-polarity surge whose slope (converted from sine latitude to latitude)
changed from 12 m s-1 to 8 m s-1 as it moved northward during 1970–1972.
Now we know that diffusion and flow both contribute to the transport (Devore
et al. 1984; Wang et al. 1989) and that both terms must be considered when
interpreting the slopes of the surges (Wang et al. 2009). As described by
Sheeley et al. (1989), supergranular diffusion provides an effective flow that
is proportional to the surface gradient of magnetic flux density. Thus, in
Figure 10, the initial slope of the surge marked by a dotted line is 12 m s-1,
but diffusion may contribute $\sim$30$\%$ of this value, depending on the
magnitude of the flux gradients in the sunspot belts. At higher latitudes the
gradients are smaller and the 8 m s-1 slope may more closely represent the
true speed of meridional flow.
The flow speed can be inferred through its influence on the magnetic field
distribution. At low latitudes, the competition between poleward flow and
equatorward diffusion from the sunspot belts determines the amount of leading-
polarity flux that reaches the equator and is annihilated by its counterpart
in the other hemisphere. Consequently, this competition also determines the
amount of unbalanced trailing-polarity flux that remains in each hemisphere
for reversing the polar field. At high latitudes, the competition between
poleward flow from the sunspot belts and equatorward diffusion from the polar
region determines the latitudinal shape of the polar field. The observed
topknots of flux (Svalgaard et al. 1978) are consistent with a flow profile
that decreases linearly or quadratically toward the poles (Devore et al. 1984;
Sheeley et al. 1989).
A new era of understanding occurred when Wang et al. (2002) found that a small
variation of flow speed would be sufficient to regulate the polar-field
reversal if the speed were correlated with the strength of the sunspot cycle.
In a strong cycle, a faster flow would produce less unbalanced trailing-
polarity flux and cause the polar fields to fluctuate until the unbalanced
flux arrived there late in the cycle. In a weak cycle, a slower flow would
compensate by producing more unbalanced trailing-polarity flux.
Recent simulations suggest that a small ($\sim$$15\%$) increase of flow speed
may be responsible for the currently weakened polar and interplanetary fields
(Schrijver & Liu 2008; Wang et al. 2009). On the other hand, the increased
depth and length of this minimum probably result from changes in the
subsurface return flow, as suggested by the reduced migration speed of the
torsional oscillations prior to the start of cycle 24 (Howe et al. 2009).
## 7\. It’s your grandfather’s solar minimum!
Having examined the present solar minimum and compared it with previous
minima, we find differences which suggest that ‘It’s not your father’s solar
minimum’ (to paraphrase the Oldsmobile advertisement). However, this minimum
does contain similarities to older minima during the past 100 years or more.
Perhaps it’s your grandfather’s solar minimum.
### Acknowledgments.
I am grateful to Y.-M. Wang (NRL) for several useful discussions and for
providing material for Figures 6, 8, and 10. Sunspot numbers in Figure 3 came
from www.ngdc.noaa.gov/stp/SOLAR/ftpsunspotnumber.html. I thank David Hathaway
(NASA/MSFC) for permission to use his butterfly diagram
(solarscience.msfc.nasa.gov). I thank Rachel Howe (NSO) for providing material
for Figure 5 and for several useful discussions. Frank Hill, Rudi Komm, and
Irene González-Hernández (all at NSO) provided helpful comments about
helioseismology, and Roger Ulrich (UCLA) provided MWO measurements of
torsional oscillations and episodic poleward surges which I continue to
appreciate. Financial support was provided by NASA and NRL.
## References
* Devore et al. (1984) Devore, C. R., Sheeley, N. R., Jr., & Boris, J. P. 1984, Solar Phys., 92, 1
* Eddy (1976) Eddy, J. A. 1976, Science, 192, 1189
* Howard & Labonte (1980) Howard, R., & Labonte, B. J. 1980, ApJ, 239, L33
* Howard & Labonte (1981) Howard, R., & Labonte, B. J. 1981, Solar Phys., 74, 131
* Howe et al. (2009) Howe, R., Christensen-Dalsgaard, J., Hill, F., Komm, R., Schou, J., & Thompson, M. J. 2009, ApJ, 701, L87
* Leighton (1964) Leighton, R. B. 1964, ApJ, 140, 1547
* Schrijver & Liu (2008) Schrijver, C. J., & Liu, Y. 2008, Solar Phys., 252, 19
* Sheeley (1967) Sheeley, N. R., Jr. 1967, ApJ, 147, 1106
* Sheeley (2008) Sheeley, N. R., Jr. 2008, ApJ, 680, 1553
* Sheeley et al. (1989) Sheeley, N. R., Jr., Wang, Y., & Devore, C. R. 1989, Solar Phys., 124, 1
* Svalgaard et al. (1978) Svalgaard, L., Duvall, T. L., Jr., & Scherrer, P. H. 1978, Solar Phys., 58, 225
* Ulrich & Boyden (2005) Ulrich, R. K., & Boyden, J. E. 2005, ApJ, 620, L123
* Wang et al. (2002) Wang, Y., Lean, J., & Sheeley, N. R., Jr. 2002, ApJ, 577, L53
* Wang et al. (1989) Wang, Y., Nash, A. G., & Sheeley, N. R., Jr. 1989, Science, 245, 712
* Wang et al. (2009) Wang, Y.-M., Robbrecht, E., & Sheeley, N. R., Jr. 2009, ApJ (in press)
* Wilson (1978) Wilson, O. C. 1978, ApJ, 226, 379
|
arxiv-papers
| 2010-05-20T20:32:35 |
2024-09-04T02:49:10.570537
|
{
"license": "Public Domain",
"authors": "N. R. Sheeley Jr",
"submitter": "Neil Sheeley Jr.",
"url": "https://arxiv.org/abs/1005.3834"
}
|
1005.3861
|
hep-th/??????
CECS-PHY-10/08
Kauffman Knot Invariant from
SO(N) or Sp(N) Chern-Simons theory
and the Potts Model
Marco Astorino***marco.astorino@gmail.com
Instituto de Física,
Pontificia Universidad Católica de Valparaíso
and
Centro de Estudios Científicos (CECS), Valdivia,
Chile
Abstract
The expectation value of Wilson loop operators in three-dimensional SO(N)
Chern-Simons gauge theory gives a known knot invariant: the Kauffman
polynomial. Here this result is derived, at the first order, via a simple
variational method. With the same procedure the skein relation for Sp(N) are
also obtained. Jones polynomial arises as special cases: Sp(2), SO(-2) and
SL(2,$\mathbb{R}$).
These results are confirmed and extended up to the second order, by means of
perturbation theory, which moreover let us establish a duality relation
between SO($\pm$N) and Sp($\mp$N) invariants.
A correspondence between the firsts orders in perturbation theory of SO(-2),
Sp(2) or SU(2) Chern-Simons quantum holonomies and the partition function of
the $Q=4$ Potts Model is built.
## 1 Introduction
In a milestone work [1] Witten realised that the expectation value of a Wilson
loop, computed with a three-dimensional Chern-Simons action measure, was a
knot invariant. This is due to the fact that the Wilson loops are observables
for Chern-Simons theories, having therefore diffeomorphism invariant
expectation values. More in general this feature stems from the property that
such a quantum field theory manifests general covariance, which in turn is a
consequence of the metric independent structure: any physical quantity
computed in this framework is a topological invariant.
In practise, for SU(N) Chern-Simons field theory, the resulting knot invariant
is the HOMFLY polynomial, which in particular specialises into the Jones
polynomial in the case of SU(2). These outcomes were derived through both
conformal field theory (as in [1]) or perturbative quantum field theory (see
for instance [2]). But a simpler heuristic derivation was proposed in [3] and
[4] (for reviews see also [5] and [6]), at least up to the first order in the
inverse coupling constant of the theory. It is based on a variational
approach: it studies the behaviour in the expectation value of the Wilson loop
when one performs small geometric deformation.
In the conformal field theory scheme similar results have been found in [9],
[10] and [11] for several other groups: SO(N), Sp(N), SU(n$|$m) and
OSp(m$|$2n).
It would be interesting to test whether the variational procedure, which is
expressly realised to reproduce the HOMFLY polynomial from SU(N) gauge theory,
may apply also in different contexts. In section 3 are studied the SO(N),
SL(N,$\mathbb{R}$) and Sp(N) cases.
The results obtained are moreover analysed in section 4 by means of the more
rigorous standard perturbation theory and extended up to the subsequent order,
the second.
Finally in section 5 we try to interpret these results from the statistical
mechanic point of view, trying to connect the holonomies’s first order
expansion to one of the more famous lattice statistical system: the Q-Potts
Model111We are referring to the standard two dimensional Potts model, not to
some variant with multiple Boltzmann weights, which in much literature are
misleading called in same way.; which at the moment remains unsolved apart for
its easiest personification when Q=2, the Ising model. We start (section 2)
introducing the notation and summarising the fundamental properties of Chern-
Simons theory and Kauffman polynomial that are useful in derivation of skein
relations.
## 2 Chern-Simons theory and Kauffman polynomial
Let’s consider a Chern-Simons theory for a gauge field connection one-form
$A=A^{a}_{\ \mu}(x)T^{a}dx^{\mu}$ valued in a generic semi-simple Lie algebra
$\mathfrak{g}$, with action:
$\mathcal{L_{CS}}[A]=\frac{k}{4\pi}\int_{\mathcal{M}^{3}}d^{3}x\
\frac{\epsilon^{\mu\nu\rho}}{2}\ \left(A^{a}_{\ \mu}\partial_{\nu}A^{a}_{\
\rho}-\frac{1}{3}A^{a}_{\ \mu}A^{b}_{\ \nu}A^{c}_{\ \rho}f^{abc}\right)$
where $\mathcal{M}^{3}$ is a compact three-dimensional manifold whose
coordinates are labelled by Greek letters ($\mu,\nu,\rho,...$); while the
internal group indices will be denoted by Latin letters ($a,b,c,...$). The Lie
algebra is spanned by generators $T^{a},T^{b},\dots$, obeying the commutation
relations $[T^{a},T^{b}]=if^{abc}T^{c}$ and normalised as follows: ${\rm
Tr}\;(T^{a}T^{b})=\frac{1}{2}\delta^{ab}$.
This action got several notable properties: $(i)$ it changes by $2\pi kn_{g}$
under a gauge transformation $A_{\mu}\rightsquigarrow
A^{\prime}_{\mu}=g^{-1}A_{\mu}g-ig^{-1}(\partial_{\mu}g)$ ($n_{g}$ is the
degree of the mapping $g:\mathcal{M}^{3}\rightarrow\mathcal{G}$); thus,
$\forall\ k\in\mathbb{Z}$, $\mathrm{exp}(i\mathcal{L_{CS}})$ is a complete
gauge invariant quantity that will play the rôle of the path integral measure.
$(ii)$ The curvature of the gauge field at the point $x\in\mathcal{M}^{3}$ is
given by:
$F^{a}_{\ \mu\nu}(x)=\frac{4\pi}{k}\epsilon_{\mu\nu\lambda}\
\frac{\delta\mathcal{L_{CS}}[A(x)]}{\delta A^{a}_{\ \lambda}(x)}$
We will interested in computing expectation values $\langle W(\gamma)\rangle$
for Wilson loops $W_{\gamma}[A]$ along closed paths $\gamma$, that in fact may
be thought as a knot on $\mathcal{M}^{3}$, defined as follows:
$\displaystyle W_{\gamma}[A]$ $\displaystyle=$ $\displaystyle{\rm
Tr}\;\Big{[}\mathrm{P}\
\mathrm{exp}\Big{(}i\oint_{\gamma}A_{\mu}dx^{\mu}\Big{)}\Big{]}$
$\displaystyle\langle W(\gamma)\rangle$ $\displaystyle=$
$\displaystyle\mathcal{Z}^{-1}\ \int\mathscr{D}A\ \mathrm{exp}\left(i\
\mathcal{L_{CS}}[A]\right)\ W_{\gamma}[A]$
In this notation $\gamma$ represents both common knots
$\gamma(t):I\rightarrow\mathcal{M}^{3}$ and n-component knots, also called
knot-links,
$\gamma(t_{1},t_{2},\dots,t_{n})=(\gamma_{1}(t_{1}),\gamma_{2}(t_{2}),\dots,\gamma_{n}(t_{n})):I_{1}\times
I_{2}\times\dots I_{n}\rightarrow\mathcal{M}^{3}$. In the latter case $\langle
W(\gamma)\rangle=\langle W(\gamma_{1})W(\gamma_{2})\dots
W(\gamma_{n})\rangle$. Without losing generality one may think the compact
interval $I_{i}=[0,1]$ and $\gamma(0)=\gamma(1)$ in order to have closed
paths. The fact that Chern-Simons action is independent of the particular
choice of a metric on the three-manifold suggests that the Wilson loop
expectation values may capture some invariant or topological characteristic of
the system’s geometry: either that of the knots or of the manifold itself.
Now we introduce the Kauffman polynomial which is a regular isotopy invariant
of knots and, if suitably normalised, becomes an ambient isotopy invariant.
Actually we will deal with its equivalent Dubrovnik version. To each knot-link
there is associated a finite Laurent polynomial $D_{K}=D_{K}(a,z)$ of two
variables with integer coefficients, such that if $K_{1}\sim K_{2}$, then
$D_{K_{1}}=D_{K_{2}}$ (while the reverse is not necessary true). The
polynomial can be constructed, as in [7] or [14], by the following
rules222Sometimes, as in [5], can be found a different normalisation for
$D_{K}$: $iii)^{\prime}\ \ D(\bigcirc)=1+\frac{a-a^{-1}}{z}$; in our notation
$1+\frac{a-a^{-1}}{z}$ will result the $\langle\bigcirc\rangle$ ’s
normalisation. (see figure 1 for notation, $\bigcirc$ stands for the unknotted
circle):
$\displaystyle i)$ $\displaystyle\qquad D(L_{+})-D(L_{-})=z\
[D(L_{0})-D(L_{\infty})]$ $\displaystyle ii)$ $\displaystyle\qquad
D(\hat{L}_{\pm})=a^{\pm}D(\hat{L}_{0})$ $\displaystyle iii)$
$\displaystyle\qquad D(\bigcirc)=1$
Figure 1: Different crossing configurations involved in the skein relations.
Dealing with unoriented links, arrows can be ignored because they carry no
sensitive information.
In $i)$ and $ii)$ the small diagrams $\\{L_{k}\\}_{k=\pm,0,\infty}$ stand for
larger link diagrams that differ only as indicated by the smaller ones.
Starting from any knot-links K and using recursively Reidemeister moves and
the skein relations (2) at each diagram’s crossing, one can obtain uniquely
its regular isotopy invariant $D_{K}(a,z)$. It is possible to normalise
$D_{K}$ by a factor that take into account also eventual contributions of
twists. For this purpose is used the writhe $w(K)=\sum_{p}\epsilon(p)$, where
$p$ runs over all crossing in $K$ and $\epsilon(L_{\pm})=\pm 1$ is the sign of
the type of crossing. So finally we are able to define a genuine ambient
isotopy invariant: the normalised Kauffman-Dubrovnik polynomial333While
$D_{k}$ is defined for unoriented knots, to calculate the writhe in $Y_{K}$
one needs to define an orientation. At the end the orientation does not affect
the result for knots but it affects the invariant polynomial in case of proper
links. Thus $Y_{K}$ is said to be defined for semi-oriented knot-links.:
$Y_{K}(a,z)=(a)^{-w(K)}D_{K}(a,z)$
## 3 Variational derivation of the skein relation
It’s well known (see [5] for details) that the Wilson loops satisfy the
following differential equations:
$\displaystyle\delta_{A}W_{\gamma}[A]$ $\displaystyle=$
$\displaystyle\frac{\delta W_{\gamma}[A]}{\delta A^{a}_{\ \mu}(x)}=i\ T^{a}\
dx^{\mu}\ W_{\gamma}[A]$ $\displaystyle\delta_{\gamma_{x}}W_{\gamma}[A]$
$\displaystyle=$ $\displaystyle iF^{a}_{\ \mu\nu}T^{a}\
dx^{\mu}dx^{\nu}W_{\gamma}[A]$
where $\delta_{\gamma_{x}}$ is the variation corresponding to an infinitesimal
deformation of the loop $\gamma$ in the neighbourhoods of a point $x$. It’s
then possible to compute this variation for an expectation value of a Wilson
line along a knotted path $\gamma$ and to use it to obtain a formula for the
switching identity $\langle W(\hat{L}_{+})\rangle-\langle
W(\hat{L}_{-})\rangle$ as444Proposition 17.4 and theorem 17.5 of [5]. follows:
$\delta_{\gamma_{x}}\langle W(\gamma)\rangle=-\frac{4\pi
i}{k}\frac{1}{\mathcal{Z}}\int\mathscr{D}A\ \mathrm{exp}\ \left(i\
\mathcal{L_{CS}}[A]\right)\
\left[\epsilon_{\mu\nu\lambda}dx^{\mu}dx^{\nu}dy^{\lambda}\right][\sum_{a}T^{a}T^{a}]W_{\gamma}[A]$
(3.1)
Note that studying the formal properties of this integral three assumptions
are always used: $i)$ the limits of differentiation and integration commute:
$\delta_{\gamma_{x}}\langle
W_{\gamma}[A]\rangle=\langle\delta_{\gamma_{x}}W_{\gamma}[A]\rangle$; $ii)$
integrating by parts it’s possible to discard the boundary term; $iii)$ the
existence of an appropriate functional measure on this moduli space.
From the previous equation one is able to write the switching identity
$\langle W(\hat{L}_{+})\rangle-\langle W(\hat{L}_{-})\rangle$. The quantity
$\left[\epsilon_{\mu\nu\lambda}dx^{\mu}dx^{\nu}dy^{\lambda}\right]$ is
dimensionless and, whether properly normalised, can be thought -1,0 or 1. Then
(3.1) has a standard interpretation (we follow [5]) if one calls the operator,
which in some sense enclose the loop’s small deformation,
$C=\sum_{a}T^{a}T^{a}$:
$\langle W(\hat{L}_{+})\rangle-\langle W(\hat{L}_{-})\rangle=-\frac{4\pi
i}{k}\langle C\ W(\gamma)\rangle$ (3.2)
Graphically $\langle C\ W(\gamma)\rangle$ is represented in the l.h.s of
figure’s 2 equation. Note that the sign is a convention which may be reversed
exchanging $\hat{L}_{+}\leftrightarrow\hat{L}_{-}$.
Till this point the whole model has been valid for a generic gauge group
$\mathcal{G}$. In particular was successfully used in the literature to
reproduce the Witten’s result for HOMFLY polynomials from the SU(N) group.
Instead in this paper we specialise our study to two particular algebras which
have simple Fierz identities: the ones associated to the orthogonal group
SO(N) and the symplectic group Sp(N), for a generic N.
### 3.1 SO(N) and Kauffman polynomial
Here the features of the algebra under consideration begin to play an
important rôle. In fact to evaluate the operator $C$ one needs to use the
Fierz identity; in particular we have for SO(N) in the fundamental
representation (in [8] Fierz identities are presented for almost all semi-
simple Lie groups):
$\sum_{a}(T^{a})^{i}_{\ j}(T^{a})^{k}_{\ l}=\frac{1}{4}\left(\delta^{i}_{\
l}\delta^{k}_{\ j}-\delta^{ik}\delta_{jl}\right)$
This expression in the Baxter’s abstract tensor notation (see [5]) reads as
the diagrammatic relation drawn in figure 2.
Figure 2: Abstract diagrammatic representation of Fierz identity for SO(N)
Hence, substituting in (3.2) the Fierz identity we have:
$\langle W(L_{+})\rangle-\langle W(L_{-})\rangle=-\frac{\pi
i}{k}\big{[}\langle W(L_{0})\rangle-\langle W(L_{\infty})\rangle\big{]}$ (3.3)
To get in touch with the known results, one has to take the limit of $k>>1$,
namely the analogous of the first order perturbation expansion, thus the
previous expression reduces to:
$\langle W(L_{+})\rangle-\langle W(L_{-})\rangle=\big{(}q-q^{-1}\big{)}\
\big{[}\langle W(L_{0})\rangle-\langle W(L_{\infty})\rangle\big{]}$
These are exactly the skein relations that are found by means of the original
Witten’s method based on conformal field theory arguments (see [9] and [10]),
once $q:=\mathrm{exp}(-\frac{\pi i}{2k})$ is defined555[10] uses a different
killing metric normalisation for the Lie algebra generators; in order to
compare with it one has to define a slightly different
$q:=\textrm{exp}(-\frac{\pi i}{k})$. [9] uses an inverse definition of the
writhe and of the crossing diagrams, so what they call $\alpha=a^{-1}$ and
their $q$ is our $q^{-1}$.. So is not difficult to see that $D_{K}=\langle
W(K)\rangle/\langle W(\bigcirc)\rangle$ fulfils the definition of Dubrovnik
polynomial (normalised as in [7] and [14]666Clearly if write-normalised by a
factor $a^{-w(K)}$ (where $w(L_{\pm})=\pm 1$) $D_{K}(a,z)$ became an ambient
isotopy invariant.), with $z=(q-q^{-1})$. The only thing that remains to fix
is the value of $a$ such that $\langle W(\hat{L}_{+})\rangle=a\langle
W(\hat{L}_{0})\rangle$. This can be done considering the closure of the path
in the skein relation (3.3), as shown in the figure below:
Figure 3: Diagrammatic closure of the SO(N) skein relation (3.3)
$\displaystyle\langle W(\hat{L}_{+})\rangle-\langle W(\hat{L}_{-})\rangle$
$\displaystyle=$ $\displaystyle-\frac{\pi i}{k}\big{[}\langle W(\bigcirc\
\hat{L}_{0})\rangle-\langle W(\hat{L}_{0})\rangle\big{]}$ $\displaystyle
a\langle W(\hat{L}_{0})\rangle-a^{-1}\langle W(\hat{L}_{0})\rangle$
$\displaystyle=$ $\displaystyle-\frac{\pi i}{k}\big{[}(N-1)\langle
W(\hat{L}_{0})\rangle\big{]}$ (3.4)
Solutions for (3.1) are $a=q^{N-1}$ or $a=-q^{1-N}$, which however gives rise
at an equivalent $D_{K}$ polynomials777Just redefine
$q\rightarrow\tilde{q}=-q^{-1}$ to verify the second root branch redundancy..
The factor $N$ comes from the diagrammatic tensor interpretation of the unknot
circle, that is $\delta_{i}^{\ i}=N$. It’s worth to observe that these
Dubrovnik-Kauffman polynomials $D_{K}(a=-q^{1-N},z=q-q^{-1})$ do not run out
all the original ones, but constitute a smaller subset depending on the fact
that $a$ assumes only discrete values depending on $N$ (which generally is
thought in $\mathbb{N}$).
The consistency check up to the $1/k$ order proposed in [3] is intrinsically
satisfied using the quadratic Casimir operator of
$\mathfrak{so}(N):\mathbbm{1}(N-1)/4$. Moreover the variational first order
approach, can be generalised to subsequents orders with the same arguments
presented in [12] and [13] for SU(N) groups. But we will prefer explore the
subsequent order of the expansion (see section 4) through a different method
based on the standard quantum field theory of perturbations.
Finally note that the original Jones polynomial
$a^{-w(K)}D_{K}(\bar{a}=-q^{3},\bar{z}=q-q^{-1})$ is not included in this sub-
class of Kauffman polynomial, unless choosing unconventionally $N=-2$ (once
the polynomial is analytic continued for all integers values of N).
Negative dimensions group theory is a powerful technique, first introduced by
Penrose, to calculate algebraic invariants (see [15], [16] and [17]). In that
case it relates the Casimirs and Young tableau of SO(-2) to the ones of Sp(2).
Some speculation about this possibility are done in the next subsection, while
a more rigorous treatment is done on section 4.
One may be puzzled not to come across Jones polynomial for the SO(3) group
which is locally isomorphic to SU(2) where this relation holds. The reason for
this mismatch is based on the fact that in this context, more than groups
similarities, the Lie algebras invariants play a key rôle.
Actually, as also for SL(2,$\mathbb{R}$) generators the same SU(2) Fierz
identity for the $C$ operator holds, Jones polynomial can be recovered with
the same procedure of [3]. It is not surprising because
$\mathfrak{sl}(2,\mathbb{R})$ is the real split form of the $A_{1}$ algebra
(known also as the $\mathfrak{sl}(2,\mathbb{C})$ algebra by an abuse of
notation), while $\mathfrak{su}(2)$ is the real compact one.
### 3.2 Sp(N) skein relations and Jones Polynomial for Sp(2)
In this section we consider the Symplectic group Sp(N), for even N; apart from
the relation with SO(-N) it is an interesting case for itself. Its Fierz
identity (see again [8]) for the generators in the fundamental representation
is:
$\sum_{a}(T_{a})^{i}_{\ j}(T_{a})^{k}_{\ l}=\frac{1}{4}\left(\delta^{i}_{\
l}\delta^{k}_{\ j}+f^{ik}f_{jl}\right)$
where $f^{ij}=-f^{ji}\ ,\ f^{ij}f_{jk}=\delta^{i}_{\ k}$. As the fundamental
representation of this group is pseudoreal, unlike SO(N), the orientation
should not be neglected as it is shown in figure 4.888In [10] another approach
(which has the advantage that leaves the Wilson lines unoriented) is also
presented, but not preferred as requires the specific choice of a ”time”
direction, which breaks the topological invariance because it is no longer
possible to freely rotate the Wilson lines. Plugging this Fierz identity for
Sp(N) into eq. (3.2) one fits the same skein relation of [10] which is
obtained by a totally different approach.999We refer to the one drawn in
figure 17 of [10]
Figure 4: Fierz identity for Sp(N), dots represent points where orientations
of the line change.
There is a particular case where those computation are easily101010Even
without the oriented diagram notation which is unnecessary heavy for Sp(2).
One might work, in a complete compatible way, with the arrowed diagrams but
paying the price of redefining appropriate oriented Reidemeister moves and
oriented Kauffman state bracket as described in cap $6^{0}$ of [5] and [10].
carried on till get its knot invariant: N=2, just the one suspected to be
related to the Jones polynomial, as we saw in section 3.1. In fact for Sp(2)
the antisymmetric matrix $f^{ij}$ may be straight interpreted, without losing
generality, as the Levi-Civita tensor $\epsilon^{ij}$ and its inverse
$f_{ij}=-\epsilon_{ij}$ . Hence the algebraic (eq. (3.5)) and diagrammatic
(fig. 5) representations of the C operator appear respectively as follows:
$\displaystyle\sum_{a}(T_{a})^{i}_{\ j}(T_{a})^{k}_{\
l}=\frac{1}{4}\left(\delta^{i}_{\ l}\delta^{k}_{\
j}-\epsilon^{ik}\epsilon_{jl}\right)=\frac{1}{4}\left(2\delta^{i}_{\
l}\delta^{k}_{\ j}-\delta^{i}_{\ j}\delta^{k}_{\ l}\right)$ (3.5)
Figure 5: Diagrammatic representation of Fierz identity for Sp(2)
Now substituting the Fierz identity for Sp(2) into (3.2) we have:
$\displaystyle\langle W(L_{+})\rangle-\langle W(L_{-})\rangle$
$\displaystyle=$ $\displaystyle-\frac{2\pi i}{k}\langle
W(L_{0})\rangle+\frac{\pi i}{2k}\langle W(L_{+})\rangle+\frac{\pi
i}{2k}\langle W(L_{-})\rangle$ $\displaystyle\left(1-\frac{\pi
i}{2k}\right)\langle W(L_{+})\rangle$ $\displaystyle-$
$\displaystyle\left(1+\frac{\pi i}{2k}\right)\langle W(L_{-})=-\frac{2\pi
i}{k}\langle W(L_{0})\rangle$ $\displaystyle q\langle
W(L_{+})\rangle-q^{-1}\langle W(L_{-})\rangle$ $\displaystyle=$
$\displaystyle\tilde{z}\langle W(L_{0})\rangle$
Where $q$ is the same of section 3.1, while it is defined
$\tilde{z}:=-\frac{2\pi i}{k}=x-x^{-1}$ if we call $x:=\textrm{exp}(-\frac{\pi
i}{k})$. Again we are considering at this stage $k>>1$, i.e these equalities
hold up to first order in the inverse coupling constant of the theory111111The
first order consistency check proposed in [3] is trivially satisfied using,
this time, the quadratic Casimir operator of $\mathfrak{sp}(2):\
3\mathbbm{1}/4$. Closing the path in the previous skein relation as done for
SO(N) we will be able to get a constraint that reduces one variable
dependence:
$\displaystyle q\langle W(\hat{L}_{+})\rangle-q^{-1}\langle
W(\hat{L}_{-})\rangle$ $\displaystyle=$ $\displaystyle\tilde{z}\langle
W(\hat{L}_{0}\ \bigcirc)\rangle$ $\displaystyle aq\langle
W(\hat{L}_{0})\rangle-a^{-1}q^{-1}\langle W(\hat{L}_{0})\rangle$
$\displaystyle=$ $\displaystyle x^{2}-x^{-2}\langle W(\hat{L}_{0})\rangle$
$\displaystyle\Longrightarrow\qquad aq$ $\displaystyle=$ $\displaystyle x^{2}$
As before the second root $aq=-x^{-2}$ leads exactly to the same results. So
at large values of $k$ for a normalised (to be a) expectation value
$P(K)=a^{-w(K)}\langle W(K)\rangle/\langle W(\bigcirc)\rangle$ the original
one variable Jones polynomial follows directly:
$x^{2}P(L_{+})-x^{-2}P(L_{-})=(x-x^{-1})P(L_{0})$
So actually the estimation suggested by negative dimension group theory seems
to work reliably. As it’s here proved the Sp(2) Chern-Simons expectation
values of a Wilson knot-link gives the Jones polynomial invariant for the same
link.
## 4 Perturbative Quantum Field approach
It’s worth analysing the heuristic previous section’s results in a more
carefully way. We opt for the standard quantum field theory of perturbation as
developed for the SU(N) group in [2], which maybe got the disadvantage of
being less qualitative from a geometrical point of view but got the benefit of
being more analytically quantitative. The fact of being, in principle, a
different approach also adds some guaranties on the consistency of the check.
Not least this method let us push the expansion, in the inverse coupling
constant $k$, to one order further.
Note that for this procedure a framing of the knot is needed; in this paper is
always used the _vertical frame_ defined as the one that got linking number
equal to the writhe of the knot $\varphi_{f}(K)=w(K)$. Framed knots can be
thought as bands, so in this picture a writhe for a knot represents a band
twist. As Kauffman polynomial are regular isotopy invariant, twisted bands are
the most suitable objects to be described with. The expectation value for the
Wilson loop computed along a vertical framed, m-component
($C_{1},C_{2},...,C_{m}$) knot-link $K$ in a Chern-Simons theory for a generic
semisimple group $\mathcal{G}$ is given at second order by:
$\displaystyle\langle W(K)\rangle$ $\displaystyle=$
$\displaystyle\Big{(}\prod_{k=1}^{m}\textrm{dim}\
T_{k}\Big{)}\Big{\\{}1-i\Big{(}\frac{2\pi}{k}\Big{)}\sum_{k=1}^{m}c_{2}(T_{k})\varphi_{f}(C_{k})$
$\displaystyle-$ $\displaystyle\Big{(}\frac{2\pi}{k}\Big{)}^{2}\
\sum_{k=i}^{m}\Big{[}\frac{1}{2}c_{2}^{2}(T_{k})\varphi^{2}_{f}(C_{k})-c_{v}c_{2}(T_{k})\rho(C_{k})\Big{]}$
$\displaystyle-$
$\displaystyle\Big{(}\frac{2\pi}{k}\Big{)}^{2}\sum_{k\neq\ell}c_{2}(T_{k})c_{2}(T_{\ell})\Big{[}\varphi_{f}(C_{k})\varphi_{f}(C_{\ell})+\frac{\chi^{2}(C_{k},C_{\ell})}{\textrm{dim}\
\mathcal{G}}\Big{]}+O\Big{(}\frac{1}{k^{3}}\Big{)}\Big{\\}}$
where $T$ stands for the fundamental representation, $\chi(C_{k},C_{\ell})$ is
the Gauss linking number between the two curves $C_{k}$ and $C_{\ell}$,
$\big{(}c_{2}(T)\big{)}_{i}^{\ j}=\sum_{a}(T^{a})_{i}^{\ k}\ (T^{a})_{k}^{\
j}$ is the quadratic Casimir in the fundamental representation, $c_{v}$ the
quadratic Casimir in the adjoint representation, $\rho(C)$ is an ambient
isotopy invariant characteristic of the knot under consideration. $\rho(C)$
represents the second coefficient of the Alexander-Conway polynomial and is
related with Arf- and Casson-invariants; in practise it is not easy to compute
apart from small knots.
Our aim is now, with the help of (4), to find the value of $a$ appearing in
(2-$ii$) in terms of its expansion in $(2\pi/k)$. The effect of changing the
frame of a link component $C_{i}$ by $\Delta\varphi_{f}(C_{i})=\Delta
w(C_{i})=\pm 1$ (or adding a twist in the band picture) reflects in the entire
Wilson loop expectantion value by:
$\langle W(K_{\varphi\pm 1})\rangle=\alpha^{(\pm)}\langle
W(K_{\varphi})\rangle$ $\alpha^{(\pm)}=1\mp
i\Big{(}\frac{2\pi}{k}\Big{)}c_{2}(T)-\frac{1}{2}\Big{(}\frac{2\pi}{k}\Big{)}^{2}c_{2}^{2}(T)+O\Big{(}\frac{1}{k^{3}}\Big{)}$
(4.2)
So we find $a^{\pm 1}=\alpha^{(\pm)}$, taking into account $D_{K}=\langle
W(K)\rangle/\langle W(\bigcirc)\rangle$ as previously defined on section 3.1.
While (2-$iii$) is trivially satisfied, is possible to extract the value of
$z$ from (2-$i$), for instance applying it to the Hopf-link $\mathcal{HL}$.
Figure 6: Skein relation 2-$i)$ applied to the the upper $\mathcal{HL}$
crossing
That is closing the skein relation (2)-$i$ as shown above one gets the
following expression:
$D_{\mathcal{HL}}-D_{\bigcirc\bigcirc}=z(a-a^{-1})D_{\bigcirc}$
written in term of relatively easy objects that can be computed directly from
(4), using as in [2], $\rho(\bigcirc)=-1/12$:
$\displaystyle D_{\bigcirc\bigcirc}$ $\displaystyle=$ $\displaystyle
N\left[1-\frac{1}{12}\left(\frac{2\pi}{k}\right)^{2}c_{v}c_{2}(T)+O\left(\frac{1}{k^{3}}\right)\right]$
(4.3) $\displaystyle D_{\mathcal{HL}}$ $\displaystyle=$ $\displaystyle
N\left[1-\frac{1}{12}\left(\frac{2\pi}{k}\right)^{2}c_{v}c_{2}(T)-\left(\frac{2\pi}{k}\right)^{2}c_{2}^{2}(T)\frac{2}{\textrm{dim}\mathcal{G}}+O\left(\frac{1}{k^{3}}\right)\right]$
An alternative way to find $z$ is imposing the equality between Kauffman
$D_{K}(a,z)$ polynomials obtained from the skein relations (2) with the
expansion of $\langle W(K)\rangle/\langle W(\bigcirc)\rangle$ coming from (4).
But this could be done just for the few simple knots where $\rho(K)$ can be
calculated, so may be here regarded as a self-consistency check.
That’s the point where the algebraic properties of the gauge groups come out;
for the groups we are interested in, they are summarised in the following
table:
| dim $\mathcal{G}$ | dim $T$ | $c_{2}$ | $c_{v}$
---|---|---|---|---
SO(N) | $N(N-1)/2$ | $N$ | $(N-1)/4$ | $(N-2)/2$
Sp(N) | $N(N+1)/2$ | $N$ | $(N+1)/4$ | $(N+2)/2$
SU(N) | $N^{2}-1$ | $N$ | $(N^{2}-1)/2N$ | $N$
hence, from (4.2), we get respectively for SO(N) and Sp(N) the following
values for $a$
$\displaystyle a_{SO(N)}$ $\displaystyle=$ $\displaystyle
1-i\left(\frac{2\pi}{k}\right)\frac{N-1}{4}-\frac{1}{2}\left(\frac{2\pi}{k}\right)^{2}\left(\frac{N-1}{4}\right)^{2}+O\left(\frac{1}{k^{3}}\right)$
(4.4) $\displaystyle a_{Sp(N)}$ $\displaystyle=$ $\displaystyle
1-i\left(\frac{2\pi}{k}\right)\frac{N+1}{4}-\frac{1}{2}\Big{(}\frac{2\pi}{k}\Big{)}^{2}\left(\frac{N+1}{4}\right)^{2}+O\left(\frac{1}{k^{3}}\right)$
while for both orthogonal and symplectic groups the value found for z is:
$z=-\frac{i\pi}{k}+O\Big{(}\frac{1}{k^{3}}\Big{)}$ (4.5)
These results are consistent with the ones found in the previous section by
means of the variational method both for SO(N) and Sp(2). Moreover (4.4) and
(4.5) extend the series expansion in $2\pi/k$ up the second order. The fact
that $z$ has not the quadratic contribution could be guessed from the very
beginning because of the peculiar property of the Chern-Simons Lagrangian: the
inversion symmetry. This implies that a change in the sign of the coupling
constant $k$ is compensated by the inversion of the orientating of the
manifold. When a knot $K$ is embedded in $\mathcal{M}^{3}$ the change of
orientation of the manifold corresponds to a $\pi$ rotation or its mirror
image $\tilde{K}$, so $\langle W(K)\rangle\big{|}_{k}=\langle
W(\tilde{K})\rangle\big{|}_{-k}$. On the other hand from skein relations (2)
is easy to see that $D_{K}(a,z)=D_{\tilde{K}}(a^{-1},-z)$; combining it with
the inversion symmetry one gets some restriction on the k-functional
dependence of the variables $a$ and $z$:
$a(k)=a^{-1}(-k)\qquad z(k)=-z(-k)$ (4.6)
So even powers of $k$ were not expected in the $z$ expansion; as one can see
(4.4) and (4.5) fulfil the constraints (4.6). The easiest functions that are
compatible with the series expansions (4.4)-(4.5), their restrictions (4.6)
and the samples (4.3) are:
$a=\textrm{exp}\left[-i\frac{2\pi}{k}c_{2}(T)\right]\qquad z=-2\ i\
\textrm{sin}\left(\frac{\pi}{2k}\right)$
Furthermore observe that in the groups table there is a value of N for whom
two lines match: for $N=2$ all the values for Sp(2) and SU(2) coincide. So the
expectation value of a Wilson loop along a generic knot K agrees in both
cases. This special point is the one where the HOMFLY and Kauffman polynomials
overlap to give the Jones polynomial. This is exactly the same result we have
found with the variational approach in section 3.2, but now extended to the
second order. Another interesting feature that can be read from the table is
the analogy between the quantities of SO(-N) and Sp(N), in particular one can
note in (4) as Wilson loop expectation values of a SO(-N)-Chern-Simons theory
for a knot $K$ correspond to the ones of its mirror image $\tilde{K}$ for a
Sp(N)-CS theory:
$\langle W(K)\rangle\Big{|}_{SO(-N)}=(-1)^{m}\ \langle
W(\tilde{K})\rangle\Big{|}_{Sp(N)}$ (4.7)
For odd-multicomponent knots-links the correspondence hold up to a global
sign, where m is the number of components. The mirror image $\tilde{K}$ is
needed in order to have opposite the chirality in framing that compensate a
sign in the odd terms expansion. In terms of Dubrovnik polynomial (4.7) became
$D_{K}|_{SO(-N)}=D_{\tilde{K}}|_{Sp(N)}$, at least for proper knots. So again
what suggested by the variational approach can be coherently recovered and
extended by the perturbative one.
The ambient isotopic Dubrovnik-Kauffman polynomial is obtained, as usual, from
the regular one thanks to a writhe normalisation: $a^{-w(K)}D_{K}$.
Another remarkable feature of the variational and perturbative approaches is
that allow us to generalise at once the present treatment also to the non-
compact groups such as SO(m,n), which are the more interesting ones for
describe general relativity in 2+1 dimensions by the Chern-Simons theory.
Although from a classical point of view locally isomorphic groups represent
the same gauge theory, we have seen as at the quantum level expectation values
even of simple knots differ. Thus in case one wants to take advance of the
Chern-Simons formalism to study quantum properties of gravity he will have to
consider the issue of which is the ”good” group election. Actually the values
of the fundamental quantities as the Casimirs $c_{2},c_{v}$, the group’s
dimension dim$\mathcal{G}$ and the fundamental representation dimension
dim($T$) of SO(m,n) are not different from the SO(N) ones, whenever $m+n=N$.
Hence the topological quantity $\langle W(K)\rangle$ (4) is not affected by
the signature change of the Cartan-Killing metric121212Of course a gauge
description of gravity needs a further step: also a signature’s change in the
space-time coordinates, this is more problematic because all the treatment
done in this paper is for compact manifolds $\mathcal{M}^{3}$.. Up the author
knowledge invariant knot polynomials for SO(m,n) groups are not found by means
of any other methods; could be interesting to verify it with the help of more
rigorous mathematical tools such as quantum groups. Moreover the SO(m,n)
Chern-Simons theory got a richer structure than the SU(N) one. In fact others
non-equivalent Chern-Simons Lagrangian can be built from their Chern’s
characteristic classes apart from the Pontryagin; for instance is possible to
use also the Euler or Nieh-Yan topological invariants (see [23] for a review).
The expectation values of knotted Wilson loops weighted by this Chern-Simons
density remains a topological invariant, but possibly of different kind.
## 5 Correspondence with the Potts Model
In this section we try to build a bridge between the previous results about
first order expectation values of quantum holonomies along a knotted path and
some statistical system such as the Potts Model. Of course it is clear that an
exact equality can not hold since the Chern-Simons observables are knot
invariants while the Potts partition functions are not. Nevertheless something
can be said, but at the price of renouncing to the knot topological
invariance. First let us remind some fundamental facts about the Potts model
that we will be used afterwords.
It is found in [19] that the partition function of the Q-Potts Model of a
statistical lattice represented by a graph G is the _Potts state bracket_
$\\{K(G)\\}$ of the knot-link K dual to the graph G. That’s because this state
bracket expansion coincides exactly with the dichromatic polynomial, or the
Tutte polynomial, of the graph G. We remember the definition of the Potts
state bracket:
$\displaystyle i)$
$\displaystyle\qquad\\{\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{A.eps}}\\}=Q^{-1/2}v\\{\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{C.eps}}\\}+\\{\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{D.eps}}\\}$
(5.1) $\displaystyle ii)$ $\displaystyle\qquad\\{\bigcirc\ K\\}=Q^{1/2}\
\\{K\\}$ $\displaystyle iii)$ $\displaystyle\qquad\\{\bigcirc\\}=Q^{1/2}$
To be more precise for any alternating knot or link K it is possible to
construct a graph lattice G(K) checkerboard shading its planar diagram and
assigning to each shadow a vertex and for each crossing a bound, as shown in
figure 7. Vice-versa for any two dimensional graph G one can associate its
dual knot K(G). Note that this is a one-to-one131313When the white region is
left outside. mapping between planar graphs and alternate knots and note that
any knot got its alternate representative, that is can be drawn as an
alternate planar diagram.
Figure 7: K(G) $\longleftrightarrow$ shading of K(G) $\longleftrightarrow$
emerging of lattice graph G inside K $\longleftrightarrow$ G(K)
Thus the Q-Potts partition function for a certain statistical lattice
$P_{G}(Q,t)$ is given by the dichromatic polynomial $Z_{G}(Q,v)$ of its graph
G (whenever $v=e^{J/kt}-1$) or by the Potts state bracket of its associated
knot $\\{K\\}$ as follows:
$P_{G(K)}(Q,t)\ =\
\sum_{\sigma}e^{\frac{J}{k_{B}t}\sum_{<i,j>}\delta(\sigma_{i},\sigma_{j})}\ =\
Q^{V/2}\\{K\\}(Q,v=e^{J/k_{B}t}-1)\ ,$ (5.2)
where $V$ is the number of vertex of the graph (i.e. the number of the
lattice’s sites or rather the number of shaded region of the knot), $t$ is the
temperature, $k_{B}$ the Boltzmann’s constant, $\sigma_{n}$ is one of the Q
possible states of the nth vertex and $J=\pm 1$ according to the ferromagnetic
or anti-ferromagnetic case.
### 5.1 SO(-2) & Sp(2) Holonomies and Q=4 Potts Model
First we consider a special case, that is when the Kauffman polynomial reduces
to the Kauffman state bracket $[K](q)$ (or to the Jones Polynomial whether
writhe normalised), which occurs for the SO(-2), Sp(2) 141414Correlated by
(4.7) or SU(2) Chern-Simons theory, as we have seen in section 3.2 and 4:
$\langle W(K)\rangle(z=q-q^{-1},a=-q^{3})=[K](q)\ .$
Then we perform a shift in the q-variable: $[K]\rightsquigarrow q^{c(K)}[K]$,
where $c(K)$ is the number of crossing in the knot K diagram. This shift is
the point where regular isotopical invariance of the Kauffman polynomial is
broken. So focusing just on the first order approximation, one gets the
following bracket $q^{c(K)}[K](q)\big{|}_{1^{st}-order}:=\ \ll
K\gg(1-i\pi/2k)$:
$\displaystyle i)$
$\displaystyle\quad\ll\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{A.eps}}\gg\
=q^{2}\ll\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{C.eps}}\gg+\ll\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{D.eps}}\gg\
=\Big{[}1-\frac{i\pi}{k}+O\Big{(}\frac{1}{k^{2}}\Big{)}\Big{]}\
\ll\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{C.eps}}\gg+\ll\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{D.eps}}\gg$
(5.3) $\displaystyle ii)$ $\displaystyle\quad\ll\bigcirc\ K\gg\
=N+O\Big{(}\frac{1}{k^{2}}\Big{)}\ \ll K\gg$ $\displaystyle iii)$
$\displaystyle\quad\ll\bigcirc\gg\ =N+O\Big{(}\frac{1}{k^{2}}\Big{)}$
The analogy with the Potts state bracket (5.1) is now evident:
$\\{K\\}(Q,v)=\ \ll K\gg(\pm v^{1/2}Q^{-1/4}).$ (5.4)
Let now concentrate on the SO(-2) case, such that once the q-shift is
reabsorbed one recovers knot invariance, so $Q=N^{2}=4$. Using (5.2) and (5.4)
it is easy to see that $-2^{V}\ll K\gg$ represents the Q=4 Potts partition
function for the lattice graph associated to the knot K. In terms of the first
order Wilson loops expansion it reads:
$P_{G(K)}=Q^{V/2}\\{K\\}=N^{V}\ q^{c(K)}\ \langle
W(K)\rangle\Big{|}_{1^{st}-\mathrm{order}}$ (5.5)
An example may get things clearer: consider a 2x2 lattice graph G of figure 7
and its dual knot-link $K(G)$ (with $V=4$). From skein relations (5.1) (or
equally from the deletion-contraction rule that define the dichromatic
polynomial $Z_{G}(4,v)$) one gets the Q=4 Potts partition function for the
graph $G(K)$:
$Z_{G}(4,v)=4^{V/2}\\{K\\}=4^{2}(4^{2}+4\cdot 4v+6v^{2}+4\cdot
4^{-1}v^{3}+4^{-1}v^{4})$ (5.6)
while from the skein relations (5.3) one get the expectation value of the
holonomy along the knot $K(G)$, up to $O(1/k^{2})$:
$-2^{V}\ q^{c(K)}\ \langle
W(K)\rangle\Big{|}_{1^{st}-\mathrm{ord}}=2^{4}\big{(}1-\frac{i2\pi}{k}\big{)}\Big{[}16\big{(}1+\frac{i2\pi}{k}\big{)}-32\big{(}1+\frac{i\pi}{k}\big{)}+24-8\big{(}1-\frac{i\pi}{k}\big{)}+4\big{(}1-\frac{i2\pi}{k}\big{)}\Big{]}$
It’s easy to see that (5.5) is fullfilled imposing $v=-2+i2\pi/k$ in (5.6). So
the first order expectation value of the Wilson loop along a knotted path K
for a SO(-2)/Sp(2) Chern Simons theory can be extracted from the partition
function of a Q=4 Potts model of a lattice graph $G(K)$ dual to the knot $K$,
and vice-versa. This correspondence works well for any two dimensional lattice
graph, not just for regular ones like the sample presented in figure 7.
Even thought $\langle W(K)\rangle\big{|}_{1^{st}-\mathrm{order}}$ and
$P_{G}(K)$ are not exactly the same they share some features, for instance
their zeroes. So $\langle W(K)\rangle\big{|}_{1^{st}}$’s zeros can be
interpreted as the Fisher zeros of the statistical lattice associated to $K$,
which encode many important physical properties of the system. Also the
critical temperature $t_{c}$ (when the statistical system acquires conformal
invariance) of the Potts model can be easily read: In the knot formalism it
occurs where $\langle W(K)\rangle=\langle W(\tilde{K})\rangle$, that is when
$1-i\pi/k=1$, so in the limit $k\rightarrow\infty$, which means
$t_{c}=\frac{J}{k_{B}}\frac{1}{\textrm{ln}(\sqrt{Q}+1)}$.
It’s worth remark at this point that the SO(-2)/Sp(2) group (or even SU(2))
gives rise to the Jones polynomial too. This polynomial (at the non-
perturbative level) is known to describe the partition function of a
particular kind of Potts model with two Boltzmann factor, which is of
different kind respect to the standard Potts model considered here (see [14]
and [20]).
The correspondence holds also at the following orders of the perturbative
expansion, basically in the same way it works at the first order. For instance
one can obtain $\langle W(K)\rangle\big{|}_{2^{sd}-order}$ from the Q=4 Potts
partition function identifying $v$ and $Q$ as follows:
$\displaystyle v$ $\displaystyle\leftrightsquigarrow$
$\displaystyle-2\left[1-\frac{i\pi}{k}-\left(\frac{\pi}{k}\right)^{2}+O\left(\frac{1}{k^{3}}\right)\right]$
$\displaystyle Q^{\frac{1}{2}}$ $\displaystyle\leftrightsquigarrow$
$\displaystyle-2\left[1-\frac{1}{2}\left(\frac{\pi}{k}\right)^{2}+O\left(\frac{1}{k^{3}}\right)\right]$
The simple relation between $Q$ and $N$ is now spoiled and moreover this fact
makes the analogy between the two models purely formal because choosing a
particular Q imply fixing at the same time the temperature to a constant
value.
### 5.2 Sp(N) holonomies and Q-Potts Model
We would like to do something similar to previous subsection, but for generic
$N$. Now that procedure is less direct because the Kauffman polynomial can not
be cast in a simple form such as the state bracket [K]. To connect the two
theories, in particular to give the Q-Potts partition function a similar
structure to the Dubrovnik polynomial one, we can introduce a new bracket
polynomial $\|K\|(Q,v)$ defined by the following skein relations:
$\displaystyle i)$
$\displaystyle\quad\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{A.eps}}\|-\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{B.eps}}\|\
=(Q^{-1/4}v^{1/2}-Q^{1/4}v^{-1/2})\big{[}\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{C.eps}}\|-\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{D.eps}}\|\big{]}$
(5.7) $\displaystyle ii)$
$\displaystyle\quad\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{Rcurl.eps}}\|=(Q^{1/4}v^{1/2}+Q^{1/4}v^{-1/2})\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{Arc.eps}}\|\
\
,\quad\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{Lcurl.eps}}\|=(Q^{-1/4}v^{1/2}+Q^{3/4}v^{-1/2})\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{Arc.eps}}\|$
$\displaystyle iii)$ $\displaystyle\quad\|\bigcirc\|=Q^{1/2}$ $\displaystyle
iv)$
$\displaystyle\quad\|\raisebox{-0.31pt}{\includegraphics[width=8.5359pt]{2cross+.eps}}\|=(Q^{-1/2}v+Q^{1/2}+Q^{1/2}v^{-1})\
\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{C.eps}}\|+\
\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{D.eps}}\|$
The Q-Potts partition function, in character of the dichromatic polynomial
$Z_{G(K)}(Q,v)$, has the following form in term of $\|K\|$:
$Z_{G}(Q,T)=Q^{V/2}[Q^{-1/4}v^{1/2}]^{c(K)}\|K\|.$
Even in this form $\|K\|$ is not a isotopical invariant of the knots, as
$\langle W(K)\rangle$ because the two coefficients in (5.7-$ii$) are not
reciprocal and (5.7-$iv$) does not satisfy the second Reidemeister move.
However there is a point where both (5.7-$ii,iv$) becomes invariant, that is
for $v=(-Q\pm\sqrt{Q^{2}-4Q})/2$. This value of the temperature is exactly the
one that relates the Potts model to the Khovanov homology [21]. Comparing the
$\|K\|(Q,v)$ bracket with the first order expectation value of the holonomy
$\langle W(K)\rangle\big{|}_{1^{st-ord}}$ one has to impose $Q=N^{2}$ and
$v=N(1-i\pi/k)$. So the $\|K\|(N,k)$ invariance occurs, in terms of the Chern-
Simons coupling constant $k$ and the fundamental representation dimension $N$,
just for $N=-2$, i.e the previous case we analysed in section 5.1.
Therefore for a generic $Q=N^{2}\neq 4$ is not possible to pass from the Potts
partition function to the first order Wilson loop expectation value as we did
for the $SO(N)/Sp(2)$ case. What can be done at most is define a generic
bracket polynomial which include both $P_{G}$ and $\langle W(K)\rangle$ and
specialises to one or the another for some values of its variables. This is
done in appendix A.
## 6 Comments and Conclusions
In this paper is analysed the relation between expectation values of Wilson
loop in three-dimensional SO(N) Chern-Simons field theory and an isotopic
invariant of knots, the Kauffman polynomial. This equivalence is achieved in a
simple intuitive knot variational approach borrowed by [3]’s and [5]’s scheme
which was elaborated for obtaining the Witten result: HOMFLY polynomial from
the SU(N) gauge group. The key point of this construction is based on the
existence of a Fierz identity for the infinitesimal generators of the group in
certain representations. With precisely the same interpretation of the
expectation value’s path variations and no other extra assumptions respect to
the original work, here we exactly get the conformal field theory known result
for SO(N): Kauffman polynomial. It suggests that the easy variational knot
approach, expressly built for SU(N), works well also for different gauge group
theories as SO(N). So its heuristic geometrical assumptions are endorsed.
Convinced of all that and encouraged by negative dimension group theory
suggestion we explored also the Sp(N) group getting the exact skein relation.
In particular in the simple Sp(2) case we are able to find its isotopic
invariant: the original Jones Polynomial.
Furthermore to enforce and extend those results, an independent procedure has
been performed, the quantum field theory method can not only full recover the
variational approach but also: improve its outcomes precision of an order of
magnitude, extend to groups with semi-definite Cartan-Killing metric as well
Sp(N) with $N\neq 2$ and most of all prove, up to $O(1/k^{3})$, the
correspondence between isotopy invariant polynomials from SO(N) and Sp(-N)
Chern-Simons theories.
To sum up, these procedures give for SU(N), SO(N)/Sp(N) and Sp(2) the famous
HOMFLY, Kauffman and Jones polynomials respectively. Hence they may be used
for other groups or representations to find new link invariants, both based on
skein relations or not. This could give new insights into knots theory, which
is still looking for a link invariant able to distinguish conclusively knots
isotopic equivalence.
From a physical point of view it’s interesting to note that not only the Jones
polynomial, at non perturbative level, correspond to the partition function of
the Potts model with two Boltzmann weight factors, but also its first order
perturbation expansion, in the realm of the Chern-Simons theory, gives the
standard Q=4 Potts partition function (and vice-versa). Moreover the
connection between the quantum holonomies of Sp(2) Chern-Simons theory and the
$Q=4$ Potts partition function opens the possibility to relate apparently
disconnected physical systems. This is actually the main motivation of the
author. In fact, since [22], it is well known that Sp(2)$\times$Sp(2) Chern-
Simons theory describes 2+1 gravity with negative cosmological constant.
Furthermore the first terms in the Kauffman bracket expansion give states of
3+1 quantum gravity in the loop representation [6]. This feature of knot
theory may represent the tip of an iceberg that links discrete statistical
models with the expectation value of holonomies of gravitational theories.
Work in this direction is in progress.
## Acknowledgements
I would like to thank Louis Kauffman, Roberto Troncoso, Steven Willison and
Jorge Zanelli for fruitful discussions.
The Centro de Estudios Científicos (CECS) is funded by the Chilean Government
through the Millennium Science Initiative and the Centers of Excellence Base
Financing Program of Conicyt and by Conicyt grant ”Southern Theoretical
Physics Laboratory” ACT-91. CECS is also supported by a group of private
companies which at present includes Antofagasta Minerals, Arauco, Empresas
CMPC, Indura, Naviera Ultragas and Telefónica del Sur.
## Appendix A General Potts-Dubrovnik polynomial $M_{K}$
Define the following bracket polynomial $M_{K}(a,b,c,d,z)$:
$\displaystyle i)$ $\displaystyle\quad
M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{A.eps}})-M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{B.eps}})\
=z\big{[}M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{C.eps}})-M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{D.eps}})\big{]}$
$\displaystyle ii)$ $\displaystyle\quad
M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{Rcurl.eps}})=a\
M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{Arc.eps}})\quad,\quad
M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{Lcurl.eps}})=b\
M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{Arc.eps}})$
$\displaystyle iii)$ $\displaystyle\quad M(\bigcirc)=d$ $\displaystyle iv)$
$\displaystyle\quad
M(\raisebox{-0.31pt}{\includegraphics[width=8.5359pt]{2cross+.eps}})=c\
M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{C.eps}})+\
M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{D.eps}})$
$M_{K}$ reduces to the Kauffman-Dubrovnik polynomial when $b=a^{-1},\ c=0,\
d=1$; while to $\langle W(K)\rangle$ when $d=(a-a^{-1})/z+1$. So for those
values of the variables it is an invariant of regular isotopy. But the Potts
partition function is not invariant so this latter has $b\neq a^{-1}$ and $c$
switched on, as can see in the following table, where two different
specialisations of the $M_{K}$ polynomial are shown:
$M_{K}$ | $a$ | $b$ | $c$ | $d$ | $z$
---|---|---|---|---|---
$\langle W(K)\rangle$ | $\alpha$ | $\alpha^{-1}$ | $0$ | $(a-a^{-1})/z+1$ | $-i\pi/k$
$\|K(G)\|$ | $Q^{\frac{1}{4}}v^{\frac{1}{2}}+Q^{\frac{1}{4}}v^{\frac{-1}{2}}$ | $Q^{\frac{-1}{4}}v^{\frac{1}{2}}+Q^{\frac{3}{4}}v^{\frac{-1}{2}}$ | $Q^{-\frac{1}{2}}v+Q^{\frac{1}{2}}+Q^{\frac{1}{2}}v^{-1}$ | $Q^{\frac{1}{2}}$ | $Q^{-\frac{1}{4}}v^{\frac{1}{2}}-Q^{\frac{1}{4}}v^{-\frac{1}{2}}$
## References
* [1] E. Witten, “Quantum field theory and the Jones polynomial,” Commun. Math. Phys. 121 (1989) 351.
* [2] E. Guadagnini, M. Martellini and M. Mintchev, “Wilson Lines in Chern-Simons Theory and Link Invariants,” Nucl. Phys. B 330 (1990) 575.
* [3] P. Cotta-Ramusino, E. Guadagnini, M. Martellini and M. Mintchev, “Quantum field theory and link invariants,” Nucl. Phys. B 330, 557 (1990).
* [4] L. Smolin, “Invariants of links and critical points of the Chern-Simons path integrals,” Mod. Phys. Lett. A 4 (1989) 1091.
* [5] L. H. Kauffman, “Knots and physics,” Singapore: World Scientific (1991) 538 p.
* [6] R. Gambini and J. Pullin, “Loops, Knots, Gauge Theories And Quantum Gravity,” Cambridge, UK: Univ. Pr. (1996) 321 p
* [7] L. H. Kauffman, ”An invariant of regular isotopy,” Trans. Amer. Math. Soc. 318 , 417 (1990) [http://math.uic.edu/`~`kauffman/IRH.pdf].
* [8] P. Cvitanovic, “Group theory for Feynman diagrams in non-Abelian gauge theories,” Phys. Rev. D 14 (1976) 1536.
* [9] T. W. Kim, B. H. Cho and S. U. Park, “Chern-Simons theories on SO(N) and Sp(2N) and link polynomials,” Phys. Rev. D 42, 4135 (1990).
* [10] J. H. Horne, “Skein Relations And Wilson Loops In Chern-Simons Gauge Theory,” Nucl. Phys. B 334 (1990) 669.
* [11] Y. S. Wu and K. Yamagishi, “Chern-Simons theory and Kauffman polynomials,” Int. J. Mod. Phys. A 5, 1165 (1990).
* [12] B. Bruegmann, “Witten’s identity for Chern-Simons theory,” Int. J. Theor. Phys. 34, 145 (1995), [arXiv:hep-th/9401055].
* [13] R. Gambini and J. Pullin, “Variational derivation of exact skein relations from Chern–Simons theories,” Commun. Math. Phys. 185, 621 (1997) [arXiv:hep-th/9602165].
* [14] F. Y. Wu, “Knot theory and statistical mechanics,” Rev. Mod. Phys. 64 (1992) 1099 [Erratum-ibid. 65 (1993) 577].
* [15] P. Cvitanovic, “Group theory,” Princeton University Press, 2008, [http://www.birdtracks.eu/version8.9/GroupTheory.pdf]
* [16] N. Maru and S. Kitakado, “Negative dimensional group extrapolation and a new chiral-nonchiral duality in N = 1 supersymmetric gauge theories,” Mod. Phys. Lett. A 12 (1997) 691 [arXiv:hep-th/9609230].
* [17] G. Parisi and N. Sourlas, “Random Magnetic Fields, Supersymmetry And Negative Dimensions,” Phys. Rev. Lett. 43 (1979) 744.
* [18] S. Carlip, “Quantum gravity in 2+1 dimensions,” Cambridge, UK: Univ. Pr. (1998)
* [19] L. H. Kauffman, “Statistical Mechanics And The Jones Polynomial,” 1988;
in ”New Developments in the thory of knots” (pp 278-312), World Scientific.
* [20] F.Y.Wu, “The Potts Model”, Rew. Mod. Phys, Vol. 54, No 1, 1982.
* [21] L. H. Kauffman , “Remarks on Khovanov Homology and the Potts Model” , http://arxiv.org/abs/0907.3178 .
* [22] A. Achucarro and P. K. Townsend, “A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories,” Phys. Lett. B 180 (1986) 89.
* [23] J. Zanelli, “Lecture notes on Chern-Simons (super-)gravities,” arXiv:hep-th/0502193.
|
arxiv-papers
| 2010-05-21T00:03:38 |
2024-09-04T02:49:10.577179
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marco Astorino",
"submitter": "Marco Astorino",
"url": "https://arxiv.org/abs/1005.3861"
}
|
1005.3979
|
Zbigniew Fiedorowicz http://www.math.ohio-state.edu/people/fiedorow/view
Steven Gubkin http://www.math.ohio-state.edu/people/gubkin/view Rainer Vogt
http://www.mathematik.uni-osnabrueck.de/staff/phpages/vogtr.rdf.shtml
primarymsc201018D10 secondarymsc201055P48 secondarymsc201006A07
# Associahedra and Weak Monoidal Structures on Categories
Z. Fiedorowicz Department of Mathematics, The Ohio State University
Columbus, OH 43210-1174, USA fiedorow@math.ohio-state.edu S. Gubkin
Department of Mathematics, The Ohio State University
Columbus, OH 43210-1174, USA gubkin@math.ohio-state.edu R.M. Vogt
Universität Osnabrück, Fachbereich Mathematik/Informatik
Albrechtstr. 28a, 49069 Osnabrück, Germany rainer@mathematik.uni-
osnabrueck.de
###### Abstract
This paper answers the following question: what algebraic structure on a
category corresponds to an $A_{n}$ structure (in the sense of Stasheff) on the
geometric realization of its nerve?
###### keywords:
monoidal categories
###### keywords:
operads
###### keywords:
Tamari lattice
In his trailblazing paper [12], Stasheff constructed an infinite hierarchy of
higher homotopy associativity conditions for an H-space $X$. These conditions
are parametrized by a family $\\{K_{n}\\}_{n\geq 2}$ of polyhedra, which came
to be known as associahedra. The vertices of $K_{n}$ are in 1-1 correspondence
with all possible ways of associating an $n$-fold product $x_{1}x_{2}\dots
x_{n}$, and an H-space $X$ is said to be an $A_{n}$-space if there is a map
$K_{n}\times X^{n}\longrightarrow X$ whose restriction to the vertices
enumerates all possible ways of associating the binary multiplication on $X$
into an $n$-fold multiplication. An $A_{\infty}$-space is known to be
equivalent to a strict monoid $MX$ and hence, up to group completion, to a
loop space.
At the same time111Stasheff informs us that, although [12] and [9] both
appeared in 1963, Mac Lane’s work preceded his and influenced his thinking. He
further informs us (cf. [14]) that the associahedra were implicitly defined in
the even earlier work of Tamari [15], [16]. Mac Lane [9] analyzed higher
associativity conditions for monoidal structures on categories. He formulated
analogs of Stasheff’s $A_{n}$ conditions for categories. For $n=2,3,4$ the
analogy is perfect. In particular Mac Lane’s $A_{4}$ condition is that a
pentagonal diagram commute, whereas Stasheff’s $K_{4}$ is a pentagon. However
for $n\geq 5$ the analogy breaks down. Mac Lane’s coherence theorem states
that the $A_{4}$ condition implies all the higher $A_{n}$ conditions for
$n\geq 5$. By contrast for any $n\geq 2$ one can construct H-spaces $X$ which
satisfy the $A_{n}$ condition but not the $A_{n+1}$ (or any higher) condition.
In this paper we show how Mac Lane’s notion of a monoidal structure on a
category can be weakened so as to obtain a full hierarchy of $A_{n}$
conditions. The paper is similar in spirit to [1] where an $E_{n}$ hierarchy
of commutativity conditions on categories was considered, analogous to those
on $n$-fold loop spaces. Similarly to the case of associativity for
categories, Joyal and Street [4, Proposition 5.4] showed that if these
commutativity conditions are required to hold up to natural isomorphisms, then
the $E_{3}$ condition implies all higher $E_{n}$ conditions. In [1] we
demonstrated that we could recover the entire $E_{n}$ hierarchy for categories
by weakening these commutativity conditions to hold up to natural
transformations instead. This strategy does not work for associativity, since
LaPlaza [6, Theorem 5] showed that even if the associativity conditions are
weakened to hold up to natural transformations, instead of isomorphisms, this
laxened form of Mac Lane’s $A_{4}$ condition still implies all higher $A_{n}$
conditions. Thus a different strategy for weakening Mac Lane’s $A_{n}$
conditions for categories is required.
In sections 1 and 2 we develop this strategy: we define the category
theoretical analogues of Stasheff’s associahedra in section 1 and
$A_{n}$-monoidal categories in section 2. In section 3 we relate our work to
that of LaPlaza (and implicitly to that of Tamari) and give a simpler proof of
his coherence result. In section 4 we prove a rectification result for
$A_{\infty}$-monoidal categories, similar in spirit to Mac Lane’s
rectification of a monoidal category to a strictly monoidal one, by
translating the rectification of an $A_{\infty}$-space to a monoid into
category theory.
This paper presupposes some familiarity with the notion of operad and related
concepts. A précis of the relevant definitions may be found in [10] and some
historical context in [13]. Since we will be dealing exclusively with
noncommutative operations, we will be using the non-$\Sigma$ forms of operads
throughout.
To forestall any possible misunderstanding, it should be pointed out that this
paper is not related in any significant way to the notion of
$A_{\infty}$-category as developed by Fukaya, Kontsevich, Soibelman and others
(c.f. [5] for an overview). In this paper we are discussing ordinary
categories with weak monoidal structures, not some notion of a weak higher
category.
We would like to take this opportunity to thank Jim Stasheff and Stefan Forcey
for some helpful suggestions and references to previous work in this area.
## 1 The associahedra as an operad in $CAT$
In order to keep track of associativity data for our weakly monoidal
categories, we will need a categorical equivalent of the associahedron
$K_{m}$. To begin with we formalize the notion of a parenthesized word:
###### Definition 1.1
A parenthesized word $(W,P)$is a finite linear order $W$ together with a
(possibly empty) collection of closed intervals $P=\\{p_{i}=[a_{i},b_{i}]\\}$
subject to the following requirements.
* •
The cardinality of each $p_{i}$ is at least 2 and is strictly smaller than the
cardinality of $W$.
* •
For any $i,j$, either $p_{i}\subset p_{j}$, $p_{j}\subset p_{i}$ or $p_{i}\cap
p_{j}=\varnothing.$
A parenthesized word $(W,P)$ can be converted into a parenthesized string of
characters by putting as many left parentheses in front of an element $a\in W$
as $a$ is an initial element of some $p_{i}\in P$ and as many right
parentheses after an element $b\in W$ as $b$ is a final element in some
$p_{i}\in P$, and concatenating the resulting characters. For instance
$\left\\{x_{1}<x_{2}<x_{3}<x_{4}<x_{5}<x_{6},\\{[x_{2},x_{6}],[x_{2},x_{4}],[x_{5},x_{6}]\\}\right\\}\mapsto
x_{1}((x_{2}x_{3}x_{4})(x_{5}x_{6})).$
It is clear that $(W,P)$ can be recovered from the parenthesized string and we
will often find it convenient to represent $(W,P)$ in this way. In most cases
we will use the standard linear orders $W_{m}=\\{x_{1}<x_{2}\dots<x_{m}\\}$.
In some induction arguments however we will need to consider subintervals of
the $W_{m}$.
###### Definition 1.2
We define $\mathfrak{K}_{m}$ to be the poset of parenthesized words on the
linear order $W_{m}$, where $(W_{m},P_{2})\leq(W_{m},P_{1})$ iff $P_{1}\subset
P_{2}$. The minimal elements in this order are called the fully parenthesized
words of length $m$. In the degenerate cases $m=1$ and $m=0$, the poset
$\mathfrak{K}_{1}$ consists of the single parenthesized word
$id=(W_{1},\emptyset)$, and $\mathfrak{K}_{0}$ consists of the single
parenthesized word $0=(\emptyset,\emptyset)$. The string
$x_{1}x_{2}...x_{m}=(W_{m},\emptyset)$ is the terminal object in
$\mathfrak{K}_{m}$. As noted above, sometimes it will be convenient to use
some other linear order $W^{\prime}$ of the same cardinality $m$. In that case
the unique order isomorphism between $W^{\prime}$ and $W_{m}$ specifies a
canonical isomorphism between $\mathfrak{K}_{m}$ and the corresponding poset
of parenthesized words on $W^{\prime}$.
###### Example 1.3
The poset $\mathfrak{K}_{4}$:
| | |
---|---|---|---
| | |
$\textstyle{(x_{1}x_{2})(x_{3}x_{4})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(x_{1}x_{2})x_{3}x_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{1}x_{2}(x_{3}x_{4})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{((x_{1}x_{2})x_{3})x_{4}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{1}(x_{2}(x_{3}x_{4}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{1}x_{2}x_{3}x_{4}}$$\textstyle{(x_{1}x_{2}x_{3})x_{4}}$$\textstyle{x_{1}(x_{2}x_{3}x_{4})}$$\textstyle{\ignorespaces(x_{1}(x_{2}x_{3}))x_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{1}(x_{2}x_{3})x_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces
x_{1}((x_{2}x_{3})x_{4})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
The $m$-th associahedron is defined to be the polytope which has one vertex
for every fully parenthesized word of length $m$. Two vertices $(W,P_{i})$ and
$(W,P_{j})$ are on the same $k$-dimensional face if they share at least $m-k$
parentheses, i.e. $P_{i}\cap P_{j}$ has cardinality at least $m-k$. Thus our
poset $\mathfrak{K}_{m}$ is exactly the face poset of the $m$-th
associahedron, and so the geometric realization of the nerve of
$\mathfrak{K}_{m}$ is simply the barycentric subdivision of the $m$-th
associahedron. Note that we are using a Fraktur font to distinguish the poset
$\mathfrak{K}_{m}$ from the associahedron $K_{m}$ which is the geometric
realization of its nerve (as a topological space).
The following lemma will prove to be surprisingly useful:
###### Lemma 1.4
Let $(W_{k},P)<(W_{k},P^{\prime})$ in $\mathfrak{K}_{k}$. Then the subposet
$[(W_{k},P),(W_{k},P^{\prime})]=\\{(W_{k},P^{\prime\prime})\in\mathfrak{K}_{k}|(W_{k},P)\leq(W_{k},P^{\prime\prime})\leq(W_{k},P^{\prime})\\}$
is isomorphic to the poset $\mathcal{I}^{m}$ where $\mathcal{I}$ is the poset
$1<0$ and $m$ is the number of parentheses in $(W_{k},P)$ which are not in
$(W_{k},P^{\prime})$. In other words, the factorizations of a fixed morphism
in $\mathfrak{K}_{k}$ form a commutative cubical diagram.
###### Proof 1.1.
We can uniquely associate to each element $(W_{k},P^{\prime\prime})$ in
$[(W_{k},P),(W_{k},P^{\prime})]$ a characteristic function on the set of
parentheses in $(W_{k},P)$ which are not in $(W_{k},P^{\prime})$ by giving the
value 1 to each parenthesis which occurs in $(W_{k},P^{\prime\prime})$ and 0
to any which do not so occur. But such a characteristic function is evidently
the same thing as an object of $\mathcal{I}^{m}$ and it is clear that order
relations match.
If we take $(W_{k},P^{\prime})=(W_{k},\emptyset)$, then geometrically this
gives a decomposition of the associahedra into cubes. The decomposition of
$K_{4}$ into 5 squares looks like this:
| | |
---|---|---|---
| | |
$\textstyle{(x_{1}x_{2})(x_{3}x_{4})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(x_{1}x_{2})x_{3}x_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{1}x_{2}(x_{3}x_{4})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{((x_{1}x_{2})x_{3})x_{4}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{1}(x_{2}(x_{3}x_{4}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{1}x_{2}x_{3}x_{4}}$$\textstyle{(x_{1}x_{2}x_{3})x_{4}}$$\textstyle{x_{1}(x_{2}x_{3}x_{4})}$$\textstyle{\ignorespaces(x_{1}(x_{2}x_{3}))x_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{1}(x_{2}x_{3})x_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces
x_{1}((x_{2}x_{3})x_{4})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
The decomposition of $K_{5}$ into 14 cubes can be found here:
http://arxiv.org/src/1005.3979v4/anc/cubical.flv.
There is a one-to-one correspondence between parenthesized words and stable
rooted trees. Briefly these are planar rooted trees where each node has at
least two input edges. We refer to [7] for a formal definition. The
correspondence is given by labelling the leaves of such a tree with the labels
$x_{1},x_{2},\dots x_{n}$ in left to right order. [In the degenerate cases
$n=1$ and $n=0$, the identity $id\in\mathfrak{K}_{1}$ corresponds to the tree
with a single edge and no nodes and $\emptyset\in\mathfrak{K}_{0}$ corresponds
to the empty tree with no edges and no nodes.] Then for each node of the tree,
except for the bottom root node, one takes the set of labels sitting over that
node as one of the intervals $p_{i}\in P$ in the collection $P$, thus giving
us a parenthesized word $(W_{n},P)$. For example, here are all of the
parenthesized words on the linear order $W_{4}$ and their corresponding stable
rooted trees:
###### Example 1.5
The following trees
---
$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$, | | | | | |
---|---|---|---|---|---|---
$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$, | | |
---|---|---|---
$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$, | | |
---|---|---|---
$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$, | |
---|---|---
$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$, | | | |
---|---|---|---|---
$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$, | | | |
---|---|---|---|---
| | |
$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$, | | | |
---|---|---|---|---
$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$, | |
---|---|---
$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$,
---
$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$, | | | | |
---|---|---|---|---|---
$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
represent $(x_{1}x_{2})(x_{3}x_{4})$, $((x_{1}x_{2})x_{3})x_{4}$,
$(x_{1}(x_{2}x_{3}))x_{4}$, $x_{1}((x_{2}x_{3})x_{4})$,
$x_{1}(x_{2}(x_{3}x_{4}))$, $(x_{1}x_{2}x_{3})x_{4}$,
$x_{1}(x_{2}x_{3}x_{4})$, $(x_{1}x_{2})x_{3}x_{4}$, $x_{1}(x_{2}x_{3})x_{4}$,
$x_{1}x_{2}(x_{3}x_{4})$, $x_{1}x_{2}x_{3}x_{4}$ respectively.
The poset structure on $\mathfrak{K}_{n}$ of Definition 1.2 can be described
in terms of trees as follows: $T<T^{\prime}$ if $T^{\prime}$ can be obtained
from $T$ by shrinking some of the edges of $T$. It is of course more
convenient to use parenthesized words when describing $\mathfrak{K}_{n}$ as a
poset. However the language of trees is more convenient to describe the operad
structure on the $\mathfrak{K}_{n}$.
The $\\{\mathfrak{K}_{i}\\}_{i\geq 0}$ form an operad in $CAT$, the category
of small categories. Given stable rooted trees $S\in\mathfrak{K}_{m}$ and
$T_{i}\in\mathfrak{K}_{k_{i}}$ for $i=1,2,\dots,m$, we obtain a new tree by
grafting the root of $T_{i}$ to the $i$-th leaf of $S$. In terms of
parenthesized words we are substituting the word for $T_{i}$ in place of the
$i$-th character of the word for $S$, and reindexing to insure that all
characters in the resulting word are distinct. This only makes sense if $m\geq
2$ and all $k_{i}\geq 2$. If $S=id\in\mathfrak{K}_{1}$, we define the composed
tree to be $T_{1}$. If $T_{i}=id\in\mathfrak{K}_{1}$, then we leave the $i$-th
leaf of $S$ unchanged. If $T_{i}=0\in\mathfrak{K}_{0}$, then we delete the
$i$-th leaf of $S$. If this leaves only one input edge for the node below, we
delete that node as well. If it leaves no input edges for the node below, we
delete both that node and the edge below. We apply this algorithm recursively:
if the next node below receives only one input edge or no input edges, we
delete that node or that node together with the edge below, and so on. In the
special case when $\sum k_{i}=1$ or $\sum k_{i}=0$, the resulting degenerate
trees are defined to be $id$ or $0$ respectively.
This process is is clearly functorial in each of
$\mathfrak{K}_{m},\mathfrak{K}_{k_{1}},\mathfrak{K}_{k_{2}},\dots\mathfrak{K}_{k_{m}}$,
and so we obtain a functor
$\gamma_{m,k_{1},k_{2},\dots
k_{m}}:\mathfrak{K}_{m}\times\prod_{1}^{m}\mathfrak{K}_{k_{i}}\to\mathfrak{K}_{\sum_{1}^{m}k_{i}}.$
These functors define a categorical operad
$\mathfrak{K}=\\{\mathfrak{K}_{i}\\}_{i\geq 0}$.
The associahedral operad $\mathfrak{K}=\\{\mathfrak{K}_{i}\\}_{i\geq 0}$ has
an operadic filtration
$\mathfrak{K}^{(2)}\subset\mathfrak{K}^{(3)}\subset\mathfrak{K}^{(4)}\dots,$
where $\mathfrak{K}^{(n)}_{i}$ is the subposet of $\mathfrak{K}_{i}$
consisting of trees where each node has input valence $\leq n$ (i.e has at
most $n$ incoming edges). We define $\mathfrak{K}^{(\infty)}=\mathfrak{K}$. We
note for future reference that if an element $(W,P)$ of $\mathfrak{K}_{k}$
lies in filtration $n$ and $(W^{\prime},P^{\prime})<(W,P)$ then
$(W^{\prime},P^{\prime})$ also lies in filtration $n$. This follows from our
description above of the poset structure in terms of trees.
###### Proposition 1.6
The poset $\mathfrak{K}^{(n)}_{i}$ is the face poset of a subcomplex of the
(unsubdivided) associahedron $K_{i}$. This subcomplex contains all cells of
$K_{i}$ of dimension $\leq n-2$. Consequently the nerve of
$\mathfrak{K}^{(n)}_{i}$ is $(n-3)$-connected. In particular if $n\geq 4$ the
nerve of $\mathfrak{K}^{(n)}_{i}$ is simply connected.
###### Proof 1.2.
We note that if an element $(W,P)$ of $\mathfrak{K}_{i}$ is in
$\mathfrak{K}^{(n)}_{i}$ and $(W^{\prime},P^{\prime})<(W,P)$ then
$(W^{\prime},P^{\prime})$ is also contained in $\mathfrak{K}^{(n)}_{i}$, since
the tree representing $(W,P)$ is obtained from the tree representing
$(W^{\prime},P^{\prime})$ by shrinking internal edges. It follows that
$\mathfrak{K}^{(n)}_{i}$ is the face poset of a subcomplex of $K_{i}$. Now the
vertices of $K_{i}$ are parametrized by the elements of
$\mathfrak{K}^{(2)}_{i}$, which are represented by binary trees. It follows
that the cells of $K_{i}$ of dimension $j$ are obtained by shrinking $j$
internal edges of a binary tree. It easily follows that the dimension of the
cell parametrized by a given tree is the sum over all nodes of the incoming
valence of that node minus 2. Thus the maximal possible incoming valence of a
node in a tree parametrizing a cell of dimension $j$ is $j+2$. Hence the
subcomplex of $K_{i}$ parametrized by $\mathfrak{K}^{(n)}_{i}$ contains all
cells of $K_{i}$ of dimension $\leq n-2$.
Now the nerve of $\mathfrak{K}^{(n)}_{i}$ is the barycentric subdivision of
this subcomplex of $K_{i}$. Moreover $K_{i}$ is obtained from this subcomplex
by adding cells of dimensions $\geq n-1$. Since $K_{i}$ is contractible, it
follows that the complex and hence the nerve of $\mathfrak{K}^{(n)}_{i}$ is
$(n-3)$-connected.
###### Remark 1.7
$\mathfrak{K}^{(n)}_{i}$ is generally larger than the face poset of the
$(n-2)$-skeleton of $K_{i}$. For instance $\mathfrak{K}^{(3)}_{5}$ is the face
poset of the subcomplex of the 3-dimensional associahedron $K_{5}$ consisting
of all the edges together with the three square faces.
###### Remark 1.8
Our categorical operad $\mathfrak{K}$ is almost the same as Leinster’s $StTr$
([7, pages 233-234]). The only difference is that he has $StTr(0)=\emptyset$,
whereas we have $\mathfrak{K}_{0}=\\{0\\}$. So our approach encodes the notion
of a unit for algebras over $\mathfrak{K}$. Leinster expected that the nerve
of $StTr(k)=\mathfrak{K}_{k}$ is homeomorphic to the associahedron , which we
prove. Thus Leinster’s topological operad is precisely the same as Stasheff’s.
The tree description of a $CAT$-operad containing $\mathfrak{K}$ appears in
[3].
###### Remark 1.9
Since the nerve of a product in $CAT$ is a product in $TOP$, it follows that
the nerve of a $\mathfrak{K}^{(n)}$ algebra is an $A_{n}$-space in the sense
of Stasheff.
## 2 $A_{n}$-monoidal categories and coherence
###### Definition 2.1
For $n=2,3,\dots,\infty$, an $A_{n}$-monoidal category is a category
$\mathcal{C}$ together with multiplications
$\mu_{k}:\mathcal{C}^{k}\rightarrow\mathcal{C}$ for $0\leq k<n+1$ such that
1. 1.
$\mu_{1}:\mathcal{C}\rightarrow\mathcal{C}$ is the identity functor.
2. 2.
$\mu_{0}:*\rightarrow\mathcal{C}$ is an object $0\in\mathcal{C}$ that acts as
a strict unit in the sense that
$\mu_{k}(Id_{\mathcal{C}}^{\,i}\times 0\times
Id_{\mathcal{C}}^{\,j})=\mu_{k-1}$
for any $i,j$ such that $i+j=k-1$.
$\mathcal{C}$ is also equipped with natural transformations (associators)
$\alpha^{i,j,k}:\mu_{i+1+k}\circ(Id_{\mathcal{C}}^{\,i}\times\mu_{j}\times
Id_{\mathcal{C}}^{\,k})\longrightarrow\mu_{i+j+k},$
for $0\leq i+j+k<n+1$, satisfying
$\displaystyle(i)$ α^i,0,k, α^i,1,k and α^0,j,0 are the identity
and the coherence conditions specified by the following commutative diagrams
(ii)
$\textstyle{\mu_{a+b+d+2}(\overline{A},0,\overline{B},\mu_{c}(\overline{C}),\overline{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{a+b+1,c,d}_{(\overline{A},0,\overline{B}),\overline{C},\overline{D}}}$$\textstyle{\mu_{a+b+d+1}(\overline{A},\overline{B},\mu_{c}(\overline{C}),\overline{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{a+b,c,d}_{(\overline{A},\overline{B}),\overline{C},\overline{D}}}$$\textstyle{\mu_{a+b+c+d+1}(\overline{A},0,\overline{B},\overline{C},\overline{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mu_{a+b+c+d}(\overline{A},\overline{B},\overline{C},\overline{D})}$
(iii)
$\textstyle{\mu_{a+c+d+2}(\overline{A},\mu_{b}(\overline{B}),\overline{C},0,\overline{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{a,b,c+d+1}_{\overline{A},\overline{B},(\overline{C},0,\overline{D})}}$$\textstyle{\mu_{a+c+d+1}(\overline{A},\mu_{b}(\overline{B}),\overline{C},\overline{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{a,b,c+d}_{\overline{A},\overline{B},(\overline{C},\overline{D})}}$$\textstyle{\mu_{a+b+c+d+1}(\overline{A},\overline{B},\overline{C},0,\overline{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mu_{a+b+c+d}(\overline{A},\overline{B},\overline{C},\overline{D})}$
(iv)
$\textstyle{\mu_{a+d+1}(\overline{A},\mu_{b+c+1}(\overline{B},0,\overline{C}),\overline{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{a,b+c+1,d}_{\overline{A},(\overline{B},0,\overline{C}),\overline{D}}}$$\textstyle{\mu_{a+d+1}(\overline{A},\mu_{b+c}(\overline{B},\overline{C}),\overline{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{a,b+c,d}_{\overline{A},(\overline{B},\overline{C}),\overline{D}}}$$\textstyle{\mu_{a+b+c+d+1}(\overline{A},\overline{B},0,\overline{C},\overline{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mu_{a+b+c+d}(\overline{A},\overline{B},\overline{C},\overline{D})}$
(v)
$\textstyle{\mu_{a+c+e+2}(\overline{A},\mu_{b}(\overline{B}),\overline{C},\mu_{d}(\overline{D}),\overline{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{a+c+1,d,e}_{(\overline{A},\mu_{b}(\overline{B}),\overline{C}),\overline{D},\overline{E}}}$$\scriptstyle{\alpha^{a,b,c+e+1}_{\overline{A},\overline{B},(\overline{C},\mu_{d}(\overline{D}),\overline{E})}}$$\textstyle{\mu_{a+b+c+e+1}(\overline{A},\overline{B},\overline{C},\mu_{d}(\overline{D}),\overline{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{a+c+1,d,e}_{(\overline{A},\overline{B},\overline{C}),\overline{D},\overline{E}}}$$\textstyle{\mu_{a+c+d+e+1}(\overline{A},\mu_{b}(\overline{B}),\overline{C},\overline{D},\overline{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{a,b,c+d+e}_{\overline{A},\overline{B},(\overline{C},\overline{D},\overline{E})}}$$\textstyle{\mu_{a+b+c+d+e}(\overline{A},\overline{B},\overline{C},\overline{D},\overline{E})}$
(vi)
$\textstyle{\mu_{a+e+1}(\overline{A},\mu_{b+d+1}(\overline{B},\mu_{c}(\overline{C}),\overline{D}),\overline{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{a+e+1}(id_{\overline{A}},\alpha^{b,c,d}_{\overline{B},\overline{C},\overline{D}},id_{\overline{E}})}$$\scriptstyle{\alpha^{a,b+d+1,e}_{\overline{A},(\overline{B},\mu_{c}(\overline{C}),\overline{D}),\overline{E}}}$$\textstyle{\mu_{a+e+1}(\overline{A},\mu_{b+c+d}(\overline{B},\overline{C},\overline{D}),\overline{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{a,b+c+d,e}_{\overline{A},(\overline{B},\overline{C},\overline{D}),\overline{E}}}$$\textstyle{\mu_{a+b+d+e+1}(\overline{A},\overline{B},\mu_{c}(\overline{C}),\overline{D},\overline{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{a+b,c,d+e}_{(\overline{A},\overline{B}),\overline{C},(\overline{D},\overline{E})}}$$\textstyle{\mu_{a+b+c+d+e}(\overline{A},\overline{B},\overline{C},\overline{D},\overline{E})}$
Here $\overline{A}$, $\overline{B}$, $\overline{C}$, $\overline{D}$,
$\overline{E}$ are taken to be objects of $\mathcal{C}^{a}$,
$\mathcal{C}^{b}$, $\mathcal{C}^{c}$, $\mathcal{C}^{d}$, $\mathcal{C}^{e}$,
respectively.
Essentially coherence conditions (i)-(iv) require the associators to be
compatible with the strict unit 0, while (v) and (vi) just say that if we are
removing two pairs of matching parentheses in a multiplication, it doesn’t
matter which we remove first.
###### Remark 2.2
In [7, pages 93-94], Leinster defines the notion of a lax monoidal category,
which is similar in spirit to the above definition, but there are some crucial
differences. A lax monoidal category in his sense, has multiplications
$\mu_{k}(A_{1},A_{2},\dots,A_{k})=(A_{1}\otimes A_{2}\otimes\dots\otimes
A_{k})$
for all $k\in\mathbb{N}$ together with natural transformations
$\gamma^{k_{1},\ldots,k_{n}}:\mu_{n}\circ(\mu_{k_{1}}\otimes\ldots\otimes\mu_{k_{n}})\to\mu_{k_{1}+\ldots+k_{n}}$
and a natural transformation
$\iota_{A}:A\longrightarrow\mu_{1}(A)=(A).$
The natural transformations $\gamma$ satisfy a coherence condition which is
essentially our coherence conditions (v) and (vi) combined into a single
diagram. There is no unit condition for $\mu_{0}$ (so one might as well
require the existence of $\mu_{k}$ for $k>0$ only). Moreover his natural
transformation $\iota$ is not the identity. Thus a lax monoidal category in
his sense possesses arbitrarily long nondegenerate strings of composable
natural transformations between unary multiplications
$A\stackrel{{\scriptstyle\iota_{A}}}{{\longrightarrow}}(A)\stackrel{{\scriptstyle\iota_{(A)}}}{{\longrightarrow}}((A))\stackrel{{\scriptstyle\iota_{((A))}}}{{\longrightarrow}}\dots$
It follows that the operad controlling such a structure has an infinite
dimensional nerve.
The main result of this paper is:
###### Theorem 2.3
A category $\mathcal{C}$ is a $\mathfrak{K}^{(n)}$-algebra iff it is an
$A_{n}$-monoidal category.
###### Proof 2.1.
Given an action,
$\theta_{i}:\mathfrak{K}^{(n)}_{i}\times\mathcal{C}^{i}\longrightarrow\mathcal{C}$,
define $\mu_{i}:\mathcal{C}^{i}\longrightarrow\mathcal{C}$ to be the
restriction of this action to $\\{x_{1}x_{2}\dots
x_{i}\\}\times\mathcal{C}^{i}$, where $x_{1}x_{2}\dots
x_{i}=(W_{i},\emptyset)$ is the terminal object of $\mathfrak{K}_{i}$. This
makes sense for $0\leq i<n+1$, since in those cases $x_{1}x_{2}\dots x_{i}$ is
contained in the $n$-th filtration $\mathfrak{K}^{(n)}$. We then define
$\alpha^{i,j,k}$ to be the restriction of $\theta_{i+j+k}$ to
$\left\\{\left(W_{i+j+k},\\{[x_{i+1},x_{i+j}]\\}\right)\longrightarrow(W_{i+j+k},\emptyset)\right\\}\times\mathcal{C}^{i+j+k}$,
for $0\leq i+j+k<n+1$.
Conditions (1), (2) and (i) follow from the fact that
$(x_{1})\in\mathfrak{K}^{(n)}_{1}$ is the identity of the operad and composing
the constant $0\in\mathfrak{K}^{(n)}_{0}$ into any input of
$\\{x_{1}x_{2}\dots x_{i}\\}\in\mathfrak{K}^{(n)}_{i}$ gives
$\\{x_{1}x_{2}\dots x_{i-1}\\}\in\mathfrak{K}^{(n)}_{i-1}$. Conditions
(ii)-(iv) also follow from the latter fact. Finally conditions (v) and (vi)
follow from the restriction of $\theta_{a+b+c+d+e}$ to
$\mathcal{D}\times\mathcal{C}^{a+b+c+d+e}$ and
$\mathcal{D}^{\prime}\times\mathcal{C}^{a+b+c+d+e}$, where $\mathcal{D}$ and
$\mathcal{D}^{\prime}$ are the following commutative diagrams in
$\mathfrak{K}^{(n)}_{a+b+c+d+e}$:
$\textstyle{\left(W_{a+b+c+d+e},\left\\{[x_{a+1},x_{a+b}],[x_{a+b+c+1},x_{a+b+c+d}]\right\\}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(W_{a+b+c+d+e},\left\\{[x_{a+1},x_{a+b}]\right\\}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(W_{a+b+c+d+e},\left\\{[x_{a+b+c+1},x_{a+b+c+d}]\right\\}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(W_{a+b+c+d+e},\emptyset)}$
$\textstyle{\left(W_{a+b+c+d+e},\left\\{[x_{a+1},x_{a+b+c+d}],[x_{a+b+1},x_{a+b+c}]\right\\}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(W_{a+b+c+d+e},\left\\{[x_{a+b+1},x_{a+b+c}]\right\\}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(W_{a+b+c+d+e},\left\\{[x_{a+1},x_{a+b+c+d}]\right\\}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(W_{a+b+c+d+e},\emptyset)}$
respectively.
Conversely suppose that $\mathcal{C}$ is an $A_{n}$-monoidal category. Then we
define
$\theta_{i}:\mbox{Obj}(\mathfrak{K}^{(n)}_{i})\times\mathcal{C}^{i}\longrightarrow\mathcal{C}$
by induction on $i$ as follows. We define $\theta_{0}$ to be $\mu_{0}$ and
$\theta_{1}$ to be $\mu_{1}=id_{\mathcal{C}}$. Having defined $\theta_{j}$ for
$j<i$, consider an object $T$ in $\mathfrak{K}^{(n)}_{i}$ represented by a
tree
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|
$\textstyle{T_{1}}$$\textstyle{T_{2}}$$\textstyle{\dots}$$\textstyle{\
T_{k}}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
with $k<n+1$ and where $T_{j}$ has $m_{j}$ input edges, so that
$m_{1}+m_{2}+\dots+m_{k}=i$. Let
$(\overline{A}_{1},\overline{A}_{2},\dots,\overline{A}_{k})$ represent an
object in $\mathcal{C}^{i}$, with
$\overline{A}_{j}\in\mbox{Obj}\left(\mathcal{C}^{m_{j}}\right)$,
$j=1,2,\dots,k$. By induction $\theta_{m_{j}}(T_{j},\overline{A}_{j})$ are
already defined for $j=1,2,\dots,k$. We then define
$\theta_{i}\left(T,\overline{A}_{1},\overline{A}_{2},\dots,\overline{A}_{k})=\mu_{k}(\theta_{m_{1}}(T_{1},\overline{A}_{1}),\theta_{m_{2}}(T_{2},\overline{A}_{2}),\dots,\theta_{m_{k}}(T_{k},\overline{A}_{k})\right)$
[Here we use implicitly the canonical isomorphisms between the associahedral
posets based on subintervals of $W_{i}$ with the associahedral posets based on
the standard linear orders $W_{m_{i}}$ of the same cardinality, c.f. 1.2.] We
define $\theta_{i}$ for morphisms in $\mathcal{C}^{i}$ similarly. This
completes the induction.
Next we extend the definition of $\theta_{i}$ to define natural
transformations
$\theta_{i}:\mbox{IMor}(\mathfrak{K}^{(n)}_{i})\times\mbox{Obj}(\mathcal{C}^{i})\longrightarrow\mbox{Mor}(\mathcal{C})$
where $\mbox{IMor}(\mathfrak{K}^{(n)}_{i})$ are the indecomposable morphisms
in $\mathfrak{K}^{(n)}_{i}$, i.e. morphisms which can’t be factored
nontrivially (or equivalently morphisms given by dropping a single pair of
matching parentheses in a parenthesized word). Again we proceed by induction
on $i$, starting with $i=0$ and $i=1$ where these are vacuously defined. Now
consider an indecomposable morphism $\lambda:T\to T^{\prime}$ in
$\mathfrak{K}^{(n)}_{i}$, where $T$ has the form
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---|---|---
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$\textstyle{T_{1}}$$\textstyle{T_{2}}$$\textstyle{\dots}$$\textstyle{\
T_{k}}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
with $k<n+1$ and where $T_{j}$ has $m_{j}$ input edges, so that
$m_{1}+m_{2}+\dots+m_{k}=i$. Then $\lambda$ is obtained by shrinking a single
interior edge in $T$. There are two possibilities: (1) an interior edge of
some tree $T_{j}$ is shrunk or (2) an edge below some $T_{j}$ is shrunk. Now
let $(\overline{A}_{1},\overline{A}_{2},\dots,\overline{A}_{k})$ represent an
object in $\mathcal{C}^{i}$, with
$\overline{A}_{j}\in\mbox{Obj}\left(\mathcal{C}^{m_{j}}\right)$,
$j=1,2,\dots,k$. In the first case we define
$\theta_{i}(\lambda,\overline{A}_{1},\overline{A}_{2},\dots,\overline{A}_{k})=$
$\mu_{k}\left(id_{\theta_{m_{1}}(T_{1},\overline{A}_{1})},id_{\theta_{m_{2}}(T_{2},\overline{A}_{2})},\dots,id_{\theta_{m_{j-1}}(T_{j-1},\overline{A}_{j-1})},\theta_{m_{j}}(\lambda^{\prime},\overline{A_{j}}),id_{\theta_{m_{j+1}}(T_{j+1},\overline{A}_{j+1})},\dots,id_{\theta_{m_{k}}(T_{k},\overline{A}_{k})}\right)$
where $\lambda^{\prime}$ is the indecomposable morphism in
$\mathfrak{K}^{(n)}_{m_{j}}$ given by shrinking that particular edge. In the
second case we define
$\theta_{i}(\lambda,\overline{A}_{1},\overline{A}_{2},\dots,\overline{A}_{k})=\alpha^{m_{1}+m2+\dots
m_{j-1},m_{j},m_{j+1}+m_{j+2}+\dots+m_{k}}_{(\overline{A}_{1},\overline{A}_{2},\dots,\overline{A}_{j-1}),\overline{A}_{j},(\overline{A}_{j+1},\dots,\overline{A}_{k})}.$
This completes the induction.
Finally to extend $\theta_{i}$ to all morphisms in $\mathfrak{K}_{i}$, we must
show that for any factorization of a morphism in $\mathfrak{K}^{(n)}_{i}$ into
indecomposable morphisms, the corresponding composition of natural
transformations defines the same morphism in $\mathcal{C}$. But according to
Lemma 1.4, the factorizations of any morphism in $\mathfrak{K}_{i}$ give rise
to a cubical diagram in $\mathcal{C}$. According to coherence conditions (v)
and (vi) of an $A_{n}$-monoidal category, all the 2-dimensional faces of this
cubical diagram commute. It is an elementary consequence that the entire
cubical diagram in $\mathcal{C}$ commutes, c.f. Lemma 1 below. It follows that
there are well defined functors:
$\theta_{i}:\mathfrak{K}^{(n)}_{i}\times\mathcal{C}^{i}\longrightarrow\mathcal{C}$
for all $i\geq 0$. The fact that $\theta_{i}$ are compatible with the operadic
compositions
$\gamma_{m,k_{1},k_{2},\dots
k_{m}}:\mathfrak{K}^{(n)}_{m}\times\prod_{1}^{m}\mathfrak{K}^{(n)}_{k_{i}}\to\mathfrak{K}^{(n)}_{\sum_{1}^{m}k_{i}}$
follows from the inductive construction of $\theta_{i}$ if all the $k_{i}>1$.
If $k_{i}\leq 1$ or $m=1$, the compatibility follows from conditions (1), (2)
and (i)-(iv) of the definition of an $A_{n}$-monoidal category.
###### Lemma 1.
A cubical diagram in any category commutes iff each of its 2-dimensional faces
commutes.
###### Proof 2.2.
We proceed by induction on the dimension of the cube. The statement is
vacuously true if the dimension is $\leq 2$. Suppose it is true for all
cubical diagrams of dimension $<m$, and suppose we are given an
$m$-dimensional cubical diagram. Consider two edge paths from the initial
object $A$ of the diagram to $Z$, the terminal object. Let these edge paths
factor as
$A\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}B\stackrel{{\scriptstyle
f}}{{\longrightarrow}}Z,\qquad
A\stackrel{{\scriptstyle\beta}}{{\longrightarrow}}C\stackrel{{\scriptstyle
g}}{{\longrightarrow}}Z$
respectively, where $\alpha$ and $\beta$ are edges of the diagram and $f$ and
$g$ are composites of the remainders of these edge paths. If $\alpha=\beta$,
then by induction $f=g$ and we are done. Otherwise let
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{D}$
be the 2-dimensional face spanned by $\alpha$ and $\beta$. Pick any edge path
$h:D\longrightarrow Z$, and consider the diagram
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$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{f}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h\quad}$$\textstyle{Z}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{g}$
By hypothesis the square commutes and by induction the two triangles commute.
Hence $f\alpha=g\beta$. This completes the induction and proof.
## 3 Relation to coherence theorems for monoidal categories
###### Definition 3.1
We say that an $A_{n}$-monoidal category is undirected if all the
associativity natural transformations $\alpha^{i,j,k}$ are isomorphisms.
###### Proposition 3.2
An undirected $A_{n}$-monoidal category is a monoidal category if $n\geq 4$.
###### Proof 3.1.
If $\mathcal{C}$ is an undirected $A_{n}$-monoidal category, then the
corresponding action functors
$\theta_{i}:\mathfrak{K}^{(n)}_{i}\times\mathcal{C}^{i}\longrightarrow\mathcal{C}$
extend to
$\theta_{i}:\overline{\mathfrak{K}}^{(n)}_{i}\times\mathcal{C}^{i}\longrightarrow\mathcal{C}$,
where $\overline{\mathfrak{K}}^{(n)}_{i}$ is obtained from
$\mathfrak{K}^{(n)}_{i}$ by formally inverting all the morphisms. By
Proposition 1.6 the nerve of $\mathfrak{K}^{(n)}_{i}$ is simply connected. Now
recalling that inverting all the morphisms in a connected category has the
effect of killing off the higher homotopy groups of its nerve (c.f. [11,
Proposition 1]), we see that the nerve of $\overline{\mathfrak{K}}^{(n)}_{i}$
is contractible, and it follows that the objects of $\mathfrak{K}^{(2)}_{i}$
are connected to each other by uniquely defined isomorphisms in
$\overline{\mathfrak{K}}^{(n)}_{i}$. The images of these isomorphisms under
$\theta_{i}$ specify uniquely defined natural isomorphisms connecting all
possible different ways of associating the binary product
$\mu_{2}:\mathcal{C}^{2}\longrightarrow\mathcal{C}$ into an $i$-fold product
$\mathcal{C}^{i}\longrightarrow\mathcal{C}$ so that all diagrams involving
them commute. Thus $\mathcal{C}$ is a monoidal category, in the classical
sense of Mac Lane.
Next we derive LaPlaza’s coherence theorem [6], which generalizes Mac Lane’s
coherence theorem to the case where the associativity natural transformation
for a monoidal structure on a category is not required to be an isomorphism.
We begin with a preliminary version of this result.
###### Theorem 3.3
Let $(\mathcal{C},\Box,0,\eta)$ be a directed monoidal category with a strict
unit. That is, $\Box:\mathcal{C}\times\mathcal{C}\longrightarrow\mathcal{C}$
is a bifunctor and 0 is an object of $\mathcal{C}$ which serves as a strict
unit for $\Box$, i.e. the restrictions of $\Box$ to $0\times\mathcal{C}$ and
$\mathcal{C}\times 0$ are the identity. Finally $\eta_{A,B,C}:(A\Box B)\Box
C\longrightarrow A\Box(B\Box C)$ is a natural transformation (not necessarily
an isomorphism) such that $\eta_{A,B,C}$ is the identity whenever one of $A$,
$B$, $C$ is 0 and such that the pentagonal diagram
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---|---
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$\textstyle{(A\Box B)\Box(C\Box
D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{A,B,C\Box
D}}$$\textstyle{((A\Box B)\Box C)\Box
D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{A\Box
B,C,D}}$$\scriptstyle{\eta_{A,B,C}\Box id_{D}}$$\textstyle{A\Box(B\Box(C\Box
D))}$$\textstyle{(A\Box(B\Box C))\Box
D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{A,B\Box
C,D}}$$\textstyle{A\Box((B\Box C)\Box
D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{A}\Box\eta_{B,C,D}}$
commutes. Then $\mathcal{C}$ can be endowed with the structure of an
$A_{\infty}$-monoidal category.
###### Proof 3.2.
We define $\mu_{0}(*)=0$, $\mu_{1}$ to be the identity, $\mu_{2}=\Box$ and
then we inductively define $\mu_{i}$ to be the composite
$\textstyle{\mathcal{C}^{i}=\mathcal{C}\times\mathcal{C}^{i-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{\mathcal{C}}\times\mu_{i-1}}$$\textstyle{\mathcal{C}\times\mathcal{C}\stackrel{{\scriptstyle\Box}}{{\longrightarrow}}\mathcal{C}}$
Thus
$\mu_{i}(A_{1},A_{2},A_{3},\dots,A_{i})=A_{1}\Box(A_{2}\Box(A_{3}\Box(\dots\Box(A_{i-1}\Box
A_{i})\dots)))$ and we have
$\textstyle{(*)}$$\textstyle{\mu_{a+b}(\overline{A},\overline{B})=\mu_{a+1}(\overline{A},\mu_{b}(\overline{B}))}$
for any objects $\overline{A}\in\mathcal{C}^{a}$,
$\overline{B}\in\mathcal{C}^{b}$.
Now let $\overline{B}\in\mathcal{C}^{b}$ and $\overline{C}\in\mathcal{C}^{c}$.
We define the associativity $\alpha^{0,b,c}_{\overline{B},\overline{C}}$
inductively on $b$. We assume $c>0$, since $\alpha^{0,b,0}$ is required by
definition to be the identity. For $b=1$, we also require $\alpha^{0,1,c}$ to
be the identity. So suppose $b>1$ and
$\overline{B}=(B_{1},\overline{B}^{\prime})$. Then we define $\alpha^{0,b,c}$
to be the composite
$\mu_{c+1}(\mu_{b}(\overline{B}),\overline{C})=\mu_{2}(\mu_{b}(\overline{B}),\mu_{c}(\overline{C}))=(B_{1}\Box\mu_{b-1}(\overline{B}^{\prime}))\Box\mu_{c}(\overline{C})$
$\scriptstyle{\eta_{B_{1},\mu_{b-1}(\overline{B}^{\prime}),\mu_{c}(\overline{C})}}$$\textstyle{B_{1}\Box(\mu_{b-1}(\overline{B}^{\prime})\Box\mu_{c}(\overline{C}))}$
followed by the composite
$B_{1}\Box(\mu_{b-1}(\overline{B}^{\prime})\Box\mu_{c}(\overline{C}))=B_{1}\Box\mu_{2}(\mu_{b-1}(\overline{B}^{\prime}),\mu_{c}(\overline{C}))=B_{1}\Box\mu_{c+1}(\mu_{b-1}(\overline{B}^{\prime}),\overline{C})$
$\scriptstyle{id_{B_{1}}\Box\alpha^{0,b-1,c}_{\overline{B}^{\prime},\overline{C}}\qquad\qquad}$$\textstyle{B_{1}\Box\mu_{b+c-1}(\overline{B}^{\prime},\overline{C})=\mu_{b+c}(\overline{B},\overline{C}).}$
This completes the inductive definition of
$\alpha^{0,b,c}_{\overline{B},\overline{C}}$. We then define
$\alpha^{a,b,c}_{\overline{A},\overline{B},\overline{C}}$ to be the composite
$\mu_{a+c+1}(\overline{A},\mu_{b}(\overline{B}),\overline{C})=\mu_{a+1}(\overline{A},\mu_{c+1}(\mu_{b}(\overline{B}),\overline{C}))$
$\scriptstyle{\mu_{a+1}(id_{\overline{A}},\alpha^{0,b,c}_{\mu_{b}(\overline{B}),\overline{C}})\qquad\qquad\qquad\qquad}$$\textstyle{\mu_{a+1}(\overline{A},\mu_{b+c}(\overline{B},\overline{C}))=\mu_{a+b+c}(\overline{A},\overline{B},\overline{C}).}$
Note that this implies that
$\alpha^{a,b,c}_{\overline{A},\overline{B},\overline{C}}$ is the identity if
$c=0$, and that
$\textstyle{(**)}$$\textstyle{\alpha^{a_{1}+a_{2},b,c}_{(\overline{A}_{1},\overline{A}_{2}),\overline{B},\overline{C}}=\mu_{a_{1}+1}(id_{A_{1}},\alpha^{a_{2},b,c}_{\overline{A}_{2},\overline{B},\overline{C}}).}$
Conditions (1), (2), (i)-(iv) for an $A_{\infty}$-monoidal category are either
true by construction or follow by a straight forward induction argument using
the hypotheses that 0 is a strict unit for $\Box$ and that $\eta_{A,B,C}$ is
the identity whenever one of $A$, $B$ or $C$ is 0.
By (**) the verification of condition (v) reduces to the special case of the
diagram
$\textstyle{\mu_{c+e+2}(\mu_{b}(\overline{B}),\overline{C},\mu_{d}(\overline{D}),\overline{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{c+1,d,e}_{(\mu_{b}(\overline{B}),\overline{C}),\overline{D},\overline{E}}}$$\scriptstyle{\alpha^{0,b,c+e+1}_{\overline{B},(\overline{C},\mu_{d}(\overline{D}),\overline{E})}}$$\textstyle{\mu_{b+c+e+1}(\overline{B},\overline{C},\mu_{d}(\overline{D}),\overline{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{c+1,d,e}_{(\overline{B},\overline{C}),\overline{D},\overline{E}}}$$\textstyle{\mu_{c+d+e+1}(\mu_{b}(\overline{B}),\overline{C},\overline{D},\overline{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{0,b,c+d+e}_{\overline{B},(\overline{C},\overline{D},\overline{E})}}$$\textstyle{\mu_{b+c+d+e}(\overline{B},\overline{C},\overline{D},\overline{E})}$
since the general diagram for (v) can be obtained from this one by applying
the functor $\mu_{a+1}(\overline{A},-)$ to it. By (*) and (**), this diagram
in turn is the same as the diagram
$\textstyle{\mu_{2}(\mu_{b}(\overline{B}),\mu_{c+e+1}(\overline{C},\mu_{d}(\overline{D}),\overline{E}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}(id_{\mu_{b}(\overline{B})},\alpha^{c,d,e}_{\overline{C},\overline{D},\overline{E}})}$$\scriptstyle{\alpha^{0,b,1}_{\overline{B},\mu_{c+e+1}(\overline{C},\mu_{d}(\overline{D}),\overline{E})}}$$\textstyle{\mu_{b+1}(\overline{B},\mu_{c+e+1}(\overline{C},\mu_{d}(\overline{D}),\overline{E}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{b+1}(id_{\overline{B}},\alpha^{c,d,e}_{\overline{C},\overline{D},\overline{E}})}$$\textstyle{\mu_{2}(\mu_{b}(\overline{B}),\mu_{c+d+e}(\overline{C},\overline{D},\overline{E}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{0,b,1}_{\overline{B},\mu_{c+d+e}(\overline{C},\overline{D},\overline{E})}}$$\textstyle{\mu_{b+1}(\overline{B},\mu_{c+d+e}(\overline{C},\overline{D},\overline{E}))}$
This last diagram in turn commutes because
$\alpha^{0,b,1}_{\overline{B},X}:\mu_{2}(\mu_{b}(\overline{B}),X)\longrightarrow\mu_{b+1}(\overline{B},X)$
is a natural transformation.
By similar reasoning, the verification of condition (vi) reduces to the
special case of the diagram
$\textstyle{\mu_{e+1}(\mu_{b+d+1}(\overline{B},\mu_{c}(\overline{C}),\overline{D}),\overline{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{e+1}(\alpha^{b,c,d}_{\overline{B},\overline{C},\overline{D}},id_{\overline{E}})}$$\scriptstyle{\alpha^{0,b+d+1,e}_{(\overline{B},\mu_{c}(\overline{C}),\overline{D}),\overline{E}}}$$\textstyle{\mu_{e+1}(\mu_{b+c+d}(\overline{B},\overline{C},\overline{D}),\overline{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{0,b+c+d,e}_{(\overline{B},\overline{C},\overline{D}),\overline{E}}}$$\textstyle{\mu_{b+d+e+1}(\overline{B},\mu_{c}(\overline{C}),\overline{D},\overline{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{b,c,d+e}_{\overline{B},\overline{C},(\overline{D},\overline{E})}}$$\textstyle{\mu_{b+c+d+e}(\overline{B},\overline{C},\overline{D},\overline{E})}$
By (*) it follows that this diagram is unchanged if we replace $E$ throughout
by $\mu_{e}(E)$. Hence we may as well suppose that $e=1$ and $\overline{E}=E$
is an object of $\mathcal{C}$. Then by the inductive definition of
$\alpha^{0,i,j}$ and (**), we can factor the diagram as follows:
$\textstyle{(B_{1}\Box\mu_{b+d}(\overline{B}^{\prime},\mu_{c}(\overline{C}),\overline{D}))\Box
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(id_{B_{1}}\Box\alpha^{b-1,c,d}_{\overline{B}^{\prime},\overline{C},\overline{D}})\Box
id_{E}}$$\scriptstyle{\eta_{B_{1},\mu_{b+d}(\overline{B}^{\prime},\mu_{c}(\overline{C}),\overline{D}),E}}$$\textstyle{(B_{1}\Box\mu_{b+c+d-1}(\overline{B}^{\prime},\overline{C},\overline{D}))\Box
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{B_{1},\mu_{b+c+d-1}(\overline{B}^{\prime},\overline{C},\overline{D}),E}}$$\textstyle{B_{1}\Box(\mu_{b+d}(\overline{B}^{\prime},\mu_{c}(\overline{C}),\overline{D})\Box
E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{B_{1}}\Box(\alpha^{b-1,c,d}_{\overline{B}^{\prime},\overline{C},\overline{D}}\Box
id_{E})}$$\scriptstyle{id_{B_{1}}\Box\alpha^{0,b+d,1}_{(\overline{B}^{\prime},\mu_{c}(\overline{C}),\overline{D}),\overline{E}}}$$\textstyle{B_{1}\Box(\mu_{b+c+d-1}(\overline{B}^{\prime},\overline{C},\overline{D})\Box
E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{B_{1}}\Box\alpha^{0,b+c+d-1,1}_{(\overline{B},\overline{C},\overline{D}),E}}$$\textstyle{B_{1}\Box\mu_{b+d+1}(\overline{B}^{\prime},\mu_{c}(\overline{C}),\overline{D},E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{B_{1}}\Box\alpha^{b-1,c,d+1}_{\overline{B},\overline{C},(\overline{D},E)}}$$\textstyle{B_{1}\Box\mu_{b+c+d}(\overline{B}^{\prime},\overline{C},\overline{D},E)}$
The upper square commutes by naturality of $\eta$. The commutativity of the
lower square corresponds to a reduction of the problem from $b$ to $b-1$.
Recursing on this reduction we reduce to the case $b=0$, i.e. showing that the
diagram
$\textstyle{\mu_{d+1}(\mu_{c}(\overline{C}),\overline{D})\Box
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{0,c,d}_{\overline{C},\overline{D}}\Box
id_{E}}$$\scriptstyle{\alpha^{0,d+1,1}_{(\mu_{c}(\overline{C}),\overline{D}),E}}$$\textstyle{\mu_{c+d}(\overline{C},\overline{D})\Box{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{0,c+d,1}_{(\overline{C},\overline{D}),E}}$$\textstyle{\mu_{d+2}(\mu_{c}(\overline{C}),\overline{D},E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{0,c,d+1}_{\overline{C},(\overline{D},E)}}$$\textstyle{\mu_{c+d+1}(\overline{C},\overline{D},E)}$
commutes. By (*) and the inductive definition of $\alpha^{0,i,j}$ this diagram
can be replaced and expanded into the following diagram
$\textstyle{\mu_{2}(\mu_{c}(\overline{C}),\mu_{d}(\overline{D}))\Box
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{0,c,1}_{\overline{C},\mu_{d}(\overline{D})}\Box
id_{E}}$$\scriptstyle{\alpha^{0,2,1}_{(\mu_{c}(\overline{C}),\mu_{d}(\overline{D})),E}}$$\textstyle{\mu_{c+1}(\overline{C},\mu_{d}(\overline{D}))\Box{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{0,c+1,1}_{(\overline{C},\mu_{d}(\overline{D})),E}}$$\textstyle{\mu_{3}(\mu_{c}(\overline{C}),\mu_{d}(\overline{D}),E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{0,c,2}_{\overline{C},(\mu_{d}(\overline{D}),E)}}$$\scriptstyle{\mu_{2}(id_{\mu_{c}(\overline{C})},\alpha^{0,d,1}_{\overline{D},E})}$$\textstyle{\mu_{c+2}(\overline{C},\mu_{d}(\overline{D}),E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{c+1}(id_{\overline{C}},\alpha^{0,d,1}_{\overline{D},E})}$$\textstyle{\mu_{2}(\mu_{c}(\overline{C}),\mu_{d+1}(\overline{D},E))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{0,c,1}_{\overline{C},\mu_{d+1}(\overline{D},E)}}$$\textstyle{\mu_{c+1}(\overline{C},\mu_{d+1}(\overline{D},E))}$
The lower square commutes by naturality of $\alpha$. So it suffices to show
the upper square commutes. This is just the previous diagram with
$\overline{D}$ replaced by $\mu_{d}(\overline{D})$. Thus we have reduced to
the case $d=1$. We will find it convenient to display this diagram in
reflected form:
$\textstyle{\mu_{2}(\mu_{c}(\overline{C}),D)\Box
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{0,c,1}_{\overline{C},D}\Box
id_{E}}$$\scriptstyle{\alpha^{0,2,1}_{(\mu_{c}(\overline{C}),D),E}}$$\textstyle{\mu_{3}(\mu_{c}(\overline{C}),D,E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{0,c,2}_{\overline{C},(D,E)}}$$\textstyle{\mu_{c+1}(\overline{C},D)\Box{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{0,c+1,1}_{(\overline{C},D),E}}$$\textstyle{\mu_{c+2}(\overline{C},D,E)}$
We have to show this diagram commutes, where $D$ and $E$ are objects of
$\mathcal{C}$ and $\overline{C}$ is an object of $\mathcal{C}^{c}$.
This diagram commutes trivially if $c\leq 1$. So assume $c>1$ and
$\overline{C}=(C_{1},\overline{C}^{\prime})$. Again using (*) and the
inductive definition of $\alpha^{0,i,j}$, we can expand this diagram into
$\textstyle{\mu_{2}(\mu_{2}(C_{1},\mu_{c-1}(\overline{C}^{\prime})),D)\Box
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{0,2,1}_{\mu_{2}(C_{1},\mu_{c-1}(\overline{C}^{\prime})),D}\Box
id_{E}}$$\scriptstyle{\alpha^{0,2,1}_{(\mu_{2}(C_{1},\mu_{c-1}(\overline{C}^{\prime}),D),E}}$$\textstyle{\mu_{3}(\mu_{2}(C_{1},\mu_{c-1}(\overline{C}^{\prime})),D,E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{0,2,2}_{(C_{1},\mu_{c-1}(\overline{C}^{\prime})),(D,E)}}$$\textstyle{\mu_{3}(C_{1},\mu_{c-1}(\overline{C}^{\prime}),D)\Box{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\alpha^{0,3,1}_{(C_{1},\mu_{c-1}(\overline{C}^{\prime}),D),E}}$$\textstyle{\mu_{4}(C_{1},\mu_{c-1}(\overline{C}^{\prime}),D,E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mu_{2}(C_{1},\mu_{2}(\mu_{c-1}(\overline{C}^{\prime}),D))\Box{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}(id_{C_{1}},\alpha^{0,1,1}_{\mu_{c-1}(\overline{C}^{\prime}),D})\Box
id_{E}}$$\textstyle{\mu_{2}(C_{1},\mu_{3}(\mu_{c-1}(\overline{C}^{\prime}),D,E))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}(id_{C_{1}},\alpha^{0,1,2}_{\mu_{c-1}(\overline{C}^{\prime}),(D,E)})}$$\textstyle{\mu_{2}(C_{1},\mu_{c}(\overline{C}^{\prime},D))\Box
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mu_{2}(C_{1},\mu_{c+1}(\overline{C}^{\prime},D,E))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mu_{c+1}(\overline{C},D)\Box{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{0,c+1,1}_{(\overline{C},D),E}}$$\textstyle{\mu_{c+2}(\overline{C},D,E)}$
The top square in this diagram is the original diagram with $\overline{C}$
replaced by $(C_{1},\mu_{c-1}(\overline{C}^{\prime}))$, thus reducing it to
the case $c=2$. This top square can be expanded into the pentagonal diagram of
the hypothesis of the theorem and thus commutes. It remains to show that the
bottom square commutes.
After rewriting the bottom square in reflected form and applying the inductive
definition of $\alpha^{0,i,j}$ we obtain the following expanded diagram
$\textstyle{(C_{1}\Box(\mu_{c-1}(\overline{C}^{\prime})\Box
D))\Box{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad(id_{C_{1}}\Box\alpha^{0,1,1}_{\mu_{c-1}(\overline{C}^{\prime}),D})\Box
id_{E}}$$\scriptstyle{\eta_{C_{1},\mu_{c-1}(\overline{C}^{\prime})\Box
D,E}}$$\textstyle{(C_{1}\Box\mu_{c}(\overline{C}^{\prime},D))\Box
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{C_{1},\mu_{c}(\overline{C}^{\prime},D),E}}$$\textstyle{C_{1}\Box((\mu_{c-1}(\overline{C}^{\prime})\Box
D)\Box
E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad
id_{C_{1}}\Box(\alpha^{0,1,1}_{\mu_{c-1}(\overline{C}^{\prime}),D}\Box
id_{E})}$$\scriptstyle{id_{C_{1}}\Box\alpha^{0,2,1}_{(\mu_{c-1}(\overline{C}^{\prime}),D),E}}$$\textstyle{C_{1}\Box(\mu_{c}(\overline{C}^{\prime},D)\Box
E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{C_{1}}\Box\alpha^{0,c,1}_{(\overline{C}^{\prime},D),E}}$$\textstyle{C_{1}\Box\mu_{3}(\mu_{c-1}(\overline{C}^{\prime}),D,E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\qquad
id_{C_{1}}\Box\alpha^{0,c-1,2}_{\overline{C}^{\prime},(D,E)}}$$\textstyle{C_{1}\Box\mu_{c+1}(\overline{C}^{\prime},D,E)}$
The top square commutes by naturality of $\eta$ and the bottom square by
naturality of $\alpha$.
This completes the verification that we have constructed an
$A_{\infty}$-monoidal structure on $\mathcal{C}$.
To obtain the full version of LaPlaza’s coherence theorem, we start with an
operadic reformulation of Theorem 3.3.
###### Definition 3.4
The LaPlaza operad $\mathcal{L}=\\{\mathcal{L}_{m}\\}_{m\geq 0}$ is the operad
in $CAT$ which acts on directed monoidal categories as in the hypothesis of
Theorem 3.3. Specifically $\mathcal{L}_{m}$ can be described as a full
subcategory of the free directed monoidal category on $m$ generating objects
$\\{x_{1},x_{2},\dots,x_{m}\\}$, whose objects look like $x_{1}\Box
x_{2}\Box\dots\Box x_{m}$ after removing all parentheses. Thus
$\mathcal{L}_{0}=\\{0\\}$, $\mathcal{L}_{1}=\\{x_{1}\\}$, and for $m\geq 2$
the objects of $\mathcal{L}_{m}$ are in bijective correspondence with planar
binary trees with $m$ input edges.
###### Remark 3.5
$\mathcal{L}_{2}$ is the trivial poset $\\{x_{1}\Box x_{2}\\}$,
$\mathcal{L}_{3}$ is the poset
$\eta_{x_{1},x_{2},x_{3}}:(x_{1}\Box x_{2})\Box x_{3}\longrightarrow
x_{1}\Box(x_{2}\Box x_{3}),$
isomorphic to $\mathcal{I}$, while $\mathcal{L}_{4}$ is the pentagonal poset
generated by the labelled arrows shown below.
$\textstyle{(x_{1}\Box x_{2})\Box(x_{3}\Box
x_{4})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{x_{1},x_{2},x_{3}\Box
x_{4}}}$$\textstyle{((x_{1}\Box x_{2})\Box x_{3})\Box
x_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{x_{1}\Box
x_{2},x_{3},x_{4}}\quad}$$\scriptstyle{\eta_{x_{1},x_{2},x_{3}}\Box
id_{x_{4}}}$ $\textstyle{x_{1}\Box(x_{2}\Box(x_{3}\Box
x_{4}))}$$\textstyle{(x_{1}\Box(x_{2}\Box x_{3}))\Box
x_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{x_{1},x_{2}\Box
x_{3},x_{4}}}$$\textstyle{x_{1}\Box((x_{2}\Box x_{3})\Box
x_{4})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\
id_{x_{1}}\Box\eta_{x_{2},x_{3},x_{4}}}$
LaPlaza’s coherence theorem states that $\mathcal{L}_{m}$ is a poset for all
$m$.
###### Remark 3.6
LaPlaza works with natural transformations $\eta_{A,B,C}:A\Box(B\Box
C)\longrightarrow(A\Box B)\Box C$. Moreover, he does not consider units. So,
if $\mathcal{L}^{\ast}\subset\mathcal{L}$ is the suboperad obtained from
$\mathcal{L}$ by dropping the unit, the operad $\mathcal{L}^{\ast}$ is dual to
LaPlaza’s original one. Note also that
$\mathcal{L}_{m}=\mathcal{L}^{\ast}_{m}$ for $m\geq 1$. Our poset
$\mathcal{L}_{m}$, $m\geq 3$ is precisely the poset considered by Tamari [16],
and is now commonly called the Tamari lattice [17].
With this notation, we can reformulate Theorem 3.3 as follows.
###### Theorem 3.7
There is a map of $CAT$-operads
$\Lambda:\mathfrak{K}\longrightarrow\mathcal{L}$
which is a surjection.
The existence of $\Lambda$ is clear from the statement of Theorem 3.3.
Surjectivity follows from the proof of Theorem 3.3, where it is shown that
$(x_{1}x_{2})x_{3}\longrightarrow x_{1}x_{2}x_{3}$
maps via $\Lambda$ to
$\eta_{x_{1},x_{2},x_{3}}:(x_{1}\Box x_{2})\Box x_{3}\longrightarrow
x_{1}\Box(x_{2}\Box x_{3}),$
and the fact that $\eta_{x_{1},x_{2},x_{3}}$ generates $\mathcal{L}$ as a
$CAT$-operad.
LaPlaza’s coherence theorem is not immediately apparent from Theorem 3.7,
since a quotient category of a poset need not be a poset. We need the
following additional observation.
###### Lemma 2.
For any object $T\in\mathcal{L}_{m}$, the inverse image under $\Lambda$ of the
subcategory $\\{T\\}$ is a subposet of $\mathfrak{K}_{m}$ containing both a
minimal and a maximal object.
###### Proof 3.3.
For $m=0,1,2$, the functor $\Lambda$ is an isomorphism and there is nothing to
prove. For $m\geq 3$, we may regard $T$ as a planar binary tree. Clearly the
minimal object of $\Lambda^{-1}\\{T\\}$ is $T$ regarded as an object of
$\mathfrak{K}_{m}$. The maximal object of $\Lambda^{-1}\\{T\\}$ is obtained
from $T$ by successively shrinking the rightmost incoming edge to every node
of $T$, with the exception of those edges which are leaves, till the rightmost
edge of each node is a leaf.
###### Example 3.8
The inverse images in Lemma 2 for
$\Lambda:\mathfrak{K}_{4}\longrightarrow\mathcal{L}_{4}$ are as follows:
$\displaystyle\Lambda^{-1}\\{((x_{1}\Box x_{2})\Box x_{3})\Box x_{4}\\}$
$\displaystyle=$
$\displaystyle\left\\{((x_{1}x_{2})x_{3})x_{4}=\raisebox{10.0pt}{$\lx@xy@svg{\hbox{\raise
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###### Corollary 3.9
(LaPlaza Coherence Theorem) For all $m$, the category $\mathcal{L}_{m}$ is a
poset (known as the Tamari lattice for $m\geq 3$).
###### Proof 3.4.
Let
$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{T}$
be morphisms in $\mathcal{L}_{m}$. Since $\Lambda$ is surjective, we may find
preimages
$f^{\prime}:S^{\prime}\longrightarrow T^{\prime},\quad
g^{\prime}:S^{\prime\prime}\longrightarrow T^{\prime\prime}$
under $\Lambda$ of $f$ and $g$ respectively. Now let $S^{\prime\prime\prime}$
be the minimal element of $\Lambda^{-1}\\{S\\}$ and let
$T^{\prime\prime\prime}$ be the maximal element of $\Lambda^{-1}\\{T\\}$. Then
since $\mathfrak{K}_{m}$ is a poset, we have a commutative diagram
---
$\textstyle{S^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{T^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{\prime\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T^{\prime\prime\prime}}$$\textstyle{S^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\textstyle{T^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Applying $\Lambda$ to this diagram and noting that $\Lambda$ sends the
unlabelled arrows to identities, we obtain $f=g$.
We will now give an explicit description of the posets $\mathcal{L}_{m}$ for
$m\geq 4$. Similar considerations apply to $\mathfrak{K}_{m}$ and give an
alternative description of those posets.
###### Definition 3.10
Let $\mathcal{O}$ be a $CAT$-operad with a single nullary operation
$\mathcal{O}_{0}=\\{0\\}$ (such as $\mathfrak{K}$ or $\mathcal{L}$). Suppose
$m\geq 4$ and let $\\{a<b<c\\}\subset\\{1,2,3,\dots,m\\}$. We define the
functor $\pi_{a,b,c}:\mathcal{O}_{m}\longrightarrow\mathcal{O}_{3}$ to be the
composite
$\mathcal{O}_{m}\longrightarrow\mathcal{O}_{m}\times\prod_{i=1}^{m}\mathcal{O}_{k_{i}}\longrightarrow\mathcal{O}_{3}.$
Here $k_{i}=0$ if $i\not\in\\{a,b,c\\}$, $k_{a}=k_{b}=k_{c}=1$, the first map
takes $\phi\in\mathcal{O}_{m}$ to
$(c;\epsilon_{1},\epsilon_{2},\dots,\epsilon_{m})$, where
$\epsilon_{a}=\epsilon_{b}=\epsilon_{c}=id\in\mathcal{O}_{1}$ with all other
$\epsilon_{i}=0\in\mathcal{O}_{0}$, and the second map is composition in
$\mathcal{O}$.
###### Proposition 3.11
There is a commutative diagram
$\textstyle{\mathfrak{K}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\\{\pi_{a,b,c}\\}}$$\scriptstyle{\Lambda}$$\textstyle{\prod_{\\{1\leq
a<b<c\leq
m\\}}\mathfrak{K}_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\prod\Lambda}$$\textstyle{\mathcal{L}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\\{\pi_{a,b,c}\\}}$$\textstyle{\prod_{\\{1\leq
a<b<c\leq
m\\}}\mathcal{L}_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\prod_{\\{1\leq
a<b<c\leq m\\}}\mathcal{I}}$
with the horizontal arrows being full imbeddings of posets.
The proof is straight forward and left as an exercise for the reader.
## 4 Rectification of $A_{\infty}$-monoidal categories
It is well known that a monoidal category is equivalent to a strictly monoidal
category, c.f. [9, pages 257-259]. [Recall that a monoidal category is strict
if the associativity natural transformations $\eta_{A,B,C}$ of Theorem 3.3 are
the identities.] We establish an analogous result for $A_{\infty}$-monoidal
categories.
We first need a preliminary construction.
###### Definition 4.1
For $k\geq 2$ we define the poset $\widehat{\mathfrak{K}}_{k}$ to have as
objects combinatorial trees as defined in [7, Appendix E] with $k$ input
edges. All nodes except the root node, i.e. the node at the output of the
tree, are required to have more than one incoming edge. The root node may have
zero, one, or more incoming edges. We define $T<T^{\prime}$ if $T^{\prime}$
can be obtained from $T$ by shrinking some internal edges. We define
$\widehat{\mathfrak{K}}_{1}$ to consist of the single tree:
$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
and $\widehat{\mathfrak{K}}_{0}$ to consist of the single tree
$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
###### Example 4.2
---
$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
is allowed in $\widehat{\mathfrak{K}}_{2}$, while
---
$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
is not allowed.
The collection $\widehat{\mathfrak{K}}=\\{\widehat{\mathfrak{K}}_{k}\\}_{k\geq
0}$ is a right module over the associahedral operad, that is there are maps of
posets
$\widehat{\mathfrak{K}}_{m}\times\prod_{i=1}^{m}\mathfrak{K}_{k_{i}}\longrightarrow\widehat{\mathfrak{K}}_{k_{1}+k_{2}+\dots+k_{m}}$
satisfying the usual associativity and unit conditions. This right action is
defined in exactly the same way as we defined the operad structure on
$\mathfrak{K}$, with the single exception that when we compose with
$0\in\mathfrak{K}_{0}$, we never delete the root node. Moreover,
$\widehat{\mathfrak{K}}$ is also a left module over $Ass$, the trivial operad
parametrizing strictly monoidal structures. The left action
$Ass(m)\times\prod_{i=1}^{m}\widehat{\mathfrak{K}}_{k_{i}}\cong\prod_{i=1}^{m}\widehat{\mathfrak{K}}_{k_{i}}\longrightarrow\widehat{\mathfrak{K}}_{k_{1}+k_{2}+\dots+k_{m}}$
is given by
$\left(\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr\\\\\\\\\\\\\\\\\\\\\\\\\\\\}}}\ignorespaces{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 1.0pt\raise-5.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 1.0pt\raise-10.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 1.0pt\raise-15.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 1.0pt\raise-20.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 1.0pt\raise-25.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 1.0pt\raise-30.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\right.$ | |
---|---|---
|
$\textstyle{T_{i1}}$$\textstyle{T_{i2}}$$\textstyle{\dots}$$\textstyle{\ T_{ik_{i}}}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$ $\left.\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr\\\\\\\\\\\\\\\\\\\\\\\\\\\\}}}\ignorespaces{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 1.0pt\raise-5.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 1.0pt\raise-10.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 1.0pt\raise-15.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 1.0pt\raise-20.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 1.0pt\raise-25.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 1.0pt\raise-30.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\right)_{i=1}^{m}\qquad\mapsto\qquad$ | | | | |
---|---|---|---|---|---
| |
$\textstyle{T_{11}}$$\textstyle{\dots}$$\textstyle{T_{1k_{1}}}$$\textstyle{\dots}$$\textstyle{T_{m1}}$$\textstyle{\dots}$$\textstyle{T_{mk_{m}}}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
It is clear that the left and right actions commute with each other, so
$\widehat{\mathfrak{K}}$ is an $Ass$-$\mathfrak{K}$-bimodule.
###### Theorem 4.3
There is a functorial construction $\mathcal{C}\mapsto\mathcal{M}\mathcal{C}$
together with functors $I:\mathcal{C}\longrightarrow\mathcal{M}\mathcal{C}$
and $E:\mathcal{M}\mathcal{C}\longrightarrow\mathcal{C}$ which associates to
each $A_{\infty}$-monoidal category $\mathcal{C}$ a strictly monoidal category
$\mathcal{M}\mathcal{C}$ such that
1. 1.
the induced maps by $I$ and $E$ on the nerves of the categories are mutually
inverse homotopy equivalences,
2. 2.
the functor $I$ induces a lax homomorphism of $A_{\infty}$-spaces in the sense
of [2],
3. 3.
if $\mathcal{C}$ is strictly monoidal then $E$ is a strictly monoidal functor.
###### Proof 4.1.
Let
$\mathcal{M}\mathcal{C}=\widehat{\mathfrak{K}}\otimes_{\mathfrak{K}}\mathcal{C}=\left(\coprod_{k\geq
0}\widehat{\mathfrak{K}}_{k}\times\mathcal{C}^{k}\right)/\approx$
where the equivalence relation is given by
$\left(T\circ(S_{1},S_{2},\dots,S_{m}),(\overline{A}_{1},\overline{A}_{2},\dots,\overline{A}_{m}\right)\approx\left(T,S_{1}(\overline{A}_{1}),S_{2}(\overline{A}_{2}),\dots,S_{m}(\overline{A}_{m})\right),$
where $T\in\widehat{\mathfrak{K}}_{m}$, $S_{i}\in\mathfrak{K}_{k_{i}}$,
$\overline{A}_{i}\in\mathcal{C}^{k_{i}}$, for $i=1,2,\dots,m$. The left action
of $Ass$ on $\widehat{\mathfrak{K}}$ then induces a strict monoidal structure
on $\mathcal{M}\mathcal{C}$.
There are functors $I:\mathfrak{K}\longrightarrow\widehat{\mathfrak{K}}$ and
$J:\widehat{\mathfrak{K}}\longrightarrow\mathfrak{K}$. The functor $I$ takes a
tree $S\in\mathfrak{K}$ to the tree
$\textstyle{S}$$\textstyle{{\scriptscriptstyle\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
The functor $E$ takes a tree in $\widehat{\mathfrak{K}}_{k}$, deletes the root
vertex if it has only one or no incoming edges, and regards it as a tree in
$\mathfrak{K}$. The composite $EI$ is the identity of $\mathfrak{K}$. There is
a natural transformation from the composite $IE$ to the identity of
$\widehat{\mathfrak{K}}$, given by shrinking the edge above the root vertex.
The functors $I$, $E$ and this natural transformation are compatible with the
right actions of $\mathfrak{K}$ on $\widehat{\mathfrak{K}}$ and on itself.
Hence there are induced functors
$I:\mathcal{C}\longrightarrow\mathcal{M}\mathcal{C},\qquad
E:\mathcal{M}\mathcal{C}\longrightarrow\mathcal{C}$
such that $EI$ is the identity of $\mathcal{C}$ and we have an induced natural
transformation from $IE$ to the identity of $\mathcal{M}\mathcal{C}$. It
follows that the maps induced by $I$ and $E$ on the nerves of these categories
are mutually inverse equivalences.
Moreover the nerve of $\mathcal{M}\mathcal{C}$ is homeomorphic to the
topological construction described in [2, Theorem 1.26] applied to the nerve
of $\mathcal{C}$. There it is shown that the map induced by $I$ is a lax
homomorphism of $A_{\infty}$-spaces. The fact that $E$ is strictly monoidal,
when $\mathcal{C}$ is strictly monoidal, is straight forward.
## References
* [1] C. Balteanu, Z. Fiedorowicz, R. Schwänzl, and R. M. Vogt, Iterated monoidal categories, Adv. in Math. 176 (2003), 277–349.
* [2] M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Springer Lecture Notes in Math. 347 (1973).
* [3] V. Ginzburg and M. Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), 203–272.
* [4] A. Joyal and R. Street, Braided tensor categories, Adv. in Math. 102 (1993), 20–78.
* [5] B. Keller, Introduction to A-infinity algebras and modules, Homology, Homotopy and Applications 3 (2001), 1–35.
* [6] M. L. LaPlaza, Coherence for associativity not an isomorphism, J. Pure Appl. Algebra 2(1972), 107–120.
* [7] T. Leinster, Higher Operads, Higher Categories, Cambridge Univ. Press, 2004.
* [8] S. Mac Lane, Categories for the Working Mathematician, 2nd. ed., Springer-Verlag, 1998.
* [9] S. Mac Lane, Natural associativity and commutativity, Rice Univ. Studies 49 (1963), 28–46.
* [10] J. P. May, Definitions: operads, algebras and modules, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), 1–7, Contemp. Math., 202, Amer. Math. Soc., Providence, RI, 1997\.
* [11] D. Quillen, Higher algebraic $K$-theory I, Higher $K$-Theories, Battelle Institute Conference 1972, Springer Lecture Notes in Math. 341 (1973), 85-147.
* [12] J. D. Stasheff, Homotopy associativity of H-spaces, I, Trans. Am. Math. Soc. 108 (1963), 275–292.
* [13] J. D. Stasheff, The prehistory of operads, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), 9–14, Contemp. Math., 202, Amer. Math. Soc., Providence, RI, 1997\.
* [14] J. D. Stasheff, How I ‘met’ Dov Tamari, preprint.
* [15] D. Tamari, Monoides préordonnés et chaînes de Malcev, Doctorat ès-Sciences Mathématiques Thèse de Mathématique, Paris, 1951.
* [16] D. Tamari, The algebra of bracketings and their enumeration, Nieuw Archief voor Wiskunde, Ser. 3, 10 (1962), 131–146.
* [17] Wikipedia, Tamari lattice, http://en.wikipedia.org/wiki/Tamari_lattice (as of Sept. 9, 2011, 16:24 GMT).
|
arxiv-papers
| 2010-05-21T15:15:47 |
2024-09-04T02:49:10.591098
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zbigniew Fiedorowicz, Steven Gubkin, and Rainer M. Vogt",
"submitter": "Zbigniew Fiedorowicz",
"url": "https://arxiv.org/abs/1005.3979"
}
|
1005.4002
|
# Implicit particle filters for data assimilation
Alexandre J. Chorin
Department of Mathematics
University of California at Berkeley
and Lawrence Berkeley National Laboratory
Matthias Morzfeld
Department of Mechanical Engineering
University of California at Berkeley
and Lawrence Berkeley National Laboratory
Xuemin Tu
Department of Mathematics
University of California at Berkeley
and Lawrence Berkeley National Laboratory
###### Abstract
Implicit particle filters for data assimilation update the particles by first
choosing probabilities and then looking for particle locations that assume
them, guiding the particles one by one to the high probability domain. We
provide a detailed description of these filters, with illustrative examples,
together with new, more general, methods for solving the algebraic equations
and with a new algorithm for parameter identification.
## 1 Introduction
There are many problems in science, for example in meteorology and economics,
in which the state of a system must be identified from an uncertain equation
supplemented by noisy data (see e.g. [7, 15]). A natural model of this
situation consists of an Ito stochastic differential equation (SDE):
$dx=f(x,t)\,dt+g(x,t)\,dw,$ (1)
where $x=(x_{1},x_{2},\dots,x_{m})$ is an $m$-dimensional vector, $f$ is an
$m$-dimensional vector function, $g(x,t)$ is an $m$ by $m$ matrix, and $w$ is
Brownian motion which encapsulates all the uncertainty in the model. In the
present paper we assume for simplicity that the matrix $g(x,t)$ is diagonal.
The initial state $x(0)$ is given and may be random as well.
The SDE is supplemented by measurements $b^{n}$ at times $t^{n},n=0,1,\dots$.
The measurements are related to the state $x(t)$ by
$b^{n}=h(x^{n})+GW^{n},$ (2)
where $h$ is a $k$-dimensional, generally nonlinear, vector function with
$k\leq m$, $G$ is a matrix, $x^{n}=x(t^{n})$, and $W^{n}$ is a vector whose
components are independent Gaussian variables of mean 0 and variance 1,
independent also of the Brownian motion in equation (1). The independence
requirements can be greatly relaxed but will be observed in the present paper.
The task of a filter is to assimilate the data, i.e., estimate $x$ on the
basis of both equation (1) and the observations (2).
If the system (1) and equation (2) are linear and the data are Gaussian, the
solution can be found in principle via the Kalman-Bucy filter (see e.g. [12]).
In the general case, one often estimates $x$ as a statistic (often the mean)
of a probability density function (pdf) evolving under the combined effect of
equations (1) and (2). The initial state $x^{0}$ being known, all one has to
do is evaluate sequentially the pdfs $P_{n+1}$ of the variables $x^{n+1}$
given the equations and the data. In a “particle” filter this is done by
following “particles” (replicas of the system) whose empirical distribution at
time $t^{n}$ approximates $P_{n}$. One may for example (see e.g. [1, 7, 8,
12]) use the pdf $P_{n}$ and equation (1) to generate a prior density (in the
sense of Bayes) , and then use the data $b^{n+1}$ to generate sampling weights
which define a posterior density $P_{n+1}$. In addition, one has to sample
backward to take into account the information each measurement provides about
the past. This process can be very expensive because in most weighting
schemes, most of the weights tend to zero fast and the number of particles
needed can grow catastrophically (see e.g. [14, 2]); various strategies have
been proposed to ameliorate this problem.
Our remedy is implicit sampling [5, 6]. The number of particles needed in a
filter remains moderate if one can find high probability particles; to this
end, implicit sampling works by first picking probabilities and then looking
for particles that assume them, so that particles are guided efficiently to
the high probability region one at a time, without needing a global guess of
the target density. In the present paper we provide an expository account of
particle filters, separating clearly the general principles from details of
implementation; we provide general solution algorithms for the resulting
algebraic equations, in particular for nonconvex cases which we had not
considered in our previous publications, as well as a new algorithm for
parameter identification based on an implicit filter. We also provide
examples, in particular of nonconvex problems.
Implicit filters are a special case of chainless/Markov field sampling methods
[3, 4]; a key connection was made in [16, 17], where it was observed that in
the sampling of stochastic differential equations, the marginals needed in
Markov field sampling can be read off the equations and need not be estimated
numerically.
## 2 The mathematical framework
The conditional probability density $P_{n}(x)$ at time $t^{n}$, determined by
the SDE (1) given the observations (2), satisfies the recurrence relation (see
e.g. [7]):
$P_{n+1}(x^{n+1})=P_{n}(x^{n})P(x^{n+1}|x^{n})P(b^{n+1}|x^{n+1})/Z_{0},$ (3)
where $P_{n+1}(x^{n+1})$ is the probability of the sample $x^{n+1}$ at time
$t^{n+1}$ given the observations $b^{j}$ for $j\leq{n+1}$, $P_{n}(x^{n})$ is
the probability of a the sample $x^{n}$ at time $t^{n}$ given the observations
$b^{j}$ for $j\leq n$, $P({x}^{n+1}|x^{n})$ is the probability of a sample
$x^{n+1}$ at time $t^{n+1}$ given a sample $x^{n}$ at time $t^{n}$,
$P(b^{n+1}|x^{n+1})$ is the probability of the observations $b^{n+1}$ given
the sample $x^{n+1}$ at time $t^{n+1}$, and $Z_{0}$ is a normalization
constant independent of $x^{n}$ and $x^{n+1}$. This is Bayes’ theorem.
We estimate $P_{n+1}$ with the help of M particles, with positions $X_{i}^{n}$
at time $t^{n}$ and $X^{n+1}_{i}$ at time $t^{n+1}$ ($i=1,\dots,M$), which
define empirical densities $\hat{P}_{n},\hat{P}_{n+1}$ that approximate
$P_{n},P_{n+1}$. We do this by requiring that, when a particle moves from
$X_{i}^{n}$ to $X^{n+1}_{i}$ the probability of $X^{n+1}_{i}$ be
$P(X^{n+1}_{i})=P(X^{n}_{i})P(X^{n+1}_{i}|X^{n}_{i})P(b^{n+1}|X^{n+1}_{i})/Z_{0},$
(4)
where the hats have been omitted, $P(X^{n}_{i})$, the probability of
$X^{n}_{i}$, is assumed given, the pdf $P(X^{n+1}_{i}|X^{n}_{i})$, the
probability of $X^{n+1}_{i}$ given $X^{n}_{i}$, is determined by the SDE (1),
the pdf $P(b^{n+1}|X^{n+1}_{i})$, the probability of the observations
$b^{n+1}$ given the new positions $X^{n+1}_{i}$, is determined by the
observation equation (2). We shall see below that one can set $P(X^{n}_{i})=1$
without loss of generality.
Equation (4) defines the pdf we need to sample for each particle; this pdf is
known, in the sense that once one has a sample, one can evaluate its
probability (up to a constant); the difficulty is to find high probability
samples, especially when the number of variables is large. The idea in
implicit sampling is to define probabilities first, and then look for
particles that assume them; this is done by choosing once and for all a fixed
reference random variable, say $\xi$, with a given pdf, say a Gaussian
$\exp(-\xi^{T}\xi/2)/(2\pi)^{m/2})$, which one knows how to sample so that
most samples have high probability, and then making $X^{n+1}_{i}$ a function
of $\xi$, a different function of each particle and each step, each function
designed so that the map $\xi\rightarrow X^{n+1}_{i}$ connects highly probable
values of $\xi$ to highly probable values of $X^{n+1}_{i}$. To that end, write
$P(X^{n+1}_{i}|X^{n}_{i})P(b^{n+1}|X^{n+1}_{i})=\exp(-F_{i}(X)),$
where on the right-hand side $X$ is a shorthand for $X^{n+1}_{i}$ and all the
other arguments are omitted. This defines a function $F_{i}$ for each particle
$i$ and each time $t^{n}$. For each $i$ and $n$, $F_{i}$ is an explicitly
known function of $X=X^{n+1}_{i}$. Then solve the equation
$F_{i}(X)-\phi_{i}=\xi^{T}\xi/2,$ (5)
where $\xi$ is a sample of the fixed reference variable and $\phi_{i}$ is an
additive factor needed to make the equation solvable. The need for $\phi_{i}$
becomes obvious if one considers the case of a linear observation function $h$
in equation (2), so that the right side of equation (5) is quadratic but the
left is a quadratic plus a constant. It is clear that setting $\phi=\min F$
will do the job, but this is not necessarily the best choice (see below). We
also require that for each particle, the function $X^{n+1}_{i}=X=X(\xi)$
defined by (5) be one-to-one so that the correct pdf is sampled, in
particular, it must have distinct branches for $\xi>0$ and $\xi<0$. The
solution of (5) is discussed in the next section. From now on we omit the
index $i$ in both $F$ and $\phi$, but it should not be forgotten that these
function vary from particle to particle and from one time step to the next.
Once the function $X=X(\xi)$ is determined, each value of $X^{n+1}=X$ (the
subscript $i$ is omitted) appears with probability
$\exp(-\xi^{T}\xi/2)J^{-1}/(\pi)^{m/2}$, where $J$ is the Jacobian of the map
$X=X(\xi)$, while the product $P(X^{n+1}|X^{n})P(b^{n+1}|X^{n+1})$ evaluated
at $X^{n+1}$ equals $\exp(-\xi^{T}\xi/2)\exp(-\phi)/(2\pi)^{m/2}$. The
sampling weight for the particle is therefore $\exp(-\phi)J$. If the map
$\xi\rightarrow X$ is smooth near $\xi=0$, so that $\phi$ and $J$ do not vary
rapidly from particle to particle, and if there is an easy way to compute $J$
(see the next section), then we have an effective way to sample $P_{n+1}$
given $P_{n}$. It is important to note that though the functions $F$ and
$\phi$ vary from particle to particle, the probabilities of the various
samples are expressed in terms of the fixed reference pdf, so that they can be
compared with each other.
The weights can be eliminated by resampling. A standard resampling algorithm
goes as follows [7]: let the weight of the $i$-th particle be
$W_{i},i=1,\dots,M$. Define $A=\sum W_{i}$; for each of $M$ random numbers
$\theta_{k},k=1,\dots,M$ drawn from the uniform distribution on $[0,1]$,
choose a new ${\widehat{X}}^{n+1}_{k}=X^{n+1}_{i}$ such that
$A^{-1}\sum_{j=1}^{i-1}W_{j}<\theta_{k}\leq A^{-1}\sum_{j=1}^{i}W_{j}$, and
then suppress the hat. This justifies the statement following equation (4)
that one can set $P(X_{n})=1$.
To see what has been gained, compare our construction with the usual
“Bayesian” particle filter, where one samples
$P(X^{n+1}|X^{n})P(b^{n+1}|X^{n+1})$ by first finding a “prior” density
$Q(X^{n+1})$ (omitting all arguments other than $X^{n+1}$), such that the
ratio $W=P(X^{n+1})/Q(X^{n+1})$ is close to a constant, and then assigning to
the $i$-th particle the importance weight $W=W_{i}$ evaluated at the location
of the particle. The pdf defined by the set of positions and weights is the
density $P_{n+1}$ we are looking for. An important special case is the choice
$Q(X^{n+1})=P(X^{n+1}|X^{n})$; the prior is then defined by the equation of
motion alone and the posterior is obtained by using the observations to weight
the particles. We shall refer to this special case as “standard importance
sampling” or “standard filter”. Of course, once the positions and the weights
of the particles have been determined, one should resample as above.
The catch in these earlier constructions is that the prior density $Q$ and the
desired posterior can come close to being mutually singular, and the number of
particles needed may become catastrophically large, especially when the number
of variables $m$ is large. To avoid this catch one has to make a good guess
for the pdf $Q$, which may not be easy because $Q$ should approximate the
density $P_{n+1}$ one is looking for- this is the basic conundrum of Monte
Carlo methods, in which one needs a good estimate to get a good estimate. In
contrast, in implicit sampling one does a separate calculation for each sample
and there is no need for prior global information. One can of course still
identify the pdf defined by the positions of the particles at time $t^{n+1}$
as a “prior” and the pdf defined by both the positions and the weights as a
“posterior” density.
Finally, implicit sampling can be viewed as an implicit Monte Carlo scheme for
solving the Zakai equation [18], which describes the evolution of the
(unnormalized) conditional distribution for a SDE conditioned by observations.
This should be contrasted with the procedure in the popular ensemble Kalman
filter (see e.g. [9]), where a Gaussian approximation of the pdf defined by
the SDE is extracted from a Monte Carlo solution of the corresponding Fokker-
Planck equation, a Gaussian approximation is made for the pdf
$P(b^{n+1}|x^{n+1})$, and new particle positions are obtained by a Kalman
step. Our replacement of the Fokker-Planck equation that corresponds to the
SDE alone by a Zakai equation that describes the evolution of the unnormalized
conditional distribution does away with the need for the approximate and
expensive extraction of Gaussians and consequent Kalman step.
## 3 Solution of the algebraic equation that defines a new sample
We now explain how to solve equation (5), $F(X)-\phi=\xi^{T}\xi/2$, under
several sets of assumptions which are met in practice. Note the great latitude
this equation provides in linking the $\xi$ variables to the $X$ variables;
equation (5) is a single equation that connects $2m$ variables (the $m$
components of $\xi$ and the $m$ components of $X$) and can be satisfied by
many maps $\xi\rightarrow X$; these are useful as long as (i) they are one-to-
one, (ii) they map the neighborhood of $0$ into a set that contains the
minimum of $F$, (iii) they are smooth near $\xi=0$ so that the weights
$\exp(-\phi)$ and the Jacobian $J$ not vary unduly from particle to particle
in the target area, and (iv) they allow the Jacobian $J$ to be calculated
easily. The solution methods presented here are far from exhaustive; further
examples and refinements in the context of specific applications.
Algorithm (A) (presented in [5, 6]) : Assume the function $F$ is convex
upwards. For each particle, we set up an iteration, with iterates $X^{n+1,j}$,
$j=0,1,\dots$, ($X^{j}$ for brevity), with $X^{0}=0$, that converge to the
next position $X^{n+1}$ of that particle. The index $i$ that identifies the
particle is omitted again. We write the equations as if the system were one-
dimensional; the multidimensional case was presented in detail in [6]. First
we sample the reference variable $\xi$. The iteration is defined when one
knows how to find $X^{j+1}$ given $X^{j}$.
Expand the observation function $h$ in equation (2) around $X^{j}$:
$h(X^{j+1})=h(X^{j})+(Dh)^{j}(X^{j+1}-X^{j}),$ (6)
where $(Dh)^{j}$ is the derivative of $h$ evaluated at $X^{j}$. The
observation equation (2) is now approximated as a linear function of
$X^{j+1}$, and the function $F$ is the sum of two Gaussians in $X^{j+1}$.
Completing a square yields a single Gaussian with a remainder $\phi$, i.e.,
$F(X)=(x-{\bar{a}})^{2}/(2{\bar{v}})+\phi(X^{j})$, where the parameters
$\phi,{\bar{a}},{\bar{v}}$ are functions of $X^{j}$ (this is what we called in
[5] a “pseudo-Gaussian”). The next iterate is now
$X^{j}={\bar{a}}+\sqrt{\bar{v}}\xi$. In the multidimensional case, when each
component of the function $h$ in equation (2) depends on more than one
variable, finding $X$ as a function of $\xi$ may require the solution of a
linear system of equations, which can be performed e.g. by a Choleski
factorization, as in [6], or by a rotation, as in [5]. If the iteration
converges, it converges to the exact solution of equation (5), with $\phi$ the
limit of the $\phi(X^{j})$. Its convergence can be accelerated by Aitken’s
extrapolation [10]. The Jacobian $J$ can be evaluated either by an implicit
differentiation of equation (5) or numerically, by perturbing $\xi$ in
equation (5) and solving the perturbed equation (which should not require more
than a single additional iteration step). It is easy to see that this
iteration, when it converges, produces a mapping $\xi\rightarrow X$ that is
one to one and onto.
An important special case occurs when the observation function $h$ is linear
in $X$; it is immaterial whether the SDE (1) is linear. In this case the
iteration converges in one step; the Jacobian $J$ is easy to find; if in
addition the function $g(x,t)$ in equation (1) is independent of $x$, then $J$
is independent of the particle and need not be evaluated; the additive term
$\phi$ can be written explicitly as a function of the previous position
$X^{n}$ of the particle and of the observation $b^{n+1}$. We recover an easy
implementation of optimal sequential importance sampling (see e.g. [1, 7, 8]).
This iteration has been used in [6]. It may fail to converge if the function
$F$ is not convex (as happens in particular when the observation function $h$
is highly nonlinear). One may resort then to the next construction.
Algorithm (B). Assume the function $F$ is $U$-shaped, i.e., in the scalar
case, it is at least piecewise differentiable, $F^{\prime}$ vanishes at a
single point which is a minimum, $F$ is strictly decreasing on one side of the
minimum and strictly increasing on the other, with $F(X)=\infty$ when
$X=\pm\infty$. In the $m$-dimensional case, assume that $F$ has a single
minimum and that each intersection of the graph of the function $y=F(X)$ with
a vertical plane through the minimum is $U$-shaped in the scalar sense (note
that a function may be $U$-shaped without being convex).
Find $z$, the minimum of $F$ (note that this is the minimum of a given real
valued function, not a minimum of a possibly multimodal pdf generated by the
SDE; finding this minimum is not equivalent to the difficult problem of
finding a maximum likelihood estimate of the state of the system). The minimum
$z$ can be found by standard minimization algorithms.
Again we are solving the equations by finding iterates $X^{j}$ that converge
to $X^{n+1}$. In the scalar case, given a sample of the reference variable
$\xi$, find first $X^{0}$ such that $X^{0}-z$ has the sign of $\xi$, and then
find the next iterates $X^{j}$ by standard tools (e.g. by Newton iteration),
modified so that the $X^{j}$ are prevented from leaping over $z$.
In the vector case, if the observation function is diagonal, i.e. each
component of the observation is a function of a single component of the
solution $X$, then the scalar algorithm can be used component by component. In
more complicated situations one can take advantage of the freedom in
connecting $\xi$ to $X$.
Here is an interesting example of the use of this freedom, which we present in
the case of a multidimensional problem where the observation function is
linear but need not be diagonal. Set $\phi=\min F$. The function $F(X)-\phi$
can now be written as $(X-a)^{T}A(X-a)/2$, where $a$ is a known vector, $T$
denotes a transpose as before, and $A$ is a positive definite symmetric
matrix. Write further $y=X-a$. Equation (5) becomes
$y^{T}Ay=|\xi|^{2},$ (7)
where $|\xi|$ is the length of the vector $\xi$. Make the ansatz:
$y=\lambda\eta,$
where $\lambda$ is a scalar, $\eta=\xi/|\xi|$ is a random unit vector and
$\xi$ is a sample of of the reference density. Substitution into (7) yields
$\lambda^{2}(\eta^{T}A\eta)=|\xi|^{2}.$ (8)
It is easy to see that $E[\eta_{i}\eta_{j}]=\delta_{ij}/m$, where $E[\cdot]$
denotes an expected value, the $\eta_{i}$ are the components of $\eta$, $m$ is
the number of variables, and $\delta_{ij}$ is the Kronecker delta, and hence
$E[\eta^{T}A\eta]={\rm trace}(A)/m.$
Replace equation (8) by
$\lambda^{2}\Lambda=|\xi|^{2}.$ (9)
where $\Lambda={\rm trace}(A)/m$. This equation has the solution
$\lambda=|\xi|/\sqrt{\Lambda}$, and substitution into the ansatz leads to
$y_{i}=\xi_{i}/\sqrt{\Lambda}$, a transformation with Jacobian
$J=\Lambda^{-m/2}$. The difference between equations (8) and (9) can be
compensated for by adding to $\phi$ the term
$\lambda^{2}[(\eta^{T}A\eta)-\Lambda]$. Notice now that as
$m\rightarrow\infty,(\eta^{T}A\eta)\rightarrow\Lambda$ (a stochastic weak law
of large numbers), so that when the number of variables is sufficiently large,
the perturbation one has to compensate for becomes negligible. Generalizations
and applications of this construction will be given elsewhere in the context
of specific applications.
One can readily devise algorithms also for cases where $F$ is not $U$-shaped,
for example, by dividing $F$ into monotonic pieces and sampling each of these
pieces with its predetermined probability. An alternative that is usually
easier is to replace the non-$U$-shaped function $F$ by a suitable $U$-shaped
function $F_{0}$ and make up for the bias by adding $F_{0}(X)-F(X)$ to the
additive term $\phi$; see the examples below.
## 4 Backward sampling and sparse observations
The algorithms of the previous sections are sufficient to create a filter, but
accuracy may require an additional step. Every observation provides
information not only about the future but also about the past- it may, for
example, tag as improbable earlier states that had seemed probable before the
observation was made; in general one has to go back and correct the past after
every observation (this backward sampling is often misleadingly motivated
solely by the need to create greater diversity among the particles in a
Bayesian filter). A detailed construction has been presented in [6]; the
examples in the present paper are simple enough so that backward sampling does
not significantly enhance their performance, so we will be content here with
presenting the construction in principle, without much detail; it is a
straightforward extension of the work above.
Consider the $i$-th particle, and suppose we have sampled its positions
$X^{n-1}$, $X^{n},X^{n+1}$ at times $t^{n-1},t^{n},t^{n+1}$. Now we would like
to go back and resample a new position $X^{n}$ at time $t^{n}$, given
$X^{n-1}$ and $X^{n+1}$. The probability density of $X=X^{n}$ is proportional
to $P(X)=P(X|X^{n-1})P(b^{n}|X)P(X^{n+1}|X)$. Write $P(X)=\exp(-F(X))$, sample
a Gaussian reference variable $\xi$, solve $F(X)-\phi=\xi^{T}\xi/2$ as above,
and you are done. If need be, one can then go further back and resample
$X^{n-1},X^{n-2},\dots$ Note that backward sampling relates $P_{n+1}$ to
$P_{n-k}$ for $k\geq 0$.
A similar construction can be used when the observations are sparse in time,
for example, if the time step needed to discretize the SDE accurately is
shorter than the time interval between observations. Suppose we have sampled
$X^{n-1}$, have an observation at time $t^{n+1}$ but not at time $t^{n}$, so
that we have to sample simultaneously $X^{n}$ and $X^{n+1}$ from the SDE and
the observation $b^{n+1}$. The joint probability of $X=(X^{n},X^{n+1})$ is
proportional to $P(X^{n}|X^{n-1})P(X^{n+1}|X^{n})P(b^{n+1}|X^{n+1})$. Again,
write this probability as $\exp(-F(X))$ and equate $F(X)-\phi$ to
$\xi^{T}\xi/2$, where $\xi$ is a $2M$-dimensional reference variable. Detailed
expression for the vector case, as well as examples, can be found in [6].
## 5 Examples
We now present examples that illustrate the algorithms we have just described.
For more examples, see [5, 6].
We begin with a response to a comment we have often heard: ”this is nice, but
the construction will fail the moment you are faced with potentials with
multiple wells”. This is not so- the function $F$ depends on the nature of the
noise in the SDE and on the function $h=h(x)$ in the observation equation (2),
but not on the potential. Consider for example a one dimensional particle
moving in the potential $V(x)=2.5(x^{2}-0.5)^{2}$, (see Figure 1), with the
force $f(x)=-\nabla V=-10x(x^{2}-1)$ and the resulting SDE $dx=f(x)dt+\sigma
dw$, where $\sigma=.1$ and $w$ is Brownian motion with unit variance; with
this choice of parameters the SDE has an invariant density concentrated in the
neighborhoods of $x=\pm\sqrt{1/2}$. We consider linear observations
$b^{n}=x(t^{n})+W$, where $W$ is a Gaussian variable with mean zero and
variance $s=.025$. We approximate the SDE by an Euler scheme [11] with time
step $\delta=0.01$, and assume observations are available at all the points
$n\delta$. The particles all start at $x=0$. We produce data $b^{n}$ by
running a single particle and adding to its positions errors drawn from the
assumed error density in equation (2), and then attempt to reconstruct this
path with our filter.
Figure 1: The potential in the first example.
Figure 2: A random path (broken line) and its reconstruction by our filter
(solid line).
For the $i$-the particle located at time $n\delta$ at $X^{n}_{i}$ the function
$F(X)$ is
$F(X)=(X-X^{n}_{i})^{2}/(2\sigma\delta)+(X-b^{n+1})^{2})^{2}/(2s),$
which is always convex. A completion of a square yields $\min
F=\phi=(1/2)(X^{n}_{i}-b^{n+1})^{2}/(\sigma\delta+s)$; the Jacobian $J$ is
independent of the particle and need not be evaluated. In Figure 2 we display
a particle run used to generate data and its reconstruction by our filter with
$50$ particles. This figure is included for completeness but both of these
paths are random, their difference varies from realization to realization, and
may be large or small by accident. To get a quantitative estimate of the
performance of the filter, we repeated this calculation $10^{4}$ times and
computed the mean and the variance of the difference $\Delta$ between the run
that generated the data and its reconstruction at time $t=1$, see Table I.
This Table shows that the filter is unbiased and that the variance of $\Delta$
is comparable to the variance of the error in the observations $s=0.025$. Note
that even with one single particle (and therefore no resampling) the results
are still acceptable.
Table I
Mean and variance of the discrepancy between the observed path and the
reconstructed path in example 1 as a function of the number of particles M,
with $s=0.025$.
M | mean | variance
---|---|---
100 | -.0001 | .021
50 | -.0001 | .022
20 | -.0001 | .023
10 | .0001 | .024
5 | -.0001 | .027
1 | -.0001 | .038
We now discuss the relation between the posterior we wish to sample and the
prior in several special cases, including non-convex situations. We want to
produce samples of the pdf $P(x)=\exp(-F(x))/Z$, where $Z$ is a normalization
constant and
$F(x)=x^{2}/(2\sigma)+(h(x)-b)^{2}/(2s)$ (10)
and $h(x)$ is a given function of $x$ (as in equation (2)) and $\sigma,s,b$
are given parameters. This can be viewed as a the first time step in time for
a filtering problem where all the particles start from the same point so that
$\exp(-F(x))/Z=P_{1}$, or as an analysis of the sampling for one particular
particle in a general filtering problem, or as an instance of the more general
problem of sampling a given pdf when the important events may be rare. In
standard Bayesian sampling one samples the variable with pdf
$\exp(-x^{2})/(2\sigma))/\sqrt{2\pi\sigma}$ and then one attaches to the
sample at $x$ the weight $\exp(-(h(x)-b)^{2}/(2s))$; in an implicit sampler
one finds a sample $x$ by solving $F(x)-\phi=\xi^{2}/2$ for a suitable $\phi$
and $\xi$ and attaching to the sample the weight $\exp(-\phi)J$. For given
$\sigma,s$, the problem becomes more challenging as $|b|$ increases.
In both the standard and the implicit filters one can view the empirical pdf
generated by the unweighted samples as a “prior” and the one generated by the
weighted samples as the “posterior”. The difficulty with standard filters is
that the prior and posterior densities may approach being mutually singular,
so it is of interest to estimate the Radon-Nikodym derivative of one of these
with respect to the other. If that derivative is a constant, we have achieved
perfect importance sampling, as every neighborhood in the sample space is
visited with a frequency proportional to its density. We estimate the Radon-
Nikodym derivative of the prior with respect to the posterior as follows. In
this simple problem one can evaluate the probability of any interval with
respect to the posterior we wish to sample by quadratures. We divide the
interval $[0,1]$ into $K$ pieces of equal lengths $1/K$, then find numerically
points $Y_{1},Y_{2},\dots,Y_{K-1},$ with $Y_{K}=+\infty$, such that the
posterior probability of the interval $[-\infty,Y_{k}]$ is $k/K$ for
$k=1,2,\dots,K$. We then find $L=10^{5}$ samples of the prior and plot of a
histogram of the frequencies with which these samples fall into the posterior
equal probability intervals $(Y_{k-1},Y_{k})$. The more this histogram departs
from being a constant independent of $k$, the more samples are needed to
calculate the statistics of the posterior.
If $h(x)$ is linear, the weights in the implicit filter are all equal and the
histogram is constant for all values of $b$. This remains true for all values
of $b$, i.e., however far the observation $b$ is from what one may expect from
the SDE alone. This is not the case with a standard Bayesian filter, where
some parts of the sample space that have non-zero probability are visited very
rarely. In Table II we list the histogram of frequencies for a linear
observation function $h(x)=x$ and $b=2$ in a standard Bayesian filter, with
K=10. We used $10^{4}$ samples; the fluctuations in the implicit case measure
only the accuracy with which the histogram is computed with this number of
samples.
Table II
Histogram of the Radon-Nikodym derivative of the prior with respect to the
posterior, standard Bayesian filter vs. the implicit filter, 10000 particles,
$b=2$, $\sigma=s=0.1$, $h(x)=x$.
k | standard | implicit
---|---|---
1 | .987 | .099
2 | .006 | .108
3 | .002 | .097
4 | .001 | .099
5 | .004 | .101
6 | .003 | .099
7 | .001 | .101
8 | .001 | .101
9 | .000 | .102
10 | .000 | .093
As a consequence, estimates obtained with the implicit filter are much more
reliable than the ones obtained with the standard Bayesian filter. In Table
III we list the estimates of the mean position of the linear problem as a
function of b, with 30 particles, $\sigma=s=0.1$, for the standard Bayesian
and the implicit filters, compared with the exact result. The standard
deviations are not displayed, they are all near 0.01.
Table III
Comparison of the the estimates of the means, implicit vs. standard filter,
$30$ particles, together with the exact results, linear case, as explained in
the text.
b | exact | standard | implicit
---|---|---|---
0 | 0 | -.05 | .02
0.5 | .25 | .10 | .27
1. | .5 | .18 | .51
1.5 | .75 | .23 | .76
2. | 1. | .26 | 1.01
The results in this one-dimensional problem mirror the situation with the
example of Bickel et al. [2, 14], designed to display the breakdown of the
standard Bayesian filter when the number of dimension is large; what happens
there is that one particle hogs almost the whole weight, so that the number of
particles needed grows catastrophically; in contrast, the implicit filter
assigns equal weights to all the particles in any number of dimensions, so
that the number of particles needed is independent of dimension, see also [6].
Figure 3: A non-convex function $F$ (solid line) and a $U$-shaped substitute
(broken line).
We now turn to nonlinear and nonconvex examples. Let the observation function
$h$ be strongly nonlinear: $h(x)=x^{3}$. With $\sigma=s=0.1$; the pdf (10)
becomes non-$U$-shaped for $|b|\geq.77$. In Figure 3 we display the function
$F$ for $b=1$ (the solid curve). To use the algorithms above we need a
substitute function $F_{0}$ that is $U$-shaped; we also display in Figure 3
(the broken line) the function $F_{0}$ we used; the recipe here is to link a
point above the local minimum on the left to the absolute minimum on the right
by a straight line. There are many other possible constructions; the only
general rule is to make the minimum of $F_{0}$ equal the absolute minimum of
$F$, for obvious reasons. As described above, we solve
$F_{0}(x)-\phi=\xi^{T}\xi/2$ and set $\phi=\min F_{0}+F_{0}(x)-F(x)$. It is
important to note that this construction does not introduce any bias. The
function $F_{0}$ constructed in this way is $U$-shaped but need not be convex,
so that one needs algorithm (B) described above. In Table IV we compare the
Radon-Nikodym derivatives of the prior with respect to the posterior for the
resulting implicit sampling and for standard Bayesian sampling with
$\sigma=s=0.1,b=1.5$.
Table IV
Radon-Nikodym derivatives of the prior with respect to the posterior,
$h(x)=x^{3},\sigma=s=0.1,b=1.5,$ $10000$ samples, $F_{0}$ as in the text.
k | standard | explicit
---|---|---
1 | .9948 | .0899
2 | .0028 | .0537
3 | .0011 | .0502
4 | .0004 | .0563
5 | .0003 | .0696
6 | .0002 | .1860
7 | .0001 | .1107
8 | .0001 | .1194
9 | .0001 | .1196
10 | 0. | .1446
The histogram for the implicit filter is no longer perfectly balanced. The
asymmetry in the histogram reflects the asymmetry of $F_{0}$ and can be
eliminated by biasing $\xi$, but there is no reason to do so; there is enough
importance sampling without this extra step.
In Table V we display the estimates of the means of the density for the two
filters with 1000 particles for various values of $b$, compared with the exact
results (the number of particles is relatively large because with $h(x)=x^{3}$
and our parameter choices the variance of the conditional density is
significant, and this number of particles is needed for meaningful comparisons
of either algorithm with the exact result).
Table V
Comparison of the the estimates of the means, implicit vs. standard filter,
$1000$ particles, together with the exact result, when $h(x)=x^{3}$, as
explained in the text.
b | exact | standard | implicit
---|---|---|---
0. | 0. | -0.00 $\pm$.01 | -.00 $\pm$.01
.5 | .109 | .109 $\pm$.01 | .109$\pm$.01
1.0 | .442 | .394 $\pm$ .04 | .451$\pm$.02
1.5 | .995 | .775$\pm$.09 | .995$\pm$.01
2.0 | 1.18 | .875$\pm$.05 | 1.18$\pm$.01
2.5 | 1.30 | .895 $\pm$.02 | 1.29$\pm$.02
As mentioned in the previous section, there are alternatives to the
replacement of $F$ by $F_{0}$; the point is that for each particle the
function $F$ is an explicitly known non-random function, and this fact can be
used in multiple ways.
## 6 Parameter identification
One important application of particle filters is to parameter identification,
where the SDE contains an unknown parameter and the data are used to find this
parameter’s value. One of the standard ways of doing this (see e.g [7]) is
system augmentation: one adds to the SDE the equation $d\sigma=0$ for the
unknown parameter $\sigma$, one offers $\sigma$ a gamut of possible values,
and one relies on the resampling process that eliminate the values that do not
fit the data. With the implicit filter this procedure fails, because the
particles are not eliminated fast enough. The alternative we are proposing is
finding the unknown parameter $\sigma$ by stochastic approximation.
Specifically, Find a statistic $T$ of the output of the filter which is a
function of $\sigma$, such that the expected value $E[T]$ vanishes when
$\sigma$ has the right value $\sigma^{*}$, and then solve the equation
$E[T]=E[T(\sigma)]=0$ by the Robbins-Monro algorithm [13], in which the
equation $E[T]=0$ is solved by the iteration:
$\sigma_{n+1}=\sigma_{n}-\alpha_{n}T(\sigma_{n}),$ (11)
where which converges when the coefficients $\alpha_{n}$ are such that
$\sum\alpha_{n}\rightarrow\infty$ while $\sum\alpha_{n}^{2}$ remains bounded.
As a concrete example, consider the SDE $dx=dW$, where $W$ is Brownian motion
with variance $\sigma$, discretized with time steps $\delta$, with
observations $b^{n}=x^{n}+\eta$, where $\eta$ is a Gaussian with mean zero and
variance $s$. Data are generated by running the SDE once with the true value
$\sigma^{*}$ of $\sigma$, adding the appropriate noise, and registering the
result at time $n\delta$ as $b^{n}$ for $n=1,2,\dots,N$. For the functional
$T$ we choose
$T(\sigma)=C\sum(\Delta_{i}\Delta_{i-1})/\left((\sum\Delta_{i}^{2})(\sum\Delta^{2}_{i-1})\right)^{1/2},$
(12)
where the summations are over $i$ between $2$ and $N$, $\Delta_{i}$ is the
estimate of the increment of $x$ in the $i$-the step and $C$ is a scaling
constant. Clearly if the $\sigma$ used in the filtering equals $\sigma^{*}$
then by construction the successive values of $\Delta_{i}$ are independent and
$E[T]=0.$. We picked the parameters
$N=100,\sigma=10^{-2},s=10^{-4},\delta=0.01$ (so that that the increment of
$W$ in one step has variance $10^{-4}$).
Our algorithm is as follows: We make a guess $\sigma_{1}$, run the filter for
$N$ steps, evaluate $T$, and make a new guess for $\sigma$ using equation (11)
and $a_{1}=1$, rerun the filter, etc., with the $a_{n}$, the coefficient in
equation (11) at the $n$-th step, equal to $1/n$. The scaling factor in (11)
was found by trial and error: if it is too large the iteration becomes
unstable, if it is too small the convergence is slow; we settled on $C=4$.
This algorithm requires that the filter be run without either resampling or
backward sampling, because resampling and backward sampling introduce
correlations between successive values of the $\Delta_{i}$ and bias the values
of $T$. In a long run, in particular in a strongly nonlinear setting, one may
need resampling for the filter to stay on track, and this can be done by
segmentation: divide the run of the filter into segments of some moderate
length $L$, perform the summations in the definition of $T$ over that segment,
then go back and run that segment with resampling, then proceed to the next
segment, etc.
The first question is, how well is it possible in principle to reconstruct an
unknown value of $\sigma$ from $N$ observations; this issue was already
discussed in [5]. Given $100$ samples of a Gaussian variable of mean $0$ and
variance $\sigma$, the variance reconstructed from the observations is a
random variable of mean $\sigma$ and variance $.16\cdot\sigma$; $100$
observations do not contain enough information to reconstruct $\sigma$
perfectly. A good way to estimate the best result that can be achieved is to
run the algorithm with the guess $\sigma_{1}$ equal to the exact value
$\sigma^{*}$ with which the data were generated. When this was done, the
estimate of $\sigma$ was $1.27\sigma^{*}$. This result indicates the
achievable accuracy.
In Table VI we display the result of our algorithm when we start with the
starting value $\sigma_{1}=10\sigma^{*}$ and with $50$ particles. Each
iteration requires that one run the filter once.
Table VI
Convergence of the parameter identification algorithm.
Iteration | new estimate $\sigma/\sigma^{*}$
---|---
0 | 10.
1 | .819
2 | .943
3 | 1.02
4 | 1.05
5 | 1.08
6 | 1.10
7 | 1.13
8 | 1.15
9 | 1.16
10 | 1.17
11 | 1.18
12 | 1.18
13 | 1.18
## 7 Conclusions
We have presented the implicit filter for data assimilation, together with
several algorithms for the solution of the algebraic equations, including
cases with non-convex functions $F$, as well as an algorithm for parameter
identification. The key idea in implicit sampling is to solve an algebraic
equation of the form $F(X)-\phi=\xi^{T}\xi/2$ for every particle, where the
function $F$ is explicitly known, $X$ is the new position of the particle,
$\phi$ is an additive factor, and $\xi$ is a sample of a fixed reference pdf;
$F$ varies from particle to particle and step to step. This construction makes
it possible to guide the particles to the high-probability area one by one
under a wide variety of circumstances. It is important to note that the
equation that links $\xi$ to $X$ is underdetermined and its solution can be
adapted for each particular problem.
Implicit sampling is of interest in particular because of its potential uses
in high dimensional problems, which are only briefly alluded to in the present
paper. The effectiveness of implicit sampling in high-dimensional settings
depends on one’s ability to design maps $\xi\rightarrow x$ that satisfy the
criteria above and are computationally efficient. The design of such maps is
problem dependent and we will present examples in the context of specific
applications.
Acknowledgements We would like to thank Prof. Jonathan Weare for asking
penetrating questions and for making very useful suggestions, Prof. Robert
Miller for good advice and encouragement, and Mr. G. Zehavi for performing
some of the preliminary computations. This work was supported in part by the
Director, Office of Science, Computational and Technology Research, U.S.
Department of Energy under Contract No. DE-AC02-05CH11231, and by the National
Science Foundation under grants DMS-0705910 and OCE-0934298.
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2024-09-04T02:49:10.602581
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alexandre J. Chorin, Matthias Morzfeld, Xuemin Tu",
"submitter": "Xuemin Tu",
"url": "https://arxiv.org/abs/1005.4002"
}
|
1005.4201
|
# The stability of Einstein static universe in the DGP braneworld
Kaituo Zhang, Puxun Wu, Hongwei Yu 111Corresponding author Department of
Physics and Institute of Physics,
Hunan Normal University, Changsha, Hunan 410081, China
Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of
Ministry of Education, Hunan Normal University, Changsha, Hunan 410081, China
###### Abstract
The stability of an Einstein static universe in the DGP braneworld scenario is
studied in this letter. Two separate branches denoted by $\epsilon=\pm 1$ of
the DGP model are analyzed. Assuming the existence of a perfect fluid with a
constant equation of state, $w$, in the universe, we find that, for the branch
with $\epsilon=1$, there is no a stable Einstein static solution, while, for
the case with $\epsilon=-1$, the Einstein static universe exists and it is
stable when $-1<w<-\frac{1}{3}$. Thus, the universe can stay at this stable
state past-eternally and may undergo a series of infinite, non-singular
oscillations. Therefore, the big bang singularity problem in the standard
cosmological model can be resolved.
###### pacs:
98.80.Cq, 04.50.Kd
## I Introduction
Although most of the problems in the standard cosmological model can be
resolved by the inflation theory, the resolution of the existence of a big
bang singularity in the early universe is still elusive. Based upon the
string/M-theory, the pre-big bang Gasperini2003 and ekpyrotic/cyclic
Khoury2001 scenarios have been proposed to address the issue. Recently, Ellis
et al proposed, in the context of general relativity, a new scenario, called
an emergent universe Ellis2004a ; Ellis2004b to avoid this singularity. In
this scenario, the space curvature is positive, which is supported by the
recent observation from WMAP7 Komatsu2010 where it was found that a closed
universe is favored at the $68\%$ confidence level, and the universe stays,
past-eternally, in an Einstein static state and then evolves to a subsequent
inflationary phase. So, in an emergent theory, the universe originates from an
Einstein static state rather than from a big bang singularity. However, the
Einstein static universe in the classical general relativity is unstable,
which means that it is extremely difficult for the universe to remain in such
an initial static state in a long time due to the existence of perturbations,
such as the quantum fluctuations. Therefore, the original emergent model does
not seem to resolve the big bang singularity problem successfully as expected.
However, in the early epoch, the universe is presumably under extreme physical
conditions, the realization of the initial state may be affected by novel
physical effects, such as those stemming from quantization of gravity, or a
modification of general relativity or even other new physics. As a result, the
stability of the Einstein static state has been examined in various cases
Carneiro2009 ; Mulryne2005 ; Parisi2007 ; Wu2009 ; Lidsey2006 ; Bohmer2007 ;
Seahra2009 ; Bohmer2009 ; Barrow2003 ; Barrow2009 ; Clifton2005 ; Boehmer2010
; Boehmer20093 ; Wu20092 ; Odrzywolek2009 , from loop quantum gravity
Mulryne2005 ; Parisi2007 ; Wu2009 to modified gravity (for a review see Ref.
Boehmer2010 ), from Horava-Lifshitz gravity Boehmer20093 ; Wu20092 to
Shtanov-Sahni braneworld scenario Lidsey2006
In this paper, we plan to examine the stability of the Einstein static
universe in the DGP brane-world model Dvali2000 . In this braneworld, the
whole energy-momentum is confined on a three dimensional brane embedded in a
five-dimensional, infinite-volume Minkowski bulk. Since there are two
different ways to embed the 4-dimensional brane into the 5-dimensional
spacetime, the DGP model has two separate branches denoted by $\epsilon$ with
distinct features. The $\epsilon=+1$ branch can explain the present
accelerating cosmic expansion without the introduction of dark energy
Deffayer2001 , while for the $\epsilon=-1$ branch, dark energy is needed in
order to yield an accelerating expansion, as is the case in the LDGP model
Lue2004 and the QDGP model Chimento2006 . Using the $H(z)$, CMB shift and Sne
Ia observational data, Lazkoz and Majerotto Lazkoz2007 found that the LDGP
and QDGP are slightly more favored than the self-accelerating DGP model. Let
us also note that a crossing of a phantom divide line, which is favored by the
recent various observational data Alam2004 , is possible with a single scalar
field Zhang20062 ; Chimento2006 in the $\epsilon=-1$ branch. In addition,
inflation in the DGP model displays some new characteristics. It should be
noted, however, that only in the $\epsilon=-1$ case can inflation exit
spontaneously Bouhmadi-Lopez2004 ; Cai2004 ; Papantonopoulos2004 ; Zhang2004 ;
Zhang2006 ; Campo2007 . Also, in contrast to the Randall-Sundrum Randall1999
and Shtanov-Sahni Shtanov2003 braneworld scenarios with high energy
modifications to general relativity, the DGP brane produces a low energy
modification (for a review of the phenomenology of the DGP model, see Ref.
Lue2006 ).
## II The friedmann equation in DGP braneworld
For a homogeneous and isotropic universe which is described by the Friedmann-
Robertson-Walker (FRW) metric.
$\displaystyle
ds^{2}=-dt^{2}+a^{2}(t)\bigg{(}\frac{dr^{2}}{1-kr^{2}}+r^{2}d^{2}\Omega\bigg{)}\;,$
(1)
the Friedmann equation on the warped DGP brane can be written as Maeda2003
$\displaystyle
H^{2}+\frac{k}{a^{2}}=\frac{1}{3\mu^{2}}[\rho+\rho_{0}(1+\epsilon\mathcal{A}(\rho,a))]\;,$
(2)
where $H$ is the Hubble parameter, $k$ is the constant curvature of the three-
space of the FRW metric, $\rho$ is the total energy density and $\mu$ is a
parameter denoting the strength of the induced gravity on the brane. For
$\epsilon=-1$, the brane tension can be assumed to be positive, while for
$\epsilon=+1$, it is negative. $\mathcal{A}$ is given by
$\displaystyle\mathcal{A}=\bigg{[}\mathcal{A}_{0}^{2}+\frac{2\eta}{\rho_{0}}\bigg{(}\rho-\mu^{2}\frac{\mathcal{E}_{0}}{a^{4}}\bigg{)}\bigg{]}^{1/2}\;,$
(3)
where
$\displaystyle\mathcal{A}_{0}=\sqrt{1-2\eta\frac{\mu^{2}\Lambda}{\rho_{0}}},\quad\eta=\frac{6m_{5}^{6}}{\rho_{0}\mu^{2}}\;\;\;(0<\eta\leq
1),\quad\rho_{0}=m_{\lambda}^{4}+6\frac{m_{5}^{6}}{\mu^{2}}\;,$ (4)
with $\Lambda$ defined as
$\displaystyle\Lambda=\frac{1}{2}(^{(5)}\Lambda+\frac{1}{6}\kappa_{5}^{6}\lambda^{2})\;.$
(5)
Here $\kappa_{5}$ is the 5-dimensional Newton constant, ${}^{(5)}\Lambda$ the
5-dimensional cosmological constant in the bulk, $\lambda$ the brane tension,
and $\mathcal{E}_{0}$ a constant related to Weyl radiation. For simplicity, we
will neglect the dark radiation term and restrict ourselves to the Randall-
Sundrum critical case, i.e. $\Lambda=0$, then Eq.(2) simplifies to
$\displaystyle
H^{2}+\frac{k}{a^{2}}=\frac{1}{3\mu^{2}}\bigg{(}\rho+\rho_{0}+\epsilon\rho_{0}\sqrt{1+\frac{2\eta\rho}{\rho_{0}}}\bigg{)}.$
(6)
Since in the very early era of the universe, the total energy density should
be very high. Thus, we will, in the following, only consider the ultra high
energy limit, $\rho$$\gg$$\rho_{0}$. In addition, we are interested in a
closed universe, so we set the constant curvature $k$ to be $+1$. As a result,
the Friedmann equation reduces to
$\displaystyle
H^{2}=\frac{1}{3\mu^{2}}(\rho+\epsilon\sqrt{2\rho\rho_{0}})-\frac{1}{a^{2}}.$
(7)
This describes a 4-dimensional gravity with minor corrections, which implies
that $\mu$ must have an energy scale as the Planck scale in the DGP model.
The energy density $\rho$ of a perfect fluid in the universe satisfies the
conservation equation
$\displaystyle\dot{\rho}=-3H(1+w)\rho,$ (8)
where $w=\frac{p}{\rho}$ is the equation of state of the perfect fluid. A
constant $w$ is considered in the present paper, which is a good approximation
if the perfect fluid is a scalar field and the variation of the potential of
scalar field is very slow with time.
Differentiating Eq. (7) with respect to cosmic time, one gets
$\displaystyle\dot{H}=-\frac{1}{2\mu^{2}}(\rho+p)\bigg{(}1+\epsilon\sqrt{\frac{\rho_{0}}{2\rho}}\bigg{)}+\frac{1}{a^{2}},$
(9)
Combining this equation with the Friedmann equation given in Eq. (7), we have
$\displaystyle\frac{\ddot{a}}{a}=-\frac{1}{2\mu^{2}}(\rho+p)\bigg{(}1+\epsilon\sqrt{\frac{\rho_{0}}{2\rho}}\bigg{)}+\frac{1}{3u^{2}}(\rho+\epsilon\sqrt{2\rho\rho_{0}}).$
(10)
## III The Einstein static solution
The Einstein static solution is given by the conditions $\dot{a}=0$ and
$\ddot{a}=0$, which imply
$\displaystyle a=a_{Es},\qquad H(a_{Es})=0\;.$ (11)
At the critical point determined by above conditions, we find, using Eq. (10)
$\displaystyle\sqrt{\rho_{Es}}=\frac{\epsilon\sqrt{2\rho_{0}}(1-3\omega)}{2(1+3\omega)},$
(12)
which means that in this dynamical system, there is only one Einstein static
state solution. In order to guarantee the physical meaning of $\rho_{Es}$, it
is necessary that
$\displaystyle\frac{\epsilon(1-3\omega)}{1+3\omega}\geq 0.$ (13)
Substituting Eq. (12) into the Friedmann equation, we obtain at the critical
point
$\displaystyle\frac{1}{a^{2}_{Es}}=\frac{\rho_{0}(1-3\omega)(1+\omega)}{2\mu^{2}(1+3\omega)^{2}},$
(14)
with the requirement $(1-3\omega)(1+\omega)>0$.
Before analyzing the stability of the critical point, we want to express Eq.
(10) in terms of $a$ and $H$. To do so, we put the Friedmann equation in a
different way
$\displaystyle\sqrt{\rho}=\frac{\sqrt{2}}{2}\bigg{(}-\epsilon\sqrt{\rho_{0}}+\sqrt{\rho_{0}+6\mu^{2}\bigg{(}H^{2}+\frac{1}{a^{2}}\bigg{)}}\bigg{)}.$
(15)
Thus Eq. (10) can be re-written as
$\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle=$
$\displaystyle-\frac{1}{4\mu^{2}}(1+\omega)\rho_{o}+\frac{1}{4\mu^{2}}\epsilon(1+\omega)\sqrt{\rho_{0}^{2}+6\mu^{2}\rho_{0}\bigg{(}H^{2}+\frac{1}{a^{2}}\bigg{)}}$
(16)
$\displaystyle\quad-\frac{1}{2}(1+3\omega)\bigg{(}H^{2}+\frac{1}{a^{2}}\bigg{)}.$
Now we study the stability of the critical point. For convenience, we
introduce two variables
$\displaystyle x_{1}=a,\quad x_{2}=\dot{a}.$ (17)
It is then easy to obtain the following equations
$\displaystyle\dot{x_{1}}=x_{2},$ (18)
$\displaystyle\dot{x_{2}}=-\frac{1}{4\mu^{2}}\rho_{0}(1+w)x_{1}+\frac{1}{4\mu^{2}}\epsilon(1+\omega)\sqrt{\rho_{0}^{2}x_{1}^{2}+6\mu^{2}\rho_{0}(1+x_{2}^{2})}-\frac{1}{2}(1+3\omega)\frac{x_{2}^{2}+1}{x_{1}}.$
(19)
In these variable, the Einstein static solution corresponds to the fixed
point, $x_{1}=a_{Es},\;x_{2}=0$. The stability of the critical point is
determined by the eigenvalue of the coefficient matrix resulting from
linearizing the system described by above two equations near the critical
point. Using $\lambda^{2}$ to denote the eigenvalue, we have
$\displaystyle\lambda^{2}=\frac{\rho_{0}}{8\mu^{2}}\epsilon(1+\omega)|1+3\omega|-\frac{3\rho_{0}}{2\mu^{2}}\frac{\omega(1+\omega)}{1+3\omega}$
(20)
If $\lambda^{2}<0$, the corresponding equilibrium point is a center point
otherwise it is a saddle one. In order to analyze the stability of the
critical point in detail, we now divide our discussions into two cases, i.e.,
$\epsilon=-1$ and $\epsilon=1$.
A. $\epsilon=1$
In this case, $-\frac{1}{3}<\omega<\frac{1}{3}$ is required to ensure that the
critical point is physically meaningful. It then follows that $\lambda^{2}>0$,
which means that this critical point is a saddle point. Thus, there is no
stable Einstein static solution, and an emergent universe is not realistic in
this case.
B. $\epsilon=-1$
Now the requirement for the critical point to be physically meaningful is
$-1<\omega<-\frac{1}{3}$. This exactly agrees with the condition of stability
($\lambda^{2}<0$). Hence, as long as the critical point exists, it is always
stable. So, if the scale factor satisfies the condition given in Eq. (14)
initially and $w$ is within the region of stability, the universe can stay at
this stable state past-eternally, and may undergo a series of infinite, non-
singular oscillations, as shown in Fig. (1). As a result, the big bang
singularity can be avoided successfully.
Figure 1: The evolutionary curve of the scale factor with time (left) and the
phase diagram in space ($a$, $\dot{a}$) (right) for the case $\epsilon=-1$ in
Planck unit and with $w=-0.50$.
## IV Leaving the Einstein static state
Now, we have shown that an stable Einstein static state exists in the
$\epsilon=-1$ branch. However, in order to have a successful cosmological
scenario, a graceful exit to an inflationary epoch is needed. This is possible
in the following sense. In the analysis carried out in the present paper, the
equation of state $w$ of the perfect fluid in the universe is assumed to be a
constant, and this is a good approximation if the energy component in the
early universe is only that of a minimally coupled scalar field with a self-
interaction potential. One can show that the kinetic energy and potential
energy of this scalar field should be both non-zero constants for an Einstein
static solution Ellis2004a ; Ellis2004b ; Mulryne2005 ; Lidsey2006 . That is
to say, the scalar field rolls along a plateau potential. However a realistic
inflationary model clearly requires the potential to vary as the scalar field
evolves. Thus, the constant potential is merely a past-asymptotic limit of a
smoothly varying one, as pointed out in Refs. Ellis2004a ; Ellis2004b ;
Mulryne2005 ; Lidsey2006 ; Carneiro2009 . So, the essentially slowly varying
potential will eventually break the equilibrium of the Einstein static state
and lead to an exit from the initial Einstein phase to an inflationary one.
Some specific forms of such a potential that implements these features have
been constructed in Refs Ellis2004a ; Ellis2004b ; Mulryne2005 .
## V Conclusions
In this paper, we have studied the existence and stability of the Einstein
static universe in the DGP braneworld scenario. By assuming the existence of a
perfect fluid with a constant equation of state, which is a good approximation
if the perfect fluid is a scalar field and the variation of the potential of
scalar field is very slow with time, we have shown that for the branch with
$\epsilon=1$, there is no stable Einstein static universe, whereas, for the
branch with $\epsilon=-1$, the Einstein static universe exists and it is
stable if the equation of state $w$ satisfies $-1<\omega<-\frac{1}{3}$. Thus,
the universe can stay at this stable state past-eternally, and may undergo a
series of infinite, non-singular oscillations. Hence, in the $\epsilon=-1$
branch of the DGP model, the universe can originate from an Einstein static
state and then enter an inflation era. Furthermore, the universe can exit,
spontaneously, this inflation phase to a radiation dominated era, as shown in
previous studies Bouhmadi-Lopez2004 ; Cai2004 ; Papantonopoulos2004 ;
Zhang2004 ; Zhang2006 ; Campo2007 . As a result, the big bang singularity
problem in the standard cosmological scenario can be resolved successfully.
###### Acknowledgements.
This work was supported in part by the National Natural Science Foundation of
China under Grants Nos. 10775050, 10705055 and 10935013, the SRFDP under Grant
No. 20070542002, the FANEDD under Grant No. 200922, the National Basic
Research Program of China under Grant No. 2010CB832803, the NCET under Grant
No.09-0144, the PCSIRT under Grant No. IRT0964, and the Programme for the Key
Discipline in Hunan Province.
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|
arxiv-papers
| 2010-05-23T13:37:25 |
2024-09-04T02:49:10.617611
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kaituo Zhang, Puxun Wu, Hongwei Yu",
"submitter": "Puxun Wu",
"url": "https://arxiv.org/abs/1005.4201"
}
|
1005.4582
|
# Axigluon on like-sign charge asymmetry ${\cal A}^{b}_{s\ell}$, FCNCs and CP
asymmetries in $B$ decays
Chuan-Hung Chen1,2111Email: phychen@mail.ncku.edu.tw and Gaber
Faisel3,4222Email:gfaisel@cc.ncu.edu.tw 1Department of Physics, National
Cheng-Kung University, Tainan, 701 Taiwan
2National Center for Theoretical Sciences, Hsinchu 300, Taiwan
3 Egyptian Center for Theoretical Physics, Modern University for Information
and Technology, Cairo, Egypt
4Physics Department, Faculty of Education, Thamar University, Thamar ,Yemen
###### Abstract
A non-universal axigluon in generalized chiral color models leads to flavor
changing neutral currents (FCNCs) at tree level. We analyze phenomenologically
the new contributions to $B_{q}$ (q=d, s) mixing and the related CP
asymmetries (CPAs) that are generated by axigluon exchange. We find that
although $\Delta m_{B_{q}}$ can give a strict constraint on the parameters of
$b\to q$ transition, the precise measurement of $\sin 2\beta_{J/\Psi K^{0}}$
can further exclude the parameter space of $b\to d$ transition. The axigluon-
mediated effects can enhance the like-sign dimuon charge asymmetry ${\cal
A}^{b}_{s\ell}$ by one order of magnitude larger than the standard model
prediction. Accordingly, large CPA $\sin 2\beta^{J/\Psi\phi}_{s}$ and CPA
difference $\sin 2\beta_{J\Psi K^{0}}-\sin 2\beta_{\phi K^{0}}$ are achieved.
## I Introduction
In the standard model (SM), with three families of quarks, the unique CP
violating phase of the Cabibbo-Kobayashi-Maskawa (CKM) matrix can explain some
of the observed CP violating phenomena in $K$ and $B$ systems. However, the
failure of the KM phase in explaining the matter-antimatter asymmetry and some
recent measurements of CP violating observations in $B$ meson mixings and
decays motivates the search for new source of CP violation (CPV). Therefore,
it is an important issue to explore and to find new CP violating effects in
various systems, such as cosmos, Large Hadron Collider (LHC), Tevatron, $B$
factories etc.
Recently, several hints for the existence of new CP violating sources are
revealed in experiments. The first hint is observed in the CP asymmetries
(CPAs) of $B\to\pi K$ decays where by naive SM estimation, one expects that
$\bar{B}_{d}\to\pi^{+}K^{-}$ and $B^{-}\to\pi^{0}K^{-}$ decays have similar
CPAs. However, it is surprising that the world average difference between the
two CPAs contradicts the expectation as the experimental result is
TheHeavyFlavorAveragingGroup:2010qj
$\displaystyle\Delta
A_{CP}=A_{CP}(\pi^{+}K^{-})-A_{CP}(\pi^{0}K^{-})=-(14.8_{-1.4}^{+1.3})\%\,,$
(1)
whereas the SM prediction is $\Delta A_{CP}(SM)=0.025\pm 0.015$ Beneke:2003zv
. The large deviation from the SM prediction indicates a puzzle in the
asymmetries and it is introduced in the literature as $B\to\pi K$ puzzle
pikpuz . The second hint is observed in the time-dependent CPA of $B_{s}$
system, where CDF and DØ have shown an unexpected large CP phase in the
mixing-induced CPA for $B_{s}\to J/\Psi\phi$ and the two possible solutions
are given by TheHeavyFlavorAveragingGroup:2010qj
$\displaystyle 2\beta^{J/\Psi\phi}_{s}=2\beta_{s}+2\phi^{\rm
NP}_{s}=-0.75^{+0.32}_{-0.21}\ {\texttt{or}}\ -2.38^{+0.25}_{-0.34}$ (2)
at $90\%$ confidence level (CL). Here, $\beta_{s}\approx-0.019$ Chen:2008ug
is the SM CP violating phase and $\phi^{\rm NP}_{s}$ is the CP violating phase
of new physics. The significant deviation from the SM prediction could be
speculated by the contributions of new physics.
The third hint is observed in the like-sign charge asymmetry which is defined
as Abazov:2010hv
$\displaystyle{\cal A}^{b}_{s\ell}$ $\displaystyle=$
$\displaystyle\frac{N^{++}_{b}-N^{--}_{b}}{N^{++}_{b}+N^{--}_{b}}\,,$ (3)
where $N^{++(--)}_{b}$ denotes the number of events that $b$\- and
$\bar{b}$-hadron semileptonically decay into two positive (negative) muons.
Recently, DØ Collaboration has announced the measurement on ${\cal
A}^{b}_{s\ell}$ in the dimuon events Abazov:2010hv with
$\displaystyle{\cal A}^{b}_{s\ell}=\left(-9.57\pm 2.51({\rm stat})\pm
1.46({\rm syst})\right)\times 10^{-3}\,.$ (4)
The SM prediction is ${\cal A}^{b}_{s\ell}=(-2.3^{+0.5}_{-0.6})\times 10^{-4}$
Abazov:2010hv ; Lenz:2006hd . If the semileptonic b-hadron decays do not
involve CP violating phase, then the charge asymmetry is directly related to
the mixing-induced CPAs in $B_{d}$\- and $B_{s}$-meson oscillations. Although
the errors of the data are still large, however the $3.2$ standard deviations
from the SM prediction can be attributed to new CP violating phases in $b\to
d$ and $b\to s$ transitions Randall:1998te ; Dighe:2010nj ; Dobrescu:2010rh ;
Choudhury:2010ya .
In order to explore the new physics and to avoid the uncontrollable QCD
uncertainties, we will concentrate our study on the mixing parameter $\Delta
m_{B_{q}}$, the charge asymmetry ${\cal A}^{b}_{s\ell}$ and the time-dependent
CPA in $B_{q}$ oscillation, where QCD effects can be controlled well by
Lattice QCD.
In the literature, many extensions of the SM such as chiral color models
chiral ; nonuni ; Sehgal:1987wi ; Doncheski:1997yj ; Giordani:2003ib ;
Choudhury:2007ux , $Z^{\prime}$ models LL_PRD45 ; BZ1 ; BZ2 etc have been
proposed. The flavor non-universal axigluon in the generalized chiral color
models AKR ; Frampton:2009rk has been studied for solving the anomalous
forward-backward asymmetry (FBA) in the top-quark pair production at the
Tevatron D0_PRL100 ; CDF_PRL101 . Although other models such as Z’, diquarks
models Arhrib:2009hu etc may have significant contributions to the FBA,
however, large gauge couplings and flavor changing effects should be
introduced in which chiral color model does not need. Inspired by the effects
of the axigluon on the top-quark FBA, we study the axigluon-mediated phenomena
in $B$-meson system.
A flavor universal axigluon has flavor-conserving effects only. For non-
universal axigluon which has different couplings to different quarks, flavor
changing neutral currents (FCNCs) can be generated at tree level. This is
achieved after transforming the weak eigenstates of the quarks into their
physical eigenstates. As a consequence, many phenomena will be affected by
these FCNC effects. In this paper, we analyze in detail the non-universal
axigluon contributions to the time-dependent CPAs in $B_{q}$ oscillation after
taking into account the constraint from the mixing parameter $\Delta
m_{B_{q}}$.
This paper is organized as follows. In Sec. II, we formulate the interactions
of $b\to q$ transitions which are induced by flavor non-universal axigluon
exchange. Accordingly, we derive the corresponding effective Hamiltonian for
$\Delta B=1,2$ processes. Furthermore, we discuss the contributions of the
axigluon to the charge asymmetry ${\cal A}^{b}_{s\ell}$ and the time-dependent
CPAs for $B_{d}\to J/\psi K^{0}$, $B_{d}\to\phi K^{0}$, $B_{s}\to J/\Psi\phi$
decays. The detailed numerical analysis is presented in Sec. III. We give the
conclusion in Sec. IV.
## II Formalism
In order to study the contributions of the non-universal axigluon to the FCNC
processes, we start by writing the interactions of the massive color-octet
gauge boson with quarks as
$\displaystyle{\cal L}_{A}$ $\displaystyle=$ $\displaystyle
g_{V}\bar{q}^{\prime}\gamma_{\mu}T^{b}q^{\prime}G^{b\mu}_{A}+g_{A}\bar{q}^{\prime}\gamma_{\mu}\gamma_{5}{\bf
Z}T^{b}q^{\prime}G^{b\mu}_{A}\,,$ (5)
where we have suppressed the flavor and color indices, $g_{V,A}$ are the gauge
couplings of the new gauge group $SU(3)_{A}\times SU(3)_{B}$, $T^{b}$ are the
Gell-Mann matrices which are normalized by $Tr(T^{b}T^{c})=\delta^{ac}/2$ and
$\bf Z$ is $3\times 3$ diagonalized matrix with diag(Z)=(1, 1, $\zeta$). Here
$\zeta=\tilde{g}_{A}/g_{A}$ where $\tilde{g}_{A}$ denotes the gauge coupling
of the third-generation quark and its value depends on a specific model, e.g.
$\zeta=-1$ in Ref. Frampton:2009rk . For simplicity, we assume that the new
exotic quarks which are required for anomaly free are very heavy and their
effects are negligible. Hence, we still focus on three flavors for each up and
down type quarks. Following the scenario in Refs. AKR ; Frampton:2009rk for
solving the large top-quark FBA, we assume that the axigluon couplings to the
third generation are different from their couplings to the first two
generations. The left- and right-handed quarks are $SU(2)$ doublet and singlet
respectively. Thus, after spontaneous symmetry breaking, the interacting and
physical eigenstates can be related by unitary matrices as
$q_{\chi}=V^{Q}_{\chi}q^{\prime}$ with $\chi$ being the chiralities $L$ and
$R$ and $Q$ being up or down type quarks. Since $\bf Z$ is not a unit matrix,
the FCNCs are arisen from the axial-vector currents and the corresponding
Lagrangian is given by
$\displaystyle{\cal L}_{FCNC}$ $\displaystyle=$ $\displaystyle
g_{A}\bar{q}\gamma_{\mu}(V^{Q}_{R}{\bf Z}V^{Q\dagger}_{R}P_{R}-V^{Q}_{L}{\bf
Z}V^{Q\dagger}_{L}P_{L})T^{b}qG^{b\mu}_{A}$ (6)
with $P_{L(R)}=(1\mp\gamma_{5})/2$. Since $V^{Q}_{\chi}$ are unknown matrices,
the FCNCs are associated with left and right-handed currents generally.
Nevertheless, if $V^{Q}_{R}=V^{Q}_{L}$, from Eq. (6) we see that the FCNCs are
only associated with axial-vector currents. In terms of the flavor indices,
the matrix $V^{q}_{\chi}{\bf Z}V^{q\dagger}_{\chi}$ can be decomposed as
$\displaystyle\left(V^{Q}_{\chi}{\bf Z}V^{Q\dagger}_{\chi}\right)_{ij}$
$\displaystyle=$ $\displaystyle\delta_{ij}+\left(V^{Q}_{\chi}({\bf
Z-1})V^{Q\dagger}_{\chi}\right)_{ij}=\delta_{ij}+(\zeta-1)(V^{Q}_{\chi})_{i3}(V^{Q*}_{\chi})_{j3}\,.$
(7)
Therefore, the Lagrangian of $b\to q$ transition can be written as
$\displaystyle{\cal L}_{b\to q}$ $\displaystyle=$ $\displaystyle
g_{A}\bar{q}\gamma_{\mu}(F^{QR}_{qb}P_{R}-F^{QL}_{qb}P_{L})T^{b}bG^{b\mu}_{A}$
(8)
with $F^{Q\chi}_{qb}=(\zeta-1)(V^{Q}_{\chi})_{i3}(V^{Q*}_{\chi})_{33}$ where
$i=(1,2,3)$ denotes the family order of the same type $Q$ quark. Based on Eq.
(8), we study the impacts of non-universal axigluon exchange on $\Delta B=2$
processes and the time-dependent CPAs in $B_{q}$ system.
By Eq. (8), the effective Hamiltonian for $\Delta B=2$ transitions which is
generated by the tree-level axigluon mediation can be written as
$\displaystyle{\cal H}^{A}_{\Delta B=2}$ $\displaystyle=$
$\displaystyle\frac{g^{2}_{A}}{4m^{2}_{V}}\left[-\frac{1}{N_{C}}\left(\bar{q}\gamma_{\mu}(F^{DR}_{qb}P_{R}+F^{DL}_{qb}P_{L})b\right)^{2}\right.$
(9) $\displaystyle+$
$\displaystyle\left.\bar{q}_{\alpha}\gamma_{\mu}\left(F^{DR}_{qb}P_{R}+F^{DL}_{qb}P_{L}\right)b_{\beta}\bar{q}_{\beta}\gamma^{\mu}\left(F^{DR}_{qb}P_{R}+F^{DL}_{qb}P_{L}\right)b_{\alpha}\right]\,,$
where $N_{C}$ denotes the number of colors and we have used the identity
$\displaystyle
T^{b}_{ij}T^{b}_{k\ell}=-\frac{1}{2N_{C}}\delta_{ij}\delta_{k\ell}+\frac{1}{2}\delta_{i\ell}\delta_{jk}\,.$
(10)
In order to calculate the $B_{q}-\bar{B}_{q}$ mixing, we write the relevant
hadronic matrix elements to be
$\displaystyle\langle
B_{q}|\bar{q}\gamma_{\mu}P_{L(R)}b\bar{q}\gamma_{\mu}P_{L(R)}b|\bar{B}_{q}\rangle=\frac{1}{3}m_{B_{q}}f^{2}_{B_{q}}\hat{B}_{q}\,,$
$\displaystyle\langle
B_{q}|\bar{q}\gamma_{\mu}P_{R}b\bar{q}\gamma_{\mu}P_{L}b|\bar{B}_{q}\rangle=-\frac{5}{12}m_{B_{q}}f^{2}_{B_{q}}\hat{B}^{RL}_{1q}\,,$
$\displaystyle\langle
B_{q}|\bar{q}_{\alpha}\gamma_{\mu}P_{L}b_{\beta}\bar{q}_{\beta}\gamma^{\mu}P_{R}b_{\alpha}|\bar{B}_{q}\rangle=-\frac{7}{12}m_{B_{q}}f^{2}_{B_{q}}\hat{B}^{RL}_{2q}\,.$
(11)
To estimate the new physics effects, we employ the vacuum insertion method to
calculate the above matrix elements, i.e.
$\hat{B}_{q}\sim\hat{B}^{RL}_{1q}\sim\hat{B}^{RL}_{2q}\sim 1$ Gabbiani:1996hi
; Badin:2007bv . Additionally, in the heavy quark limit, we take $m_{b}\sim
m_{B_{q}}$. As a result, the transition matrix element for $B_{q}-\bar{B}_{q}$
oscillation mediated by axigluon exchange becomes
$\displaystyle M^{A,q}_{12}$ $\displaystyle=$ $\displaystyle\langle
B_{q}|{\cal H}^{A}_{\Delta
B=2}|\bar{B}_{q}\rangle=\frac{g^{2}_{A}}{18m^{2}_{V}}m_{B_{q}}f^{2}_{B_{q}}U^{D}_{qb}\,,$
$\displaystyle U^{D}_{qb}$ $\displaystyle=$
$\displaystyle(F^{DR}_{qb})^{2}+(F^{DL}_{qb})^{2}+4F^{DR}_{qb}F^{DL}_{qb}\,.$
(12)
For reducing the number of free parameters, we will take the approximation
$V^{Q}_{R}\approx V^{Q}_{L}=V^{D}$ in our analysis, i.e. $F^{DR}_{qb}\approx
F^{DL}_{qb}=F^{D}_{qb}$, then $U^{D}_{qb}=6(F^{D}_{qb})^{2}$. We note that the
approximation $V^{Q}_{R}\approx V^{Q}_{L}$ can be realized in hermitian Yukawa
matrices Chen:2001cv .
By combining the contributions of SM and axigluon, the transition matrix
element for $\Delta B=2$ can be formulated as
$\displaystyle M^{B_{q}}_{12}$ $\displaystyle=$ $\displaystyle|M^{\rm
SM,q}_{12}|R^{q}_{A}e^{i2(\beta_{q}+\phi^{\rm NP}_{q})}\,,$ (13)
where the new parameters are defined by
$\displaystyle R^{q}_{A}$ $\displaystyle=$
$\displaystyle\left(1+(r^{q}_{A})^{2}+2r^{q}_{A}\cos 2(\beta^{\rm
NP}_{q}-\beta_{q})\right)^{1/2}\,,$ $\displaystyle 2\beta^{\rm NP}_{q}$
$\displaystyle=$ $\displaystyle{\rm arg}(M^{A,q}_{12})\,,$ $\displaystyle
r^{q}_{A}$ $\displaystyle=$ $\displaystyle\frac{|M^{A,q}_{12}|}{|M^{\rm
SM,q}_{12}|}\,,$ $\displaystyle\tan 2\phi^{\rm NP}_{q}$ $\displaystyle=$
$\displaystyle\frac{r^{q}_{A}\sin 2(\beta^{\rm
NP}_{q}-\beta_{q})}{1+r^{q}_{A}\cos 2(\beta^{\rm NP}_{q}-\beta_{q})}\,,$ (14)
and $M^{SM,q}_{12}$ is given by BBL
$\displaystyle
M^{SM,q}_{12}=\frac{G^{2}_{F}m^{2}_{W}}{12\pi^{2}}\eta_{B}m_{B_{q}}f^{2}_{B_{q}}\hat{B}_{q}(V^{*}_{tq}V_{tb})^{2}S_{0}(x_{t})$
(15)
with $S_{0}(x_{t})=0.784x_{t}^{0.76}$, $x_{t}=(m_{t}/m_{W})^{2}$ and
$\eta_{B}\approx 0.55$ is the QCD correction to $S_{0}(x_{t})$ Hence, the mass
difference between heavy and light $B_{q}$ is $\Delta
m_{B_{q}}=2|M^{B_{q}}_{12}|=\Delta m^{\rm SM}_{B_{q}}R^{q}_{A}$. After
obtaining $M^{B_{q}}_{12}$, the time-dependent CPA through inclusive
semileptonic decays can be defined as Nakamura:2010zzi
$\displaystyle a^{q}_{s\ell}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\bar{B}_{q}(t)\to\ell^{+}X)-\Gamma(B_{q}(t)\to\ell^{-}X)}{\Gamma(\bar{B}_{q}(t)\to\ell^{+}X)+\Gamma(B_{q}(t)\to\ell^{-}X)}\,,$
(16) $\displaystyle=$ $\displaystyle\frac{1-|q/p|^{4}}{1+|q/p|^{4}}$
with
$\displaystyle\left(\frac{q}{p}\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{M^{B_{q}^{*}}_{12}-i\Gamma^{B_{q}^{*}}_{12}/2}{M^{B_{q}}_{12}-i\Gamma^{B_{q}}_{12}/2}\,,$
(17)
where $\Gamma^{B_{q}}_{12}$ denotes the absorptive part of
$B_{q}\leftrightarrow\bar{B}_{q}$ transition. Due to $\Gamma^{B_{q}}_{12}\ll
M^{B_{q}}_{12}$, the wrong-sign charge asymmetry can be simplified as
$\displaystyle a^{q}_{s\ell}$ $\displaystyle=$ $\displaystyle
Im\left(\frac{\Gamma^{B_{q}}_{12}}{M^{B_{q}}_{12}}\right)\approx\frac{\Delta\Gamma^{\rm
SM}_{B_{q}}}{\Delta m_{B_{q}}}\sin(2\beta_{q}+2\phi^{\rm
NP}_{q}-\theta^{\Gamma}_{q})\,.$ (18)
Here, $\theta^{\Gamma}_{q}$ stands for the phase of $\Gamma^{B_{q}}_{12}$.
Since the absorptive part is dominated by the SM contribution, we will assume
that $\Gamma^{B_{q}}_{12}=\Gamma^{q,SM}_{12}$ in our numerical analysis. A
detailed discussions about new physics effects on $\Gamma^{B_{q}}_{12}$ can be
found in Refs. Dighe:2010nj ; Choudhury:2010ya . Since $a^{q}_{s\ell}$ is
associated with the CP phases directly, a non-zero charge asymmetry will be an
indication of CP violation. Accordingly, the like-sign charge asymmetry
defined in Eq. (3) can be written as Abazov:2010hv ; Grossman:2006ce
$\displaystyle{\cal A}^{b}_{s\ell}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(b\bar{b}\to\ell^{+}\ell^{+}X)-\Gamma(b\bar{b}\to\ell^{-}\ell^{-}X)}{\Gamma(b\bar{b}\to\ell^{+}\ell^{+}X)+\Gamma(b\bar{b}\to\ell^{-}\ell^{-}X)}\,,$
(19) $\displaystyle=$
$\displaystyle\frac{f_{d}Z_{d}a^{d}_{s\ell}+f_{s}Z_{s}a^{s}_{s\ell}}{f_{d}Z_{d}+f_{s}Z_{s}}\,,$
where $f_{q}$ is the production fraction of $B_{q}$ and
$\displaystyle Z_{q}$ $\displaystyle=$
$\displaystyle\frac{1}{1-y^{2}_{q}}-\frac{1}{1-x^{2}_{q}}\,,$ $\displaystyle
y_{q}$ $\displaystyle=$
$\displaystyle\frac{\Delta\Gamma_{B_{q}}}{2\Gamma_{B_{q}}}\,,\ \ \
x_{q}=\frac{\Delta m_{B_{q}}}{\Gamma_{B_{q}}}\,.$ (20)
Using $f_{d}=0.323(37)$, $f_{s}=0.118(15)$, $x_{d}=0.774(37)$, $y_{d}\sim 0$,
$x_{s}=26.2(5)$ and $y_{s}=0.046(27)$, the asymmetry can be rewritten as
$\displaystyle{\cal A}^{b}_{s\ell}=c_{d}a^{d}_{s\ell}+c_{s}a^{s}_{s\ell}$ (21)
with $c_{d}=0.506(43)$ and $c_{s}=0.494(43)$ Abazov:2010hv .
Another important time dependent CPA can be defined by Nakamura:2010zzi
$\displaystyle A_{f_{CP}}(t)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\bar{B}_{q}(t)\to f_{CP})-\Gamma(B_{q}(t)\to
f_{CP})}{\Gamma(\bar{B}_{q}(t)\to f_{CP})+\Gamma(B_{q}(t)\to f_{CP})}\,,$
$\displaystyle=$ $\displaystyle S_{f_{CP}}\sin\Delta
m_{B_{q}}t-C_{f_{CP}}\cos\Delta m_{B_{q}}t\,,$ $\displaystyle S_{f_{CP}}$
$\displaystyle=$
$\displaystyle\frac{2Im\lambda_{f_{CP}}}{1+|\lambda_{f_{CP}}|^{2}}\,,\ \ \
C_{f_{CP}}=\frac{1-|\lambda_{f_{CP}}|^{2}}{1+|\lambda_{f_{CP}}|^{2}}$ (22)
with
$\displaystyle\lambda_{f_{CP}}$ $\displaystyle=$
$\displaystyle-\left(\frac{M^{B_{q}^{*}}_{12}}{M^{B_{q}}_{12}}\right)^{1/2}\frac{A(\bar{B}\to
f_{CP})}{A(B\to f_{CP})}=-e^{-2i(\beta_{q}+\phi^{\rm
NP}_{q})}\frac{\bar{A}_{f_{CP}}}{A_{f_{CP}}}\,,$ (23)
where $f_{CP}$ denotes the final CP eigenstate, $S_{f_{CP}}$ and $C_{f_{CP}}$
are the so-called mixing-induced and direct CPAs, $A_{f_{CP}}$ and
$\bar{A}_{f_{CP}}$ are the amplitudes of $B$ and $\bar{B}$ mesons decaying to
$f_{CP}$ and
$\bar{A}_{f_{CP}}/A_{f_{CP}}=-\eta_{f_{CP}}A_{f_{CP}}(\theta_{W}\to-\theta_{W})/A_{f_{CP}}(\theta_{W})$
with $\eta_{f_{CP}}$ and $\theta_{W}$ are the CP eigenvalue of $f_{CP}$ and
the weak CP phase respectively. Clearly, besides $\Delta B=2$ effects, the
mixing-induced CPA is also related to the $\Delta B=1$ process. In this paper,
we will concentrate on $f_{CP}=J/\Psi K_{S}$ and $\phi K_{S}$ for $q=d$ and on
$f_{CP}=J/\Psi\phi$ for $q=s$.
To calculate the decay amplitude of $B(\bar{B})\to f_{CP}$, we need to discuss
the interactions of $\Delta B=1$ processes. With the approximation
$V^{Q}_{R}\approx V^{Q}_{L}$, the effective Hamiltonian of $b\to
qq^{\prime}q^{\prime}$ can be expressed as
$\displaystyle{\cal H}_{b\to qq^{\prime}q^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{g_{A}}{m^{2}_{V}}F^{D}_{qb}\bar{q}\gamma_{\mu}\gamma_{5}T^{b}b\sum_{q^{\prime}=u,d,s,c}\bar{q}^{\prime}\gamma^{\mu}\left(g_{+}P_{R}+g_{-}P_{L}\right)T^{b}q^{\prime}$
(24)
with $g_{\pm}=g_{V}\pm g_{A}$. Using Eq. (10), we can rewrite the last
equation as
$\displaystyle{\cal H}^{A}_{b\to qq^{\prime}q^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V^{*}_{tq}V_{tb}\left[C^{\prime}_{q3}O^{q}_{3}+C^{\prime}_{q4}O^{q}_{4}+C^{R}_{q3}O^{qR}_{3}+C^{R}_{q4}O^{qR}_{4}\right.$
(25) $\displaystyle+$
$\displaystyle\left.C^{\prime}_{q5}O^{q}_{5}+C^{\prime}_{q6}O^{q}_{6}+C^{L}_{q5}O^{qL}_{5}+C^{qL}O^{qL}_{6}\right]$
in which the new Wilson coefficients are expressed by
$\displaystyle C^{\prime}_{q3}$ $\displaystyle=$
$\displaystyle\frac{1}{8N_{C}}\frac{\sqrt{2}F^{D}_{qb}}{G_{F}V^{*}_{tq}V_{tb}}\frac{g_{A}g_{-}}{m^{2}_{V}}\,,\
\ \ C^{\prime}_{q4}=-N_{C}C^{\prime}_{q3}\,,$ $\displaystyle C^{L}_{q5}$
$\displaystyle=$ $\displaystyle-C^{\prime}_{q3}\,,\ \ \
C^{L}_{q6}=-N_{C}C^{L}_{q5}\,,$ $\displaystyle C^{\prime}_{q5}$
$\displaystyle=$
$\displaystyle\frac{1}{8N_{C}}\frac{\sqrt{2}F^{D}_{qb}}{G_{F}V^{*}_{tq}V_{tb}}\frac{g_{A}g_{+}}{m^{2}_{V}}\,,\
\ \ C^{\prime}_{q6}=-N_{C}C^{\prime}_{q5}\,,$ $\displaystyle C^{R}_{q3}$
$\displaystyle=$ $\displaystyle-C^{\prime}_{q5}\,,\ \ \
C^{R}_{q4}=-N_{C}C^{R}_{q3}$ (26)
and the associated operators are
$\displaystyle O^{q}_{3}$ $\displaystyle=$
$\displaystyle(\bar{q}b)_{V-A}\sum_{q^{\prime}}(\bar{q}^{\prime}q^{\prime})_{V-A}\,,\
\ \
O^{q}_{4}=(\bar{q}_{\alpha}b_{\beta})_{V-A}\sum_{q^{\prime}}(\bar{q}^{\prime}_{\beta}q^{\prime}_{\alpha})_{V-A}\,,$
$\displaystyle O^{q}_{5}$ $\displaystyle=$
$\displaystyle(\bar{q}b)_{V-A}\sum_{q^{\prime}}(\bar{q}^{\prime}q^{\prime})_{V+A}\,,\
\ \
O^{q}_{6}=(\bar{q}_{\alpha}b_{\beta})_{V-A}\sum_{q^{\prime}}(\bar{q}^{\prime}_{\beta}q^{\prime}_{\alpha})_{V+A}\,,$
$\displaystyle O^{qR}_{3}$ $\displaystyle=$
$\displaystyle(\bar{q}b)_{V+A}\sum_{q^{\prime}}(\bar{q}^{\prime}q^{\prime})_{V+A}\,,\
\ \
O^{qR}_{4}=(\bar{q}_{\alpha}b_{\beta})_{V+A}\sum_{q^{\prime}}(\bar{q}^{\prime}_{\beta}q^{\prime}_{\alpha})_{V+A}\,,$
$\displaystyle O^{qL}_{5}$ $\displaystyle=$
$\displaystyle(\bar{q}b)_{V+A}\sum_{q^{\prime}}(\bar{q}^{\prime}q^{\prime})_{V-A}\,,\
\ \
O^{qR}_{6}=(\bar{q}_{\alpha}b_{\beta})_{V+A}\sum_{q^{\prime}}(\bar{q}^{\prime}_{\beta}q^{\prime}_{\alpha})_{V-A}$
(27)
with $(\bar{f}^{\prime}f)_{V\pm
A}=\bar{f}^{\prime}\gamma_{\mu}(1\pm\gamma_{5})f$. Besides the new free
parameters that are introduced earlier, the non-leptonic $B$ decays suffer
from large uncertain QCD effects such as $\langle f_{CP}|{\cal H}_{b\to
qq^{\prime}q^{\prime}}|B\rangle$. For estimating the new physics effects, we
employ the naive factorization approach (NFA). Under the NFA, we find that the
related effective Wilson coefficients for $\bar{B}_{d}\to J/\Psi\bar{K}^{0}$
and $\bar{B}_{s}\to J/\Psi\phi$ are
$\displaystyle
C^{\prime}_{s3}+\frac{C^{\prime}_{s4}}{N_{C}}+C^{L}_{s5}+\frac{C^{R}_{s6}}{N_{C}}+C^{\prime}_{s5}+\frac{C^{\prime}_{s6}}{N_{C}}+C^{R}_{s3}+\frac{C^{R}_{s4}}{N_{C}}\,.$
(28)
With the results in Eq. (26), we clearly see that the influence of axigluon-
mediated effects on $J/\Psi(\bar{K}^{0},\phi)$ modes vanishes. In our analysis
we neglect the nonfactorizable contributions as they are subleading and
difficult to estimate. Now, only $B\to\phi K$ can display the axigluon-
mediated effects. Using NFA and the interactions of Eq. (25), the total decay
amplitude of $B\to\phi K$ is written as
$\displaystyle\bar{A}_{\phi\bar{K}^{0}}$ $\displaystyle=$
$\displaystyle\langle\phi\bar{K}^{0}|{\cal H}_{b\to
ss\bar{s}}|\bar{B}^{0}\rangle\,,$ (29) $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V^{*}_{ts}V_{tb}(a^{\rm
SM}+a^{\prime}_{s4}+a^{R}_{s4})\langle\phi|\bar{s}\gamma_{\mu}s|0\rangle\langle\bar{K}^{0}|\bar{s}\gamma^{\mu}b|\bar{B}\rangle$
where ${\cal H}_{b\to ss\bar{s}}$ is the sum of the SM and axigluon effective
Hamiltonian and $a^{\rm SM}=a_{3}+a_{4}+a_{5}$ with
$\displaystyle a_{3}$ $\displaystyle=$ $\displaystyle
C_{3}+\frac{C_{4}}{N_{C}}\,,\ \ \ a_{4}=C_{4}+\frac{C_{3}}{N_{C}}\,,\ \ \
a_{5}=C_{5}+\frac{C_{6}}{N_{C}}\,,$ $\displaystyle
a^{\prime}_{s4}=C^{\prime}_{s4}+C^{\prime}_{s3}/N_{C}\,,\ \ \
a^{R}_{s4}=C^{R}_{s4}+C^{R}_{s3}/N_{C}\,.$
Here, $C_{3-6}$ are the effective Wilson coefficients from the gluon penguin
in the SM BBL . We note that the electroweak penguin contributions are very
small and thus we neglect them. Using $V_{ts}=-|V_{ts}|e^{-i\beta_{s}}$
Nakamura:2010zzi , we can write
$\displaystyle\frac{\bar{A}_{\phi\bar{K}^{0}}}{A_{\phi
K^{0}}}=-e^{2i\beta_{s}}\frac{a^{SM}+a^{R}_{s4}}{a^{SM}+a^{R^{*}}_{s4}}=-e^{2i(\beta_{s}+\theta^{\rm
NP}_{s})}$ (30)
with
$\displaystyle\tan\theta^{\rm NP}_{s}$ $\displaystyle=$
$\displaystyle\frac{|a^{R}_{s4}|\sin(\beta^{\rm
NP}_{s}-\beta_{s})}{a^{SM}+|a^{R}_{s4}|\cos(\beta^{\rm NP}_{s}-\beta_{s})}\,.$
By Eqs. (22) and (23), the mixing-induced CPA via $B_{d}\to\phi K^{0}$ decay
is obtained as
$\displaystyle S_{\phi K^{0}}$ $\displaystyle\equiv$ $\displaystyle\sin
2\beta_{\phi K^{0}}=\sin 2(\beta_{d}+\phi^{\rm NP}_{d}-\beta_{s}-\theta^{\rm
NP}_{s})\,,$ (31)
while the CPAs through $B_{d,s}\to J/\Psi(K_{S},\phi)$ decays are given by
$\displaystyle S_{J/\Psi K^{0}}$ $\displaystyle\equiv$ $\displaystyle\sin
2\beta_{J/\Psi K^{0}}=\sin 2(\beta_{d}+\phi^{\rm NP}_{d})\,,$ $\displaystyle
S_{J/\Psi\phi}$ $\displaystyle\equiv$ $\displaystyle\sin
2\beta^{J/\Psi\phi}_{s}=\sin 2(\beta_{s}+\phi^{\rm NP}_{s})\,.$ (32)
Although the measurement of $\sin 2\beta_{J/\Psi K^{0}}$ has approached to the
precision level, however, it might be difficult to tell if there exists new
physics by measuring $\sin 2\beta_{J/\Psi K^{0}}$ only. Nevertheless, one can
investigate a new asymmetry defined by Grossman:1996ke
$\displaystyle\Delta_{\beta_{d}}=\sin 2\beta_{J/\Psi K^{0}}-\sin 2\beta_{\phi
K^{0}}$ (33)
which is less than $5\%$ in the SM Grossman:1996ke . If a large value of
$\Delta_{\beta_{d}}$ is measured, it will be a strong hint for new physics
beyond SM.
## III Numerical Analysis
So far, we have introduced seven new free parameters in the general chiral
color models and they are: two gauge couplings $g_{V,A}$, four parameters in
the two complex quantities $F^{D}_{qb}$ and $m_{V}$. In order to display the
dependence of $\Delta_{\beta_{d}}$ on $m_{V}$, we use the results in Ref.
Frampton:2009rk and take $g_{V}=-0.577g_{s}$ and $g_{A}=-1.155g_{s}$ with
$\alpha_{s}=g_{s}^{2}/4\pi=0.119$. Thus, the five remaining parameters are
$|F^{D}_{qb}|$, $\beta^{\rm NP}_{q}$ for q=d, s and $m_{V}$. We list the input
values used for numerical calculations in Table 1, where the relevant CKM
matrix elements $V_{tq}=\bar{V}_{tq}\exp(-i\beta_{q})$ are obtained from the
UTfit Collaboration Bona:2009tn , the decay constant of $B_{q}$ is referred to
the result given by HPQCD Collaboration Gamiz:2009ku , the CDF and D$\O$
average value of $\Delta m_{B_{s}}$ is from Ref.
TheHeavyFlavorAveragingGroup:2010qj and the SM Wilson coefficients of $b\to
qq^{\prime}\bar{q}^{\prime}$ are obtained from Ref. BBL . Other inputs are
obtained from particle data group (PDG) Nakamura:2010zzi .
Table 1: Numerical inputs for the parameters in the SM. $\bar{V}_{td}$ | $\beta_{d}$ | $\bar{V}_{ts}$ | $\beta_{s}$ | $m_{B_{d}}$ | $m_{B_{s}}$
---|---|---|---|---|---
$8.51(22)\times 10^{-3}$ | $(22\pm 0.8)^{\circ}$ | $-4.07(22)\times 10^{-2}$ | $-(1.03\pm 0.06)^{\circ}$ | 5.28 GeV | 5.37 GeV
$f_{B_{d}}\sqrt{\hat{B}}_{d}$ [MeV] | $f_{B_{s}}\sqrt{\hat{B_{s}}}$ [MeV] | $f_{B_{d}}$ [MeV] | $f_{B_{s}}$ [MeV] | $S^{\rm exp}_{J/\Psi K^{0}}$ | $\bar{m}_{t}(\bar{m}_{t})$
$216\pm 15$ | $266\pm 18$ | $190\pm 13$ | $231\pm 15$ | $0.655\pm 0.024$ | 163.8 GeV
$(\Delta m_{B_{d}})^{\rm exp}$ | $(\Delta m_{B_{s}})^{\rm exp}$ | $C_{3}$ | $C_{4}$ | $C_{5}$ | $C_{6}$
$0.507\pm 0.005$ ps-1 | $17.78\pm 0.12$ ps-1 | $0.013$ | $-0.0335$ | 0.0095 | $-0.0399$
After setting up the inputs, we study the contributions of the axigluon to
FCNC processes and their associated CPAs that are defined earlier. We start by
exploring the allowed parameter space. Since the non-universal axigluon
induces FCNCs at tree level, the observed $B_{q}-\bar{B}_{q}$ mixing parameter
$\Delta m_{B_{q}}$ will give a strict constraint on the parameter space. In
Fig. 1(a)[(b)], the allowed range for $\beta^{\rm NP}_{d[s]}$ and
$|F^{D}_{d(s)b}|/m_{V}$ (in units of $10^{-6}$) is drawn by the down-left
hatched lines where we have taken the SM contributions ($\Delta m^{\rm
SM}_{B_{d}}$, $\Delta m^{\rm SM}_{B_{s}}$) to be $(0.506,\,17.76)$ ps-1.
Furthermore, since the observed $S_{J/\Psi K^{0}}$ has been a precise
measurement, it is plausible that the current data can further exclude the
values of the parameter space which are allowed by $\Delta m_{B_{d}}$. Taking
$2\sigma$ errors of $S^{\rm exp}_{J/\Psi K^{0}}$ as the experimental bound,
the allowed region for $\beta^{\rm NP}_{d}$ and $|F^{D}_{db}|/m_{V}$ sketched
by down-right hatched lines is plotted in Fig. 1(a). Clearly, $S^{\rm
exp}_{J/\Psi K^{0}}$ gives a strong constraint on the parameters that
contribute to $M^{B_{d}}_{12}$. From Fig. 1, we see that, except the two
narrow regions correspond to $|F^{D}_{db}|/m_{V}>1\times 10^{-6}$ GeV-1, the
allowed values of $|F^{D}_{db}|/m_{V}$ are limited to be
$|F^{D}_{db}|/m_{V}\leq 0.4\times 10^{-6}$ GeV-1, whereas the allowed values
of $|F^{D}_{sb}|/m_{V}$ can be one order of magnitude larger than those of
$|F^{D}_{db}|/m_{V}$. In general, the range of the CP violating phase
$\beta^{\rm NP}_{q}$ is $[-\pi,\pi]$, for illustration, we just show the
results within $[-\pi,0]$. The pattern of the constraint in $[0,\pi]$ is
similar to that in $[-\pi,0]$. In order to illustrate the influence of the
uncertainties of the SM on the free parameters, in Fig. 2 we plot the allowed
values of $|F^{D}_{sb}|/m_{V}$ and $\beta^{\rm NP}_{s}$ by including the
errors of $f_{B_{s}}\sqrt{\hat{B}_{s}}$ and $V_{ts}$. Comparing with Fig.
1(b), we see that the allowed range is extended slightly. We note that due to
the strict constraint of $S^{\rm exp}_{J/\Psi K^{0}}$, the bounds on the
parameters for $b\to d$ transition are not changed significantly, therefore,
we don’t show the corresponding diagram for $b\to d$ transition.
Figure 1: (a)[(b)] Constraints on $\beta^{\rm NP}_{d[s]}$ and
$|F^{D}_{d[s]b}|/m_{V}$ (in units of $10^{-6}$) obtained from
$B_{d[s]}-\bar{B}_{d[s]}$ mixing (down-left hatched lines) and $\sin
2\beta_{J/\Psi K^{0}}$ (down-right hatched lines). Figure 2: Legend is the
same as Fig. 1(b), but the errors of $\Delta m_{B_{s}}$ in the SM are
included.
According to Eq. (19), if we assume no new CP violating phase in semi-leptonic
decays, we will see that the charge asymmetry ${\cal A}^{b}_{s\ell}$ depends
on two kinds of CP violating phases. One of the two phases is originated from
$B_{d}-\bar{B}_{d}$ mixing which is a $b\to d$ transition, and the other phase
is originated from $B_{s}-\bar{B}_{s}$ which is associated with $b\to s$
transition. In other words, we have to consider four parameters $\beta^{\rm
NP}_{(d,s)}$ and $|F^{D}_{(d,s)b}|/m_{V}$ simultaneously. However, if we
consider $b\to(d,s)$ transitions at the same time, we may induce a large
effect on $s\to d$ because the $\Delta K=2$ process is associated with
$F^{D}_{ds}=(\zeta-1)V^{D}_{13}V^{D*}_{23}$, i.e. $B_{d}-\bar{B}_{d}$,
$B_{s}-\bar{B}_{s}$ and $K^{0}-\bar{K}^{0}$ mixings have strong correlations.
In order to avoid inducing a large $K^{0}-\bar{K}^{0}$ mixing, we set a small
value for $V^{D}_{13}$. This is consistent with the results shown in Fig. 1(a)
where $\Delta m_{B_{d}}$ and $S_{J/\Psi K^{0}}$ strongly constrain
$|F^{D}_{db}|/m_{V}$. Hence, we assume that $a^{d}_{s\ell}$ is dominated by
the SM contribution where $a^{d}_{s\ell}(SM)=-4.8\times 10^{-4}$ Lenz:2006hd .
Consequently, the enhanced $|{\cal A}^{b}_{s\ell}|$ can be attributed to $b\to
s$ transition. With Eqs. (14), (18) and (21) and the values given in Table 1,
the contours of ${\cal A}^{b}_{s\ell}$ as a function of $\beta^{\rm NP}_{s}$
and $|F^{D}_{sb}|/m_{V}$ are shown in Fig. 3(a) where the values of the
contours are in units of $10^{-4}$. As can be seen from the figure, not only
the sign of ${\cal A}^{b}_{s\ell}$ can fit the data, but also its magnitude
can be enhanced by axigluon-mediated effects. By combining with the constraint
of $\Delta m_{B_{s}}$, the region of $\beta^{\rm NP}_{s}$ for large $|{\cal
A}^{b}_{s\ell}|$ is limited. In Fig. 3(b), we display ${\cal A}^{b}_{s\ell}$
as a function of $\beta^{\rm NP}_{s}$ where the solid, dashed and dash-dotted
line represents $|F^{D}_{sb}|/m_{V}=(3,4,5)\times 10^{-6}$ GeV-1,
respectively. As shown in the figure, negative and positive values of
$\beta^{\rm NP}_{s}$ can enhance ${\cal A}^{b}_{s\ell}$. It should be noted
that, although the axigluon-mediated effect can not enhance the like-sign
charge asymmetry to be as large as the central value of DØ data, however,
$|{\cal A}^{b}_{s\ell}|$ is enhanced by one order of magnitude larger than the
SM prediction.
Figure 3: (a) Contours of ${\cal A}^{b}_{s\ell}$ as a function of $\beta^{\rm
NP}_{s}$ and $|F^{D}_{sb}|/m_{V}$ (in units of $10^{-6}$). (b) ${\cal
A}^{b}_{s\ell}$ as a function of $\beta^{\rm NP}_{s}$, where the solid, dashed
and dash-dotted line stands for $|F^{D}_{sb}|/m_{V}=(3,4,5)\times 10^{-6}$,
respectively. The values on the plot (a) are ${\cal A}^{b}_{s\ell}$ in units
of $10^{-4}$.
Unlike the case of the charge asymmetry, the time-dependent CPA of $B_{s}\to
J/\Psi\phi$ decay depends only on the CP phase in $b\to s$ transition. As a
consequence, when the new CP violating effects are small in $M^{B_{d}}_{12}$,
${\cal A}^{b}_{s\ell}$ and $S_{J/\Psi\phi}$ defined in Eq. (22) can have a
strong correlation. By using Eq. (32), the contours of $S_{J/\Psi\phi}$ are
plotted as a function of $\beta^{\rm NP}_{s}$ and $|F^{D}_{sb}|/m_{V}$ in Fig.
4(a). From the figure, we find that large $S_{J/\Psi\phi}$ can be archived
when ${\cal A}^{b}_{s\ell}$ is one order of magnitude larger than the SM
prediction. Moreover, we also plot $S_{J/\Psi\phi}$ as a function of
$\beta^{\rm NP}_{s}$ in Fig. 4(b), where the solid, dashed and dash-dotted
line denotes $|F^{D}_{sb}|/m_{V}=(3,4,5)\times 10^{-6}$ GeV-1, respectively.
Clearly, a large ${\cal A}^{b}_{s\ell}$ indicates a large $S_{J/\Psi\phi}$.
Although the measured values of ${\cal A}^{b}_{s\ell}$ and $S_{J/\Psi\phi}$
contain large errors, however, a few sigma deviations from the SM prediction
can be considered as a hint for new physics effect.
Figure 4: (a) Contours of $S_{J/\Psi\phi}$ as a function of $\beta^{\rm
NP}_{s}$ and $|F^{D}_{sb}|/m_{V}$ (in units of $10^{-6}$). (b)
$S_{J/\Psi\phi}$ as a function of $\beta^{\rm NP}_{s}$, where the solid,
dashed and dash-dotted line represents $|F^{D}_{sb}|/m_{V}=(3,4,5)\times
10^{-6}$, respectively.
It is well known that the golden process to measure the angle $\beta_{d}$ in
the SM is $B_{d}\to J/\Psi K^{0}$ which is dominated by tree diagram. Although
new physics can also affect this decay mode via $b\to sc\bar{c}$ transition,
however, as discussed in Eq. (28), the axigluon contributions to $B_{d}\to
J/\Psi K^{0}$ vanish. Hence, the source of the time-dependent CPA in $B_{d}\to
J/\Psi K^{0}$ decay is only originated from the $B_{d}$ oscillation. Since
$\beta_{d}$ is also a parameter in the SM, a single measurement of $S_{J/\Psi
K^{0}}$ or $\sin 2\beta_{J/\Psi K^{0}}$ is hard to uncover the new physics. To
probe the new physics, the best way is to compare the CPA of $J/\Psi K^{0}$
with that of $\phi K_{S}$. Therefore, we do not discuss each of $S_{J/\Psi
K^{0}}$ and $S_{\phi K^{0}}$ separately. Instead, we focus on the CPA
difference $\Delta_{\beta_{d}}$ which is defined in Eq. (33) and it is only
few percent in the SM. By Eqs. (31) and (33), we see that although
$\Delta_{\beta_{d}}$ is insensitive to $F^{D}_{db}$ however it is strongly
dependent on $F^{D}_{sb}$. To see the contributions of the axigluon to
$\Delta_{\beta_{d}}$, we present the contours of $\Delta_{\beta_{d}}$ as a
function of $\beta^{\rm NP}_{s}$ and $|F^{D}_{sb}|/m_{V}$ in Fig. 5(a)[(b)],
where we have set $|F^{D}_{db}|/m_{V}=0$ and figure (a)[(b)] corresponds to
$m_{V}=0.5[1]$ TeV. Since the decay amplitude of $B\to\phi K$ depends on
$F^{D}_{sb}/m^{2}_{V}$ while $\Delta m_{B_{s}}$ is $(F^{D}_{sb}/m_{V})^{2}$,
thus a specific value for $m_{V}$ has to be given when calculating the
contours of $\Delta_{\beta_{d}}$. For further understanding the $\beta^{\rm
NP}_{s}$-dependence, we display $\Delta_{\beta_{d}}$ as a function of
$\beta^{\rm NP}_{s}$ in Fig. 6, where figure (a)[(b)] is for $m_{V}=0.5[1]$
TeV and the solid, dashed and dash-dotted line stands for
$|F^{D}_{sb}|/m_{V}=(3,4,5)\times 10^{-6}$ GeV-1, respectively. It is clear
that the axigluon contributions to $\Delta_{\beta_{d}}$ are larger than that
of the SM.
Figure 5: Contours of $\Delta_{\beta_{d}}$ as a function of $\beta^{\rm
NP}_{s}$ and $|F^{D}_{sb}|/m_{V}$ (in units of $10^{-6}$) with (a) $m_{V}=0.5$
TeV and (b) $m_{V}=1$ TeV. Figure 6: $\Delta_{\beta_{d}}$ as a function of
$\beta^{\rm NP}_{s}$ with (a) $m_{V}=0.5$ TeV and (b) $m_{V}=1$ TeV, where the
solid, dashed and dash-dotted line represents
$|F^{D}_{sb}|/m_{V}=(3,4,5)\times 10^{-6}$, respectively.
In order to comprehend further the correlations among various physical
observables under the influence of the axigluon, we display the scatter plots
of ${\cal A}^{b}_{s\ell}$, $S_{J/\Psi\phi}$ and $\Delta_{\beta_{d}}$ with
$m_{V}=0.5(1)$ TeV versus $\Delta m_{B_{s}}$ in Fig. 7, where we have chosen
the range of $\beta^{\rm NP}_{s}$ to be $[-\pi,0]$. As an illustration, we
also show the scatter plots of ( ${\cal A}^{b}_{s\ell}$, $S_{J/\Psi\phi}$) and
(${\cal A}^{b}_{s\ell}$, $\Delta_{\beta_{d}}$) with $m_{V}=1$ TeV in Fig. 8,
in which the constraint of $\Delta m_{B_{s}}$ has been included and
$\beta^{\rm NP}_{s}$ belongs to $[-\pi,0]$. By Fig. 8(a), we see that the
correlation between ${\cal A}^{b}_{s\ell}$ and $S_{J/\Psi\phi}$ is linear,
where this behavior can be understood by the linear dependence between the
like-sign charge asymmetry and the mixing-induced CPA of $B_{s}$. Due to the
linearity, we expect that the correlation between $S_{J/\Psi\phi}$ and
$\Delta_{\beta_{d}}$ should be similar to that between ${\cal A}^{b}_{s\ell}$
and $\Delta_{\beta_{d}}$. Therefore, we just show the latter case in Fig.
8(b).
Figure 7: Correlations between $\Delta m_{B_{s}}$ and (a) ${\cal
A}^{b}_{s\ell}$, (b) $S_{J/\Psi\phi}$, (c)[(d)] $\Delta_{\beta_{d}}$ with
$m_{V}=0.5[1]$ TeV, where the angle $\beta^{\rm NP}_{s}$ belongs to
$[-\pi,0]$. Figure 8: (a) Correlation between ${\cal A}^{b}_{s\ell}$ and
$S_{J/\Psi\phi}$ and (b) correlation between ${\cal A}^{b}_{s\ell}$ and
$\Delta_{\beta_{d}}$ with $m_{V}=1$ TeV, where the constraint of $\Delta
m_{B_{s}}$ has been included and the angle $\beta^{\rm NP}_{s}$ belongs to
$[-\pi,0]$.
## IV Conclusion
In general, a flavor non-universal axigluon in generalized chiral color models
can induce FCNCs at tree level. We study phenomenologically the axigluon-
mediated effects on $\Delta B=2$ FCNC processes and the associated CPAs. We
find that although $\Delta m_{B_{q}}$ strongly constrain the free parameters,
the precise measurement of $S_{J/\Psi K^{0}}$ can further exclude the
parameter space of $b\to d$ transition. Furthermore, for avoiding inducing
large $K^{0}-\bar{K}^{0}$ mixing, the parameter $V^{D}_{13}$ is chosen to be
small so that the like-sign charge asymmetry ${\cal A}^{b}_{s\ell}$ and
$\Delta_{\beta_{d}}$ are insensitive to the parameters of $b\to d$ transition.
As a result, the CP violating observables ${\cal A}^{b}_{s\ell}$,
$S_{J\Psi\phi}$ and $\Delta_{\beta_{d}}$ are strongly correlated and are only
sensitive to the parameters of $b\to s$ transition.
By the study, we find that the axigluon effects do not only preserve the
negative sign in ${\cal A}^{b}_{s\ell}$, but also enhance its magnitude by one
order of magnitude larger than the SM prediction. Subsequently, the associated
values of the parameters can also enhance the CPA $S_{J/\Psi\phi}$ and the CPA
difference $\Delta_{\beta_{q}}$ largely although they are only few percent in
the SM.
## Acknowledgement
This work is supported by the National Science Council of R.O.C. under Grant
No. NSC-97-2112-M-006-001-MY3. The author C.H.C would like to thank Prof.
Young-Chung Hsue for his help on using plot tool. G. Faisel would like thank
the National Center for Theoretical Sciences (NCTS) at Cheng Kung University
for the hospitality where this work has been done.
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|
arxiv-papers
| 2010-05-25T14:12:17 |
2024-09-04T02:49:10.630850
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chuan-Hung Chen, Gaber Faisel",
"submitter": "Chuan Hung Chen",
"url": "https://arxiv.org/abs/1005.4582"
}
|
1005.4752
|
# A database approach to information retrieval: The remarkable relationship
between language models and region models
Djoerd Hiemstra and Vojkan Mihajlović
University of Twente
Centre for Telematics and Information Technology
P.O. Box 217, 7500 AE Enschede, The Netherlands
{d.hiemstra,v.mihajlovic}@utwente.nl
###### Abstract
In this report, we unify two quite distinct approaches to information
retrieval: region models and language models. Region models were developed for
structured document retrieval. They provide a well-defined behaviour as well
as a simple query language that allows application developers to rapidly
develop applications. Language models are particularly useful to reason about
the ranking of search results, and for developing new ranking approaches. The
unified model allows application developers to define complex language
modeling approaches as logical queries on a textual database. We show a
remarkable one-to-one relationship between region queries and the language
models they represent for a wide variety of applications: simple ad-hoc
search, cross-language retrieval, video retrieval, and web search.
## 1 Introduction
The introduction of the relational model by Codd in 1970 [14] marks one of the
success stories of computer science. The relational model laid the path for
the development of relational database systems: general software tools for
management of data with a well-understood and well-defined behaviour. They
allow application developers to rapidly develop application programs that are
easy to understand, document and teach [17]. Indeed, saying “databases” is
saying “relational”: Virtually any introductory book or course on databases
will teach the basics of the relational data model and SQL.
It can be argued that information retrieval is still at the stage where
databases were in the 1960’s. There is no such thing as an equivalent of the
relational model for information retrieval systems. Introductory books and
courses on information retrieval [6, 46] will teach the student several
information retrieval models – mostly focusing on different ranking strategies
– each with its own strengths and weaknesses. Developing a retrieval
application or deploying a search engine requires applications to call non-
standard application program interfaces (APIs) and use non-standard query
languages.
As an example, the Terrier system, a research information retrieval system
developed by the University of Glasgow [42], is based on the so-called
divergence of randomness models [2]. Terrier provides APIs for indexing and
querying. To use the Terrier indexing API on a non-standard collection
(Terrier comes with some fully implemented APIs, for instance for HTML
documents), the application developer needs to create an object which
implements the collection interface. This will find all the files it has to
process, and opens each one to create a document object which identifies which
tags (or other byte sequences) act as document delimiters. Applications
programs that work with this setup will be logically impaired if the file
locations or document format (for instance the XML DTD) need to be changed.
Or, in analogy with Codd’s [14] analysis of the database systems from the
1960’s: The retrieval system does not provide access path independence.
As another example, the Lemur toolkit [41] is a research retrieval system that
is specifically designed to support research in language modeling [25, 37,
44]. The toolkit supports a broad range of different applications of
information retrieval such as ad hoc retrieval, distributed retrieval, cross-
language retrieval, etc. Lemur supports at least four different index types,
each supporting different kinds of queries. For instance, some indexes include
word positions to allow proximity queries, whereas others only allow very
basic functionality. Application programs that work with one kind of index
might be logically impaired if the index type is changed. In analogy with Codd
[14], the retrieval system does not provide indexing independence.111Codd
identified one more type of data independence: ordering independence. As
textual data is inherently ordered we are not concerned with ordering
independence.
In the past, we have used systems like Terrier and Lemur to research new
applications of information retrieval technology such as cross-language
retrieval [24], web retrieval [29], and video shot retrieval [26]. To develop
such retrieval approaches, it was necessary to reimplement parts of the
existing system: reimplementing APIs, introducing new APIs, introducing new
query languages, and even introducing new indexing and storage structures. In
this report, we present a framework that supports all such approaches by means
of a simple yet powerful query language (similar to SQL or relational algebra)
that hides the implementation details of retrieval approaches from the
application developer. As such, the system provides access path independence
and indexing independence.
There have been other attempts to develop approaches to information retrieval
that provide data independence. For instance, Schek [50] describes methods for
integrating databases and information retrieval systems where application
programs and queries are not aware of access paths and indexes. Fuhr [20]
describes a layered system design for information retrieval systems following
the ANSI/SPARC model [56], distinguishing a physical (internal) layer, a
conceptual layer and an external layer. The system might process queries in
several ways, such as directly by an index, or by using an index as a filter
with an additional scan of the filtered results. Probabilistic relational
algebra or probabilistic Datalog (see [19] for an overview) might serve as
conceptual query languages in such systems. An example of a system that
implements this approach is HySpirit [22]. In this report we introduce an
alternative for probabilistic relational algebra and probabilistic Datalog
that is much closer to existing models of information retrieval.
### 1.1 Region models
Motivated by the data independence issues described above, Burkowski [12]
proposes a mathematical framework which he called the containment model that
operates on sets of contiguous extents. We will call extents regions in this
report, and the model region model. A region might be a word, a phrase, a text
element such as a title, or a complete document. Burkowski’s model comes with
a small number of basic operators on sets of regions, the most important ones
being SN (select narrow) and SW (select wide). A search for chapters
containing the word “databases” would be expressed as <chapter> SW databases,
and if the application program only needs to put the chapter’s title on the
screen, the query would be <chapter_title> SN (<chapter> SW databases). In
Burkowski’s framework, the application program does not know how a text
collection and its index facilities are managed. The complexity of the
retrieval system is encapsulated in a module that only responds to simple
command strings like the ones above. Similar frameworks are introduced by
Salminen and Tompa [48], Clarke et al. [13], Baeza-Yates and Navarro [5],
Consens and Milo [15], and Jaakkola and Kilpelainen [27]. We will call the
models underlying these approaches region models in this report.
Unlike Codd’s relational model for databases, the region models above did not
have a big impact on the information retrieval research community, nor on the
development of new retrieval systems. The reason for this is quite obvious:
region models do not explain in anyway how search results should be ranked. In
fact, most region models are not concerned with ranking at all; one might say
they – like the relational model – are actually data models instead of
information retrieval models. Region model approaches that do address ranking,
like Burkowski’s model [12] and the approach by Masuda et al. [33], only
include it as an after-thought: Retrieve first, then rank with some standard
retrieval model such as a vector space model using tf.idf weights [49].
### 1.2 Language models
If anything, an approach to information retrieval has to address the ranking
of search results. Ranking is the single most important feature of a search
engine, and information retrieval modeling almost exclusively focuses on
ranking (see e.g. [6, Chapter 2]). Traditionally, developing ranking
strategies involves engineering, fitting and tuning term weighting approaches
to improve experimental results [49], although there are some notable
exceptions, for instance the probabilistic model by Robertson and Sparck-Jones
[47]. A more recent approach that does not require lots of fitting and tuning
are statistical language models for information retrieval [25, 37, 44].
Language models assign a probability to a piece of text. They are built for
each document: Each document model assigns a probability to a text query, and
documents are ranked accordingly. Language models have been applied to a wide
variety of retrieval problems, such as simple ad-hoc search [25, 28, 37],
cross-language retrieval [8, 24, 31, 57], video retrieval using speech
transcripts [16, 26], and web search [28, 29, 40]. Examples of these
applications will be shown in Section 3.
### 1.3 Unifying region models and language models
In this report we introduce an approach to information retrieval that fully
integrates region models and language models. The approach allows application
developers to define complex language modeling approaches as logical region
queries on a textual database. We show a remarkable one-to-one relation
between region queries and the language models they represent for the four
retrieval problems mentioned above: ad-hoc search, cross-language retrieval,
video retrieval, and web search. The report is organised as follows. In
Section 2 we introduce the combined region/language model. Section 3
illustrates the application of the model by relating probability measures to
region queries. Finally in Section 4 we present future work and relate the
approach to current work on XML query languages and XML database systems.
## 2 A region model for text databases and a query language
This section briefly introduces the unified region/language model. The
definitions closely follow Burkowski’s model [12], which we extend with region
scores similar to the score region algebra we used for XML information
retrieval [32].
A textual database consists of a finite sequence of words
$w_{1},w_{2},\cdots,w_{n-1}$, where $w_{i}$ is used to denote the word on
position $i$ in the database. Additionally, the textual database consists of a
hierarchy of text elements. Both words and elements are identified by the word
positions in the database. Text elements are sequences of words that have a
particular significance in the database. For example, a database with recipes
will have text elements “ingredients”, “quantities”, “instructions”, etc.,
typically marked up as XML.
A scored region $r$ is defined by two integers $r.start$ and $r.end$ ($1\leq
r.start<r.end\leq n$), and a float $r.score$ ($r.score>0$).222We intentionally
use a notation that is close to that of the relational data model; see also
Figure 1. The integers start and end represent respectively the position of
the first word that belongs to the contiguous region, and the position
directly following the last word that belongs to the region. A region might be
a text element, but also any other contiguous sequence of words. Note that the
region $(i,i+1,s)$ includes one (and only one) word $w_{i}$ with a score $s$.
Retrieval from the textual database is done with a simple query language
consisting of words, elements and five basic operators: CONTAINING,
CONTAINED_BY, SCALE, AND, and OR. The language defines an algebra on sets of
scored regions. Unlike Burkowski’s model [12], there are no additional
constraints on sets of regions. We will now one-by-one define the language
primitives in a rather informal way. For convenience, Figure 1 contains a more
formal definition of the operators using SQL.
A word
A single word, for example the query banana, produces a set of regions $R$,
where each region $r\in R$ defines a position of the word in the textual
database; $r.start$ being the position on which the word occurs,
$r.end=r.start+1$, and $r.score=1$.
An element
A single element, for instance the query <recipe> produces a set of regions
$R$, where each region $r\in R$ is tagged as “recipe”, $r.start$ being the
position of the first word of the XML element, $r.end$ being the position
following the last word of the XML element, and $r.score=1$.
$R_{1}$ CONTAINING $R_{2}$
The operator CONTAINING takes two sets of regions $R_{1}$ and $R_{2}$, and
produces the subset of regions from $R_{1}$ that contain at least one region
from $R_{2}$. For instance, the query <recipe> CONTAINING banana produces all
regions tagged as “recipe” that contain at least one occurrence of “banana”.
Inspired by language models, each “recipe” region is scored by the number of
occurrences of “banana” in the region, divided by the length of the region
(measured as $r.end-r.start$). Occurrences of “banana” are weighted by their
length and by their score (of course, in the example query both length = 1 as
well as score = 1); see Figure 1.
$R_{1}\,$CONTAINED_BY$\,R_{2}$
The operator CONTAINED_BY takes two sets of regions $R_{1}$ and $R_{2}$, and
produces the subset of regions from $R_{1}$ that are at least contained by one
region from $R_{2}$. For instance, the query <ingredient> CONTAINED_BY
<recipe> produces all ingredients that belong at least to one recipe. If a
region from the left-hand side of the expression is nested in more than one
region from the right-hand side of the expression, then the scores of those
regions are added. This will be used in the next section to express the linear
combination of several language models; see Figure 1.
$f$ SCALE $R$
The operator SCALE takes a float $f$ and a set of regions $R$ and produces all
regions from $R$ where each region $r\in R$ is scored as $f\cdot r.score$. For
instance, the query 0.2 SCALE banana produces the set of regions with the
positions of the word “banana” all with a region score of 0.2; see Figure 1.
$R_{1}$ AND $R_{2}$
The operator AND takes two sets of regions $R_{1}$ and $R_{2}$, and produces
only those regions that are both in $R_{1}$ and $R_{2}$, i.e., the
intersection of both sets when ignoring the region scores. Each region in the
result is scored by multiplying its scores in $R_{1}$ and $R_{2}$. For
instance, the query (<recipe> CONTAINING banana) AND (<recipe> CONTAINING
apple) produces all regions tagged as “recipe” that contain both the word
“banana” and the word “apple”, scored by the product of the scores of the
respective regions; see Figure 1.
$R_{1}$ OR $R_{2}$
The operator OR takes two sets of regions $R_{1}$ and $R_{2}$, and produces
those regions that either are in $R_{1}$, or in $R_{2}$, i.e., the union of
both sets when ignoring the region scores. For instance, the query (<recipe>
CONTAINING sugar) OR (<recipe> CONTAINING sweet) produces all regions tagged
as “recipe” that contain either the word “sugar” or the word “sweet” (or
both). Regions keep their score, unless both sets contain the region, in which
case the region is scored by adding its scores in $R_{1}$ and $R_{2}$; see
Figure 1.
\-- R1 CONTAINING R2
---
SELECT R1.start, R1.end, R1.score * SUM((R2.score *
(R2.end - R2.start)) / (R1.end - R1.start)) AS score
FROM R1, R2
WHERE R1.start <= R2.start AND R1.end >= R2.end
GROUP BY R1.start, R1.end, R1.score
\-- R1 CONTAINED_BY R2
SELECT R1.start, R1.end, R1.score * SUM(R2.score) AS score
FROM R1, R2
WHERE R1.start >= R2.start AND R1.end <= R2.end
GROUP BY R1.start, R1.end, R1.score
\-- f SCALE R
SELECT R.start, R.end, f * R.score AS score
FROM R
\-- R1 AND R2
SELECT R1.start, R1.end, R1.score * R2.score AS score
FROM R1, R2
WHERE R1.start = R2.start AND R1.end = R2.end
\-- R1 OR R2
SELECT R.start, R.end, SUM(R.score) AS score
FROM (SELECT * FROM R1 UNION ALL SELECT * FROM R2) AS R
GROUP BY R.start, R.end
Figure 1: Definition of operators in SQL.
Figure 1 contains a definition of the operators using SQL, as a pragmatic
means to provide a formal definition of the region algebra operators without
the need to get into specific mathematical notations. So, we show SQL
definitions here for convenience, as we assume most readers are familiar with
SQL. The definitions do not suggest in any way that the system should be
implemented on top a relational databases system. We implemented the system –
without the use of SQL – on top of MonetDB [32], but it might as well be
implemented using traditional inverted file indexes on the file system.333For
readers that do want to implement this on top of a relational DBMS, please
note that ‘R1.end’ clashes with the SQL reserved word ‘END’ in practical
systems.
A natural application of the region model, is to support structured queries in
an XML information retrieval system. The following query is an example XML
information retrieval query formulated in NEXI. NEXI [55] stands for narrowed
extended XPath, a query language that restricts XPath [9] by only allowing
descendent axis steps, and that extends XPath by a special about operator that
ranks the selected nodes by their estimated relevance to the query. NEXI is
used to evaluate XML retrieval systems in the Initiative for the Evaluation of
XML retrieval (INEX) [21]. Suppose we want to retrieve sections about
“databases” from articles that mention “book review” in either the article
title (atl) or the keywords (kwd):
//article[about(.//(atl|kwd), book review)]//sec[about(., databases)]
This can be formulated as follows as a region query:
$\begin{array}[]{l}\\!\\!\\!\mbox{\tt\small(<sec> $\\!${\mbox{\footnotesize
CONTAINING}}$\\!$ databases) $\\!${\mbox{\footnotesize CONTAINED\\_BY}}$\\!$
(<article> $\\!${\mbox{\footnotesize CONTAINING}}$\\!$}\\\
\mbox{\tt\small~{}~{}~{}~{} (((<atl> $\\!${\mbox{\footnotesize OR}}$\\!$
<kwd>)$\\!$ {\mbox{\footnotesize CONTAINING}}$\\!$ book)
$\\!${\mbox{\footnotesize CONTAINING}}$\\!$ review)) }\end{array}$
This approach is followed with success in INEX by the TIJAH system [32, 36].
The expression defines a ranking of the selected nodes. Rewriting the NEXI
query to the region expression is not trivial, but relatively easy: TIJAH has
a NEXI to region query parser.
In the next section we show the relationship between language modeling ranking
definitions and region queries, similar to the relationship between NEXI
queries and the region queries.
## 3 Logical queries for complex retrieval tasks
### 3.1 The simplest unigram language model
As said in the introduction, language models form a general approach to define
ranking formulas for retrieval applications. A language model is assigned to
every document. The language model of the document defines the probability
that the document ‘generates’ the query. Documents are ranked by this
probability. The simplest language modeling approach to information retrieval
would be defined by Equation 1.
$P(T_{1},T_{2},\cdots,T_{l}|D)=\prod_{i=1}^{l}P(T_{i}|D)$ (1)
It defines the probability of a query of length $l$ given a document $D$ as
the product of the probabilities of each term $T_{i}$ $(1\leq i\leq l)$ given
$D$. A language model that takes a simple product of terms, i.e., a model that
assumes that the probability of one term given a document does not depend on
other terms, is called a unigram language model. To make this work, we have to
define the basic probability measure $P(T|D)$; typically, it would be defined
as the number of occurrences of the term $T$ in the document $D$, divided by
the total number of terms in the document $D$. For a practical query, say,
retrieve all documents about “db” and “ir”, we would instantiate Equation 1 as
follows:
$P(T_{1}\\!=\\!\mbox{\tt\small db},T_{2}=\mbox{\tt\small
ir}|D)\;=\;P(T_{1}\\!=\\!\mbox{\tt\small
db}|D)\;\cdot\;P(T_{2}\\!=\\!\mbox{\tt\small ir}|D)$ (2)
The right-hand side of the equation corresponds to the following region
expression.
(<doc> CONTAINING db) AND (<doc> CONTAINING ir) (3)
This can be shown as follows: The region expression (<doc> CONTAINING db)
produces all documents ranked according to $P(T=\mbox{\tt\small db}|D)$, i.e.,
all regions tagged as <doc>, ranked by the number of occurrences of db in
those regions. Similarly, (<doc> CONTAINING ir) produces all documents ranked
according to $P(T=\mbox{\tt\small ir}|D)$. Finally, the operator AND results
in the regions tagged as <doc> that are in both operand sets. The score of the
result regions is defined as the product of the scores of the same regions in
the operands. Here, and in the remaining examples in this section, we assume
that <doc> regions do not nest inside each other.
We claim that there is a trivial way to rewrite the right-hand side of
Equation 2 to Equation 3 while preserving the outcome. This can be shown by
simply replacing $P(x|y)$ by (y CONTAINING x), and the multiplication in
Equation 2 by AND. Regions that are assigned zero probability by the
probability measure of Equation 2 are not retrieved by the region expression
of Equation 3. So, the region expression selects all $y$ for which $P(x|y)>0$.
If the probability measure assigns zero probability to a region then this
implies that the corresponding region expression will not retrieve it; and, if
a region is not retrieved by a region expression then this implies that its
corresponding probability function assigns zero probability to it.
### 3.2 Linear interpolation smoothing
The simple language model presented in the previous section assigns zero
probability to a document unless it contains all query terms. So, if none of
the documents contains all terms, the system does not retrieve anything. This
behaviour will be appropriate for many practical applications. In fact, it is
the default behaviour of web search engines like Google and Yahoo.
For other applications, it might be undesirable to have empty results. When
searching collections that are significantly smaller than the web, it is
likely that precise queries will not retrieve anything. In practice, language
modeling approaches therefore use a technique called “smoothing”, i.e., some
probability mass is assigned to terms that do not occur in the document. The
standard language modeling approach uses a mixture of the document model
$P(T_{i}|D)$ with a general collection model $P(T_{i}|C)$ [8, 25, 30, 37, 38,
51], called linear interpolation smoothing.
$P(T_{1},T_{2},\cdots,T_{l}|D)=\prod_{i=1}^{l}((1\\!-\\!\lambda)P(T_{i}|C)+\lambda
P(T_{i}|D))$ (4)
The document model $P(T_{i}|D)$ assigns zero probability to terms that do not
occur in the document $D$, but the collection model $P(T_{i}|C)$ assigns some
probability to any term that occurs somewhere in the collection. The
collection model probabilities are defined similar to the document model
probabilities as: The number of occurrences of the term $T$ in the total
collection $C$, divided by the total number of terms in the collection $C$.
The approach needs a parameter $\lambda$ $(0<\lambda<1)$ which is set
empirically.
For our example query, we need some value for $\lambda$ to instantiate
Equation 4. Suppose we decide $\lambda=0.8$, then we would rank documents
according to:
$\begin{array}[]{l}P(T=\mbox{\tt\small db},T=\mbox{\tt\small ir}|D)\;=\\\
\;(0.2\\!\cdot\\!P(T_{1}\\!=\\!\mbox{\tt\small
db}|C)+0.8\\!\cdot\\!P(T_{1}\\!=\\!\mbox{\tt\small db}|D))\\\ \;\cdot\\\
\;(0.2\\!\cdot\\!P(T_{2}\\!=\\!\mbox{\tt\small
ir}|C)+0.8\\!\cdot\\!P(T_{2}\\!=\\!\mbox{\tt\small ir}|D))\end{array}$ (5)
The equation corresponds to the following region expression, where the text
element <root> corresponds to the collection root, i.e., the whole database.
$\begin{array}[]{l}\mbox{\tt\small(<doc> {\mbox{\footnotesize
CONTAINED\\_BY}}}\\\ \mbox{\tt\small~{}((0.2 {\mbox{\footnotesize SCALE}}
(<root> $\\!${\mbox{\footnotesize CONTAINING}}$\\!$ db))
$\\!${\mbox{\footnotesize OR}}$\\!$ (0.8 {\mbox{\footnotesize SCALE}} (<doc>
$\\!${\mbox{\footnotesize CONTAINING}}$\\!$ db))))}\\\
\mbox{\tt\small{\mbox{\footnotesize AND}}}\\\ \mbox{\tt\small(<doc>
{\mbox{\footnotesize CONTAINED\\_BY}}}\\\ \mbox{\tt\small~{}((0.2
{\mbox{\footnotesize SCALE}} (<root> $\\!${\mbox{\footnotesize
CONTAINING}}$\\!$ ir)) $\\!${\mbox{\footnotesize OR}}$\\!$ (0.8
{\mbox{\footnotesize SCALE}} (<doc> $\\!${\mbox{\footnotesize
CONTAINING}}$\\!$ ir)))) }\\!\\!\end{array}$ (6)
This can be shown as follows: The region expression (<root> CONTAINING db)
results in a set with the single region <root> with a score equal to the
number of occurrences of db in <root>, i.e., $P(T|C)$. The SCALE operator will
multiply the region with 0.2; and the OR will union the region with all
document regions (with scores $P(T|D)$ as in the previous section), multiplied
with 0.8 by the SCALE operator. Note, that the OR operator will not actually
add $0.2\cdot P(T\\!=\\!\mbox{\tt\small db}|C)$ to $0.8\cdot
P(T\\!=\\!\mbox{\tt\small db}|D)$: This will be done by the CONTAINED_BY
operator: every document region on the left-hand side of this operator matches
(because every document region is contained by the collection root). Document
regions that are in the set 0.8 SCALE (<doc> CONTAINING db) will get as their
final score: $0.2\cdot P(T\\!=\\!\mbox{\tt\small db}|C)+0.8\cdot
P(T\\!=\\!\mbox{\tt\small db}|D)$; the others will get:
$0.2\\!\cdot\\!P(T\\!=\\!\mbox{\tt\small db}|C)$. The same line of reasoning
can be done for the part with the term ir. Finally, the AND operator combines
both parts of the query as in the previous section.
Again, we claim there is a trivial way to rewrite the right-hand side of
Equation 5 to Equation 6. This can be shown by simply replacing $P(x|y)$ by (y
CONTAINING x), the multiplication operator ‘$\cdot$’ by AND if both operands
are regions, or by SCALE if the first operand is a number; the addition
operator ‘$+$’ by OR, and by putting “z CONTAINED_BY” in front of the
expression, where $z$ defines the elements that need to be retrieved.
It might be argued that this very last step – “putting CONTAINED_BY in front”
– is not a trivial step, and we did not use it in the previous section.
However, we might as well use it in the previous section: It is easy to show
that (<doc> CONTAINING db) AND (<doc> CONTAINING ir) produces the same
regions, with the exact same scores as (<doc> CONTAINED_BY (<doc> CONTAINING
db)) AND (<doc> CONTAINED_BY (<doc> CONTAINING ir)), because the elements on
the left-hand side of both CONTAINED_BY operators all have unit score, and
because elements on the left-hand side are nested in at most one region from
the right-hand side of the CONTAINED_BY operator. So, the general procedure
that rewrites probability measures to region expressions should use the
CONTAINED_BY operator for every query term. Equivalences between region
expressions will be addressed briefly in Section 4.1.
### 3.3 Video shot retrieval using speech transcripts
Now that we showed linear interpolation smoothing, it is easy to generalise
this to any linear combination of language models. Such models have been quite
successful in spoken document retrieval for retrieving video shots [16, 26],
where videos are modeled as sequences of scenes, each consisting of sequences
of shots. The language model mixes four different levels of the video
hierarchy: shots, scenes, complete videos and the total collection as:
$\vspace{0.1cm}\begin{array}[]{l}P(T_{1},T_{2},\\!\cdots\\!,T_{l}|Shot)\,=\\\
\;\;\;{\displaystyle\prod_{i=1}^{l}(\alpha P(T_{i}|C)+\beta
P(T_{i}|Video)+\gamma P(T_{i}|Scene)+\delta P(T_{i}|Shot))}\end{array}$ (7)
where $\alpha+\beta+\gamma+\delta=1$. The main idea behind this approach is
that a good shot contains the query terms, and is part of a scene that
contains the query terms, which is part of a video that contains even more of
the query terms. Suppose we are looking for the exact shots in a collection of
videos where a knight says “ni”,444From the movie “Monty Python and the Holy
Grail” and we take $\alpha=0.18$, $\beta=0.02$, $\gamma=0.4$, and $\delta=0.4$
then the shots would be ranked according to:
$\vspace{0.1cm}\begin{array}[]{l}P(T\\!=\\!\mbox{\tt\small ni}|Shot)\;\;=\\\
\hskip 25.6073pt(0.18\\!\cdot\\!P(T\\!=\\!\mbox{\tt\small
ni}|C)\;+\;0.02\\!\cdot\\!P(T\\!=\\!\mbox{\tt\small ni}|\mathit{Video})\\\
\hskip 25.6073pt\;\;\;+\,0.4\\!\cdot\\!P(T\\!=\\!\mbox{\tt\small
ni}|\mathit{Scene})\,+\,0.4\\!\cdot\\!P(T\\!=\\!\mbox{\tt\small
ni}|\mathit{Shot}))\end{array}$ (8)
which corresponds to the following region expression.
$\vspace{0.1cm}\begin{array}[]{l}\\!\\!\\!\\!\mbox{\tt\small$\\!\\!\\!\\!$<shot>
{\mbox{\footnotesize CONTAINED\\_BY}}}\\\ \mbox{\tt\small~{} ((0.18
$\\!${\mbox{\footnotesize SCALE}}$\\!$ (<root> $\\!${\mbox{\footnotesize
CONTAINING}}$\\!$ ni)) $\\!${\mbox{\footnotesize OR}}$\\!$ (0.02
$\\!${\mbox{\footnotesize SCALE}}$\\!$ (<video> $\\!${\mbox{\footnotesize
CONTAINING}}$\\!$ ni))}\\!\\!\\!\\!\\\ \mbox{\tt\small~{}~{}~{}
$\\!${\mbox{\footnotesize OR}}$\\!$ (0.4 $\\!${\mbox{\footnotesize
SCALE}}$\\!$ (<scene> $\\!${\mbox{\footnotesize CONTAINING}}$\\!$ ni))
$\\!${\mbox{\footnotesize OR}}$\\!$ (0.4 $\\!${\mbox{\footnotesize
SCALE}}$\\!$ (<shot> $\\!${\mbox{\footnotesize CONTAINING}}$\\!$ ni)))
}\\!\\!\\!\\!\end{array}$ (9)
Showing that the region expression of Equation 9 retrieves and ranks video
shots according to Equation 8 is done as in the previous section.
### 3.4 Web retrieval with page priors
For web retrieval, non-content information like the number of hyperlinks
pointing to a web page, or the form of the URL are good indicators of the
importance of a page. Such approaches can be modeled by so-called document
priors $P(D)$ that do not depend on the query [28, 29, 40]. Document priors
are calculated once for the entire collection, stored in the system and then
used to enhance retrieval results for every query. A good example of such an
approach is Google’s PageRank algorithm [11].
Document priors are motivated as follows. Instead of ranking documents by the
probability that they generate the query, it makes more sense to rank them by
$P(D|T_{1},T_{2},\\!\cdots\\!,T_{l})$: The probability that $D$ is relevant
given the query $T_{1},T_{2},\\!\cdots\\!,T_{l}$ of length $l$. According to
Bayes’ rule:
$\vspace{0.1cm}\begin{array}[]{rcl}P(D|T_{1},T_{2},\\!\cdots\\!,T_{l})&\\!=\\!&\\!{\displaystyle\frac{P(D)\cdot
P(T_{1},T_{2},\\!\cdots\\!,T_{l}|D)}{P(T_{1},T_{2},\\!\cdots\\!,T_{l})}}\\\
&\\!\propto\\!&\\!{\displaystyle P(D)\cdot\prod_{i=1}^{l}P(T_{i}|D)}\\\
\end{array}\vspace{0.1cm}$ (10)
The denominator, $P(T_{1},T_{2},\cdots,T_{l})$, does not depend on $D$ and can
therefore be dropped, but document prior, $P(D)$, cannot be dropped unless it
is uniformly distributed over all documents. Suppose we are looking for the
entry page of Google. Documents will be ranked as follows.
$\vspace{0.1cm}\begin{array}[]{rcl}P(D|T\\!=\\!\mbox{\tt\small
google})&\\!\propto\\!&\\!{\displaystyle P(D)\cdot P(T\\!=\\!\mbox{\tt\small
google}|D)}\\\ \end{array}\vspace{0.1cm}$ (11)
To follow this approach, the system needs to have some means to store text
elements with their prior probability. Suppose an application program
calculated the PageRank of each crawled web page resulting in probabilities
$P(D)$ (or any number proportional to the probabilities, see [11]) for each
document region, which is stored as $PageRank. The dollar sign is used to
denote a region set that is stored by the system for later use. The set is
used in the query as follows.
$PageRank AND (<doc> CONTAINING google) (12)
We believe the correspondence between Equation 11 and 12 is obvious. As
before, the query $PageRank AND (<doc> CONTAINED_BY (<doc> CONTAINING google))
would be a more general query that produces the exact same results.
### 3.5 Cross-language information retrieval
In cross-language information retrieval, a collection in one language, e.g.
English, is searched by querying it in another language, e.g. Dutch. A
language modeling approach to cross-language retrieval ranks documents by the
probability $P(S_{1},S_{2},\cdots,S_{l}|D)$ of generating a Dutch query
$S_{1},S_{2},\cdots,S_{l}$ of length $l$ from the English document $D$. This
is modeled by the following procedure: first an English word $T$ is generated
from a document with probability $P(T|D)$, then the English term is translated
to Dutch independently from the document it was generated from, so with
probability $P(S|T)$, resulting in [8, 24, 57]:
$P(S_{1},S_{2},\cdots,S_{l}|D)\;=\;\prod_{i=1}^{l}\sum_{j=1}^{V}(P(S_{i}|T_{j})P(T_{j}|D))$
(13)
where $P(T_{j}|D)$ is again the document language model, and $P(S_{i}|T_{j})$
is a translation model defining the probabilities of the source language words
(for instance Dutch in case of a Dutch query) given the target language words
(English if the collection being searched is English), and where $V$ is the
size of the target language vocabulary. Such a model is used as follows: Given
a Dutch query $S_{1},S_{2},\cdots,S_{l}$, every word might have several
possible translations in English. Suppose we want to use the Dutch query
gebroken hart (English: “broken heart”) to search for English documents. The
application program would consult its dictionary to determine that there are
two possible English translations for the Dutch word “gebroken”: “broken” and
“fractured”. The probability of translating “broken” to “gebroken”, i.e.
$P(S=\mbox{\tt\small gebroken}|T=\mbox{\tt\small broken})$ might be estimated
as 1.0, for instance because from example texts we know that the English word
“broken” is always translated to “gebroken”; and the probability of
translating “fractured” to “gebroken”, i.e. $P(S=\mbox{\tt\small
gebroken}|T=\mbox{\tt\small fractured})$ might be estimated as 0.2 (note that
the two probabilities do not need to sum up to 1). In this case, an
instantiation of Equation 13 would be:
$\\!\\!\\!\\!\begin{array}[]{l}P(S_{1}\\!=\\!\mbox{\tt\small
gebroken},S_{2}\\!=\\!\mbox{\tt\small hart}|D)\;=\\\ \;\;(1.0\cdot
P(T_{1}\\!=\\!\mbox{\tt\small broken}|D)\,+\,0.2\cdot
P(T_{1}\\!=\\!\mbox{\tt\small fractured}|D))\\\ \;\;\cdot\\\ \;\;(0.5\cdot
P(T_{2}\\!=\\!\mbox{\tt\small heart}|D)\,+\,0.1\cdot
P(T_{2}\\!=\\!\mbox{\tt\small ticker}|D))\\\ \end{array}\\!\\!\\!\\!$ (14)
So, the sum over the whole target language vocabulary will in practice be a
sum over the possible translations only (those for which $P(S|T)>0$). The
probability function corresponds to the following region expression.
$\vspace{0.1cm}\\!\\!\\!\\!\begin{array}[]{l}\mbox{\tt\small((1.0
$\\!${\mbox{\footnotesize SCALE}}$\\!$ (<doc> $\\!${\mbox{\footnotesize
CONTAINING}}$\\!$ broken)) $\\!${\mbox{\footnotesize OR}}$\\!$ (0.2
$\\!${\mbox{\footnotesize SCALE}}$\\!$ (<doc> $\\!${\mbox{\footnotesize
CONTAINING}}$\\!$ fractured)))}\\\ \mbox{\tt\small{\mbox{\footnotesize
AND}}}\\\ \mbox{\tt\small((0.5 $\\!${\mbox{\footnotesize SCALE}}$\\!$ (<doc>
$\\!${\mbox{\footnotesize CONTAINING}}$\\!$ heart)) $\\!${\mbox{\footnotesize
OR}}$\\!$ (0.1 $\\!${\mbox{\footnotesize SCALE}}$\\!$ (<doc>
$\\!${\mbox{\footnotesize CONTAINING}}$\\!$ ticker)))
}\end{array}\\!\\!\\!\\!\vspace{0.1cm}$ (15)
Equation 15 can be generated from 14 as shown in the previous sections.
## 4 Discussion, open issues and future work
In this report, we presented a unified region model / language model approach
and showed its expressiveness for a wide range of applications of language
modeling: ad-hoc retrieval, smoothing, video retrieval, web search and cross-
language retrieval. In the past, we have developed separate prototype
retrieval systems for these approaches. Developing these prototype systems
meant we had to reimplement parts of our system: reimplementing APIs,
introducing new APIs, introducing new query languages, introducing new
indexes, introducing new storage structures, etc. This report shows that such
approaches can be supported by a single retrieval system that responds to a
simple query language that hides implementation details of information
retrieval approaches from the application developer.
The relationship between the region queries and the language modeling
probability functions might seem trivial because we “hard-wired” the language
modeling probability definition in the CONTAINING operator, but we believe it
is remarkable: Note that the language modeling probability functions are
arithmetic expressions that define the probability of a single document $D$.
However, the region queries are algebraic expressions for processing sets of
documents (regions) instead of single documents. Since the region query
language forms a “bulk algebra”, experiences from relational database system
design can be used to develop efficient implementations of such a system,
possibly up to a point where applications run as fast as, or possibly even
faster than, the dedicated prototypes we developed in the past.
### 4.1 Query optimization
The queries presented in Section 2 are close to the language modeling
probability functions. However, there exist alternative expressions of the
queries that produce equivalent results but that might be easier to process by
the system. Based on a study into equivalence relations for region models
[35], we conjecture that the following expressions are alternatives for the
expressions presented in Section 2: (<doc> CONTAINING db) CONTAINING ir is an
alternative for Equation 3; (<doc> CONTAINED_BY (((0.2 SCALE <root>) OR (0.8
SCALE <doc>)) CONTAINING db)) CONTAINED_BY (((0.2 SCALE <root>) OR (0.8 SCALE
<doc>)) CONTAINING ir) is an alternative for Equation 6; <shot> CONTAINED_BY
(((0.18 SCALE <root>) OR (0.02 SCALE <video>) OR (0.4 SCALE <scene>) OR (0.4
SCALE <shot>)) CONTAINING ni) is an alternative for Equation 9; $PageRank
CONTAINING google is an alternative for Equation 12; finally (<doc> CONTAINING
(broken OR (0.2 SCALE fractured))) CONTAINING ((0.5 SCALE heart) OR (0.1 SCALE
ticker)) is an alternative for Equation 15.
Additionally, query optimization would involve choosing concrete evaluation
methods attached to each operation, estimating the costs of each method, and
choosing the fastest plan. Ramírez and De Vries [45] present preliminary
results.
### 4.2 Towards existing XML query languages
It can be argued that region models are simple predecessors of models
underlying XML query languages like XPath [9] and XQuery [10]. That is,
operators like CONTAINED_BY and CONTAINING can be seen as ancestor and
descendent axis steps, as well as the function fn:contains in XPath. It would
be relatively easy to add other XPath axis steps to the query language if we
specify how regions are nested, for instance by requiring that a region has a
level (the depth in the XML tree) as well as a start, end, and score.
XML and its subsequent standards like XPath and XQuery have initiated a lot of
research into XML database systems with dedicated workshops and symposia like
DataX [34] and XSym [7]. Our implementation of the region approach is quite
similar to implementations of XML databases that use relational database
technology and a numbering of the XML nodes [53]. Interestingly, the word
positions that belong to the region start and region end of an XML element are
respectively in pre-order and post-order as in the XML database implementation
proposed by Grust [23]. Our prototype system TIJAH uses part of the code of
the PathFinder XML database system [54]. In the future, both systems might be
integrated following the XQuery full-text standard [3, 4].
### 4.3 Towards new applications of XML
Some people have argued that existing XML query languages like XPath [9] and
XQuery [10] are too powerful for simple XML information retrieval
functionality [55]. Others have argued that existing query languages are not
powerful enough. For instance Ogilvie [39] illustrates a system that answers
queries like “Who killed Abraham Lincoln” by a query that returns those
<person> elements that directly precede the word killed, which directly
precedes another <person> element containing lincoln. Such a query would be
hard, if not impossible, to express in existing XML query languages. A
solution might be the introduction of a special gluing operator in our region
model approach, let’s call it ADJ for “adjacent”, which can glue regions to
form bigger regions. Such an operator might be used for phrases, but also to
glue for instance two paragraphs together to form a region that spans two
paragraphs. We have implemented such a gluing operator in our video retrieval
system that, lacking a reliable scene detector, glues adjacent shots together
to represent a scene [26].
### 4.4 Beyond XML
Ogilvie [39] also makes a case for allowing several hierarchies of possibly
overlapping elements which combined would no longer form a tree. This need is
illustrated as well by Burkowski [12], by people studying the bible [18], and
it is picked up by several initiatives to extend XML [43, 52]. The region
approach described here would support querying of such representations quite
naturally.
## Acknowledgements
Djoerd Hiemstra was supported by the Dutch BSIK program MultimediaN: Semantic
Multimedia Access. Vojkan Mihajlović was supported by the Netherlands
Organisation for Scientific Research (NWO project 612.061.210). We like to
thank Henk Ernst Blok for fruitful discussions on region algebras, and Maarten
Fokkinga and Thijs Westerveld (CWI, Amsterdam) for helpful comments on the
report.
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|
arxiv-papers
| 2010-05-26T08:04:33 |
2024-09-04T02:49:10.641504
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Djoerd Hiemstra and Vojkan Mihajlovic",
"submitter": "Djoerd Hiemstra",
"url": "https://arxiv.org/abs/1005.4752"
}
|
1005.5011
|
Correlated imaging through atmospheric turbulence
Pengli Zhang, Wenlin Gong, Xia Shen and Shensheng Han∗
Key Laboratory for Quantum Optics and Center for Cold Atom Physics, Shanghai
Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai
201800, China
∗Corresponding author: sshan@mail.shcnc.ac.cn
###### Abstract
Correlated imaging through atmospheric turbulence is studied, and the
analytical expressions describing turbulence effects on image resolution are
derived. Compared with direct imaging, correlated imaging can reduce the
influence of turbulence to a certain extent and reconstruct high-resolution
images. The result is backed up by numerical simulations, in which turbulence-
induced phase perturbations are simulated by random phase screens inserting
propagation paths.
OCIS codes: 270.0270, 010.1330, 110.0115
As correlated imaging develops well in recent years [1, 5, 2, 3, 4], more
attention has been focused on how to apply this technique to practical
applications to overcome the limits in conventional optical systems. For an
imaging system which must look through the atmosphere, turbulence-induced
wavefront variations distort the point spread function (PSF) of the system
from its ideal diffraction-limited shape, which leads to the the degradation
of image resolution [6]. To mitigate turbulence effects, a number of methods,
such as speckle imaging and adaptive optics techniques [6], have been proposed
and applied in optical astronomy. Nonetheless, each of these techniques has
its own set of performance limits, hardware and software requirements. New
approaches to the problem of reducing these effects are still of much
interest. Here we investigate the performance of correlated imaging through
atmospheric turbulence and find that the influence of turbulence can be
weakened by the second-order intensity correlation.
A schematic of correlated imaging through the atmosphere is depicted in Fig.
1. The beam splitter (BS) divides thermal light into two beams propagating
through two distinct optical paths. One is test arm which includes an unknown
object and a telescope setup consisting of a lens with focal length $f$ and a
detector $D_{t}$. The object is located at a distance $d_{1}$ from the source
as well as $d_{2}$ to the telescope setup. The other is the reference arm
where another telescope setup consisting of a lens and a detector $D_{r}$ is
placed at $d_{0}=d_{1}+d_{2}$ from the source. For remote sensing (i.e.,
$d_{1},d_{2}\gg f$), the detector $D_{t}$ (or $D_{r}$) generally lies close to
the back focal plane of the lens (i.e., $d_{3}\approx f$). The test arm is
imbedded in the atmosphere, and turbulence-induced wavefront fluctuations in
propagation paths $d_{1}$ and $d_{2}$ are represented by $\Psi_{1}$ and
$\Psi_{2}$, respectively. While the reference arm is said to be a free-space
propagation through the distance $d_{0}$ by assuming that there exists no
turbulence. The assumption is based on the fact that the optical field in the
reference arm is totally predictable if the field distribution of the source
is well known [1, 2].
Fig. 1: Schematic of correlated imaging through atmospheric turbulence.
In the test arm, the field $E_{t}(x_{t})$ in the detector $D_{t}$ can be given
by
$\displaystyle E_{t}(x_{t})=\iint dxd\xi
E_{s}(x)h_{1}(\xi,x)t(\xi)h_{2}(x_{t},\xi),$ (1)
where $E_{s}(x)$ corresponds to the source field, and $t(\xi)$ denotes the
transmission function of the object. $h_{1}(\xi,x)$, $h_{2}(x_{t},\xi)$ are
the impulse response functions from the source to the object and from the
object to the detector $D_{t}$, respectively.
Furthermore, according to the extended Huygens-Fresnel integral [7],
$h_{1}(\xi,x)$ and $h_{2}(x_{t},\xi)$ have the forms
$\displaystyle h_{1}(\xi,x)=\frac{1}{\sqrt{j\lambda
d_{1}}}e^{\frac{jk}{2d_{1}}(x-\xi)^{2}+\Psi_{1}(x,\xi)},$ (2a) $\displaystyle
h_{2}(x_{t},\xi)=\frac{1}{j\lambda\sqrt{d_{2}d_{3}}}\int d\eta
e^{-\frac{jk}{d_{2}}(\xi-x_{t}/M)\eta+\Psi_{2}(\xi,\eta)},$ (2b)
where $k=2\pi/\lambda$ is the wave number with $\lambda$ being the wavelength,
and $M=-d_{3}/d_{2}$ is the magnification of the telescope setup.
$\Psi_{1}(x,\xi)$ and $\Psi_{2}(\xi,\eta)$ account for the random parts (due
to atmospheric turbulence) of the complex phases of the fields in the
propagation paths $d_{1}$ and $d_{2}$, respectively.
The field $E_{r}(x_{r})$ in the detector $D_{r}$ is connected to the source
field $E_{s}(x)$ by the Fresnel diffraction integral
$\displaystyle E_{r}(x_{r})=\frac{1}{\sqrt{j\lambda d_{1}|M|}}\int
dxE_{s}(x)e^{\frac{jk}{2d_{1}}(x-x_{r}/M)^{2}}.$ (3)
It’s worth pointing out that the apertures of the lenses are regarded as large
enough, and the diffraction limit of the lenses has been neglected here.
Performing the intensity correlation measurement between the test arm and the
reference arm, we get
$\displaystyle G(x_{t},x_{r})$ $\displaystyle=$ $\displaystyle\langle
I_{t}(x_{t})I_{r}(x_{r})\rangle-\langle I_{t}(x_{t})\rangle\langle
I_{r}(x_{r})\rangle$ (4) $\displaystyle=$ $\displaystyle c_{0}\int
dxdx^{\prime}dx^{\prime\prime}dx^{\prime\prime\prime}d\xi d\xi^{\prime}\langle
E_{s}(x)E^{\ast}_{s}(x^{\prime\prime\prime})\rangle$
$\displaystyle\times\langle
E^{\ast}_{s}(x^{\prime})E_{s}(x^{\prime\prime})\rangle\langle
h_{1}(\xi,x)h^{\ast}_{1}(\xi^{\prime},x^{\prime})\rangle$
$\displaystyle\times\langle
h_{2}(x_{t},\xi)h^{\ast}_{2}(x_{t},\xi^{\prime})\rangle
t(\xi)t^{\ast}(\xi^{\prime})$ $\displaystyle\times
e^{\frac{jk}{2d_{1}}[(x^{\prime\prime}-x_{r}/M)^{2}-(x^{\prime\prime\prime}-x_{r}/M)^{2}]},$
where $c_{0}$ is a constant $(\lambda^{3}d_{1}d_{2}d_{3}|M|)^{-1}$, and
$I_{t}(x_{t}),\ I_{r}(x_{r})$ represent the intensity distributions in $D_{t}$
and $D_{r}$, respectively. Here, we have supposed that the thermal field, and
the two turbulent regions are statistically independent of each other.
If the source is fully spatially incoherent and its intensity distribution is
of the Gaussian type, the first-order correlation function of the source has
the form
$\displaystyle\langle
E_{s}(x)E^{\ast}_{s}(x^{\prime})\rangle=I_{0}e^{-\frac{x^{2}+x^{\prime
2}}{r_{e}^{2}}}\delta(x-x^{\prime}),$ (5)
where $I_{0}$ denotes the mean intensity at the center of the source, and
$r_{e}$ is the $1/e^{2}$ intensity radius. With the help of Eqs. (2a), (2b),
and (5), Eq. (4) can be rewritten as
$\displaystyle G(x_{t},x_{r})$ $\displaystyle=$ $\displaystyle I^{2}_{0}\int
dxdx^{\prime}d\eta d\eta^{\prime}d\xi
d\xi^{\prime}t(\xi)t^{\ast}(\xi^{\prime})$ (6) $\displaystyle\times
e^{-\frac{2(x^{2}+x^{\prime
2})}{r_{e}^{2}}}e^{\frac{jk}{2d_{1}}[(x^{\prime}-x_{r}/M)^{2}-(x-x_{r}/M)^{2}]}$
$\displaystyle\times
e^{\frac{jk}{2d_{1}}[(x-\xi)^{2}-(x^{\prime}-\xi^{\prime})^{2}]}\langle
e^{\Psi_{1}(x,\xi)+\Psi^{\ast}_{1}(x^{\prime},\xi^{\prime})}\rangle$
$\displaystyle\times e^{\frac{jk}{d_{2}}[(\xi-
x_{t}/M)\eta-(\xi^{\prime}-x_{t}/M)\eta^{\prime}]}$
$\displaystyle\times\langle
e^{\Psi_{2}(\xi,\eta)+\Psi^{\ast}_{2}(\xi^{\prime},\eta^{\prime})}\rangle.$
The ensemble average of phase variations arising from turbulence can be
approximated by [7]
$\displaystyle\langle
e^{\Psi_{i}(x,\xi)+\Psi^{\ast}_{i}(x^{\prime},\xi^{\prime})}\rangle$ (7)
$\displaystyle\cong$ $\displaystyle
e^{-\frac{1}{\rho^{2}_{i}}[(x-x^{\prime})^{2}+(x-x^{\prime})(\xi-\xi^{\prime})+(\xi-\xi^{\prime})^{2}]},$
where $\rho_{i}=(0.545C^{2(i)}_{n}k^{2}d_{i})^{-3/5}$ ($i=1,2$) is the
coherence length of a spherical wave propagating in the turbulent medium and
$C^{2(i)}_{n}$ corresponds to the refractive-index structure constants
describing the strength of atmospheric turbulence in the propagation path
$d_{i}$. It’s worth emphasizing that we have adopted a quadratic approximation
of the Rytov’s phase structure function in Eq. (7) to obtain the analytical
formula, and this approximation has been used widely in literatures [7, 4].
Substituting Eq. (7) to Eq. (6) and integrating over
$\eta,\eta^{\prime},x,x^{\prime}$, we have
$\displaystyle G(x_{t},x_{r})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\pi}I^{2}_{0}c_{0}}{\sqrt{\alpha\beta_{2}(\alpha+2\beta_{1})}}\int
d\xi|t(\xi)|^{2}$ (8) $\displaystyle\times
e^{-\frac{2A^{2}}{\alpha+2\beta_{1}}(\xi-
x_{r}/M)^{2}}e^{-\frac{B^{2}}{\beta_{2}}(\xi-x_{t}/M)^{2}},$
where $A=k/2d_{1}$, $B=k/2d_{2}$, $\alpha=r_{e}^{-2}/2$,
$\beta_{i}=\rho_{i}^{-2}$ .
By making $x_{r}=x_{t}$ in Eq. (8), we carry out a special point-to-point
intensity correlation [8] and obtain the PSF of the correlated imaging system
$\displaystyle h_{g}(x_{r},\xi)=e^{-\frac{2A^{2}}{\alpha+2\beta_{1}}(\xi-
x_{r}/M)^{2}}e^{-\frac{B^{2}}{\beta_{2}}(\xi-x_{r}/M)^{2}}.$ (9)
For the sake of comparison, we also present the intensity distribution in
$D_{t}$,
$\displaystyle
I_{t}(x_{t})=\frac{\sqrt{\pi}I_{0}c_{0}}{\sqrt{\alpha\beta_{2}}}\int
d\xi|t(\xi)|^{2}e^{-\frac{B^{2}}{\beta_{2}}(\xi-x_{t}/M)^{2}},$ (10)
and the PSF of the test arm
$\displaystyle h_{t}(x_{t},\xi)=e^{-\frac{B^{2}}{\beta_{2}}(\xi-
x_{t}/M)^{2}}.$ (11)
From Eqs. (9) and (11), we can see that the full widths at half maximum (FWHM)
of $h_{g}$ and $h_{t}$ both broaden with the increase of $\beta_{i}$ (apart
from the influence of the size of the source), which indicates that the
resolution, whether for correlated imaging or direct imaging, is degraded by
atmospheric turbulence. Additionally, and most importantly, $h_{g}$ has a
narrower FWHM compared to $h_{t}$ , which means that correlated imaging is
helpful to reduce turbulent effects and achieve high-resolution images.
In simulations, we consider correlated imaging through horizontal paths in the
atmosphere, and thus $C_{n}^{2}$ can be regarded as constant in the whole
turbulent regions. The numerical model of light propagation in turbulence has
been developed well [9, 10]. The spatial power spectral density of the index
of refraction fluctuations can be described by the Von Karman spectrum [9],
$\displaystyle\Phi_{n}(K,z)=0.033C_{n}^{2}(z)(K^{2}+L_{0}^{-2})^{-11/6}e^{-(Kl_{0}/2\pi)^{2}},$
(12)
where $K^{2}=K^{2}_{x}+K^{2}_{y}+K^{2}_{y}$, $z$ is the propagation distance
from the source, $L_{0}$ and $l_{0}$ represent the outer scale and inner scale
of the turbulence, respectively. By using the spectrum in Eq. (12) to filter a
complex Gaussian pseudorandom field and inverse transforming the result, one
obtains a two-dimensional phase screen which has the same statistics as the
turbulence-induced phase variations [9]. For long atmospheric paths, the
multiple phase-screen model [10] has been used in simulations. The turbulent
region with the propagation length $d_{i}$ is broken into a number of layers
with a thickness $\Delta z$. Phase fluctuations in each layer are represented
by a phase screen inserting in the middle of the layer. The effect of field
propagation through these continuous layers can be calculated separately and
then combined to characterize propagation through the entire turbulent region,
provided the index of refraction fluctuations for each layer are statistically
independent [6].
Fig. 2: Simulated ( open circles) and theoretical (solid line) on-axis
irradiance variance versus the propagation distance. The outer scale and inner
scale of turbulence are $L_{0}=3$ m and $l_{0}=1$ cm, respectively.
First of all, to verify the computer programm, we investigate the behavior of
a Gaussian beam (waist radius $w_{0}=7$ cm and wavelength $\lambda=2\ \mu$m)
traveling through the atmosphere with a strong turbulence level
($C_{n}^{2}$=$10^{-12}\textrm{m}^{-2/3}$). The thickness of each layer is
$\Delta z=50$ m. The on-axis normalized intensity variance, defined as
$\sigma_{I}^{2}=\langle I^{2}\rangle/\langle I\rangle^{2}-1$ [9], is plotted
as a function of the propagation distance in Fig. 2. The good coincidence
between the simulated data (open circles) and the theoretical result (solid
line) predicted by [11] proves the validity of the programm.
After the validation, we apply the programm to simulate the correlated imaging
system shown in Fig. 1. The thermal source ($\lambda=0.532\ \mu$m and diameter
$D=2r_{e}=5$ cm) was described by a grid of $512\times 512$ with a sample
spacing $\Delta x=\Delta y=5$ mm. The distances were set as $d_{1}=d_{2}=10$
km and the focal length $f=1$ m. The turbulence regions in the paths
$d_{i}(i=1,2)$ were divided into 20 layers with a thickness $\Delta z=500$ m,
respectively. The turbulent parameters were assumed constant at the outer
scale $L_{0}=100$ m and the inner scale $l_{0}=5$ mm. By averaging over
$10^{4}$ samples, simulated results [see Fig. (3)] clearly show the image
resolution decrease with the increase of the turbulent strength, which accords
with the analytical calculation from Eq. (8).
Fig. 3: The reconstructed images (from left to right) via the correlation in
the atmosphere with turbulent levels $C_{n}^{2}=10^{-16}\textrm{m}^{-2/3},\
2.5\times 10^{-16}\textrm{m}^{-2/3},\ 5\times 10^{-16}\textrm{m}^{-2/3}$, and
$10^{-15}\textrm{m}^{-2/3}$, respectively.
Fig. 4: The acquired images of the double slit in the atmosphere with
turbulent level $C_{n}^{2}=10^{-15}\textrm{m}^{-2/3}$ . (a) was obtained by
the test arm, and (b) was extracted from the correlation. The normalized
horizontal sections of the images are plotted in (c), where open circles
correspond to the simulated data and solid lines show the theoretical
predictions from Eqs. (8) and (10), respectively.
To compare direct imaging and correlated imaging, a simple double slit (slit
width 10 cm and center-to-center separation 20 cm) was used. After statistics
over $10^{4}$ samples, we obtained a blurred image detected by the test arm
directly [see Fig. 4(a)] and a clear image reconstructed through the
correlation [see Fig. 4(b)]. This confirms the analytical result that ghost
imaging could reduce turbulent effects and improve resolution.
In summary, by taking advantage of the extended Huygens-Fresnel integral, we
have presented the theoretical expressions that describes how atmospheric
turbulence corrupts the image resolution. Meanwhile, the analytical
calculations and the numerical simulations have demonstrated that correlated
imaging can provide imaging performance superior to direct imaging through the
atmosphere. As an unique image-formed method, correlated imaging can be
effectively combined with conventional phase compensating techniques (e.g.,
adaptive optics) to further eliminate turbulent effects.
This work is supported by the Hi-Tech Research and development Programm of
China (Grant No. 2006AA12Z115), Shanghai Fundamental Research Project (Grant
No. 09JC1415000), and the National Natural Science Foundation of China (Grant
No. 6087709).
## References
* [1] Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
* [2] J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802(R) (2008).
* [3] Kam Wai Clifford Chan, M. N. O’ Sullivan, and R. W. Boyd, “High-order thermal ghost imaging,” Opt. Lett. 34, 3343 (2009).
* [4] Jing Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express 17, 7916-7921 (2009).
* [5] W. Gong, P. Zhang, X. Shen and S. Han, “Imaging in scattering media via the second-order correlation of light field,” arXiv.Quant-ph/0908.0185v1 (2009).
* [6] M. C. Roggemann, B. Welsh, Imaging through turbulence (CRC, USA, 1996).
* [7] J. C. Ricklin, and F. M. Davidson, “Atmospheric turbulence effect on a partially coherent Gaussian beam: implication for free-space laser communication,” J. Opt. Soc. Am. 19, 1794-1802 (2002).
* [8] Pengli Zhang, Wenlin Gong, Xia Shen, Dajie Huang, and Shensheng Han, “Improving resolution by the second-order correlation of light fields,” Opt. Lett. 34, 1222-1224 (2009).
* [9] A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. 39, 5426-5444 (2000).
* [10] D. L. Knepp, “multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE, 71, 722-736 (1983).
* [11] L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves in Random and Complex Media, 11. 271-291 (2001).
|
arxiv-papers
| 2010-05-27T09:35:30 |
2024-09-04T02:49:10.655231
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pengli Zhang, Wenlin Gong, Xia Shen and Shensheng Han",
"submitter": "Peng-Li Zhang",
"url": "https://arxiv.org/abs/1005.5011"
}
|
1005.5057
|
# Erlangen Programme at Large 3.1
Hypercomplex Representations of the Heisenberg Group and Mechanics
Vladimir V. Kisil School of Mathematics
University of Leeds
Leeds LS2 9JT
UK kisilv@maths.leeds.ac.uk http://www.maths.leeds.ac.uk/~kisilv/ Dedicated
to the memory of V.I. Arnold
(Date: 27th May 2010)
###### Abstract.
In the spirit of geometric quantisation we consider representations of the
Heisenberg(–Weyl) group induced by hypercomplex characters of its centre. This
allows to gather under the same framework, called p-mechanics, the three
principal cases: quantum mechanics (elliptic character), hyperbolic mechanics
and classical mechanics (parabolic character). In each case we recover the
corresponding dynamic equation as well as rules for addition of probabilities.
Notably, we are able to obtain whole classical mechanics without any kind of
semiclassical limit $\hslash\rightarrow 0$.
###### Key words and phrases:
Heisenberg group, Kirillov’s method of orbits, geometric quantisation, quantum
mechanics, classical mechanics, Planck constant, dual numbers, double numbers,
hypercomplex, jet spaces, hyperbolic mechanics, interference, Segal–Bargmann
representation, Schrödinger representation, dynamics equation, harmonic and
unharmonic oscillator, contextual probability, $\mathcal{PT}$-symmetric
Hamiltonian
###### 2000 Mathematics Subject Classification:
Primary 81R05; Secondary 81R15, 22E27, 22E70, 30G35, 43A65.
On leave from Odessa University.
††copyright: ©:
###### Contents
1. 1 Introduction
2. 2 Heisenberg group and $p$-mechanics
1. 2.1 The Heisenberg group and induced representations
2. 2.2 Convolutions (observables) on $\mathbb{H}^{n}{}$ and commutator
3. 2.3 States and Probability
3. 3 Elliptic characters and Quantum Dynamics
1. 3.1 Segal–Bargmann and Schrödinger Representations
2. 3.2 Commutator and the Heisenberg Equation
3. 3.3 Quantum Probabilities
4. 4 Hypercomplex Repersentations of the Heisenberg Group
1. 4.1 Hyperbolic Representations and Addition of Probabilities
1. 4.1.1 Hyperbolic Representations of the Heisenberg Group
2. 4.1.2 Hyperbolic Dynamics
3. 4.1.3 Hyperbolic Probabilities
2. 4.2 Parabolic (Classical) representations on the phase space
1. 4.2.1 Classical Non-Commutative Representations
2. 4.2.2 Hamilton Equation
3. 4.2.3 Classical probabilities
5. 5 Discussion
## 1\. Introduction
Complex valued representations of the Heisenberg group (also known as Weyl or
Heisenberg-Weyl group) provide a natural framework for quantum mechanics
[Howe80b, Folland89]. This is the most fundamental example of the Kirillov
orbit method and geometrical quantisation technique [Kirillov99, Kirillov94a].
Following the pattern we consider representations of the Heisenberg group
which are induced by hypercomplex characters of its centre. Besides complex
numbers (which correspond to the elliptic case) there are two other types of
hypercomplex numbers: dual (parabolic) and double (hyperbolic) [Yaglom79]*App.
C [Kisil09c].
To describe dynamics of a physical system we use a universal equation based on
inner derivations of the convolution algebra [Kisil00a] [Kisil02e]. The
complex valued representations produce the standard framework for quantum
mechanics with the Heisenberg dynamical equation [Vourdas06a].
The double number valued representations, with the hyperbolic unit
$\mathrm{j}^{2}=1$, is a natural source of hyperbolic quantum mechanics
developed for a while [Hudson04a, Hudson66a, Khrennikov03a, Khrennikov05a,
Khrennikov08a]. The universal dynamical equation employs hyperbolic commutator
in this case. This can be seen as a Moyal bracket based on the hyperbolic sine
function. The hyperbolic observables act as operators on a Krein space with an
indefinite inner product. Such spaces are employed in study of
$\mathcal{PT}$-symmetric Hamiltonians and hyperbolic unit $\mathrm{j}^{2}=1$
naturally appear in this setup [GuentherKuzhel10a].
The representations with values in dual numbers provide a convenient
description of the classical mechanics. For this we do not take any sort of
semiclassical limit, rather the nilpotency of the parabolic unit
($\varepsilon^{2}=0$) do the task. This removes the vicious necessity to
consider the Planck _constant_ tending to zero. The dynamical equation takes
the Hamiltonian form. We also describe classical non-commutative
representations of the Heisenberg group which acts in the first jet space.
###### Remark 1.1.
It is commonly accepted that the striking difference between quantum and
classical mechanics is non-commutativity of observables in the first case. In
particular the Heisenberg commutation relations, see (2.5), imply the
uncertainty principle, the Heisenberg equation of motion and other quantum
features. However our work shows that quantum mechanics is mainly determined
by the properties of complex numbers. Non-commutative representations of the
Heisenberg group in dual numbers implies the Poisson dynamical equation and
local addition of probabilities in Section 4.2, which are completely
classical.
###### Remark 1.2.
It is worth to note that our technique is different from contraction technique
in the theory of Lie groups [LevyLeblond65a, GromovKuratov05b]. Indeed a
contraction of the Heisenberg group $\mathbb{H}^{n}{}$ is the commutative
Euclidean group $\mathbb{R}^{2n}{}$ which does not recreate neither quantum
nor classical mechanics.
The approach provides not only three different types of dynamics, it also
generates the respective rules for addition of probabilities as well. For
example, the quantum interference is the consequence of the same complex-
valued structure, which directs the Heisenberg equation. The absence of an
interference (a particle behaviour) in the classical mechanics is again the
consequence the nilpotency of the parabolic unit. Double numbers creates the
hyperbolic law of additions of probabilities which were extensively
investigates [Khrennikov03a, Khrennikov05a]. There are still unresolved issues
with positivity of the probabilistic interpretation in the hyperbolic case
[Hudson04a, Hudson66a].
The work clarifies foundations of quantum and classical mechanics. We
recovered from the representation theory the existence of three non-isomorphic
model of mechanics already discussed in [Hudson04a, Hudson66a] from
translation invariant formulation. It also hinted that hyperbolic counterpart
is (at least theoretically) as natural as classical and quantum mechanics are.
The approach provides a framework for description of aggregate system which
have say both quantum and classical components. This can be used to model
quantum computers with classical terminals [Kisil09b].
Remarkably, simultaneously with the work [Hudson66a] group-invariant
axiomatics of geometry lead R.I. Pimenov [Pimenov65a] to description of
$3^{n}$ Cayley–Klein constructions. The connection between group-invariant
geometry and respective mechanics were explored in many works of N.A. Gromov,
see for example [Gromov90a, Gromov90b, GromovKuratov05b]. Those already
highlighted the rôle of three types of hypercomplex units for the realisation
of elliptic, parabolic and hyperbolic geometry and kinematic.
There is a further connection between representations of the Heisenberg group
and hypercomplex numbers. The symplectomorphism of phase space are also
automorphism of the Heisenberg group [Folland89]*§ 1.2. Induced representation
of the symplectic group naturally lead to hypercomplex numbers [Kisil09c].
Hamiltonians, which produce those symplectomorphism, are of interest, for
example, in quantum optic [ATorre10a]. An analysis of those Hamiltonians by
means of creation/annihilation operators recreate hypercomplex coefficients as
well [Kisil11a, Kisil11c].
###### Remark 1.3.
This work is performed within the “Erlangen programme at large” framework
[Kisil06a, Kisil05a], thus it would be suitable to explain the numbering of
various papers. Since the logical order may be different from chronological
one the following numbering scheme is used:
Prefix | Branch description
---|---
“0” or no prefix | Mainly geometrical works, within the classical field of Erlangen programme by F. Klein
“1” | Papers on analytical functions theories and wavelets
“2” | Papers on operator theory, functional calculi and spectra
“3” | Papers on mathematical physics
For example, this is the first paper in the mathematical physics area.
## 2\. Heisenberg group and $p$-mechanics
### 2.1. The Heisenberg group and induced representations
Let $(s,x,y)$, where $x$, $y\in\mathbb{R}^{n}{}$ and $s\in\mathbb{R}{}$, be an
element of the Heisenberg group $\mathbb{H}^{n}{}$ [Folland89, Howe80b]. The
group law on $\mathbb{H}^{n}{}$ is given as follows:
(2.1)
$\textstyle(s,x,y)\cdot(s^{\prime},x^{\prime},y^{\prime})=(s+s^{\prime}+\frac{1}{2}\omega(x,y;x^{\prime},y^{\prime}),x+x^{\prime},y+y^{\prime}),$
where the non-commutativity is due to $\omega$—the _symplectic form_ on
$\mathbb{R}^{2n}{}$ [Arnold91]*§ 37:
(2.2) $\omega(x,y;x^{\prime},y^{\prime})=xy^{\prime}-x^{\prime}y.$
The Heisenberg group is non-commutative Lie group with the centre
$Z=\\{(s,0,0)\in\mathbb{H}^{n}{},\ s\in\mathbb{R}{}\\}.$
The left shifts
(2.3) $\Lambda(g):f(g^{\prime})\mapsto f(g^{-1}g^{\prime})$
act as a representation of $\mathbb{H}^{n}{}$ on a certain linear space of
functions. For example, action on $L_{2}{}(\mathbb{H}{},dg)$ with respect to
the Haar measure $dg=ds\,dx\,dy$ is the _left regular_ representation, which
is unitary.
The Lie algebra $\mathfrak{h}^{n}$ of $\mathbb{H}^{n}{}$ is spanned by
left-(right-)invariant vector fields
(2.4) $\textstyle S^{l(r)}=\pm{\partial_{s}},\quad
X_{j}^{l(r)}=\pm\partial_{x_{j}}-\frac{1}{2}y_{j}{\partial_{s}},\quad
Y_{j}^{l(r)}=\pm\partial_{y_{j}}+\frac{1}{2}x_{j}{\partial_{s}}$
on $\mathbb{H}^{n}{}$ with the Heisenberg _commutator relations_
(2.5) $[X_{i}^{l(r)},Y_{j}^{l(r)}]=\delta_{ij}S^{l(r)}$
and all other commutators vanishing. We will omit the supscript $l$ for left-
invariant field sometimes.
We can construct linear representations by induction [Kirillov76]*§ 13 from a
character $\chi$ of the centre $Z$. There are several models for induced
representations, here we prefer the following one, which is presented
stripping off all generalities, cf. [Kirillov76]*§ 13 [MTaylor86]*Ch. 5. Let
$F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ be the space of functions on
$\mathbb{H}^{n}{}$ having the properties:
(2.6) $f(gh)=\chi(h)f(g),\qquad\text{ for all }g\in\mathbb{H}^{n}{},\ h\in Z$
and
(2.7) $\int_{\mathbb{R}^{2n}{}}\left|f(0,x,y)\right|^{2}dx\,dy<\infty.$
Then $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ is invariant under the left shifts and
those shifts restricted to $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ make a
representation ${\rho_{\chi}}$ of $\mathbb{H}^{n}{}$ induced by $\chi$.
If the character $\chi$ is unitary, then the induced representation is unitary
as well. However the representation ${\rho_{\chi}}$ is not necessarily
irreducible. Indeed, left shifts are commuting with the right action of the
group. Thus any subspace of null-solutions of a linear combination
$aS+\sum_{j=1}^{n}(b_{j}X_{j}+c_{j}Y_{j})$ of left-invariant vector fields is
left-invariant and we can restrict ${\rho_{\chi}}$ to this subspace. The left-
invariant differential operators define analytic condition for functions, cf.
[Vourdas06a].
###### Example 2.1.
The function $f_{0}(s,x,y)=e^{\mathrm{i}hs-h(x^{2}+y^{2})/4}$, where
$h=2\pi\hslash$, belongs to $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ for the
character $\chi(s)=e^{\mathrm{i}hs}$. It is also a null solution for all the
operators $X_{j}-\mathrm{i}Y_{j}$. The closed linear span of functions
$f_{g}=\Lambda(g)f_{0}$ is invariant under left shifts and provide a model for
Segal–Bargmann type representation of the Heisenberg group, which will be
considered below.
###### Remark 2.2.
An alternative construction of induced representations is as follow
[Kirillov76]*§ 13.2. Consider a subgroup $H$ of a group $G$. Let a smooth
section $\mathbf{s}:G/H\rightarrow G$ be a left inverse of the natural
projection $\mathbf{p}:G\rightarrow G/H$. Thus any element $g\in G$ can be
uniquely decomposed as $g=\mathbf{s}(\mathbf{p}(g))*\mathbf{r}(g)$ where the
map $\mathbf{r}:G\rightarrow H$ is defined by the previous identity. For a
character $\chi$ of $H$ we can define a _lifting_
$\mathcal{L}_{\chi}:L_{2}{}(G/H)\rightarrow L_{2}^{\chi}{}(G)$ as follows:
(2.8)
$[\mathcal{L}_{\chi}f](g)=\chi(\mathbf{r}(g))f(\mathbf{p}(g))\qquad\text{where
}f(x)\in L_{2}{}(G/H).$
The image space of the lifting $\mathcal{L}_{\chi}$ is invariant under left
shifts. We also define the _pulling_ $\mathcal{P}:L_{2}^{\chi}{}(G)\rightarrow
L_{2}{}(G/H)$, which is a left inverse of the lifting and explicitly cab be
given, for example, by $[\mathcal{P}F](x)=F(\mathbf{s}(x))$. Then the induced
representation on $L_{2}{}(G/H)$ is generated by the formula
${\rho_{\chi}}(g)=\mathcal{P}\circ\Lambda(g)\circ\mathcal{L}$.
### 2.2. Convolutions (observables) on $\mathbb{H}^{n}{}$ and commutator
Using a left invariant measure $dg=ds\,dx\,dy$ on $\mathbb{H}^{n}{}$ we can
define the convolution of two functions:
(2.9) $\displaystyle(k_{1}*k_{2})(g)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{H}^{n}{}}k_{1}(g_{1})\,k_{2}(g_{1}^{-1}g)\,dg_{1}.$
This is a non-commutative operation, which is meaningful for functions from
various spaces including $L_{1}{}(\mathbb{H}^{n}{},dg)$, the Schwartz space
$S{}$ and many classes of distributions, which form algebras under
convolutions. Convolutions on $\mathbb{H}^{n}{}$ are used as _observables_ in
$p$-mechanic [Kisil96a, Kisil02e].
A unitary representation ${\rho}$ of $\mathbb{H}^{n}{}$ extends to
$L_{1}{}(\mathbb{H}^{n}{},dg)$ by the formula:
(2.10) ${\rho}(k)=\int_{\mathbb{H}^{n}{}}k(g){\rho}(g)\,dg.$
This is also an algebra homomorphism of convolutions to linear operators.
For a dynamics of observables we need inner _derivations_ $D_{k}$ of the
convolution algebra $L_{1}{}(\mathbb{H}^{n}{})$, which are given by the
_commutator_ :
$\displaystyle D_{k}:f\mapsto[k,f]$ $\displaystyle=$ $\displaystyle k*f-f*k$
$\displaystyle=$
$\displaystyle\int_{\mathbb{H}^{n}{}}k(g_{1})\left(f(g_{1}^{-1}g)-f(gg_{1}^{-1})\right)\,dg_{1},\quad
f,k\in L_{1}{}(\mathbb{H}^{n}{}).$
To describe dynamics of a time-dependent observable $f(t,g)$ we use the
universal equation, cf. [Kisil94d, Kisil96a]:
(2.12) $S\dot{f}=[H,f],$
where $S$ is the left-invariant vector field (2.4) generated by the centre of
$\mathbb{H}^{n}{}$. The presence of operator $S$ fixes the dimensionality of
both sides of the equation (2.12) if the observable $H$ (Hamiltonian) has the
dimensionality of energy [Kisil02e]*Rem 4.1. If we apply a right inverse
$\mathcal{A}$ of $S$ to both sides of the equation (2.12) we obtain the
equivalent equation
(2.13) $\dot{f}=\left\\{\\!\left[H,f\right]\\!\right\\},$
based on the universal bracket
$\left\\{\\!\left[k_{1},k_{2}\right]\\!\right\\}=k_{1}*\mathcal{A}k_{2}-k_{2}*\mathcal{A}k_{1}$
[Kisil02e].
###### Example 2.3 (Harmonic oscillator).
Let $H=\frac{1}{2}(m\omega^{2}q^{2}+\frac{1}{m}p^{2})$ be the Hamiltonian of a
one-dimensional harmonic oscillator, where $\omega$ is a constant frequency
and $m$ is a constant mass. Its _p-mechanisation_ will be the second order
differential operator on $\mathbb{H}^{n}{}$ [BrodlieKisil03a]*§ 5.1:
$\textstyle H=\frac{1}{2}(m\omega^{2}X^{2}+\frac{1}{m}Y^{2}),$
where we dropped sub-indexes of vector fields (2.4) in one dimensional
setting. We can express the commutator as a difference between the left and
the right action of the vector fields:
$\textstyle[H,f]=\frac{1}{2}(m\omega^{2}((X^{r})^{2}-(X^{l})^{2})+\frac{1}{m}((Y^{r})^{2}-(Y^{l})^{2}))f.$
Thus the equation (2.12) becomes [BrodlieKisil03a]*(5.2):
(2.14) $\frac{\partial}{\partial s}\dot{f}=\frac{\partial}{\partial
s}\left(m\omega^{2}y\frac{\partial}{\partial
x}-\frac{1}{m}x\frac{\partial}{\partial y}\right)f.$
Of course, the derivative $\frac{\partial}{\partial s}$ can be dropped from
both sides of the equation and the general solution is found to be:
(2.15) $\textstyle f(t;s,x,y)=f_{0}\left(s,x\cos(\omega t)+m\omega
y\sin(\omega t),-\frac{x}{m\omega}\sin(\omega t)+y\cos(\omega t)\right),$
where $f_{0}(s,x,y)$ is the initial value of an observable on
$\mathbb{H}^{n}{}$.
###### Example 2.4 (Unharmonic oscillator).
We consider unharmonic oscillator with cubic potential, see
[CalzettaVerdaguer06a] and references therein:
(2.16)
$H=\frac{m\omega^{2}}{2}q^{2}+\frac{\lambda}{6}q^{3}+\frac{1}{2m}p^{2}.$
Due to absence of non-commutative products p-mechanisation is straightforward:
$H=\frac{m\omega^{2}}{2}X^{2}+\frac{\lambda}{6}X^{3}+\frac{1}{m}Y^{2}.$
Similarly to the harmonic case the dynamic equation, after cancellation of
$\frac{\partial}{\partial s}$ on both sides, becomes:
(2.17) $\dot{f}=\left(m\omega^{2}y\frac{\partial}{\partial
x}+\frac{\lambda}{6}\left(3y\frac{\partial^{2}}{\partial
x^{2}}+\frac{1}{4}y^{3}\frac{\partial^{2}}{\partial
s^{2}}\right)-\frac{1}{m}x\frac{\partial}{\partial y}\right)f.$
Unfortunately, it cannot be solved analytically as easy as the harmonic case.
### 2.3. States and Probability
Let an observable ${\rho}(k)$ (2.10) is defined by a kernel $k(g)$ on the
Heisenberg group and its representation ${\rho}$ at a Hilbert space
$\mathcal{H}$. A state on the convolution algebra is given by a vector
$v\in\mathcal{H}$. A simple calculation:
$\displaystyle\left\langle{\rho}(k)v,v\right\rangle_{\mathcal{H}}$
$\displaystyle=$
$\displaystyle\left\langle\int_{\mathbb{H}^{n}{}}k(g){\rho}(g)v\,dg,v\right\rangle_{\mathcal{H}}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{H}^{n}{}}k(g)\left\langle{\rho}(g)v,v\right\rangle_{\mathcal{H}}dg$
$\displaystyle=$
$\displaystyle\int_{\mathbb{H}^{n}{}}k(g)\overline{\left\langle
v,{\rho}(g)v\right\rangle_{\mathcal{H}}}\,dg$
can be restated as:
$\left\langle{\rho}(k)v,v\right\rangle_{\mathcal{H}}=\left\langle
k,l\right\rangle,\qquad\text{where}\quad l(g)=\left\langle
v,{\rho}(g)v\right\rangle_{\mathcal{H}}.$
Here the left-hand side contains the inner product on $\mathcal{H}$, while the
right-hand side uses a skew-linear pairing between functions on
$\mathbb{H}^{n}{}$ based on the Haar measure integration. In other words we
obtain, cf. [BrodlieKisil03a]*Thm. 3.11:
###### Proposition 2.5.
A state defined by a vector $v\in\mathcal{H}$ coincides with the linear
functional given by the wavelet transform
(2.18) $l(g)=\left\langle v,{\rho}(g)v\right\rangle_{\mathcal{H}}$
of $v$ used as the mother wavelet as well.
The addition of vectors in $\mathcal{H}$ implies the following operation on
states:
(2.19) $\displaystyle\left\langle
v_{1}+v_{2},{\rho}(g)(v_{1}+v_{2})\right\rangle_{\mathcal{H}}$
$\displaystyle=$ $\displaystyle\left\langle
v_{1},{\rho}(g)v_{1}\right\rangle_{\mathcal{H}}+\left\langle
v_{2},{\rho}(g)v_{2}\right\rangle_{\mathcal{H}}$ $\displaystyle{}+\left\langle
v_{1},{\rho}(g)v_{2}\right\rangle_{\mathcal{H}}+\overline{\left\langle
v_{1},{\rho}(g^{-1})v_{2}\right\rangle_{\mathcal{H}}}$
The last expression can be conveniently rewritten for kernels of the
functional as
(2.20) $l_{12}=l_{1}+l_{2}+2A\sqrt{l_{1}l_{2}}$
for some real number $A$. This formula is behind the contextual law of
addition of conditional probabilities [Khrennikov01a] and will be illustrated
below. Its physical interpretation is an interference, say, from two slits.
The mechanism of such interference can be both causal and local, see
[Kisil01c] [KhrenVol01].
## 3\. Elliptic characters and Quantum Dynamics
In this section we consider the representation ${\rho_{h}}$ of
$\mathbb{H}^{n}{}$ induced by the elliptic character
$\chi_{h}(s)=e^{\mathrm{i}hs}$ in complex numbers parametrised by
$h\in\mathbb{R}{}$. We also use the convenient agreement $h=2\pi\hslash$.
### 3.1. Segal–Bargmann and Schrödinger Representations
The realisation of ${\rho_{h}}$ by the left shifts (2.3) on
$L_{2}^{h}{}(\mathbb{H}^{n}{})$ is rarely used in quantum mechanics. Instead
two unitary equivalent forms are more common: the Schrödinger and
Segal–Bargmann representations.
The Segal-Bargmann representation can be obtained from the orbit method of
Kirillov [Kirillov94a]. It allows spatially separate irreducible components of
the left regular representation, each of them is located on the orbit of the
co-adjoint representation, see [Kisil02e]*§ 2.1 [Kirillov94a] for details, we
only present a brief summary here.
We identify $\mathbb{H}^{n}{}$ and its Lie algebra $\mathfrak{h}_{n}$ through
the exponential map [Kirillov76]*§ 6.4. The dual $\mathfrak{h}_{n}^{*}$ of
$\mathfrak{h}_{n}$ is presented by the Euclidean space $\mathbb{R}^{2n+1}{}$
with coordinates $(\hslash,q,p)$. The pairing $\mathfrak{h}_{n}^{*}$ and
$\mathfrak{h}_{n}$ given by
$\left\langle(s,x,y),(\hslash,q,p)\right\rangle=\hslash s+q\cdot x+p\cdot y.$
This pairing defines the Fourier transform $\hat{\
}:L_{2}{}(\mathbb{H}^{n}{})\rightarrow L_{2}{}(\mathfrak{h}_{n}^{*})$ given by
[Kirillov99]*§ 2.3:
(3.1) $\hat{\phi}(F)=\int_{\mathfrak{h}^{n}}\phi(\exp
X)e^{-2\pi\mathrm{i}\left\langle X,F\right\rangle}\,dX\qquad\textrm{ where
}X\in\mathfrak{h}^{n},\ F\in\mathfrak{h}_{n}^{*}.$
For a fixed $\hslash$ the left regular representation (2.3) is mapped by the
Fourier transform to the Segal–Bargmann type representation [Kisil02e]*(2.9)
[deGosson08a]*(1):
(3.2) $\textstyle{\rho_{\hslash}}(s,x,y):f(q,p)\mapsto
e^{-2\pi\mathrm{i}(\hslash
s+qx+py)}f\left(q-\frac{\hslash}{2}y,p+\frac{\hslash}{2}x\right).$
The collection of points $(\hslash,q,p)\in\mathfrak{h}_{n}^{*}$ for a fixed
$\hslash$ is naturally identified with the phase space of the system.
###### Remark 3.1.
It is possible to identify the case of $\hslash=0$ with classical mechanics
[Kisil02e]. Indeed, a substitution of the zero value of $\hslash$ into (3.2)
produces the commutative representation:
(3.3) ${\rho_{0}}(s,x,y):f(q,p)\mapsto
e^{-2\pi\mathrm{i}(qx+py)}f\left(q,p\right).$
It can be decomposed into the direct integral of one-dimensional
representations parametrised by the points $(q,p)$ of the phase space. The
classical mechanics, including the Hamilton equation, can be recovered from
those representations [Kisil02e]. However the condition $\hslash=0$ (as well
as $\hslash\rightarrow 0$) is not completely physical. Commutativity (and
subsequent relative triviality) of those representation is the main reason why
they are oftenly neglected. The commutativity can be outweighed by special
arrangements, e.g. an antiderivative [Kisil02e]*(4.1), but the procedure is
not straightforward, see discussion in [Kisil05c] [AgostiniCapraraCiccotti07a]
[Kisil09a]. A direct approach using dual numbers will be discussed below, cf.
Rem. 4.5.
To recover the Schrödinger representation we use Rem. 2.2, see [Kisil98a]*Ex.
4.1 for details. The subgroup
$H=\\{(s,0,y)\,\mid\,s\in\mathbb{R}{},y\in\mathbb{R}^{n}{}\\}\subset\mathbb{H}^{n}{}$
defines the homogeneous space $X=G/H$, which coincides with $\mathbb{R}^{n}{}$
as a manifold. The natural projection $\mathbf{p}:G\rightarrow X$ is
$\mathbf{p}(s,x,y)=x$ and its left inverse $\mathbf{s}:X\rightarrow G$ can be
as simple as $\mathbf{s}(x)=(0,x,0)$. For the map $\mathbf{r}:G\rightarrow H$,
$\mathbf{r}(s,x,y)=(s-xy/2,0,y)$ we have the decomposition
$(s,x,y)=\mathbf{s}(p(s,x,y))*\mathbf{r}(s,x,y)=(0,x,0)*(s-\textstyle\frac{1}{2}xy,0,y).$
For a character $\chi_{h}(s,0,y)=e^{\mathrm{i}hs}$ of $H$ the lifting
$\mathcal{L}_{\chi}:L_{2}{}(G/H)\rightarrow L_{2}^{\chi}{}(G)$ is as follows:
$[\mathcal{L}_{\chi}f](s,x,y)=\chi_{h}(\mathbf{r}(s,x,y))\,f(\mathbf{p}(s,x,y))=e^{\mathrm{i}h(s-xy/2)}f(x).$
Thus the representation
${\rho_{\chi}}(g)=\mathcal{P}\circ\Lambda(g)\circ\mathcal{L}$ becomes:
(3.4)
$[{\rho_{\chi}}(s^{\prime},x^{\prime},y^{\prime})f](x)=e^{-2\pi\mathrm{i}\hslash(s^{\prime}+xy^{\prime}-x^{\prime}y^{\prime}/2)}\,f(x-x^{\prime}).$
After the Fourier transform $x\mapsto q$ we get the Schrödinger representation
on the configuration space:
(3.5)
$[{\rho_{\chi}}(s^{\prime},x^{\prime},y^{\prime})\hat{f}\,](q)=e^{-2\pi\mathrm{i}\hslash(s^{\prime}+x^{\prime}y^{\prime}/2)-2\pi\mathrm{i}x^{\prime}q}\,\hat{f}(q+\hslash
y^{\prime}).$
Note that this again turns into a commutative representation (multiplication
by an unimodular function) if $\hslash=0$. To get the full set of commutative
representations in this way we need to use the character
$\chi_{(h,p)}(s,0,y)=e^{2\pi\mathrm{i}(\hslash+py)}$ in the above
consideration.
### 3.2. Commutator and the Heisenberg Equation
The property (2.6) of $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ implies that the
restrictions of two operators ${\rho_{\chi}}(k_{1})$ and
${\rho_{\chi}}(k_{2})$ to this space are equal if
$\int_{\mathbb{R}{}}k_{1}(s,x,y)\,\chi(s)\,ds=\int_{\mathbb{R}{}}k_{2}(s,x,y)\,\chi(s)\,ds.$
In other words, for a character $\chi(s)=e^{2\pi\mathrm{i}\hslash s}$ the
operator ${\rho_{\chi}}(k)$ depends only on
$\hat{k}_{s}(\hslash,x,y)=\int_{\mathbb{R}{}}k(s,x,y)\,e^{-2\pi\mathrm{i}\hslash
s}\,ds,$
which is the partial Fourier transform $s\mapsto\hslash$ of $k(s,x,y)$. The
restriction to $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ of the composition formula
for convolutions is [Kisil02e]*(3.5):
(3.6)
$(k^{\prime}*k)\hat{{}_{s}}=\int_{\mathbb{R}^{2n}{}}e^{{\mathrm{i}h}{}(xy^{\prime}-yx^{\prime})/2}\,\hat{k}^{\prime}_{s}(\hslash,x^{\prime},y^{\prime})\,\hat{k}_{s}(\hslash,x-x^{\prime},y-y^{\prime})\,dx^{\prime}dy^{\prime}.$
Under the Schrödinger representation (3.5) the convolution (3.6) defines a
rule for composition of two pseudo-differential operators (PDO) in the Weyl
calculus [Howe80b] [Folland89]*§ 2.3.
Consequently the representation (2.10) of commutator (2.2) depends only on its
partial Fourier transform [Kisil02e]*(3.6):
$\displaystyle[k^{\prime},k]\hat{{}_{s}}$ $\displaystyle=$ $\displaystyle
2\mathrm{i}\int_{\mathbb{R}^{2n}{}}\\!\\!\sin(\textstyle\frac{h}{2}(xy^{\prime}-yx^{\prime}))\,$
$\displaystyle\qquad\times\hat{k}^{\prime}_{s}(\hslash,x^{\prime},y^{\prime})\,\hat{k}_{s}(\hslash,x-x^{\prime},y-y^{\prime})\,dx^{\prime}dy^{\prime}.$
Under the Fourier transform (3.1) this commutator is exactly the Moyal bracket
[Zachos02a] for of $\hat{k}^{\prime}$ and $\hat{k}$.
For observables in the space $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ the action of
$S$ is reduced to multiplication, e.g. for $\chi(s)=e^{\mathrm{i}hs}$ the
action of $S$ is multiplication by $\mathrm{i}h$. Thus the equation (2.12)
reduced to the space $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ becomes the Heisenberg
type equation [Kisil02e]*(4.4):
(3.8) $\dot{f}=\frac{1}{\mathrm{i}h}[H,f]\hat{{}_{s}},$
based on the above bracket (3.2). The Schrödinger representation (3.5)
transforms this equation to the original Heisenberg equation.
###### Example 3.2.
1. (i)
Under the Fourier transform $(x,y)\mapsto(q,p)$ the p-dynamic equation (2.14)
of the harmonic oscillator becomes:
$\dot{f}=\left(m\omega^{2}q\frac{\partial}{\partial
p}-\frac{1}{m}p\frac{\partial}{\partial q}\right)f.$
The same transform creates its solution out of (2.15).
2. (ii)
Since $\frac{\partial}{\partial s}$ acts on $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$
as multiplication by $\mathrm{i}\hslash$, the quantum representation of
unharmonic dynamics equation (2.17) is:
(3.9) $\dot{f}=\left(m\omega^{2}q\frac{\partial}{\partial
p}+\frac{\lambda}{6}\left(3q^{2}\frac{\partial}{\partial
p}-\frac{\hslash^{2}}{4}\frac{\partial^{3}}{\partial
p^{3}}\right)-\frac{1}{m}p\frac{\partial}{\partial q}\right)f.$
This is exactly the equation for the Wigner function obtained in
[CalzettaVerdaguer06a]*(30).
### 3.3. Quantum Probabilities
For the elliptic character $\chi_{h}(s)=e^{\mathrm{i}hs}$ we can use the
Cauchy–Schwartz inequality to demonstrate that the real number $A$ in the
identity (2.20) is between $-1$ and $1$. Thus we can put $A=\cos\alpha$ for
some angle (phase) $\alpha$ to get the formula for counting quantum
probabilities, cf. [Khrennikov03a]*(2):
(3.10) $l_{12}=l_{1}+l_{2}+2\cos\alpha\,\sqrt{l_{1}l_{2}}$
###### Remark 3.3.
It is interesting to note that the both trigonometric functions are employed
in quantum mechanics: sine is in the heart of the Moyal bracket (3.2) and
cosine is responsible for the addition of probabilities (3.10). In the essence
the commutator and probabilities took respectively the odd and even parts of
the elliptic character $e^{\mathrm{i}hs}$.
###### Example 3.4.
Take a vector $v_{(a,b)}\in L_{2}^{h}{}(\mathbb{H}^{n}{})$ defined by a
Gaussian with mean value $(a,b)$ in the phase space for a harmonic oscillator
of the mass $m$ and the frequency $\omega$:
(3.11) $v_{(a,b)}(q,p)=\exp\left(-\frac{2\pi\omega
m}{\hslash}(q-a)^{2}-\frac{2\pi}{\hslash\omega m}(p-b)^{2}\right).$
A direct calculation shows:
$\displaystyle\left\langle
v_{(a,b)},{\rho_{\hslash}}(s,x,y)v_{(a^{\prime},b^{\prime})}\right\rangle=\frac{4}{\hslash}\exp\left(\pi\mathrm{i}\left(2s\hslash+x(a+a^{\prime})+y(b+b^{\prime})\right)\frac{}{}\right.$
$\displaystyle\left.{}-\frac{\pi}{2\hslash\omega m}((\hslash
x+b-b^{\prime})^{2}+(b-b^{\prime})^{2})-\frac{\pi\omega m}{2\hslash}((\hslash
y+a^{\prime}-a)^{2}+(a^{\prime}-a)^{2})\right)$ $\displaystyle=$
$\displaystyle\frac{4}{\hslash}\exp\left(\pi\mathrm{i}\left(2s\hslash+x(a+a^{\prime})+y(b+b^{\prime})\right)\frac{}{}\right.$
$\displaystyle\left.{}-\frac{\pi}{\hslash\omega
m}((b-b^{\prime}+{\textstyle\frac{\hslash
x}{2}})^{2}+({\textstyle\frac{\hslash x}{2}})^{2})-\frac{\pi\omega
m}{\hslash}((a-a^{\prime}-{\textstyle\frac{\hslash
y}{2}})^{2}+({\textstyle\frac{\hslash y}{2}})^{2})\right)$
Thus the kernel $l_{(a,b)}=\left\langle
v_{(a,b)},{\rho_{\hslash}}(s,x,y)v_{(a,b)}\right\rangle$ (2.18) for a state
$v_{(a,b)}$ is:
(3.12) $\displaystyle l_{(a,b)}$ $\displaystyle=$
$\displaystyle\frac{4}{\hslash}\exp\left(2\pi\mathrm{i}(s\hslash+xa+yb)\frac{}{}-\frac{\pi\hslash}{2\omega
m}x^{2}-\frac{\pi\omega m\hslash}{2\hslash}y^{2}\right)$
An observable registering a particle at a point $q=c$ of the configuration
space is $\delta(q-c)$. On the Heisenberg group this observable is given by
the kernel:
(3.13) $X_{c}(s,x,y)=e^{2\pi\mathrm{i}(s\hslash+xc)}\delta(y).$
The measurement of $X_{c}$ on the state (3.11) (through the kernel (3.12))
predictably is:
$\left\langle X_{c},l_{(a,b)}\right\rangle=\sqrt{\frac{2\omega
m}{\hslash}}\exp\left(-\frac{2\pi\omega m}{\hslash}(c-a)^{2}\right).$
###### Example 3.5.
Now take two states $v_{(0,b)}$ and $v_{(0,-b)}$, where for the simplicity we
assume the mean values of coordinates vanish in the both cases. Then the
corresponding kernel (2.19) has the interference terms:
$\displaystyle l_{i}$ $\displaystyle=$ $\displaystyle\left\langle
v_{(0,b)},{\rho_{\hslash}}(s,x,y)v_{(0,-b)}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{4}{\hslash}\exp\left(2\pi\mathrm{i}s\hslash-\frac{\pi}{2\hslash\omega
m}((\hslash x+2b)^{2}+4b^{2})-\frac{\pi\hslash\omega m}{2}y^{2}\right).$
The measurement of $X_{c}$ (3.13) on this term contains the oscillating part:
$\left\langle X_{c},l_{i}\right\rangle=\sqrt{\frac{2\omega
m}{\hslash}}\exp\left(-\frac{2\pi\omega m}{\hslash}c^{2}-\frac{2\pi}{\omega
m\hslash}b^{2}+\frac{4\pi\mathrm{i}}{\hslash}cb\right)$
Therefore on the kernel $l$ corresponding to the state $v_{(0,b)}+v_{(0,-b)}$
the measurement is
$\displaystyle\left\langle X_{c},l\right\rangle$ $\displaystyle=$
$\displaystyle 2\sqrt{\frac{2\omega m}{\hslash}}\exp\left(-\frac{2\pi\omega
m}{\hslash}c^{2}\right)\left(1+\exp\left(-\frac{2\pi}{\omega
m\hslash}b^{2}\right)\cos\left(\frac{4\pi}{\hslash}cb\right)\right).$
(a) (b)
Figure 1. Quantum probabilities: the blue (dashed) graph shows the addition of
probabilities without interaction, the red (solid) graph present the quantum
interference. Left picture shows the Gaussian state (3.11), the right—the
rational state (3.14)
The presence of the cosine term in the last expression can generate an
interference picture. In practise it does not happen for the minimal
uncertainty state (3.11) which we are using here: it rapidly vanishes outside
of the neighbourhood of zero, where oscillations of the cosine occurs, see
Fig. 1(a).
###### Example 3.6.
To see a traditional interference pattern one can use a state which is far
from the minimal uncertainty. For example, we can consider the state:
(3.14) $u_{(a,b)}(q,p)=\frac{\hslash^{2}}{((q-a)^{2}+\hslash/\omega
m)((p-b)^{2}+\hslash\omega m)}.$
To evaluate the observable $X_{c}$ (3.13) on the state $l(g)=\left\langle
u_{1},{\rho_{h}}(g)u_{2}\right\rangle$ (2.18) we use the following formula:
$\left\langle
X_{c},l\right\rangle=\frac{2}{\hslash}\int_{\mathbb{R}^{n}{}}\hat{u}_{1}(q,2(q-c)/\hslash)\,\overline{\hat{u}_{2}(q,2(q-c)/\hslash)}\,dq,$
where $\hat{u}_{i}(q,x)$ denotes the partial Fourier transform $p\mapsto x$ of
$u_{i}(q,p)$. The formula is obtained by swapping order of integrations. The
numerical evaluation of the state obtained by the addition
$u_{(0,b)}+u_{(0,-b)}$ is plotted on Fig. 1(b), the red curve shows the
canonical interference pattern.
## 4\. Hypercomplex Repersentations of the Heisenberg Group
The group of symmetries of classical mechanics—the group preserving the
symplectic form (2.2)—generates automorphisms of the Heisenberg group in a
natural way [Folland89]*§ 1.2. Those common symmetries of quantum and
classical mechanics are behind many important connections, e.g. between
classical “symplectic camel” and the Heisenberg uncertainty relations
[deGossonLuef09a].
The symplectic group of $\mathbb{R}^{2}{}$ is isomorphic to the celebrated
group $SL_{2}{}(\mathbb{R}{})$ [Lang85]. Both groups $\mathbb{H}^{n}{}$ and
$SL_{2}{}(\mathbb{R}{})$ contributes to the symmetries of the paraxial wave
equation [ATorre10a]. There are many other physical links between the
Heisenberg group and $SL_{2}{}(\mathbb{R}{})$, e.g. metaplectic representation
[Folland89]*Ch. 4.
It was demonstrated in [Kisil07a] that dual and double numbers appears very
naturally within the induced representations of the group
$SL_{2}{}(\mathbb{R}{})$. Special relativity [Ulrych05a] and global space-time
model [HerranzSantander02b, Kisil06b] also link the representation theory to
hypercomplex numbers. Physical significance of hypercomplex numbers and
representation theory of Clifford algebras was recently highlighted as well
[BocCatoniCannataNichZamp07] [Ulrych08a] [Plaksa09a] [Ulrych10a]. There is an
explicit similarity between the commutators in the Heisenberg-Weyl Lie algebra
and anticommutators defining Clifford algebra [Kisil93c] [Kisil01d], which can
be unified as a superspace [BieEelbodeSommen09a] [Berezin86]. Thus it would be
an omission to restrict linear representations of $\mathbb{H}^{n}{}$ to
complex numbers only.
### 4.1. Hyperbolic Representations and Addition of Probabilities
Now we turn to double numbers also known as hyperbolic, split-complex, etc.
numbers [Yaglom79]*App. C [Ulrych05a] [KhrennikovSegre07a]. They form a two
dimensional algebra $\mathbb{O}{}$ spanned by $1$ and $\mathrm{j}$ with the
property $\mathrm{j}^{2}=1$. There are zero divisors:
$\mathrm{j}_{\pm}=\textstyle\frac{1}{\sqrt{2}}(1\pm j),\qquad\text{ such that
}\quad\mathrm{j}_{+}\mathrm{j}_{-}=0\quad\text{ and
}\quad\mathrm{j}_{\pm}^{2}=\mathrm{j}_{\pm}.$
Thus double numbers algebraically isomorphic to two copies of $\mathbb{R}{}$
spanned by $\mathrm{j}_{\pm}$. Being algebraically dull double numbers are
nevertheless interesting as a homogeneous space [Kisil05a, Kisil09c] and they
are relevant in physics [Khrennikov05a, Ulrych05a, Ulrych08a]. The combination
of p-mechanical approach with hyperbolic quantum mechanics was already
discussed in [BrodlieKisil03a]*§ 6.
For the hyperbolic character $\chi_{\mathrm{j}h}(s)=e^{\mathrm{j}hs}=\cosh
hs+\mathrm{j}\sinh hs$ of $\mathbb{R}{}$ one can define the hyperbolic
Fourier-type transform:
$\hat{k}(q)=\int_{\mathbb{R}{}}k(x)\,e^{-\mathrm{j}qx}dx.$
It can be understood in the sense of distributions on the space dual to the
set of analytic functions [Khrennikov08a]*§ 3. Hyperbolic Fourier transform
intertwines the derivative $\frac{d}{dx}$ and multiplication by $\mathrm{j}q$
[Khrennikov08a]*Prop. 1.
###### Example 4.1.
For the Gaussian the hyperbolic Fourier transform is the ordinary function
(note the sign difference!):
$\int_{\mathbb{R}{}}e^{-x^{2}/2}e^{-\mathrm{j}qx}dx=\sqrt{2\pi}\,e^{q^{2}/2}.$
However the opposite identity:
$\int_{\mathbb{R}{}}e^{x^{2}/2}e^{-\mathrm{j}qx}dx=\sqrt{2\pi}\,e^{-q^{2}/2}$
is true only in a suitable distributional sense. To this end we may note that
$e^{x^{2}/2}$ and $e^{-q^{2}/2}$ are null solutions to the differential
operators $\frac{d}{dx}-x$ and $\frac{d}{dq}+q$ respectively, which are
intertwined (up to the factor $\mathrm{j}$) by the hyperbolic Fourier
transform. The above differential operators $\frac{d}{dx}-x$ and
$\frac{d}{dq}+q$ are images of the _ladder operators_ in the Lie algebra of
the Heisenberg group. They are intertwining by the Fourier transform, since
this is an automorphism of the Heisenberg group [Howe80a]. A careful study of
ladder operators reveals connections with hypercomplex numbers [Kisil11a,
Kisil11c].
An elegant theory of hyperbolic Fourier transform may be achieved by a
suitable adaptation of [Howe80a], which uses representation theory of the
Heisenberg group.
#### 4.1.1. Hyperbolic Representations of the Heisenberg Group
Consider the space $F_{h}^{\mathrm{j}}{}(\mathbb{H}^{n}{})$ of
$\mathbb{O}{}$-valued functions on $\mathbb{H}^{n}{}$ with the property:
(4.1) $f(s+s^{\prime},h,y)=e^{\mathrm{j}hs^{\prime}}f(s,x,y),\qquad\text{ for
all }(s,x,y)\in\mathbb{H}^{n}{},\ s^{\prime}\in\mathbb{R}{},$
and the square integrability condition (2.7). Then the hyperbolic
representation is obtained by the restriction of the left shifts to
$F_{h}^{\mathrm{j}}{}(\mathbb{H}^{n}{})$. To obtain an equivalent
representation on the phase space we take $\mathbb{O}{}$-valued functional of
the Lie algebra $\mathfrak{h}_{n}$:
(4.2)
$\chi^{j}_{(h,q,p)}(s,x,y)=e^{\mathrm{j}(hs+qx+py)}=\cosh(hs+qx+py)+\mathrm{j}\sinh(hs+qx+py).$
The hyperbolic Segal—Bargmann type representation is intertwined with the left
group action by means of the Fourier transform (3.1) with the hyperbolic
functional (4.2). Explicitly this representation is:
(4.3) ${\rho_{\hslash}}(s,x,y):f(q,p)\mapsto\textstyle
e^{-\mathrm{j}(hs+qx+py)}f\left(q-\frac{h}{2}y,p+\frac{h}{2}x\right).$
For a hyperbolic Schrödinger type representation we again use the scheme
described in Rem. 2.2. Similarly to the elliptic case one obtains the formula,
resembling (3.4):
(4.4)
$[{\rho^{\mathrm{j}}_{\chi}}(s^{\prime},x^{\prime},y^{\prime})f](x)=e^{-\mathrm{j}h(s^{\prime}+xy^{\prime}-x^{\prime}y^{\prime}/2)}f(x-x^{\prime}).$
Application of the hyperbolic Fourier transform produces a Schrödinger type
representation on the configuration space, cf. (3.5):
(4.5)
$[{\rho^{\mathrm{j}}_{\chi}}(s^{\prime},x^{\prime},y^{\prime})\hat{f}\,](q)=e^{-\mathrm{j}h(s^{\prime}+x^{\prime}y^{\prime}/2)-\mathrm{j}x^{\prime}q}\,\hat{f}(q+hy^{\prime}).$
The extension of this representation to kernels according to (2.10) generates
hyperbolic pseudodifferential operators introduced in [Khrennikov08a]*(3.4).
#### 4.1.2. Hyperbolic Dynamics
Similarly to the elliptic (quantum) case we consider a convolution of two
kernels on $\mathbb{H}^{n}{}$ restricted to
$F_{h}^{\mathrm{j}}{}(\mathbb{H}^{n}{})$. The composition law becomes, cf.
(3.6):
(4.6)
$(k^{\prime}*k)\hat{{}_{s}}=\int_{\mathbb{R}^{2n}{}}e^{{\mathrm{j}h}{}(xy^{\prime}-yx^{\prime})}\,\hat{k}^{\prime}_{s}(h,x^{\prime},y^{\prime})\,\hat{k}_{s}(h,x-x^{\prime},y-y^{\prime})\,dx^{\prime}dy^{\prime}.$
This is close to the calculus of hyperbolic PDO obtained in
[Khrennikov08a]*Thm. 2. Respectively for the commutator of two convolutions we
get, cf. (3.2):
(4.7)
$[k^{\prime},k]\hat{{}_{s}}=\int_{\mathbb{R}^{2n}{}}\\!\\!\sinh(h(xy^{\prime}-yx^{\prime}))\,\hat{k}^{\prime}_{s}(h,x^{\prime},y^{\prime})\,\hat{k}_{s}(h,x-x^{\prime},y-y^{\prime})\,dx^{\prime}dy^{\prime}.$
This the hyperbolic version of the Moyal bracket, cf. [Khrennikov08a]*p. 849,
which generates the corresponding image of the dynamic equation (2.12).
###### Example 4.2.
1. (i)
For a quadratic Hamiltonian, e.g. harmonic oscillator from Example 2.3, the
hyperbolic equation and respective dynamics is identical to quantum considered
before.
2. (ii)
Since $\frac{\partial}{\partial s}$ acts on
$F_{2}^{\mathrm{j}}{}(\mathbb{H}^{n}{})$ as multiplication by $\mathrm{j}h$
and $\mathrm{j}^{2}=1$, the hyperbolic image of the unharmonic equation (2.17)
becomes:
$\dot{f}=\left(m\omega^{2}q\frac{\partial}{\partial
p}+\frac{\lambda}{6}\left(3q^{2}\frac{\partial}{\partial
p}+\frac{\hslash^{2}}{4}\frac{\partial^{3}}{\partial
p^{3}}\right)-\frac{1}{m}p\frac{\partial}{\partial q}\right)f.$
The difference with quantum mechanical equation (3.9) is in the sign of the
cubic derivative.
#### 4.1.3. Hyperbolic Probabilities
(a) (b)
Figure 2. Hyperbolic probabilities: the blue (dashed) graph shows the addition
of probabilities without interaction, the red (solid) graph present the
quantum interference. Left picture shows the Gaussian state (3.11), with the
same distribution as in quantum mechanics, cf. Fig. 1(a). The right picture
shows the rational state (3.14), note the absence of interference oscillations
in comparison with the quantum state on Fig. 1(b).
To calculate probability distribution generated by a hyperbolic state we are
using the general procedure from Section 2.3. The main differences with the
quantum case are as follows:
1. (i)
The real number $A$ in the expression (2.20) for the addition of probabilities
is bigger than $1$ in absolute value by. Thus it can be associated with the
hyperbolic cosine $\cosh\alpha$, cf. Rem. 3.3, for certain phase
$\alpha\in\mathbb{R}{}$ [Khrennikov08a].
2. (ii)
The nature of hyperbolic interference on two slits is affected by the fact
that $e^{\mathrm{j}hs}$ is not periodic and the hyperbolic exponent
$e^{\mathrm{j}t}$ and cosine $\cosh t$ do not oscillate. It is worth to notice
that for Gaussian states the hyperbolic interference is exactly the same as
quantum one, cf. Figs. 1(a) and 2(a). This is similar to coincidence of
quantum and hyperbolic dynamics of harmonic oscillator.
The contrast between two types of interference is prominent for the rational
state (3.14), which is far from the minimal uncertainty, see the different
patterns on Figs. 1(b) and 2(b).
### 4.2. Parabolic (Classical) representations on the phase space
After the previous two cases it is natural to link classical mechanics with
dual numbers generated by the parabolic unit $\varepsilon^{2}=0$. Connection
of the parabolic unit $\varepsilon$ with the Galilean group of symmetries of
classical mechanics is around for a while [Yaglom79]*App. C.
However the nilpotency of the parabolic unit $\varepsilon$ make it difficult
if we will work with dual number valued functions only. To overcome this issue
we consider a commutative real algebra $\mathfrak{C}$ spanned by $1$,
$\mathrm{i}$, $\varepsilon$ and $\mathrm{i}\varepsilon$ with identities
$\mathrm{i}^{2}=-1$ and $\varepsilon^{2}=0$. A seminorm on $\mathfrak{C}$ is
defined as follows:
$\left|a+b\mathrm{i}+c\varepsilon+d\mathrm{i}\varepsilon\right|^{2}=a^{2}+b^{2}.$
#### 4.2.1. Classical Non-Commutative Representations
We wish to build a representation of the Heisenberg group which will be a
classical analog of the Segal–Bargmann representation (3.2). To this end we
introduce the space $F_{h}^{\varepsilon}{}(\mathbb{H}^{n}{})$ of
$\mathfrak{C}$-valued functions on $\mathbb{H}^{n}{}$ with the property:
(4.8) $f(s+s^{\prime},h,y)=e^{\varepsilon hs^{\prime}}f(s,x,y),\qquad\text{
for all }(s,x,y)\in\mathbb{H}^{n}{},\ s^{\prime}\in\mathbb{R}{},$
and the square integrability condition (2.7). It is invariant under the left
shifts and we restrict the left group action to
$F_{h}^{\varepsilon}{}(\mathbb{H}^{n}{})$.
There is an unimodular $\mathfrak{C}$-valued function on the Heisenberg group
parametrised by a point $(h,q,p)\in\mathbb{R}^{2n+1}{}$:
$E_{(h,q,p)}(s,x,y)=e^{2\pi(\varepsilon
s\hslash+\mathrm{i}xq+\mathrm{i}yp)}=e^{2\pi\mathrm{i}(xq+yp)}(1+\varepsilon
sh).$
This function, if used instead of the ordinary exponent, produces a
modification $\mathcal{F}_{c}$ of the Fourier transform (3.1). The transform
intertwines the left regular representation with the following action on
$\mathfrak{C}$-valued functions on the phase space:
(4.9) ${\rho^{\varepsilon}_{h}}(s,x,y):f(q,p)\mapsto
e^{-2\pi\mathrm{i}(xq+yp)}(f(q,p)+\varepsilon
h(sf(q,p)+\frac{y}{2\pi\mathrm{i}}f^{\prime}_{q}(q,p)-\frac{x}{2\pi\mathrm{i}}f^{\prime}_{p}(q,p))).$
###### Remark 4.3.
Comparing the traditional infinite-dimensional (3.2) and one-dimensional (3.3)
representations of $\mathbb{H}^{n}{}$ we can note that the properties of the
representation (4.9) are a non-trivial mixture of the former:
1. (i)
The action (4.9) is non-commutative, similarly to the quantum representation
(3.2) and unlike the classical one (3.3). This non-commutativity will produce
the Hamilton equations below in a way very similar to Heisenberg equation, see
Rem. 4.5.
2. (ii)
The representation (4.9) does not change the support of a function $f$ on the
phase space, similarly to the classical representation (3.3) and unlike the
quantum one (3.2). Such a localised action will be responsible later for an
absence of an interference in classical probabilities.
3. (iii)
The parabolic representation (4.9) can not be derived from either the elliptic
(3.2) or hyperbolic (4.3) by the plain substitution $h=0$.
We may also write a classical Schrödinger type representation. According to
Rem. 2.2 we get a representation formally very similar to the elliptic (3.4)
and hyperbolic versions (4.4):
$\displaystyle[{\rho^{\varepsilon}_{\chi}}(s^{\prime},x^{\prime},y^{\prime})f](x)$
$\displaystyle=$ $\displaystyle e^{-\varepsilon
h(s^{\prime}+xy^{\prime}-x^{\prime}y^{\prime}/2)}f(x-x^{\prime})$
$\displaystyle=$ $\displaystyle(1-\varepsilon
h(s^{\prime}+xy^{\prime}-\textstyle\frac{1}{2}x^{\prime}y^{\prime}))f(x-x^{\prime}).$
However due to nilpotency of $\varepsilon$ the (complex) Fourier transform
$x\mapsto q$ produces a different formula for parabolic Schrödinger type
representation in the configuration space, cf. (3.5) and (4.5):
(4.11)
$[{\rho^{\varepsilon}_{\chi}}(s^{\prime},x^{\prime},y^{\prime})\hat{f}](q)=e^{2\pi\mathrm{i}x^{\prime}q}\left(\left(1-\varepsilon
h(s^{\prime}-{\textstyle\frac{1}{2}}x^{\prime}y^{\prime})\right)\hat{f}(q)+\frac{\varepsilon
hy^{\prime}}{2\pi\mathrm{i}}\hat{f}^{\prime}(q)\right).$
This representation shares all properties mentioned in Rem. 4.3 as well.
#### 4.2.2. Hamilton Equation
The identity $e^{\varepsilon t}-e^{-\varepsilon t}=2\varepsilon t$ can be
interpreted as a parabolic version of the sine function, while the parabolic
cosine is identically equal to one [HerranzOrtegaSantander99a, Kisil07a]. From
this we obtain the parabolic version of the commutator (3.2):
$\displaystyle[k^{\prime},k]\hat{{}_{s}}(\varepsilon h,x,y)$ $\displaystyle=$
$\displaystyle\varepsilon h\int_{\mathbb{R}^{2n}{}}(xy^{\prime}-yx^{\prime})$
$\displaystyle{}\times\,\hat{k}^{\prime}_{s}(\varepsilon
h,x^{\prime},y^{\prime})\,\hat{k}_{s}(\varepsilon
h,x-x^{\prime},y-y^{\prime})\,dx^{\prime}dy^{\prime},$
for the partial parabolic Fourier-type transform $\hat{k}_{s}$ of the kernels.
Thus the parabolic representation of the dynamical equation (2.12) becomes:
(4.12) $\varepsilon h\frac{d\hat{f}_{s}}{dt}(\varepsilon h,x,y;t)=\varepsilon
h\int_{\mathbb{R}^{2n}{}}(xy^{\prime}-yx^{\prime})\,\hat{H}_{s}(\varepsilon
h,x^{\prime},y^{\prime})\,\hat{f}_{s}(\varepsilon
h,x-x^{\prime},y-y^{\prime};t)\,dx^{\prime}dy^{\prime},$
Although there is no possibility to divide by $\varepsilon$ (since it is a
zero divisor) we can obviously eliminate $\varepsilon h$ from the both sides
if the rest of the expressions are real. Moreover this can be done “in
advance” through a kind of the antiderivative operator considered in
[Kisil02e]*(4.1). This will prevent “imaginary parts” of the remaining
expressions (which contain the factor $\varepsilon$) from vanishing.
###### Remark 4.4.
It is noteworthy that the Planck constants completely disappeared from the
dynamical equation. Thus the only prediction about it following from our
construction is $h\neq 0$, which was confirmed by experiments, of course.
Using the duality between the Lie algebra of $\mathbb{H}^{n}{}$ and the phase
space we can find an adjoint equation for observables on the phase space. To
this end we apply the usual Fourier transform $(x,y)\mapsto(q,p)$. It turn to
be the Hamilton equation [Kisil02e]*(4.7). However the transition to phase
space is more a custom rather than a necessity and in many cases we can
efficiently work on the Heisenberg group itself.
###### Remark 4.5.
It is noteworthy, that the non-commutative representation (4.9) allows to
obtain the Hamilton equation directly from the commutator
$[{\rho^{\varepsilon}_{h}}(k_{1}),{\rho^{\varepsilon}_{h}}(k_{2})]$. Indeed
its straightforward evaluation will produce exactly the above expression. On
the contrast such a commutator for the commutative representation (3.3) is
zero and to obtain the Hamilton equation we have to work with an additional
tools, e.g. an anti-derivative [Kisil02e]*(4.1).
###### Example 4.6.
1. (i)
For the harmonic oscillator in Example 2.3 the equation (4.12) again reduces
to the form (2.14) with the solution given by (2.15). The adjoint equation of
the harmonic oscillator on the phase space is not different from the quantum
written in Example 3.2(i). This is true for any Hamiltonian of at most
quadratic order.
2. (ii)
For non-quadratic Hamiltonians classical and quantum dynamics are different,
of course. For example, the cubic term of $\partial_{s}$ in the equation
(2.17) will generate the factor $\varepsilon^{3}=0$ and thus vanish. Thus the
equation (4.12) of the unharmonic oscillator on $\mathbb{H}^{n}{}$ becomes:
$\dot{f}=\left(m\omega^{2}y\frac{\partial}{\partial x}+\frac{\lambda
y}{2}\frac{\partial^{2}}{\partial x^{2}}-\frac{1}{m}x\frac{\partial}{\partial
y}\right)f.$
The adjoint equation on the phase space is:
$\dot{f}=\left(\left(m\omega^{2}q+\frac{\lambda}{2}q^{2}\right)\frac{\partial}{\partial
p}-\frac{1}{m}p\frac{\partial}{\partial q}\right)f.$
The last equation is the classical Hamilton equation generated by the cubic
potential (2.16). Qualitative analysis of its dynamics can be found in many
textbooks [Arnold91]*§ 4.C, Pic. 12 [PercivalRichards82]*§ 4.4.
###### Remark 4.7.
We have obtained the Poisson bracket from the commutator of convolutions on
$\mathbb{H}^{n}{}$ without any quasiclassical limit $h\rightarrow 0$. This has
a common source with the deduction of main calculus theorems in
[CatoniCannataNichelatti04] based on dual numbers. As explained in
[Kisil05a]*Rem. 6.9 this is due to the similarity between the parabolic unit
$\varepsilon$ and the infinitesimal number used in non-standard analysis
[Devis77]. In other words, we never need to take care about terms of order
$O(h^{2})$ because they will be wiped out by $\varepsilon^{2}=0$.
An alternative derivation of classical dynamics from the Heisenberg group is
given in the recent paper [Low09a].
#### 4.2.3. Classical probabilities
It is worth to notice that dual numbers are not only helpful in reproducing
classical Hamiltonian dynamics, they also provide the classic rule for
addition of probabilities. We use the same formula (2.18) to calculate kernels
of the states. The important difference now that the representation (4.9) does
not change the support of functions. Thus if we calculate the correlation term
$\left\langle v_{1},{\rho}(g)v_{2}\right\rangle$ in (2.19), then it will be
zero for every two vectors $v_{1}$ and $v_{2}$ which have disjoint supports in
the phase space. Thus no interference similar to quantum or hyperbolic cases
(Subsection 3.3) is possible.
## 5\. Discussion
In this paper we derive mathematical models for various physical setup from
hypercomplex representations of the Heisenberg group. There are roots for such
hypercomplex characters in the structure of ladder operators associated to
three non-isomorphic quadratic Hamiltonians [Kisil11a, Kisil11c]. Such
hypercomplex representations may be also useful for many other groups as well,
see the example of the $SL_{2}{}(\mathbb{R}{})$ group in [Kisil09c]. Moreover
non-trivial parabolic characters described in [Kisil07a, Kisil09c] are
awaiting a further exploration.
There is a connection of our work with the technique of contractions and
analytic continuations of groups [Gromov90b, Gromov90a], these papers also
highlight the role of hypercomplex numbers of three types. However in our
research we do not modify the group (the Heisenberg group more specifically)
itself, we rather consider its representations in different functional spaces
created by three types of hypercomplex characters. All three cases have a lot
of algebraic similarity and can be written in a unified manner with the help
of parameter, which takes three values, say $u=\mathrm{i}$, $\varepsilon$,
$\mathrm{j}$, with $\mathrm{i}^{2}=-1$, $\varepsilon^{2}=0$,
$\mathrm{j}^{2}=1$. For example, representations (3.4), (4.4) and (4.2.1) can
be unified in:
(5.1)
$[{\rho^{u}_{h}}(s^{\prime},x^{\prime},y^{\prime})f](x)=e^{-uh(s^{\prime}+xy^{\prime}-x^{\prime}y^{\prime}/2)}f(x-x^{\prime}).$
It is noteworthy that this algebraic similarity exists along with the
significant topological and analytic differences between elliptic, parabolic
and hyperbolic cases. An illustration is the distinction of the elliptic (3.5)
and parabolic (4.11) representations in the configuration space, despite of
the fact that both representations are derived from the unified form (5.1).
The parabolic representations (4.2.1) and (4.11) of the Heisenberg group act
in the first order jet spaces. Such spaces have a well established connections
with Lagrangian and Hamiltonian formulations of quantum field theory
[GiachettaMangiarottiSardanashvily97a, Kanatchikov01b, Kisil04a], study of
aggregate quantum-classical systems [Kisil05c, Kisil09a] and spectral theory
of operators [Kisil02a]. Nevertheless the localised non-commutative
representation of $\mathbb{H}^{n}{}$ built in this paper seems to be new and
deserve detailed investigation.
We already seen that it may be useful to consider several hypercomplex units
in the same time. In the case of classical mechanics we combined $\mathrm{i}$
and $\varepsilon$. The algebra generated by $\mathrm{i}$ and $\mathrm{j}$ is
known as (commutative) Segre quaternions. Such commutative algebras with
hypercomplex units and their physical applications attracted attention of many
researchers recently [BocCatoniCannataNichZamp07] [Plaksa09a] [Ulrych05a]
[Ulrych08a].
We may even need to study an algebra which contains all three hypercomplex
units simultaneously. The most straightforward way is to take eight
dimensional commutative algebra with the basis $1$, $\mathrm{i}$,
$\varepsilon$, $\mathrm{j}$, $\mathrm{i}\varepsilon$, $\mathrm{i}\mathrm{j}$,
$\varepsilon\mathrm{j}$, $\mathrm{i}\varepsilon\mathrm{j}$. A reduction of
dimensionality from $8$ to $6$ can be achieved if we replace products
$\varepsilon\mathrm{j}$ and $\mathrm{i}\varepsilon\mathrm{j}$ through the
further identities $\varepsilon\mathrm{j}=\varepsilon$ and
$\mathrm{i}\varepsilon\mathrm{j}=\mathrm{i}\varepsilon$. This do not affect
associativity of the product.
## Acknowledgements
I am grateful to A.Yu. Khrennikov and S. Ulrych for useful discussion on
relation between double numbers and physics. S. Plaksa advised me on various
aspect of commutative hypercomplex algebras. U. Güenther draw my attention to
the connection between $\mathcal{PT}$-symmetric Hamiltonians and Krein spaces.
Prof. N.A. Gromov made several useful suggestions of methodological nature.
Constructive comments of anonymous referees provided further ground for
paper’s improvement.
## References
|
arxiv-papers
| 2010-05-27T13:07:56 |
2024-09-04T02:49:10.661737
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vladimir V. Kisil",
"submitter": "Vladimir V Kisil",
"url": "https://arxiv.org/abs/1005.5057"
}
|
1005.5155
|
# Irreducible Elements in Metric Lattices
Andreas Lochmann
###### Abstract
We describe a natural generalization of irreducibility in order lattices with
arbitrary metrics. We analyse the special cases of valuation metrics and more
general metrics for lattices.
This article is mainly based on a part of the author’s doctoral thesis, but
answers some additional questions.
## 1 Introduction
The theory of valuations and metric lattices has been mainly developed and
popularized by John von Neumann and Garrett Birkhoff. In the early years of
the 1930s, von Neumann worked on a variation of the ergodic hypothesis, and
inadvertently competed with George David Birkhoff. Only some years later, his
son Garrett Birkhoff pointed von Neumann at the use of lattice theory in
Hilbert spaces. He wrote about this in a note of the Bulletin of the AMS in
1958 [Bi2].
> John von Neumann’s brilliant mind blazed over lattice theory like a meteor,
> during a brief period centering around 1935–1937. With the aim of
> interesting him in lattices, I had called his attention, in 1933–1934, to
> the fact that the sublattice generated by three subspaces of Hilbert space
> (or any other vector space) contained 28 subspaces in general, to the
> analogy between dimension and measure, and to the characterization of
> projective geometries as irreducible, finite-dimensional, complemented
> modular lattices.
>
> As soon as the relevance of lattices to linear manifolds in Hilbert space
> was pointed out, he began to consider how he could use lattices to classify
> the factors of operator-algebras. One can get some impression of the initial
> impact of lattice concepts on his thinking about this classification problem
> by reading the introduction of […], in which a systematic lattice-theoretic
> classification of the different possibilities was initiated. […]
>
> However, von Neumann was not content with considering lattice theory from
> the point of view of such applications alone. With his keen sense for
> axiomatics, he quickly also made a series of fundamental contributions to
> pure lattice theory.
The modular law in its earliest form (as dimension function) appears in two
papers from 1936 by Glivenko and von Neumann ([Gl], [vN]). Von Neumann used it
(and lattice theory in general) in his paper to define and study Continuous
Geometry (aka. “pointless geometry”), and later applied his knowledge to found
Quantum Logic in his Mathematical Foundations of Quantum Mechanics. A later
survey about metric posets is [Mn].
The notions of join-irreducibility and join-primeness are fundamental to
Lattice Theory, in the same way as the notion of basis is fundamental to
Linear Algebra (see [Bi1]). Hence, it seems plausible to ask for an adaption
of join-irreducibility to metric lattices—the author already used this notion
in [Lo2] and [Lo1] to decompose Lipschitz functions and deduce a rigidity
theorem about Lipschitz function spaces. The aim of this article is to present
this new notion of $d$-irreducibility in Section 2 without reference to
Lipschitz function spaces. Section 3 repeats the definition of a valuation on
a lattice and its connection to metrics, Subsection 3.3 then deduces a
characterization of $d$-irreducible elements in valuation lattices. Subsection
3.2 introduces an alternative definition of valuation, which is then
generalized in Sections 4 and 5 to include further metrics on lattices, which
often are similarly natural but not based on a valuation. Subsection 5.2
finally deals with the closedness of the subset of all $d$-irreducible
elements in a lattice and in which sense they are a dense subset of each base.
### 1.1 Notation
Given an element $p$ of a lattice $L$, denote with $\Downarrow\\!p$ its
strictly lower set
$\displaystyle\Downarrow\\!p$ $\displaystyle:=$
$\displaystyle\\{f\,{\,\in\,}\,L\,:\,f\,<\,p\\}.$
Furthermore, denote with $\operatorname{\wp}(A)$ the power set of $A$.
## 2 Irreducibility Relative to a Metric
Recall the definition of a join-irreducible element $p$ in a lattice $L$:
$\displaystyle p\;=\;f\,\vee\,g$ $\displaystyle\;\Rightarrow\;$ $\displaystyle
p\;=\;f\quad\textnormal{or}\quad p\;=\;g\qquad\,\forall\,f,\,g\,{\,\in\,}\,L$
Let $L$ be equipped with the discrete metric $d_{\textnormal{dis}}$. Then the
above property is equivalent to the following:
$\displaystyle d_{\textnormal{dis}}\,(p,\,f)\quad\wedge\quad
d_{\textnormal{dis}}\,(p,\,g)$ $\displaystyle\leq$ $\displaystyle
d_{\textnormal{dis}}\,(p,\,f\,\vee\,g)\qquad\,\forall\,f,\,g\,{\,\in\,}\,L$
In the same sense, $p$ is completely join-irreducible if and only if
$\displaystyle\bigwedge_{j{\,\in\,}J}\,d_{\textnormal{dis}}\,\big{(}p,\,f_{j}\big{)}$
$\displaystyle\leq$ $\displaystyle
d_{\textnormal{dis}}\,\left(p,\;\bigvee_{j{\,\in\,}J}f_{j}\right)\qquad\,\forall\,(f_{j})_{j{\,\in\,}J}\subseteq
L,\;J\neq\emptyset.$
###### Definition 1
1
Let $L$ be a lattice with any metric $d$. We call an element $p\,{\,\in\,}\,L$
$d$-irreducible if the following holds for all $f,\,g\,{\,\in\,}\,L$:
$\displaystyle d(p,\,f)\;\wedge\;d(p,\,g)$ $\displaystyle\leq$ $\displaystyle
d(p,\,f\vee g)$
If $L$ is a complete lattice, we call $p$ completely $d$-irreducible, if the
following holds for all $(f_{j})_{j{\,\in\,}J}\subseteq L$, with $J$ an
arbitrary non-empty index set:
$\displaystyle\bigwedge_{j{\,\in\,}J}\,d\,\big{(}p,\,f_{j}\big{)}$
$\displaystyle\leq$ $\displaystyle
d\,\left(p,\;\bigvee_{j{\,\in\,}J}f_{j}\right)$
Denote the subset of $L$ of all completely $d$-irreducible elements with
$\operatorname{\textnormal{cmli}}(L)$.
###### Proposition 2
2
Let $L$ be a lattice with any metric $d$. Then each $d$-irreducible element is
join-irreducible. However, not every completely $d$-irreducible element
necessarily is completely join-irreducible.
Proof Let $p\,{\,\in\,}\,L$ be $d$-irreducible and $p\,=\,f\vee g$. Then
$d(p,\,f\vee g)\,=\,0$ and hence either $d(p,\,f)\,=\,0$ or $d(p,\,g)\,=\,0$
(or both).
For a counter-example to complete join-irreducibility, let $L\,=\,[0,\,1]$
with standard metric, supremum and infimum. Take $f_{n}\,=\,1-1/n$,
$n\,{\,\in\,}\,\mathbb{N}^{*}$, then $p\,=\,1\,=\,\bigvee f_{n}$, hence $p$ is
not completely join-irreducible. Still, it is completely $d$-irreducible: Any
sequence of real numbers $f_{n}$ with $p\,=\,\bigvee f_{n}$ must converge to
$p$ from below, hence $\bigwedge d(p,\,f_{n})\,=\,0$. $\square$
As a consequence, if $L$ is a complemented lattice, join-irreducibility,
complete join-irreducibility, $d$-irreducibility, and complete
$d$-irreducibility are all equivalent; the irreducible elements are simply
those with trivial strictly lower set.
## 3 Valuations
###### Definition 3
3
A valuation on a lattice $L$ is a function $v:L\rightarrow\mathbb{R}$ which
satisfies the modular law
$\displaystyle v(f)\;+\;v(g)$ $\displaystyle=$ $\displaystyle v(f\wedge
g)\;+\;v(f\vee g)\quad\,\forall\,f,g{\,\in\,}L.$
A valuation $v$ on $L$ is called isotone [positive] if for all $f,g{\,\in\,}L$
the relation $f<g$ implies $v(f)\leq v(g)$ [$v(f)<v(g)$].
If $L$ is totally ordered, then each function $v:L\rightarrow\mathbb{R}$ is a
valuation. It is isotone [positive] if and only if $v$ is [strictly]
monotonically increasing.
Valuations can be used to define metrics on lattices, as the following Lemma
demonstrates. It is a part of Theorem X.1 and a note in subsection X.2 of
[Bi1], and is proved there. An alternative proof is given in [Lo1].
###### Lemma 4
4
Let $v$ be an isotone valuation on the distributive lattice $L$. Then
$\displaystyle d_{v}(f,g)$ $\displaystyle:=$ $\displaystyle v(f\vee
g)\;-\;v(f\wedge g)$
defines a pseudo-metric with the following properties:
1. 1.
If there is a least element $0{\,\in\,}L$, then
$\displaystyle v(f)$ $\displaystyle=$ $\displaystyle
v(0)\;+\;d_{v}(f,0)\quad\textnormal{for all}\quad f{\,\in\,}L,$
2. 2.
$d_{v}$ is a metric if and only if $v$ is positive.
We call $d_{v}$ a valuation (pseudo-)metric. A lattice together with a
valuation metric is sometimes called a metric lattice; however, as we will
deal with lattices with non-valuation metrics as well (particularly the
supremum metric), we should better distinguish between valuation metric
lattices and non-valuation metric lattices.
### 3.1 Examples
Valuations and valuation metrics arise in a multitude of situations:
###### Example 5
5
Let $L=(\mathbb{N}^{*},\gcd,\operatorname{\textnormal{lcm}})$. Then each
logarithm is a positive valuation on $L$. The join-irreducible, completely
join-irreducible, $d$-irreducible and completely $d$-irreducible elements are
exactly the prime powers.
###### Example 6
6
Let $(X,\Sigma,\mu)$ be a probability space. The $\sigma$-algebra $\Sigma$ is
a Boolean lattice by union and intersection. Let $c{\,\in\,}\mathbb{R}$ be
arbitrary, then
$\displaystyle v(A)$ $\displaystyle:=$ $\displaystyle\mu(A)+c$
defines an isotone valuation on $\Sigma$ with $v(\emptyset)=c$. The valuation
$v$ is positive if and only if there are no null sets in $X$ other than
$\emptyset$. The distance function $d_{v}(A,B):=v(A\cup B)-v(A\cap B)$ is the
measure of the symmetric difference $A\vartriangle B$ of $A$ and $B$, if
$A\vartriangle B{\,\in\,}\Sigma$. It relates to the Hausdorff distance just as
the 1-distance of functions relates to the supremum distance.
###### Example 7
7
Let $V$ be any finite dimensional vector space, and
$L=\operatorname{\textnormal{PG}}(V)$ its lattice of subvector spaces, with
$\wedge$ the intersection and $\vee$ the span (the projective geometry of
$V$). Then the dimension function is a positive valuation on $L$. (This is the
similarity between dimension and measure mentioned before in [Bi2].) The join-
irreducible, completely join-irreducible, $d$-irreducible and completely
$d$-irreducible elements are exactly the one-dimensional subspaces and the
zero-dimensional one.
###### Example 8
8
Let $X$ be a measure space and $L$ the space of integrable Lipschitz functions
of Lipschitz constant $\leq\,1$. We may apply the Lebesgue integral to gain an
isotone valuation on $L$; as $f+g=(f\wedge g)+(f\vee g)$ holds pointwise, we
conclude
$\displaystyle\int f\,\textnormal{d}\mu+\int g\,\textnormal{d}\mu$
$\displaystyle=$ $\displaystyle\int(f\wedge g)\,\textnormal{d}\mu+\int(f\vee
g)\,\textnormal{d}\mu.$
If $X$ is a Euclidean space, or a discrete space without non-trivial null
sets, this valuation is positive, because any non-trivial non-negative
Lipschitz function has positive Lebesgue integral. Positivity fails in cases
where $X$ contains an isolated point or continuum of measure zero.
As $|f-g|=(f\vee g)-(f\wedge g)$ holds pointwise, the valuation metric $d_{v}$
equals the $L^{1}$-distance defined by
$\displaystyle d_{1}(f,g)$ $\displaystyle:=$
$\displaystyle\int|f-g|\,\textnormal{d}\mu.$
Each function $\Lambda(x,\,r):\,L\,\rightarrow\,[0,\infty)$ of the form
$\displaystyle\Lambda(x,\,r)(y)$ $\displaystyle:=$ $\displaystyle
0\,\vee\,\big{(}r\,-\,d(x,\,y)\big{)}$
with $x\,{\,\in\,}\,X$ and $r\,{\,\in\,}\,[0,\infty)$ is join-irreducible, but
not necessarily completely join-irreducible. In general, the only
$d$-irreducible function is the zero function.
The $L^{1}$-metric can be slightly modified to yield other valuation metrics:
Let $\kappa:[0,\infty)\rightarrow[0,\infty)$ be a positive valuation (i.e.,
strictly monotonically increasing), then
$\displaystyle v_{\mu,\kappa}(f)$ $\displaystyle:=$
$\displaystyle\int\kappa(f(x))\,\textnormal{d}\mu(x)$
is a positive valuation.
### 3.2 Difference Valuations
A nearly equivalent approach to valuations is to use difference valuations:
###### Definition 9
9
A difference valuation on a distributive lattice $L$ is a function $w:L\times
L\rightarrow\mathbb{R}$ which satisfies the cut law
$\displaystyle w(f,\,g)$ $\displaystyle=$ $\displaystyle w(f,\,g\vee
h)\;+\;w(f\wedge h,\,g).$
A difference valuation $w$ is called isotone if its values are non-negative,
and positive, if $w(f,\,g)\,=\,0$ implies $f\,\leq\,g$.
Given a valuation $v$ on a distributive lattice $L$,
$\displaystyle w(f,\,g)$ $\displaystyle:=$ $\displaystyle v(f)-v(f\wedge g)$
defines a difference valuation, as one can easily check. The cut law follows
from the modular equality and vice versa—it has been dubbed “cut law” because
of its appearance when applied to sets in a Venn diagram, see Figure 1. The
difference valuation is isotone/positive if and only if the valuation $v$ is
isotone/positive. If $L$ admits a least element $0$, each [isotone/positive]
difference valuation $w$ in turn defines an [isotone/positive] valuation $v$
by
$\displaystyle v(f)$ $\displaystyle:=$ $\displaystyle w(f,\,0)\;+\;c$
for any $c\,{\,\in\,}\,\mathbb{R}$, and any valuation of $L$ with difference
valuation $w$ is of this form. Finally, the distance function $d$ of a
valuation can be equally well expressed as
$\displaystyle d(f,\,g)$ $\displaystyle=$ $\displaystyle
w(f,\,g)\;+\;w(g,\,f).$
Figure 1: Visualization of the cut law of difference valuations using Venn
diagrams. Given a Stone representation $\pi$, the set $\pi(f)\setminus\pi(g)$
is cut along $\pi(h)$ to give $\pi(f)\setminus\pi(g\,\vee\,h)$ and
$\pi(f\,\wedge\,h)\setminus\pi(g)$.
Figure 2: Proof of the triangle inequality for valuation metric lattices using
difference valuations and Venn diagrams. Note that $f,g,h$ are elements of an
arbitrary distributive lattice, and represented by sets via Stone duality.
If the lattice $L$ is complemented, $w(f,\,g)$ equals $v(f\setminus g)$.
Difference valuations are easier to use in cases where a lattice is not
complemented, as they can be used as substitutes for the relative complement
operation in calculations with metrics. For example, the proof of Lemma 4 can
be seen by a simple application of Venn diagrams (see Figure 2); for details
and further examples to deduce metric inequalities in order lattices see
[Lo1].
### 3.3 $d$-Irreducible Elements
As a triviality, in the definition of a join-irreducible element,
$\displaystyle p\;=\;f\,\vee\,g$ $\displaystyle\;\Rightarrow\;$ $\displaystyle
p\;=\;f\quad\textnormal{or}\quad p\;=\;g\qquad\,\forall\,f,\,g\,{\,\in\,}\,L,$
the elements $f$ and $g$ may be chosen to be
${\,\in\,}\,\Downarrow\\!p\,\subseteq\,L$. This accounts for $d$-irreducible
elements as well, but is less trivial:
###### Lemma 10
10
Let $L$ be a distributive lattice, and $d$ a positive valuation metric on $L$.
$p\,{\,\in\,}\,L$ is $d$-irreducible if and only if
$\displaystyle d(p,\,f)\;\wedge\;d(p,\,g)$ $\displaystyle\leq$ $\displaystyle
d(p,\,f\vee g)$
holds for all $f,\,g\,{\,\in\,}\Downarrow\\!p$. In this case, “$\leq$” can be
replaced by “$=$”.
If $L$ is completely distributive, then the analog holds for complete
$d$-irreducibility as well.
Proof Let $f,\,g\,{\,\in\,}\,L$ be arbitrary and $p\,{\,\in\,}\,L$ as above.
Then holds:
$\displaystyle d(f\,\vee\,g,\,p)$ $\displaystyle=$ $\displaystyle
w(p,\,f\,\vee\,g)\;+\;w(f\,\vee\,g,\,p)$ $\displaystyle\geq$ $\displaystyle
w(p,\,f\,\vee\,g)\;+\;\big{(}w(f,\,p)\,+\,w(g,\,p)\big{)}$ $\displaystyle=$
$\displaystyle d\big{(}(f\vee
g)\,\wedge\,p,\,p\big{)}\;+\;\big{(}w(f,\,p)\,+\,w(g,\,p)\big{)}$
$\displaystyle\geq$
$\displaystyle\big{(}d(f\,\vee\,p,\,p)\,\wedge\,d(g\,\vee\,p,\,p)\big{)}\;+\;\big{(}w(f,\,p)\,+\,w(g,\,p)\big{)}$
$\displaystyle=$
$\displaystyle\big{(}w(p,\,f)\,\wedge\,w(p,\,g)\big{)}\;+\;w(f,\,p)\,+\,w(g,\,p)$
$\displaystyle\geq$
$\displaystyle\big{(}w(p,\,f)\,+\,w(f,\,p)\big{)}\;\wedge\;\big{(}w(p,\,g)\,+\,w(g,\,p)\big{)}$
$\displaystyle=$ $\displaystyle d(f,\,p)\;\wedge\;d(g,\,p)$
(1: definition, 2: by cut law, 3: definition, 4: hypothesis, 5: definition, 6:
by distributivity and positivity of $w$, 7: definition). Each step holds in
the infinite case as well, one only has to add in step 6 that $L$ is
completely distributive.
For equality, note that $d(p,\,f)\,\geq\,d(p,\,f\vee g)$ is obvious because
$f\,\leq\,f\vee g\,\leq p$; same holds for $g$ and thus
$\displaystyle d(p,\,f)\;\wedge\;d(p,\,g)$ $\displaystyle\geq$ $\displaystyle
d(p,\,f\vee g).$
$\square$
There is a characterization of join-irreducibility of an element
$p\,{\,\in\,}\,L$ in terms of its strictly lower set $\Downarrow\\!p$: $p$ is
join-irreducible if and only if for each $f,\,g\,{\,\in\,}\Downarrow\\!p$
holds $f\,\vee\,g\,{\,\in\,}\Downarrow\\!p$, i.e. if and only if
$\Downarrow\\!p$ is join-closed. Analogously, $p$ is a completely join-
irreducible element of a complete lattice $L$ if and only if $\Downarrow\\!p$
is join-complete (i.e. each supremum of elements of $\Downarrow\\!p$ again is
contained in $\Downarrow\\!p$). For valuation metrics, there is a similar
characterization of $d$-irreducibility:
###### Theorem 11
11
Let $L$ be a distributive lattice, and $d$ a positive valuation metric on $L$.
An element $p\,{\,\in\,}\,L$ is $d$-irreducible if and only if the strictly
lower set $\Downarrow\\!p$ is totally ordered.
Proof “$\Rightarrow$”: Let $f,\,g\,{\,\in\,}\Downarrow\\!p$ be arbitrary.
$\displaystyle d(f\,\vee\,g,\,p)$ $\displaystyle=$ $\displaystyle
d(f,\,p)\;\wedge\;d(g,\,p)$ $\displaystyle=$ $\displaystyle
w(p,\,f)\;\wedge\;w(p,\,g)$ $\displaystyle=$
$\displaystyle\big{(}w(g,\,f)\,+\,w(p,\,f\vee
g)\big{)}\;\wedge\;\big{(}w(f,\,g)\,+\,w(p,\,g\vee f)\big{)}$ $\displaystyle=$
$\displaystyle\big{(}w(g,\,f)\,\wedge\,w(f,\,g)\big{)}\;+\;w(p,\,g\vee f)$
$\displaystyle=$
$\displaystyle\big{(}w(g,\,f)\,\wedge\,w(f,\,g)\big{)}\;+\;d(f\,\vee\,g,\,p)$
and hence $w(g,\,f)\,\wedge\,w(f,\,g)\,=\,0$, thus one of them is zero, and we
have either $f\,\leq\,g$ or $g\,\leq\,f$.
“$\Leftarrow$”: Let $f,\,g\,{\,\in\,}\Downarrow\\!p$ be arbitrary (see Lemma
10 why we may restrict to $\Downarrow\\!p$). As $\Downarrow\\!p$ is totally
ordered, $f\vee g$ is $f$ or $g$, and hence the condition for
$d$-irreducibility is trivial.
$\square$
Theorem 11 shows that $d$-irreducibility does not depend on the concrete
choice of a valuation metric for the lattice $L$. This result resembles an
earlier connection found in Lipschitz function spaces: If $L$ is the space of
bounded non-negative Lipschitz functions of a metric space $X$ with Lipschitz
constant $\leq 1$ with pointwise supremum and infimum and supremum metric
$d_{\infty}$, then the completely $d_{\infty}$-irreducible elements are
exactly those functions of the form
$\displaystyle\Lambda(x,\,r):\;\;L$ $\displaystyle\rightarrow$
$\displaystyle[0,\,\infty)$ $\displaystyle y$ $\displaystyle\mapsto$
$\displaystyle\big{(}r\,-\,d_{X}(x,\,y)\big{)}\;\vee\;0$
with $x\,{\,\in\,}\,L$ and $r\,{\,\in\,}\,[0,\,\infty)$ (see Example 8, [Lo2],
[Lo1]). Although the supremum metric $d_{\infty}$ is not a valuation metric,
but an intervaluation metric (see Definition 18), its completely
$d_{\infty}$-irreducible elements are fully determined without any reference
to the chosen metric on $L$. One might even get rid of the metric on $X$ by
referring to minimal functions with a given function value at a single point.
## 4 Ultravaluations
One advantage of the definition of difference valuations in Subsection 3.2 is
the following alternative to valuations in lattices, which adds further
examples to our list of metrics on lattices and is easily described in terms
of a variant of Definition 9.
###### Lemma 12
12
Let $L$ be a distributive lattice, and let $w:L\times L\rightarrow[0,\infty)$
be a map which satisfies
(1) $\displaystyle w(f,g)\;\;=\;\;0\quad\textnormal{whenever}\quad f\leq
g,\quad\textnormal{and}$ (2) $\displaystyle w(f,\,g)\;\;=\;\;w(f\wedge
h,\,g)\;\vee\;w(f,\,g\vee h)\quad\,\forall\,f,g,h{\,\in\,}L.$
We call $w$ a difference ultravaluation, or just ultravaluation. Define
$\displaystyle d_{w}(f,g)$ $\displaystyle:=$ $\displaystyle
w(f,g)\;\vee\;w(g,f).$
Then $d_{w}$ is a pseudo-ultrametric. $d_{w}$ is an ultrametric if and only if
$w(f,g)=0\,\Rightarrow\,f\leq g$ holds.
Proof To get from difference valuations to ultravaluations, we just replaced
all occurences of “$+$” by “$\vee$”. As both operations are associative and
commutative, we can transfer most proofs of valuations just by replacing “$+$”
by “$\vee$”, this includes the proof of the triangle inequality:
$\displaystyle d_{w}(f,g)$ $\displaystyle=$ $\displaystyle w(f,g\vee h)\vee
w(f\wedge h,g)\vee w(g,f\vee h)\vee w(g\wedge h,f)$ $\displaystyle w(f,g\vee
h)$ $\displaystyle\leq$ $\displaystyle w(f,h)\quad\textnormal{etc.}$
$\displaystyle\Rightarrow\quad d_{w}(f,g)$ $\displaystyle\leq$ $\displaystyle
w(f,h)\vee w(h,g)\vee w(g,h)\vee w(h,f)\;=\;d_{w}(f,h)\vee d_{w}(h,g)$
On the other hand, contrary to the valuation case, the property $d_{v}(f,f)=0$
does not follow from property (2) – we have to conclude it from (1).
Assume $w(f,g)=0\,\Rightarrow\,f\leq g$ holds. Let $d_{w}(f,g)=0$. This
implies $w(f,g)=0$ and $w(g,f)=0$, and hence $f\leq g$, $g\leq f$, and $f=g$.
Now assume $d_{w}$ is a metric, $f\nleq g$, and $w(f,g)=0$. Then
$\displaystyle w(f,f\wedge g)\;=\;w(f\wedge g,f\wedge g)\;\vee\;w(f,g)\;=\;0.$
Due to $f\nleq g$, we have $f\neq f\wedge g$, hence
$\displaystyle 0\;<\;d_{w}(f,f\wedge g)\;=\;w(f,f\wedge g)\;\vee\;w(f\wedge
g,f)\;=\;w(f\wedge g,f).$
But $f\wedge g\leq f$, contradiction. $\square$
### 4.1 Examples
###### Example 13
13
Let $X$ be any set, $\kappa:X\rightarrow[0,\infty)$ arbitrary and fixed, and
$L$ a lattice of subsets of $X$. For $A,B{\,\in\,}L$ consider
$\displaystyle w(A,B)$ $\displaystyle:=$ $\displaystyle
0\;\vee\;\sup_{x{\,\in\,}A\setminus B}\;\kappa(x).$
$w$ defines an ultravaluation.
Choose $\kappa$ to be a positive constant, then the ultrametric resulting from
$w$ will be the discrete metric on $L$.
###### Example 14
14
Let $X$ be any metric space and $\operatorname{{\textnormal{Lip}}}_{0}X$ its
lattice of bounded Lipschitz function of Lipschitz constant $\leq 1$. Besides
its Stone representation, we want to provide another, more intuitive
representation of the space $\operatorname{{\textnormal{Lip}}}_{0}X$ by a
lattice of sets, using its hypograph (cp. “epigraph” in [Ro])
$\displaystyle\operatorname{\textnormal{hyp}}:\;\operatorname{{\textnormal{Lip}}}X$
$\displaystyle\rightarrow$
$\displaystyle\operatorname{\wp}\,(X\times[0,\infty))$ $\displaystyle f$
$\displaystyle\mapsto$ $\displaystyle\\{(x,r)\;:\;f(x)\leq r\\}.$
$(\operatorname{\textnormal{im}}\operatorname{\textnormal{hyp}},\,\cap,\,\cup)$
obviously is isomorphic to
$(\operatorname{{\textnormal{Lip}}}_{0}X,\,\wedge,\vee)$ as a lattice;
however, they are not yet isomorphic as complete lattices: Infinite unions of
the closed sets in
$\operatorname{\textnormal{im}}\operatorname{\textnormal{hyp}}$ are not closed
in general – we have to use the union with closure “$\ \bar{\cup}$” instead of
the traditional union. (Alternatively, we could identify subsets of
$X\times[0,\infty)$ with the same closure.)
We now apply Example 13. The most canonical $\kappa$ would be
$\kappa=\pi_{2}$, the projection onto $[0,\infty)$. The corresponding
ultrametric on $L$ is
$\displaystyle d_{\kappa}(f,g)$ $\displaystyle=$ $\displaystyle
0\;\vee\;\sup\;\\{f(x)\,\vee\,g(x)\textnormal{ with $x{\,\in\,}X$ such that
}f(x)\neq g(x)\\}.$
We shall call this metric the “peak metric” on
$\operatorname{{\textnormal{Lip}}}X$.
Another possible choice for $\kappa$ is as follows: Choose a basepoint
$x_{0}{\,\in\,}X$ and $\kappa(x,r)\;:=\;d_{X}(x,x_{0})$. Then $d_{\kappa}$
will describe the greatest distance from $x_{0}$ at which $f$ and $g$ still
differ. Finally, $\kappa(x,r)\;:=\;\exp(-d_{X}(x,x_{0}))$ will describe the
least distance from $x_{0}$ at which $f$ and $g$ differ. We will call the
first case the “outer basepoint metric” and the second case the “inner
basepoint metric”.
An application of the lower basepoint metric is as follows: Given a free group
$F$ with neutral element $x_{0}$, identify each normal subgroup
$N\trianglelefteq F$ with its characteristic function on $F$. These are
1-Lipschitz functions in the canonical word metric of $F$. $d_{\kappa}$ then
defines a topology on $\operatorname{{\textnormal{Lip}}}F$, which restricts to
the Cayley topology ([dH], V.10) on the subset of normal subgroups.
The $\Lambda$-functions defined in Example 8 are exactly the $d$-irreducible
functions of the peak metric. The $d$-irreducible functions of the outer
basepoint metric are those functions $\Lambda(x,\,r)$ with $x\,\neq\,x_{0}$,
the inner basepoint metric doesn’t admit any non-trivial $d$-irreducible
function in general. Finally, none of these three metrics admits a non-trivial
completely $d$-irreducible function.
###### Lemma 15
15
Let $X$ be finite, and let $L$ be a lattice of subsets of $X$. Then any
ultravaluation on $L$ is of the form of Example 13.
Proof For $x\,{\,\in\,}\,X$ and $A,\,B\,\subseteq\,X$ define
$\displaystyle\kappa(x)$ $\displaystyle:=$
$\displaystyle\inf\;\\{w(C,\,D)\;:\;C,\,D\,{\,\in\,}\,L\;\textnormal{with}\;x\
{\,\in\,}\ C,\;x\notin D\\}$ $\displaystyle\textnormal{and}\quad
w^{\prime}(A,\,B)$ $\displaystyle:=$ $\displaystyle
0\;\vee\;\sup_{y{\,\in\,}A\setminus B}\;\kappa(y).$
Assume $w^{\prime}(A,\,B)\,>\,w(A,\,B)$. Then there is
$y\,{\,\in\,}\,A\setminus B$ with $\kappa(y)\,\geq\,w(A,\,B)$, but this cannot
happen, as one may choose $C=A$ and $D=B$. Hence, assume
$w^{\prime}(A,\,B)\,<\,w(A,\,B)$. Then for all $y\,{\,\in\,}\,A\setminus B$
there should be $C,\,D\,{\,\in\,}\,L$ with $y\,{\,\in\,}\,C\setminus D$ and
$w(C,\,D)\,<\,w(A,\,B)$. As
$\displaystyle w(C,\,D)\,\geq\,w(C\,\wedge A,\,D\,\vee\,B),$
we might choose without loss of generality $C\subseteq A$ and $D\supseteq B$,
as choosing $C\cap A$ instead of $C$ and $D\cup B$ instead of $D$ further
decreases $w(C,\,D)$. The cut law now yields
$\displaystyle w(A,\,B)$ $\displaystyle=$ $\displaystyle w(C\wedge
D,\,B)\,\vee\,w(C,\,D)\,\vee\,w(A\vee D,\,B\vee C)\,\vee\,w(A,\,C\vee D).$
As $w(C,\,D)\,<\,w(A,\,B)$ by assumption, we find that at least one of $(C\cap
D)\setminus B$, $(A\cup D)\setminus(B\cup C)$, and $A\setminus(C\cup D)$ must
be non-empty. Choose $y^{\prime}$ out of their union and repeat the above
argument for the now smaller subset. We get an infinite sequence of different
elements from $X$, which is a contradiction because $X$ is finite. $\square$
###### Example 16
16
Not all ultravaluations are of the kind of Example 13. Let $X$ be any metric
space and $L$ the lattice of subsets of $X$. Define $w(A,\,B)$ to be the
Hausdorff dimension of $A\setminus B\,{\,\in\,}\,L$ plus $1$, and $0$ if
$A\setminus B\,=\,\emptyset$. Then $w$ is an ultravaluation and $d_{w}$ an
ultrametric.
The $d$-irreducible subsets and the completely $d$-irreducible subsets are
exactly the join-irreducible subsets, namely those with one or zero elements,
because $L$ is complemented.
Comparing Examples 7 and 16, one should note that the join operation in the
former is the span, but in the latter is the union. Thus, the first example
gives rise to a valuation, the second one to an ultravaluation.
### 4.2 $d$-Irreducible Elements
Lemma 10 can be easily adapted to the case of ultravaluations by replacing all
remaining “$+$” by “$\vee$”. Indeed, Lemma 10 holds in an even broader
generalization, what we will demonstrate in Lemma 20.
When following the proof of Theorem 11 for ultravaluation metrics (remember
that join-irreducibility is $d_{\textnormal{dis}}$-irreducibility for the
discrete metric $d_{\textnormal{dis}}$, which is an ultravaluation metric),
one ends up with the following inequality:
$\displaystyle d(f,\,g)$ $\displaystyle\leq$ $\displaystyle d(p,\,f\,\vee\,g)$
for all $d$-irreducible elements $p$ and all $f,\,g\,{\,\in\,}\Downarrow\\!p$.
If $L$ contains a least element $0\,{\,\in\,}\,L$, we conclude as special case
$\displaystyle d(0,\,g)$ $\displaystyle\leq$ $\displaystyle
d(g,\,p)\qquad\,\forall\,g\,{\,\in\,}\Downarrow\\!p.$
One would hope that there is a similar characterization of $d$-irreducible
elements in the ultravaluation case as it is in the valuation case. Starting
from the case of the discrete metric, one would ask whether join-
irreducibility is exactly this characterization, i.e. whether all join-
irreducible elements are $d$-irreducible for any ultravaluation metric $d$.
This, however, is wrong.
###### Example 17
17
We refer to Example 13. Let $X\,=\,\\{1,\,2,\,3\\}\,\subseteq\,\mathbb{Z}$ and
let $\kappa$ be the identity. Let $L$ be the lattice
$\\{\emptyset,\,\\{2\\},\,\\{3\\},\\{2,\,3\\},\,X\\}$ of subsets of $X$. Then
$X\,{\,\in\,}\,L$ is join-irreducible (because it is the only set containing
$1$), but not $d$-irreducible: $d(X,\,\\{2\\})\,=\,3$, $d(X,\,\\{3\\})\,=\,2$
and $d(X,\,\\{2,\,3\\})\,=\,1$. In particular, this example shows that
$d$-irreducibility depends on the concrete choice of $\kappa$, respectively on
the choice of the ultravaluation.
Question Is there a nice criterion to decide whether all join-irreducible
elements in an ultravaluation metric lattice are $d$-irreducible?
Lemma 15 characterizes all finite ultravaluation lattices. However, finding
the $d$-irreducible subsets in a finite ultravaluation lattice can still be
non-trivial. We demonstrate this by restating the problem as a puzzle in
Figure 3 and leave it to the reader to find any patterns.
Figure 3: We refer to Example 13. Let $L$ be the lattice of sets spanned by
the shown sets of natural numbers and let $\kappa$ be the identity. A set $A$
is not $d$-irreducible, if and only if there are subsets $B$ and
$C\,{\,\in\,}\,L$ of $A$, such that both $B$ and $C$ contain at least one
number each, which is larger than any of the remaining numbers in
$A\setminus(B\,\cup\,C)$. Which of the shown subsets are $d$-irreducible?
## 5 Intervaluations and Topological Aspects
We now present a generalized notion of valuation which includes normal
valuations and ultravaluations. In addition, this notion of intervaluations
also includes the supremum metric of function spaces, just as the
$L^{1}$-metric was found to be a valuation in Example 8.
Similar to the case of the ultravaluation, we first recognize the possibility
to replace “$+$” in the definition of a difference valuation by any
commutative and associative binary operation. But this alone will not suffice
to encompass the supremum metric, we have to weaken the main property of a
difference evaluation as well:
###### Definition 18
18
An intervaluation on a distributive lattice $(L,\wedge,\vee)$ is a map
$w:L\rightarrow[0,\infty)$ together with a commutative and associative binary
operation $\circ_{w}:[0,\infty)\times[0,\infty)\rightarrow[0,\infty)$, such
that the following properties hold:
1. 1.
$r\,\circ_{w}\,0\;=\;0\,\circ_{w}\,r\;=\;r$
2. 2.
$r\,\circ_{w}\,t\;\leq\;(r\,+\,s)\,\circ_{w}\,(t\,+\,u)\;\leq\;(r\,\circ_{w}\,t)\,+\,(s\,\circ_{w}\,u)$
3. 3.
$r\,\vee\,s\;\leq\;r\,\circ_{w}\,s$ (follows from (1) and (2))
4. 4.
$f\,\leq\,g\quad\Rightarrow\quad w(f,\,g)\,=\,0$
5. 5.
$w(f,\,g\vee h)\,\circ_{w}\,w(f\wedge
h,\,g)\;\leq\;w(f,\,g)\;\leq\;w(f,\,g\vee h)\,+\,w(f\wedge h,\,g)$
(left and right modular inequality, or cut law)
for all $f,g,h{\,\in\,}L$ and $r,s,t,u{\,\in\,}[0,\infty)$. The corresponding
intervaluation metric then is defined to be
$\displaystyle d_{w}(f,\,g)$ $\displaystyle:=$ $\displaystyle
w(f,\,g)\,\circ_{w}\,w(g,\,f).$
The intervaluation is positive if
$\displaystyle w(f,\,g)\,=\,0$ $\displaystyle\Rightarrow$ $\displaystyle
f\,\leq\,g.$
###### Proposition 19
19
An intervaluation $w$ on $L$ and its metric $d_{w}$ always fulfill:
1. 1.
$w(f,\,g)\;=\;w(f\vee g,\,g)\;=\;w(f,\,f\wedge g)\;=\;d_{w}(f\vee
g,\,g)\quad\,\forall\,f,\,g{\,\in\,}L$.
2. 2.
$d_{w}$ is a pseudo-metric .
3. 3.
$d_{w}$ is a metric if and only if $w$ is positive.
Proof (1) We choose $h=f$ or $h=g$ in both modular inequalities:
$\displaystyle 0\;\circ_{w}\;w(f,\,g)\quad\leq\quad w(f\vee
g,\,g)\quad\leq\quad 0\;+\;w(f,\,g)$ $\displaystyle
w(f,\,g)\;\circ_{w}\;0\quad\leq\quad w(f,\,g\wedge f)\quad\leq\quad
w(f,\,g)\;+\;0$ and $\displaystyle d_{w}(f\vee g,\,g)\quad=\quad w(f\vee
g,\,g)\;\circ_{w}\;0\quad=\quad w(f,\,g).$
(2) From the definition we see $d_{w}(f,\,g)\geq 0$ and $d_{w}(f,\,f)=0$ for
all $f,\,g{\,\in\,}L$. As $\circ_{w}$ is commutative, $d_{w}$ is symmetric.
$\displaystyle d_{w}(f,g)$ $\displaystyle=$ $\displaystyle
w(f,\,g)\;\circ_{w}\;w(g,\,f)$ $\displaystyle\leq$
$\displaystyle\left(w\left(f\wedge h,\,g\right)\;+\;w\left(f,\,g\vee
h\right)\right)\;\circ_{w}\;\left(w\left(g\wedge
h,\,f\right)\;+\;w\left(g,\,f\vee h\right)\right)$ $\displaystyle\leq$
$\displaystyle\left(w\left(h,\,g\right)\;+\;w\left(f,\,h\right)\right)\;\circ_{w}\;\left(w\left(h,\,f\right)\;+\;w\left(g,\,h\right)\right))$
$\displaystyle=$
$\displaystyle\left(w\left(f,\,h\right)\;+\;w\left(h,\,g\right)\right)\;\circ_{w}\;\left(w\left(h,\,f\right)\;+\;w\left(g,\,h\right)\right))$
$\displaystyle\leq$
$\displaystyle\left(w\left(f,\,h\right)\;\circ_{w}\;w\left(h,\,f\right)\right)\;+\;\left(w\left(h,\,g\right)\;\circ_{w}\;w\left(g,\,h\right)\right))$
$\displaystyle=$ $\displaystyle d_{w}(f,\,h)\;+\;d_{w}(h,\,g)$
(3, “$\Rightarrow$”) Assume $0=w(f,\,g)=w(f,\,f\wedge g)$. Then
$d_{w}(f,f\wedge g)=0+0=0$. As $d_{w}$ is a metric, we have $f=f\wedge g$, so
$f\leq g$.
(3, “$\Leftarrow$”) $d_{w}(f,\,g)=0$ implies $w(f,\,g)=0$ and $w(g,\,f)=0$,
hence $f\leq g\leq f$, and $f=g$. $\square$
We now show the generalization of Lemma 10 for intervaluations, which we
already announced in subsection 4.2.
###### Lemma 20
20
Let $L$ be a distributive lattice, and $d$ a positive intervaluation metric on
$L$. $p\,{\,\in\,}\,L$ is $d$-irreducible if and only if
$\displaystyle d(p,\,f)\;\wedge\;d(p,\,g)$ $\displaystyle\leq$ $\displaystyle
d(p,\,f\vee g)$
holds for all $f,\,g\,{\,\in\,}\Downarrow\\!p$. In this case, “$\leq$” can be
replaced by “$=$”.
If $L$ is completely distributive, then the analog holds for complete
$d$-irreducibility as well.
Proof Let $f,\,g\,{\,\in\,}\,L$ be arbitrary and $p\,{\,\in\,}\,L$ as above.
Then holds:
$\displaystyle d(f\,\vee\,g,\,p)$ $\displaystyle=$ $\displaystyle
w(p,\,f\,\vee\,g)\;\circ_{w}\;w(f\,\vee\,g,\,p)$ $\displaystyle\geq$
$\displaystyle
w(p,\,f\,\vee\,g)\;\circ_{w}\;\big{(}w(f,\,p)\,\circ_{w}\,w(g,\,p)\big{)}$
$\displaystyle=$ $\displaystyle d\big{(}(f\vee
g)\,\wedge\,p,\,p\big{)}\;\circ_{w}\;w(f,\,p)\,\circ_{w}\,w(g,\,p)$
$\displaystyle\geq$
$\displaystyle\big{(}d(f\,\vee\,p,\,p)\,\wedge\,d(g\,\vee\,p,\,p)\big{)}\;\circ_{w}\;w(f,\,p)\,\circ_{w}\,w(g,\,p)$
$\displaystyle=$
$\displaystyle\big{(}w(p,\,f)\,\wedge\,w(p,\,g)\big{)}\;\circ_{w}\;w(f,\,p)\,\circ_{w}\,w(g,\,p)$
$\displaystyle\geq$
$\displaystyle\big{(}w(p,\,f)\,\circ_{w}\,w(f,\,p)\big{)}\;\wedge\;\big{(}w(p,\,g)\,\circ_{w}\,w(g,\,p)\big{)}$
$\displaystyle=$ $\displaystyle d(f,\,p)\;\wedge\;d(g,\,p)$
(1: definition, 2: by left modular inequality, 3: definition, 4: hypothesis,
5: definition, 6: by cases and monotony of “$\circ_{w}$” (property (2) in
Definition 18), 7: definition). Each step holds in the infinite case as well.
$\square$
### 5.1 Examples
###### Example 21
21
There are several possible choices for the commutative and associative binary
operation $\circ_{w}$ in Definition 18. Choosing addition leads directly to
the definition of valuations. The next important choice is the maximum
operation: Properties (1) and (3) are obviously fulfilled, the left side of
(2) as well. (2.right) needs some short consideration: As $+$ distributes over
$\vee$, the right-hand side equals
$\displaystyle(r\,\vee\,t)\,+\,(s\,\vee\,u)$ $\displaystyle=$
$\displaystyle(r+s)\,\vee\,(r+u)\,\vee\,(t+s)\,\vee\,(t+u)$
which is greater or equal $(r+s)\,\vee\,(t+u)$ for all
$r,\,s,\,t,\,u\,{\,\in\,}[0,\,\infty)$.
Each norm $||\cdot||$ on $\mathbb{R}^{2}$ with certain normalization
properties qualifies as an operation $\circ_{w}$ via
$r\,\circ_{w}\,s\,:=\,||(r,s)||$. This accounts for the $\ell^{p}$-norms:
$\displaystyle
r\,\circ_{p}\,s\;:=\;\big{|}\big{|}(r,\,s)\big{|}\big{|}_{p}\;:=\;\sqrt[p]{r^{p}\,+\,s^{p}}$
for $p\,{\,\in\,}\,[1,\infty)$. Again, properties (1), (2.left) and (3) of
Definition 18 are trivial. Property (2.right) is the triangle inequality of
the $\ell^{p}$-norms (i.e. a special case of the Minkowski inequality [Wr]).
Given any metric $d$ on $L$ we may define $w_{d}(f,\,g)\;:=\;d(f\vee g,\,g)$
and deduce $\circ_{w}$ from $d(f,\,g)=w_{d}(f,\,g)\,\circ_{w}\,w_{d}(g,\,f)$.
The operation $\circ_{w}$ must be commutative due to the symmetry of $d_{w}$.
From the remaining properties of Definition 18, property (4) follows directly
from $d(g,\,g)=0$, while the rest is less obvious.
###### Example 22
22
The standard metric on $[0,\,\infty)$ is an intervaluation metric with
$\displaystyle w(r,\,s)$ $\displaystyle:=$ $\displaystyle 0\,\vee\,(r\,-\,s).$
However, one may freely choose $\circ_{w}$ to be addition or maximum. To prove
the cut law for both choices, it suffices to show
$\displaystyle 0\,\vee\,(r\,-\,s)$ $\displaystyle=$
$\displaystyle\left(0\,\vee\,\big{(}r\,-\,(s\,\vee\,t)\big{)}\right)\;+\;\left(0\,\vee\,\big{(}(r\,\wedge\,t)\,-\,s)\big{)}\right).$
For this, we make use of $a\,+\,b\,=\,(a\,\wedge\,b)\,+\,(a\,\vee\,b)$ with
$a\,=\,r\,\wedge\,s$ and $b\,=\,r\,\wedge\,t$, then add $r$ to both sides,
rearrange and apply $x\,-\,(x\,\wedge\,y)\,=\,0\,\vee\,(x\,-\,y)$.
###### Example 23
23
Let $(X,\,\mu)$ be a measure space, $p\,{\,\in\,}\,(1,\infty)$ arbitrary, and
$L$ the lattice of $L^{p}$-integrable non-negative Lipschitz functions of
Lipschitz constant $\leq 1$. Define
$\displaystyle r\,\circ_{w}\,s$ $\displaystyle:=$
$\displaystyle(r^{p}\,+\,s^{p})^{1/p},$ $\displaystyle\textnormal{and}\qquad
w(f,\,g)$ $\displaystyle:=$
$\displaystyle\sqrt[p]{\int\big{|}f\,-\,(f\,\wedge\,g)\big{|}^{p}\,\textnormal{d}\mu}\,.$
As $|r\,-\,(r\wedge s)|^{p}\,+\,|s\,-\,(r\wedge s)|^{p}\,=\,|r\,-\,s|^{p}$ for
all $r,\,s\,{\,\in\,}\,[0,\,\infty)$, the corresponding (pseudo-)metric is
just the $L^{p}$-metric
$\displaystyle d_{p}(f,g)$ $\displaystyle=$
$\displaystyle\sqrt[p]{\int|f\,-\,g|^{p}\,\textnormal{d}\mu}\,.$
Properties (1)-(3) of Definition 18 follow from Example 21, (4) is trivial.
The left cut law can be shown by pointwise analysis and case distinction
($h\leq g$ vs. $h>g$), the right cut law follows from Example 22 and the
Minkowski inequality. $d_{p}$ might be a pseudo-metric, depending on $\mu$.
###### Example 24
24
Here is a minimal example for a non-intervaluation metric: Take
$L=\\{a,b,c\\}$ with $a\,<\,b\,<\,c$, and $d(a,\,c)\,=\,1$, $d(a,\,b)\,=\,2$,
$d(b,\,c)\,=\,3$. Then $w(c,\,a)\,=\,1$, although $w(c\,\wedge\,b,\,a)\,=\,2$
and $w(c,\,a\,\vee\,b)\,=\,3$, which both contradict the cut law and
Proposition 19.1, no matter what $\circ_{w}$ is.
###### Example 25
25
The Lipschitz constant provides a much more interesting example for a non-
intervaluation metric. Let $X$ be an arbitrary true metric space, and $L$ a
complete lattice of functions $f:\,X\rightarrow\mathbb{R}$ with bounded
Lipschitz constant. The Lipschitz constant of a function $f{\,\in\,}L$ and the
corresponding pseudo-metric are given by
$\displaystyle\textnormal{LC}(f)$ $\displaystyle:=$
$\displaystyle\sup_{x,\,y\,{\,\in\,}\,X}{\;\frac{\,\big{|}f(x)\,-\,f(y)\big{|}\,}{d(x,\,y)}\;}$
$\displaystyle d_{\textnormal{LC}}\,(f,\,g)$ $\displaystyle:=$
$\displaystyle\textnormal{LC}(f\,-\,g).$
They are used by [Wv] as ingredient to the utilized norm, called Lipschitz
norm, which is defined as
$||f||_{L}\,:=\,||f||_{\infty}\,\vee\,\textnormal{LC}(f)$. However, neither
defines an intervaluation: Although Weaver shows in his Proposition 1.5.5 that
LC fulfills a modular inequality for ultravaluations
$\displaystyle\textnormal{LC}(f\,\vee\,g)\;\vee\;\textnormal{LC}(f\,\wedge\,g)$
$\displaystyle\leq$
$\displaystyle\textnormal{LC}(f)\;\vee\;\textnormal{LC}(g)$
the inverse inequality is wrong, as there is no bound to $\textnormal{LC}(f)$
by any combination of $\textnormal{LC}(f\wedge g)$ and $\textnormal{LC}(f\vee
g)$. To see this, consider the two-point-space $X\,=\,\\{a,b\\}$ of diameter
$l<1$, and the Lipschitz-functions $f\,=\,(0,\,l)$ and $g\,=\,(l,\,0)$. Then
$\textnormal{LC}(f)\,=\,||f||_{L}\,=\,1$, but $\textnormal{LC}(f\wedge
g)\,=\,\textnormal{LC}(f\vee g)\,=\,0$ and $||\cdot||_{L}\,=\,l$ in both
cases.
Correspondingly, the cut law is explicitly violated by $d_{\textnormal{LC}}$,
as one can see when $f$ and $g$ are two different constant functions, and $h$
crosses them both.
We now concentrate on the special case of the supremum metric.
###### Proposition 26
26
Let $Z$ be a distributive lattice with intervaluation metric $d$ (with
corresponding $w_{d}$ and $\circ_{d}$), with $r\,\circ_{d}\,s\,=\,r\,\vee\,s$
for all $r,\,s\,{\,\in\,}\,[0,\,\infty)$. Let $X$ be an arbitrary space, and
$L$ a complete lattice of functions $f:\,X\rightarrow Z$ with pointwise infima
and suprema. If
$\displaystyle w_{\infty}(f,\,g)$ $\displaystyle:=$
$\displaystyle\bigvee_{x{\,\in\,}X}w_{d}\big{(}f(x),\,g(x)\big{)}$
is bounded, it defines an intervaluation metric on $L$ with
$r\,\circ_{\infty}\,s\,=\,r\,\vee\,s$ for all
$r,\,s\,{\,\in\,}\,[0,\,\infty)$, which equals the supremum metric
$d_{\infty}$.
Proof The left inequality of the cut law is trivial. For the right side we
have to use that a supremum of sums is less than or equal to a sum of suprema,
which in turn follows from complete distributivity:
$\displaystyle\bigvee_{x{\,\in\,}X}w_{d}\big{(}fx,\,gx\big{)}$
$\displaystyle\leq$
$\displaystyle\bigvee_{x{\,\in\,}X}\left(w_{d}\big{(}fx,\,(g\vee
h)(x)\big{)}\;+\;w_{d}\big{(}(f\wedge h)(x),\,gx\big{)}\right)$
$\displaystyle\leq$ $\displaystyle\bigvee_{x{\,\in\,}X}w_{d}\big{(}fx,\,(g\vee
h)(x)\big{)}\;+\;\bigvee_{x{\,\in\,}X}w_{d}\big{(}(f\wedge h)(x),\,gx\big{)}$
$\square$
###### Corollary 27
27
Let $X$ be any metric space. The supremum metric $d_{\infty}$ is an
intervaluation metric on the space $\operatorname{{\textnormal{Lip}}}_{0}X$ of
bounded, non-negative Lipschitz functions on $X$ with Lipschitz-constant $\leq
1$.
Proof $\operatorname{{\textnormal{Lip}}}X$ is a complete lattice, as one can
easily check. We find $r\,\circ_{d_{\infty}}\,s\;=\;r\,\vee s$ and
$\displaystyle
w_{d_{\infty}}(f,\,g)\quad=\quad\bigvee_{x{\,\in\,}X}\big{|}f(x)\,-\,(f\,\wedge\,g)(x)\big{|}\quad=\quad
0\,\vee\,\bigvee_{x{\,\in\,}X}\big{(}f(x)\,-\,g(x)\big{)},$
which is the intervaluation metric of Proposition 26 applied to Example 22.
$\square$
### 5.2 Topological Aspects
We finally take a look at the subset $\operatorname{\textnormal{cmli}}(L)$ of
all completely $d$-irreducible elements of a complete lattice $L$ with
intervaluation metric $d$.
###### Proposition 28
28
Let $L$ be a complete lattice with intervaluation metric $d$, and let $L$ be
metrically complete. Then $\operatorname{\textnormal{cmli}}(L)$ is
topologically closed.
Proof Let $(p_{n})\subseteq\operatorname{\textnormal{cmli}}(L)$,
$n{\,\in\,}\mathbb{N}^{*}$ be some sequence of completely $d$-irreducible
elements converging to $p\,{\,\in\,}\,L$, and $(f_{j})_{j{\,\in\,}J}$ any non-
empty family in $L$. Then for any $n\,{\,\in\,}\,\mathbb{N}^{*}$ holds
$\displaystyle d\,\left(p,\,\bigvee f_{j}\right)$ $\displaystyle\geq$
$\displaystyle d\,\left(p_{n},\,\bigvee f_{j}\right)\;-\;d(p,\,p_{n})$
$\displaystyle\geq$ $\displaystyle\bigwedge d(p_{n},\,f_{j})\;-\;d(p,\,p_{n})$
$\displaystyle\geq$
$\displaystyle\bigwedge\big{(}d(p,\,f_{j})\,-\,d(p,\,p_{n})\big{)}\;-\;d(p,\,p_{n})$
$\displaystyle\geq$ $\displaystyle\bigwedge
d(p,\,f_{j})\;-\;\underbrace{2~{}d(p,\,p_{n})}_{\rightarrow\;0},$
i.e. the element $p$ is completely $d$-irreducible.
$\square$
###### Definition 29
29
Let $L$ be a lattice with metric $d$, $R\geq 0$ arbitrary. We define an
$R$-base of $L$ to be a subset $B\,\subseteq\,L$ such that for any
$f{\,\in\,}L$ there is $(b_{j})_{j{\,\in\,}J}\,\subseteq\,B$, $J$ an arbitrary
non-empty index set, such that $d(f,\,\bigvee_{j{\,\in\,}J}\,b_{j})\,\leq\,R$.
A base simply is a $0$-base.
###### Proposition 30
30
Consider an $R$-base $B$ of a complete lattice $L$ with intervaluation metric
$d$, $R\geq 0$. Then for each $\delta\,>\,0$,
$\operatorname{\textnormal{cmli}}(L)$ is in the $(R\,+\,\delta)$-ball around
$B$. In particular, if $R\,=\,0$, $\operatorname{\textnormal{cmli}}(L)$ lies
in the metrical closure of $B$.
Proof Let $p\,{\,\in\,}\,\operatorname{\textnormal{cmli}}(L)$ be arbitrary.
As $B$ is an $R$-base, there are $b_{j}\,{\,\in\,}\,B$,
$j\,{\,\in\,}\,J\,\neq\,\emptyset$, such that
$\displaystyle d\left(p,\,\bigvee_{j{\,\in\,}J}b_{j}\right)$
$\displaystyle\leq$ $\displaystyle R.$
From Definition 1 we infer that there is a sequence $(c_{k})\,\subseteq\,B$,
$k\,{\,\in\,}\,K\,\subseteq\,J$ whose distances to $p$ converge to $R$. If
$R\,=\,0$, the sequence $(c_{j})$ metrically converges to $p$. $\square$
Propositions 28 and 30 might help in identifying all completely
$d$-irreducible elements of a concretely given lattice.
###### Example 31
31
It is easy to see that, if $B$ is a base, and $b\,{\,\in\,}\,B$ not a join-
irreducible element, then $B\setminus\\{b\\}$ is a base as well (if
$b\,=\,f\,\vee\,g$, $f$ and $g$ are joins of elements of $B$, and as
$f,\,g\,<\,b$, $b$ is not part of these joins). Using the Lemma of Zorn, it is
possible to deduce that the subset of all join-irreducible elements
constitutes a base for any sufficiently nice lattice.
Unfortunately, this is not the case with $d$-irreducible elements: Let
$L^{\prime}$ be the completely distributive complete lattice
$[0,\,3]\times[0,\,2]$ with componentwise supremum and infimum, and with
supremum metric. Then consider the sublattice $L\subseteq L^{\prime}$ formed
by the five elements
$\displaystyle L$ $\displaystyle:=$
$\displaystyle\\{(0,\,0),\;(1,\,0),\;(0,\,1),\;(1,\,1),\;(2,\,2)\\}.$
We find
$\operatorname{\textnormal{cmli}}(L)\,=\,\\{(0,\,0),\,(1,\,0),\,(0,\,1)\\}$,
as $(1,\,1)\,=\,(1,\,0)\,\vee\,(0,\,1)$. $p\,=\,(2,\,2)$ is join-irreducible
in this lattice, but not $d$-irreducible: Take $f_{1}\,=\,(1,\,0)$,
$f_{2}\,=\,(0,\,1)$, then $\bigwedge d(p,\,f_{j})\,=\,2$, but $d(p,\,\bigvee
f_{j})\,=\,1$. Nevertheless, $(2,\,2)$ must be part of any 0-base of $L$.
## References
* [Bi1] G. Birkhoff, Lattice Theory, American Mathematical Society Colloquium Publications Vol. XXV, 2nd ed. (1948) and 3rd ed. (1960)
* [Bi2] G. Birkhoff, Von Neumann and Lattice Theory, Bull. Amer. Math. Soc. 64, Nr 3, Part 2 (1958) 50–56,
http://www.ams.org/bull/1958-64-03/S0002-9904-1958-10192-5/
S0002-9904-1958-10192-5.pdf
* [dH] P. de la Harpe, Topics in Geometric Group Theory, The University of Chicago Press (2000)
* [Gl] V. Glivenko, Géometrie des systèmes de chosen normées, Am. Jour. of Math. 58 (1936) 799–828
* [Lo1] A. Lochmann, Rough Isometries of Order Lattices and Groups, Niedersächsische Staats- und Universitätsbibliothek, Doctoral Thesis, http://webdoc.sub.gwdg.de/diss/2009/lochmann/
* [Lo2] A. Lochmann, Rough Isometries of Lipschitz Function Spaces, preprint at http://arxiv.org/abs/0710.1109
* [Mn] B. Monjardet, Metrics on partially ordered sets — a survey, Discrete Mathematics 35 (1981) 173–184
* [Ro] R. T. Rockafellar, Convex Analysis, Princeton University Press (1970)
* [vN] J. von Neumann, Lectures on continuous geometries, Princeton 1936-1937 (2 vols.), in particular chapter XVII
* [Wr] D. Werner, Funktionalanalysis, Springer (2005)
* [Wv] N. Weaver, Lipschitz Algebras, World Scientific (1999)
Georg-August-Universität Göttingen, Germany
eMail `lochmann@uni-math.gwdg.de`
|
arxiv-papers
| 2010-05-27T19:44:45 |
2024-09-04T02:49:10.674687
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andreas Lochmann",
"submitter": "Andreas Lochmann",
"url": "https://arxiv.org/abs/1005.5155"
}
|
1005.5213
|
# Comparison of different proximity potentials for asymmetric colliding nuclei
Ishwar Dutt Rajeev K. Puri rkpuri@pu.ac.in; drrkpuri@gmail.com Department of
Physics, Panjab University, Chandigarh 160 014, India
###### Abstract
Using the different versions of phenomenological proximity potential as well
as other parametrizations within the proximity concept, we perform a detailed
comparative study of fusion barriers for asymmetric colliding nuclei with
asymmetry parameter as high as 0.23. In all, 12 different proximity potentials
are robust against the experimental data of 60 reactions. Our detailed study
reveals that the surface energy coefficient as well as radius of the colliding
nuclei depend significantly on the asymmetry parameter. All models are able to
explain the fusion barrier heights within $\pm 10\%$ on the average. The
potentials due to Bass 80, AW 95, and Denisov DP explain nicely the fusion
cross sections at above- as well as below-barrier energies.
###### pacs:
24.10.-i, 25.70.Jj, 25.70.-z.
## I Introduction
The fusion of colliding nuclei with neutron -rich/ -deficient content and at
the extreme of isospin plane has attracted a large number of studies in recent
years canto06 ; Aguilera95 ; Silva97 ; Aguilera90 ; Cavallera90 ; Vega90 ;
Sonz98 ; Vinod96 ; Trotta01 ; Stefanini06 ; stefanini08 . This renewed
interest is due to the availability of radioactive-ion beams that can produce
nuclei at the extreme of isospin canto06 ; Stefanini06 ; stefanini08 .
Further, this field has also been enriched with several new phenomena that put
a stringent test on theoretical models derived to study the fusion phenomenon
in heavy-ion reactions.
As is evident from the literature, no experiment can extract information about
the fusion barriers directly. All experiments measure the fusion differential
cross sections canto06 ; Aguilera95 ; Silva97 and then with the help of
theoretical model, one extracts the fusion barriers. Theoretical models are
very helpful in understanding the nuclear interactions at a microscopic level.
A vast number of theoretical models and potentials have become available in
recent years that can explain one or the other features of fusion dynamics id1
; rkp1 ; rkp2 ; blocki77 ; wr94 ; ms2000 ; gr09 ; bass73 ; bass77 ; cw76 ;
aw95 ; ngo80 ; deni02 ; ngo75 ; deni07 .
In the galaxy of different theoretical models, proximity potential blocki77
enjoys very popular status. This phenomenological potential is a benchmark and
backbone for all microscopic/macroscopic fusion models. It is almost mandatory
to compare the potential and parametrize it within the proximity concept for
wider acceptability. In recent years, several refinements and modifications
have been proposed over original proximity potential wr94 ; ms2000 . Further
with the passage of time, different versions of the same model are also
available id1 . Many of these modifications are based on the isospin effects
either through the surface energy coefficients or via nuclear radius. It would
be of interest to test these potentials in the isospin plane and to see how
these different potentials will perform when asymmetry in the neutron/proton
content is very large.
Recently, we carried out a detailed comparative systematic study of different
fusion models for symmetric colliding nuclei id1 . Here we plan to extend this
study for those colliding nuclei that have larger neutron/proton content. In
this study, we shall compare as many as 12 proximity potentials with different
versions. This will include four versions of proximity potential, three
versions of potential due to Bass and Winther each and the latest potential
due to Ngô and a modified version of the Denisov potential. Section II, deals
with formalism in detail, Sec. III contains the results, and a summary is
presented in Sec. IV.
## II Formalism
In this section, we present the details of various proximity potentials used
for the calculation of fusion barriers. When two surfaces approach each other
within a distance of 2 - 3 fm, additional force due to the proximity of the
surface is labeled as proximity potential. Various versions of these
potentials take care of different aspects including the isospin dependence. In
the following, we discuss each of them in detail.
### II.1 $\rm Proximity~{}1977~{}(Prox~{}77)$
The basis of proximity potential is the theorem that states that _“the force
between two gently curved surfaces in close proximity is proportional to the
interaction potential per unit area between the two flat surfaces”_. According
to the original version of proximity potential 1977 blocki77 , the interaction
potential $V_{N}(r)$ between two surfaces can be written as:
$V_{N}(r)=4\pi\gamma
b\overline{R}\Phi\left(\frac{{r}-C_{1}-C_{2}}{b}\right)~{}~{}\rm MeV.$ (1)
In this, the mean curvature radius, $\overline{R}$ has the form
$\overline{R}=\frac{C_{1}C_{2}}{C_{1}+C_{2}},$ (2)
quite similar to the one used for reduced mass. Here
$C_{i}=R_{i}\left[1-\left(\frac{b}{R_{i}}\right)^{2}+\cdots\cdots\right],$ (3)
${\rm R_{i}}$, the effective sharp radius, reads as
$R_{i}=1.28A^{1/3}_{i}-0.76+0.8A^{-1/3}_{i}{~{}\rm
fm}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(i=1,2).$ (4)
In Eq. (1), $\Phi(\xi=\frac{{r}-C_{1}-C_{2}}{b})$ is a universal function that
depends on the separation between the surfaces of two colliding nuclei only.
As we see, both these factors do not depend on the isospin content. However,
$\gamma$, the surface energy coefficient, depends on the neutron/proton excess
as
$\gamma=\gamma_{0}\left[1-k_{s}\left(\frac{N-Z}{N+Z}\right)^{2}\right],$ (5)
where N and Z are the total number of neutrons and protons. In the present
version, $\gamma_{0}$ and $k_{s}$ were taken to be $0.9517~{}\rm MeV/fm^{2}$
and $1.7826$, respectively. Note that for the symmetric colliding pair i.e. (N
= Z), $\gamma=\gamma_{0}=0.9517~{}\rm MeV/fm^{2}$. If the
$\left(\frac{N-Z}{N+Z}\right)$ ratio is $0.5$, $\gamma$ reduces to
$0.5276~{}\rm MeV/fm^{2}$. Defining asymmetry parameter
$A_{s}=\left[\frac{N_{1}+N_{2}-(Z_{1}+Z_{2})}{N_{1}+N_{2}+(Z_{1}+Z_{2})}\right]$,
one notices drastic reduction in the magnitude of the potential with asymmetry
of the colliding pair. Interestingly, most of the modified proximity type
potentials use different values of the parameter $\gamma$ wr94 ; ms2000 .
The universal function $\Phi\left(\xi\right)$ was parameterized with the
following form:
$\Phi\left(\xi\right)=\left\\{\begin{array}[]{l
r}-\frac{1}{2}\left(\xi-2.54\right)^{2}-0.0852\left(\xi-2.54\right)^{3},\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{
for $\xi\leq 1.2511$ },\\\ -3.437\exp\left(-\xi/0.75\right),\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{
for $\xi\geq 1.2511$ }.\end{array}\right.$ (6)
The surface width $b$ has been evaluated close to unity. Using the above form,
one can calculate the nuclear part of the interaction potential ${V_{N}(r)}$.
This model is referred to as Prox 77 and the corresponding potential as
$V_{N}^{Prox~{}77}(r)$.
### II.2 $\rm Proximity~{}1988~{}(Prox~{}88)$
Later on, using the more refined mass formula due to Möller and Nix mn81 , the
value of coefficients $\gamma_{0}$ and $k_{s}$ were modified yielding the
values = 1.2496 $\rm MeV/fm^{2}$ and 2.3, respectively. Reisdorf wr94 labeled
this modified version as “Proximity 1988”. Note that this set of coefficients
give stronger attraction compared to the above set. Even a more recent
compilation by Möller and Nix mn95 yields similar results. We labeled this
potential as Prox 88.
### II.3 $\rm Proximity~{}2000~{}(Prox~{}00)$
Recently, Myers and Świa̧tecki ms2000 modified Eq. (1) by using up-to-date
knowledge of nuclear radii and surface tension coefficients using their
droplet model concept. The prime aim behind this attempt was to remove
descripency of the order of $4\%$ reported between the results of Prox 77 and
experimental data ms2000 . Using the droplet model ms80 , matter radius
$C_{i}$ was calculated as
$C_{i}=c_{i}+\frac{N_{i}}{A_{i}}t_{i}~{}~{}~{}~{}(i=1,2),$ (7)
where $c_{i}$ denotes the half-density radii of the charge distribution and
$t_{i}$ is the neutron skin of the nucleus. To calculate $c_{i}$, these
authors ms2000 used two-parameter Fermi function values given in Ref. dv87
and the remaining cases were handled with the help of parametrization of
charge distribution described below. The nuclear charge radius (denoted as
$R_{00}$ in Ref. bn94 ), is given by the relation:
$R_{00i}=\sqrt{\frac{5}{3}}\left<r^{2}\right>^{1/2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle=1.240A_{i}^{1/3}\left\\{1+\frac{1.646}{A_{i}}-0.191\left(\frac{A_{i}-2Z_{i}}{A_{i}}\right)\right\\}{~{}\rm
fm}$ $\displaystyle(i=1,2),$ (8)
where $<r^{2}>$ represents the mean-square nuclear charge radius. According to
Ref. bn94 , Eq. (8) was valid for the even-even nuclei with $8\leq Z<38$ only.
For nuclei with $Z\geq 38$, the above equation was modified by Pomorski _et
al_. bn94 as;
$\displaystyle
R_{00i}=1.256A_{i}^{1/3}\left\\{1-0.202\left(\frac{A_{i}-2Z_{i}}{A_{i}}\right)\right\\}{~{}\rm
fm}$ $\displaystyle~{}~{}~{}~{}~{}~{}(i=1,2).$ (9)
These expressions give good estimate of the measured mean square nuclear
charge radius $<r^{2}>$. In the present model, authors used only Eq. (8). The
half-density radius, $c_{i}$, was obtained from the relation:
$c_{i}=R_{00i}\left(1-\frac{7}{2}\frac{b^{2}}{R_{00i}^{2}}-\frac{49}{8}\frac{b^{4}}{R_{00i}^{4}}+\cdots\cdots\right)~{}~{}~{}~{}~{}~{}~{}(i=1,2).$
(10)
Using the droplet model ms80 , neutron skin $t_{i}$ reads as;
$t_{i}=\frac{3}{2}r_{0}\left[\frac{JI_{i}-\frac{1}{12}c_{1}Z_{i}A^{-1/3}_{i}}{Q+\frac{9}{4}JA^{-1/3}_{i}}\right](i=1,2).$
(11)
Here $r_{0}$ is $1.14$ fm, the value of nuclear symmetric energy coefficient
$J=32.65$ MeV and $c_{1}=3e^{2}/5r_{0}=0.757895$ MeV. The neutron skin
stiffness coefficient Q was taken to be 35.4 MeV. The nuclear surface energy
coefficient $\gamma$ in terms of neutron skin was given as;
$\gamma=\frac{1}{4\pi
r^{2}_{0}}\left[18.63{\rm(MeV)}-Q\frac{\left(t^{2}_{1}+t^{2}_{2}\right)}{2r^{2}_{0}}\right],$
(12)
where $t_{1}$ and $t_{2}$ were calculated using Eq. (11). The universal
function $\Phi(\xi)$ was reported as;
$\Phi\left(\xi\right)=\left\\{\begin{array}[]{ll}-0.1353+\sum\limits_{n=0}^{5}\left[c_{n}/\left(n+1\right)\right]\left(2.5-\xi\right)^{n+1},\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{
for \quad$0<\xi\leq 2.5$},\\\
-0.09551\exp\left[\left(2.75-\xi\right)/0.7176\right],\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{
for $\quad\xi\geq 2.5$}.\end{array}\right.$ (13)
The values of different constants $c_{n}$ were: $c_{0}=-0.1886$,
$c_{1}=-0.2628$, $c_{2}=-0.15216$, $c_{3}=-0.04562$, $c_{4}=0.069136$ and
$c_{5}=-0.011454$. For $\xi>2.74$, the above exponential expression is the
exact representation of the Thomas-Fermi extension of the proximity potential.
This potential is labeled Prox 00.
### II.4 $\rm Modified~{}Proximity~{}2000~{}(Prox~{}00DP)$
Recently, Royer and Rousseau gr09 modified Eq. (8) with slightly different
constants as;
$\displaystyle
R_{00i}=1.2332A^{1/3}_{i}\left[1+\frac{2.348443}{A_{i}}\right.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\left.-0.151541\left(\frac{A_{i}-2Z_{i}}{A_{i}}\right)\right]{~{}\rm
fm}~{}~{}(i=1,2).$ (14)
It is obtained by analyzing as many as 2027 masses with N, Z $\geq$ 8 and a
mass uncertainty $\leq$ 150 keV. Further, the accuracy of the above formula is
mainly improved by adding the Coulomb diffuseness correction or the charge
exchange correction to the mass formulas gr09 . We implement this radius in
the proximity 2000 version instead of the form given in the proximity 2000.
This new version of the proximity potential is labeled Prox 00DP id1 .
### II.5 $\rm Bass~{}1973~{}(Bass~{}73)$
This model is based on the assumption of liquid drop model bass73 . Here
change in the surface energy of two fragments due to their mutual separation
is represented by exponential factor. By multiply with geometrical arguments,
one can obtained the nuclear part of the interaction potential as
$V_{N}(r)^{Bass~{}73}=-\frac{d}{R_{12}}a_{s}A_{1}^{1/3}A_{2}^{1/3}exp(-\frac{r-R_{12}}{d}){~{}\rm
MeV},$ (15)
with ${R_{12}}=r_{0}(A_{1}^{1/3}+A_{2}^{1/3}),~{}d=1.35~{}{\rm fm}$ and
$a_{s}=17.0~{}{\rm MeV}$. The cut-off distance $R_{12}$ is chosen to yield
saturation density in the overlap region and $r_{o}=1.07~{}{\rm fm}$
corresponding half of the maximum density for individual nucleus. We labeled
this potential Bass 73.
### II.6 $\rm Bass~{}1977~{}(Bass~{}77)$
In this model, nucleus-nucleus potential is derived from the information based
on the experimental fusion cross sections by using the liquid drop model and
general geometrical arguments. The nuclear part of the potential (for
spherical nuclei with frozen densities) can be written as bass77
$\displaystyle
V_{N}\left(r\right)^{Bass~{}77}=-4\pi\gamma\frac{R_{1}R_{2}}{R_{1}+R_{2}}f\left(r-R_{1}-R_{2}\right)$
$\displaystyle\qquad\qquad=-\frac{R_{1}R_{2}}{R_{1}+R_{2}}\Phi\left(r-R_{1}-R_{2}\right)~{}{\rm
MeV},$ (16)
with
$\frac{df}{ds}=-1,\qquad\qquad{\rm for}\qquad\qquad s=0.$ (17)
Note that $f\left(s=r-R_{1}-R_{2}\right)$ and
$\Phi\left(s=r-R_{1}-R_{2}\right)$ are the universal functions. Here radius
$R_{i}$ is written as
$R_{i}=1.16A^{1/3}_{i}-1.39A^{-1/3}_{i}~{}{\rm fm}~{}~{}~{}~{}(i=1,2).$ (18)
The form of the universal function $\Phi\left(s\right)$ reads as
$\Phi\left(s\right)=\left[A\exp\left(\frac{s}{d_{1}}\right)+B\exp\left(\frac{s}{d_{2}}\right)\right]^{-1},$
(19)
with $A=0.0300$ MeV-1fm, $B=0.0061$ MeV-1fm, $d_{1}=3.30$ fm and $d_{2}=0.65$
fm. Note that where $b=1$, $\xi$ and s turn out to be the same quantities.
This model was very successful in explaining the barrier heights, positions,
and cross sections over a wide range of incident energies and masses of
colliding nuclei. We labeled this potential Bass 77.
### II.7 $\rm Bass~{}1980~{}(Bass~{}80)$
The above potential form was further improved by Bass wr94 . Here
$\Phi\left(s=r-R_{1}-R_{2}\right)$ is now given as:
$\Phi\left(s\right)=\left[0.033\exp\left(\frac{s}{3.5}\right)+0.007\exp\left(\frac{s}{0.65}\right)\right]^{-1},$
(20)
with central radius, $R_{i}$ as
$R_{i}=R_{s}\left(1-\frac{0.98}{R_{s}^{2}}\right)~{}~{}~{}~{}(i=1,2),$ (21)
where $R_{s}$ is same as given by Eq. (4). We labeled this potential as Bass
80.
### II.8 $\rm Christensen~{}and~{}Winther~{}1976~{}(CW~{}76)$
Christensen and Winther cw76 derived the nucleus-nucleus interaction
potential by analyzing the heavy-ion elastic-scattering data, based on the
semiclassical arguments and the recognition that optical-model analysis of
elastic scattering determines the real part of the interaction potential only
in the vicinity of a characteristic distance. The nuclear part of the
empirical potential due to Christensen and Winther is written as
$V_{N}^{CW~{}76}\left(r\right)=-50\frac{R_{1}R_{2}}{R_{1}+R_{2}}\Phi\left(r-R_{1}-R_{2}\right)~{}{\rm
MeV}.$ (22)
This form of the geometrical factor is similar to that of $\rm Bass~{}77$ with
different radius parameters
$R_{i}=1.233A^{1/3}_{i}-0.978A^{-1/3}_{i}~{}{\rm fm}~{}~{}~{}~{}(i=1,2).$ (23)
The universal function $\Phi(s=r-R_{1}-R_{2}$ ) has the following form
$\Phi\left(s\right)=\exp\left(-\frac{r-R_{1}-R_{2}}{0.63}\right).$ (24)
This model was tested for more than 60 reactions and we labeled it CW 76.
### II.9 $\rm Broglia~{}and~{}Winther~{}1991~{}(BW~{}91)$
A refined version of the above potential was derived by Broglia and Winther
wr94 , by taking Woods-Saxon parametrization with subsidiary condition of
being compatible with the value of the maximum nuclear force predicted by the
proximity potential Prox 77. This refined potential resulted in
$V_{N}^{BW~{}91}(r)=-\frac{V_{0}}{1+\exp\left(\frac{r-R_{0}}{0.63}\right)}~{}{\rm
MeV};$ (25) ${\rm
with},~{}V_{0}=16\pi\frac{R_{1}R_{2}}{R_{1}+R_{2}}{\gamma}{a},$ (26)
here $a=0.63$ fm and
$R_{0}=R_{1}+R_{2}+0.29.$ (27)
Here radius $R_{i}$ has the form
$R_{i}=1.233A^{1/3}_{i}-0.98A^{-1/3}_{i}~{}{\rm fm}~{}~{}~{}~{}(i=1,2).$ (28)
The form of the surface energy coefficient $\gamma$ is quite similar to the
one used in Prox 77 with slight difference
$\gamma=\gamma_{o}\left[1-k_{s}\left(\frac{N_{p}-Z_{p}}{A_{p}}\right)\left(\frac{N_{t}-Z_{t}}{A_{t}}\right)\right],$
(29)
where ${~{}\rm\gamma_{0}}$ = 0.95 $~{}\rm~{}MeV/fm^{2}$ and $k_{s}=1.8$. Note
that the second term used in this potential gives different results when the
projectile is symmetric ($N=Z$) and the target is asymmetric ($N>Z$). This
form will also give different results for larger mass asymmetry $\eta_{A}$.
Note that the radius used in this potential has same form like that of Bass
with different constants. We labeled this potential as BW 91.
### II.10 $\rm Aage~{}Winther~{}(AW~{}95)$
Winther adjusted the parameters of the above potential through an extensive
comparison with experimental data for heavy-ion elastic scattering. This
refined adjustment to slight different values of “$a$” and $R_{i}$ as aw95
${a}=\left[\frac{1}{1.17(1+0.53(A_{1}^{-1/3}+A_{2}^{-1/3}))}\right]~{}\rm fm,$
(30)
and
$R_{i}=1.20A^{1/3}_{i}-0.09~{}\rm fm~{}~{}~{}~{}(i=1,2).$ (31)
Here, $R_{0}=R_{1}+R_{2}$ only. We labeled this potential as AW 95.
### II.11 $\rm Ng$ô$~{}1980~{}(\rm Ng$ô $80)$
In earlier attempts, based on the microscopic picture of a nucleus and on the
idea of energy density formalism, the potential from Ngô and collaborators
enjoy special status ngo75 . In this model, calculations of the ion-ion
potential are performed within the framework of energy density formalism due
to Bruckener _et al_., using a sudden approximation bk68 . The need of
Hartree-Fock densities as input in this model limited its scope. This not only
made calculations tedious, but it also hindered its application to heavier
nuclei. The above-stated parametrization was improved by H. Ngô and Ch. Ngô
ngo80 , by using a Fermi-density distribution for nuclear densities as
$\rho_{n,p}(r)=\frac{\rho_{n,p}(0)}{1+\exp\left[(r-C_{n,p})/0.55\right]}~{},$
(32)
where $C$ represents the central radius of the distribution and is defined in
Prox 77 (see Eq. (3) with b = 1 fm). Here $\rho_{n,p}(0)$ is given by
$\rho_{n}(0)=\frac{3}{4\pi}\frac{N}{A}\frac{1}{r^{3}_{0_{n}}};~{}~{}~{}~{}~{}~{}~{}\rho_{p}(0)=\frac{3}{4\pi}\frac{Z}{A}\frac{1}{r^{3}_{0_{p}}}~{}.$
(33)
Ngô parameterized the nucleus-nucleus interaction potential in the spirit of
proximity concept. The interaction potential can be divided into the
geometrical factor and a universal function. The nuclear part of the
parameterized potential is written as ngo80 ;
$V_{N}^{Ngo~{}80}\left(r\right)=\overline{R}\Phi\left(r-C_{1}-C_{2}\right)~{}\rm
MeV,$ (34)
where $\overline{R}$ is defined by Eq. (2). Now the nuclear radius $R_{i}$
reads as:
$R_{i}=\frac{NR_{n_{i}}+ZR_{p_{i}}}{A_{i}}~{}~{}~{}~{}~{}(i=1,2).$ (35)
The equivalent sharp radius for protons and neutrons are given as;
$R_{p_{i}}=r_{0_{pi}}A^{1/3}_{i};~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{n_{i}}=r_{0_{ni}}A^{1/3}_{i},$
(36)
with
$r_{0_{pi}}=1.128~{}{\rm fm};~{}r_{0_{ni}}=1.1375+1.875\times
10^{-4}A_{i}~{}\rm fm.$ (37)
The above different radius formulas for the neutrons and protons take isotopic
dependence into account. The universal function $\Phi(s=r-C_{1}-C_{2})$ (in
$\rm MeV/fm$) is noted by
$\Phi\left(s\right)=\left\\{\begin{array}[]{l
r}-33+5.4\left(s-s_{0}\right)^{2},&\mbox{ for $s<s_{0}$ },\\\
-33\exp\left[-\frac{1}{5}\left(s-s_{0}\right)^{2}\right],&\mbox{ for $s\geq
s_{0}$ },\end{array}\right.$ (38)
and $s_{0}=-1.6$ fm. We labeled this potential as Ngô 80.
### II.12 New Denisov Potential (Denisov DP)
Denisov deni02 performed numerical calculations and parametrized the
potential based on 7140 pair within semi-microscopic approximation. In total,
119 spherical or near spherical nuclei along the $\beta$-stability line from
16O to 212Po were taken. The potential is evaluated for any nucleus-nucleus
combinations at 15 distances between ions around the touching point. By using
this database, a simple analytical expression for the nuclear part of the
interaction potential VN(r) between two spherical nuclei is presented as;
$\displaystyle
V_{N}\left(r\right)=-1.989843\frac{R_{1}R_{2}}{R_{1}+R_{2}}\Phi\left(r-R_{1}-R_{2}-2.65\right)$
$\displaystyle\times\left[1+0.003525139\left(\frac{A_{1}}{A_{2}}+\frac{A_{2}}{A_{1}}\right)^{3/2}\right.$
$\displaystyle\left.-0.4113263\left(I_{1}+I_{2}\right)\right],$ (39)
with
$I_{i}=\frac{N_{i}-Z_{i}}{A_{i}}~{}~{}~{}~{}(i=1,2).$ (40)
The effective nuclear radius $R_{i}$ is given as;
$\displaystyle
R_{i}=R_{ip}\left(1-\frac{3.413817}{R^{2}_{ip}}\right)+~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle 1.284589\left(I_{i}-\frac{0.4A_{i}}{A_{i}+200}\right)(i=1,2),$
(41)
where, proton radius $R_{ip}$ is given by Eq. (8) and
$\Phi\left(s=r-R_{1}-R_{2}-2.65\right)$ is given by the following complex
form:
$\Phi(s)=\left\\{\begin{array}[]{l}1-s/0.7881663+1.229218s^{2}-0.2234277s^{3}\\\
-0.1038769s^{4}\\\
-\frac{R_{1}R_{2}}{R_{1}+R_{2}}\left(0.1844935s^{2}+0.07570101s^{3}\right)\\\
+\left(I_{1}+I_{2}\right)\left(0.04470645s^{2}+0.03346870s^{3}\right),\\\
\qquad\qquad\qquad\qquad{\rm for}\quad\quad-5.65\leq s\leq 0,\\\
\left[1-s^{2}[0.05410106\frac{R_{1}R_{2}}{R_{1}+R_{2}}\exp(-\frac{s}{1.760580})\right.\\\
\left.-0.5395420(I_{1}+I_{2})\exp(-\frac{s}{2.424408})]\right]\\\
\times\exp(-\frac{s}{0.7881663}),\\\ \qquad\qquad\qquad\qquad\qquad\qquad{\rm
for}\qquad s\geq 0.\end{array}\right.$ (42)
Here $A_{i}$, $N_{i}$, $Z_{i}$, $R_{i}$, and $R_{ip}$ are, respectively, the
mass number, the number of neutrons, the number of protons, the effective
nuclear radius, and the proton radius of the target and projectile. The above
form of the universal function not only depends on the separation distance s,
but also has complex dependence on the mass as well as on the relative neutron
excess content. The above parametrization is derived for different
combinations of nuclei between 16O and 212Po.
As stated in the subsection II.4, a new radius formula has become available
recently gr09 . We here extend the above potential due to Denisov to include
this radius in its parametrization. This modified new version of the potential
is labeled as Denisov DP id1 . Note that this new implementation was reported
to yield very close agreement (within 1%) with experimental data for symmetric
colliding pairs id1 .
If one looks on the different versions of potentials (Bass 73, Bass77, Bass
80, and CW 76), one notices that although the form of the radius is different,
it is still isospin independent. Further, the corresponding universal
functions are also isospin independent. The newer versions of Winther (BW 91
and AW 95) have incorporated a $\gamma$ similar to the one used in the Prox 77
potential with a slightly different form. Here isospin content is calculated
separately for the target/ projectile. The latest version of Ngô (Ngô 80) has
some isospin dependence in the radius parameter. In most of the above
mentioned potentials, modifications are made either through the surface energy
coefficients or via nuclear radii. Both of these technical parameters can have
sizable effects on the outcome of a reaction id2 .
Using the above sets of models, the nuclear part of the interaction potential
is calculated.
By adding the Coulomb potential to a nuclear part, one can compute the total
potential $V_{T}(r)$ for spherical colliding pairs as
$\displaystyle V_{T}(r)=V_{N}(r)+V_{C}(r),$ (43)
$\displaystyle=V_{N}(r)+\frac{Z_{1}Z_{2}e^{2}}{r}.$ (44)
Since the fusion happens at a distance larger than the touching configuration
of colliding pair, the above form of the Coulomb potential is justified. One
can extract the barrier height $V^{theor}_{B}$ and barrier position
$R^{theor}_{B}$ using the following conditions
$\frac{dV_{T}(r)}{dr}|_{r=R^{theor}_{B}}=0;~{}~{}{\rm{and}}~{}~{}\frac{d^{2}V_{T}(r)}{dr^{2}}|_{r=R^{theor}_{B}}\leq
0.$ (45)
The knowledge of the shape of the potential as well as barrier position and
height, allows one to calculate the fusion cross section at a microscopic
level. To study the fusion cross sections, we shall use the model given by
Wong wg72 . In this formalism, the cross section for complete fusion is given
by
$\sigma_{fus}=\frac{\pi}{k^{2}}\sum_{l=0}^{l_{max}}\left(2l+1\right)T_{l}\left(E_{cm}\right),$
(46)
where $k=\sqrt{\frac{2\mu E}{\hbar^{2}}}$ and here $\mu$ is the reduced mass.
The center-of-mass energy is denoted by $E_{cm}$. In the above formula,
$\l_{max}$ corresponds to the largest partial wave for which a pocket still
exists in the interaction potential and T${}_{\l}\left(E_{cm}\right)$ is the
energy-dependent barrier penetration factor and is given by,
$T_{\l}\left(E_{cm}\right)=\left\\{1+\exp\left[\frac{2\pi}{\hbar\omega_{\l}}\left(V^{theor}_{B_{\l}}-E_{cm}\right)\right]\right\\}^{-1},$
(47)
where $\hbar\omega_{l}$ is the curvature of the inverted parabola. If we
assume that the barrier position and width are independent of $\l$, the fusion
cross section reduces to
$\displaystyle\sigma_{fus}(mb)=\frac{10R^{theor^{2}}_{B}\hbar\omega_{0}}{2E_{cm}}\times~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\ln\left\\{1+\exp\left[\frac{2\pi}{\hbar\omega_{0}}\left(E_{cm}-V^{theor}_{B}\right)\right]\right\\}.$
(48)
For Ecm$>>$V${}^{theor}_{B}$, the above formula reduces to well-known sharp
cut-off formula
$\sigma_{fus}(mb)=10\pi
R^{theor^{2}}_{B}\left(1-\frac{V^{theor}_{B}}{E_{cm}}\right),$ (49)
whereas for Ecm$<<$V${}^{theor}_{B}$, the above formula reduces to
$\sigma_{fus}(mb)=\frac{10R^{theor^{2}}_{B}\hbar\omega_{0}}{2E_{cm}}\exp\left[\frac{2\pi}{\hbar\omega_{0}}\left(E_{cm}-V^{theor}_{B}\right)\right].$
(50)
We used Eq. (48) to calculate the fusion cross sections.
From the above brief discussion, it is clear that the main stress is made on
the surface energy coefficients $\gamma$ and nuclear radii to incorporate the
isospin dependence in the nuclear potential. Definitely, the response of the
isospin dependent potentials will be different for asymmetric nuclei compared
to symmetric nuclei. At intermediate energies, a strong effect was reported
for the asymmetric reactions as well as for the mass dependence of the
reaction rkp .
## III Results and Discussions
The present study is conducted using a variety of the above-mentioned
potentials. In total, 60 asymmetric reactions with compound mass between 29
and 294 (that have been experimentally explored) are taken for the present
study. All nuclei considered here are assumed to be spherical in nature;
however, deformation as well as orientation of the nuclei also affect the
fusion barriers deni07 . For uniform comparison of different models, we
consider all colliding nuclei to be spherical. The lightest reaction taken is
that of 12C + 17O, whereas heaviest one is of 86Kr + 208Pb. The asymmetry
$A_{s}$ of the colliding nuclei varies between 0.02 and 0.23. The other form
of the asymmetry used in the literature is the mass asymmetry $\eta_{A}$ rkp1
; rkp2 . In the present analysis, $\eta_{A}$ varies between 0.0 and 0.97. Note
that the non zero value of $A_{s}$ will involve complex interplay of the
isospin degree of freedom which has strong role at intermediate energies as
well. The variation of $\eta$ alters the physical outcome of a reaction with
$\eta\approx 0.0$ leading to high dense matter and maximum collision volume
whereas a larger value of $\eta\approx 1.0$ will not be able to compress the
matter to higher density rkp .
As stated above, the isospin dependence of the different potentials enters via
surface energy coefficient $\gamma$. In Fig. 1, we display the variation of
$\gamma$ (in $\rm MeV~{}fm^{-2}$) with asymmetry parameter $A_{s}$. Here we
compare three versions of the surface energy coefficient $\gamma$ used in Prox
77, Prox 88, and Prox 00 potentials along with the relation suggested in AW 95
potential. For the present analysis, the mass of the reacting partner is kept
fixed equal to $A_{1}=A_{2}=40$. The $A_{s}$ was increased by increasing the
neutrons and decreasing the protons. For example, ${}^{40}_{20}Ca_{20}$ \+
${}^{40}_{20}Ca_{20}$ has $A_{s}$ = 0.0. For $A_{s}$ = 0.2, we chose the
reaction of ${}^{40}_{16}S_{24}$ \+ ${}^{40}_{16}S_{24}$ whereas for $A_{s}$ =
0.4, the reaction was ${}^{40}_{12}Mg_{28}$ \+ ${}^{40}_{12}Mg_{28}$. In all
cases, the mass of the reacting partner is kept fixed, whereas the ratio
$A_{s}$ is varied by converting the proton into neutrons. At the end of this
series, we have the reaction of ${}^{40}_{10}Ne_{30}$ \+ ${}^{40}_{10}Ne_{30}$
having $A_{s}$ = 0.5. From the figure, we see that the surface energy
coefficient $\gamma$ used in the latest proximity potentials Prox 00/ Prox
00DP as well as in original version Prox 77 is less sensitive toward the
asymmetry and isospin dependence, whereas the one used in the Prox 88
potential has a stronger dependence on the asymmetry of the reacting nuclei.
The coefficient $\gamma$ of AW 95 yields same results like Prox 77. Since
nuclear potential $V_{N}(r)$ depends directly on $\gamma$, one can conclude
that the potentials calculated within Prox 88 and Prox 77 will be far less
attractive for larger asymmetries compared to the one generated using Prox 00.
When colliding nuclei are symmetric (N = Z; $A_{s}$ = 0.0), such dependence
does not play a role. In many studies rkp1 , one finds that neutron excess,
leads to more attraction. In these studies, the total mass of the colliding
pair is not fixed and as a result, this dependence is more of mass dependence
than of isospin dependence.
In Fig. 2, we display the dependence of different nuclear radii on the
asymmetry parameter $A_{s}$. As noted above, this parameter also plays
significant role in nuclear potential and finally in the barrier calculations.
We show the dependence of different forms of nuclear radii used in various
potentials on the asymmetry parameter. We see that, the radius used in the
Prox 77 (also in Prox 88) as well as in Bass versions (i.e., Bass 73, Bass 77,
and Bass 80), and in all versions from Winther (CW 76, BW 91, and AW 95) are
independent of the asymmetry content, whereas, the one used in the Prox 00,
Prox 00DP (and Denisov DP), and Ngô 80 versions depends on the asymmetry
content of the colliding pairs.
From Figs. 1 and 2, we see that both these parameters can lead to significant
change in the nuclear potential and ultimately in the fusion barriers even if
the universal function $\Phi$(s) is kept the same.
In Fig. 3, we display the nuclear part of the interaction potential $V_{N}(r)$
at a distance of $C_{1}+C_{2}+1$ fm for the same sets of the reactions as
depicted in Figs. 1 and 2. In addition, a series of heavier reacting partners
with mass $A_{1}=A_{2}=80$ is also taken. We display four versions of
proximity potential, three versions from Bass and Winther and one each of the
latest versions of Ngô and Denisov each. We see a systematic decrease in the
attractive strength of the potentials with asymmetry content $A_{s}$. The
decrease is stronger for the Prox 88 version compared to Prox 77, Prox 00, and
Prox 00DP. The Bass 73, Bass 77, and Bass 80 versions of the potential are
independent of the asymmetry content. One also notices a very weak dependence
in the Ngô 80 potential. Two of the three versions of Winther potential have
significant dependence on the asymmetry of the reaction. The Winther 1976
potential, however, does not show such dependence due to the absence of
$\gamma$ term in the potential. The Denisov DP potential also shows a linear
decrease in the strength of the potential with asymmetry content. These
variations are stronger for heavier colliding nuclei. This figure shows true
isospin dependence of the nuclear potential as the mass of the colliding
nuclei is kept fixed. All those potentials that do not depend on the asymmetry
parameter $A_{s}$ will not show any change in the structure.
We now shift from the systematic study to the study involving real nuclei. As
stated above, here 60 reactions with $A_{s}$ between 0.02 and 0.23 and
$\eta_{A}$ between 0.0 and 0.97 are taken. For all these reactions,
experimental fusion barriers are known Aguilera95 ; Silva97 ; Aguilera90 ;
Cavallera90 ; Vega90 ; Sonz98 ; Vinod96 ; Trotta01 ; Stefanini06 ; stefanini08
; Padron02 ; tighe93 ; vaz81 ; beck03 ; rath09 ; kolata98 ; Morsad90 ;
trotta2000 ; gomes91 ; newton04 ; Liu05 ; Kolata04 ; Prasad96 ; Sinha01 ;
Szanto90 ; Stefanini02 ; quirz01 ; Stefanini2000 ; Baby2000 ; capurro02 ;
Vandana01 ; Stelson90 ; Mitsuoka07 . In Fig. 4, we display the fusion barrier
heights $V_{B}$ and barrier positions $R_{B}$ versus experimental values for
the above mentioned reactions involving 12 different potentials. For the
clarity of the figure, only 60 asymmetric reactions studied experimentally and
covering the whole range of the mass and asymmetry are displayed. We see no
clear difference with fusion barrier heights and positions. The fusion barrier
heights can be reproduced within $\pm 10\%$ in all cases on the average. Due
to the large uncertainty in the fusion barrier positions, no definite trend
and conclusion can be drawn as is observed for the symmetric colliding nuclei
id1 . To further understand the role of isospin content, we display, in Fig.
5, the percentage difference of the fusion barrier heights $\Delta V_{B}(\%)$
defined as
$\Delta V_{B}~{}(\%)=\frac{V_{B}^{theor}-V_{B}^{expt}}{V_{B}^{expt}}\times
100,$ (51)
verses asymmetry parameter $A_{s}$. In some cases, only the latest versions of
the potential are shown. Interestingly, we see that Prox 77 and Ngô 80 fail to
reproduce the barrier heights satisfactorily, whereas Prox 88, Bass 80, AW 95,
Prox 00DP, and Denisov DP do a far better job compared to other potentials. We
do not see any systematic deviation/improvement in the fusion barrier heights
with the asymmetry of the colliding nuclei. We see that the potentials Prox
88, Bass 80, AW 95, and Denisov DP can reproduce the empirical barrier heights
within $\pm 5\%$ (see the shaded regions in Fig. 5), whereas others need $\pm
10\%$ to produce the same result.
The comparison of the fusion barrier positions outcome is shown in Fig. 6. We
see that due to large uncertainty in the measurements of fusion barrier
positions, a large deviation is seen and all the models are able to reproduce
the results within $\pm 10\%$. The precise values of the fusion barrier
heights $V_{B}$ (in MeV) and positions $R_{B}$ (in fm) are shown in the Tables
1 and 2, for 60 asymmetric colliding nuclei involving significant variations
of asymmetry $A_{s}$ as well as mass asymmetry $\eta_{A}$. The experimental
(or empirical) barriers displayed in Tables 1 and 2 and in Figs. 4 - 6 are
obtained by fitting the cross sections in the approach, when shapes of both
colliding nuclei are spherical. A large number of experimental data are
available for different reactions; however, we restrict ourselves to the
latest one only.
In Figs. 7 and 8, we display the fusion cross-sections $\sigma_{fus}$ (in mb)
as a function of center-of-mass energy $E_{cm}$ for the reactions of 48Ca +
96Zr Stefanini06 , 28Si + 92Zr newton01 , 12C + 92Zr newton01 , 16O + 208Pb
Morton01 (in Fig. 7) and 16O + 50Ti Neto90 , 16O + 112Sn Vandana01 , 16O +
116Sn Vandana01 , and 16O + 120Sn Baby2000 (in Fig. 8). Here the latest
versions of proximity parametrizations along with original proximity potential
and its modifications are shown for clarity. As we see, Bass 80, Denisov DP,
and AW 95 do a better job for all the systems, whereas Prox 77 and Ngô 80 fail
to come closer to the experimental data. The above results are in agreement
with the one obtained for symmetric colliding nuclei id1 .
## IV Summary
We performed a systematic study of the role of isospin dependence on fusion
barriers by employing as many as 12 different proximity-based potentials. Some
of the potentials have isospin dependence via the surface energy coefficient
as well as via nuclear radius. We noted that the nuclear part of the potential
becomes more shallow with asymmetry of the reaction. On the other hand, a
detailed comparison of different potentials does not show any preference for
the isospin-dependent potential. Our comparison for 60 reactions reveals that
all models can explain the fusion barrier heights within $\pm 10\%$. The
potentials from Prox 88, Bass 80, AW 95, and Denisov DP perform better than
others. The fusion cross sections are nicely explained by Bass 80, AW 95, and
Denisov DP potentials at below as well as above barrier energies.
This work was supported by a research grant from the Department of Atomic
Energy, Government of India.
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Figure 1: (Color online) The variation of the surface energy coefficient
$\gamma$ $(\rm MeV~{}fm^{-2})$ with asymmetry parameter $A_{s}$. We display
the results using $\gamma$ from Prox 77, Prox 88, Prox 00, and AW 95 for
masses of reacting partner $A_{1}$ = $A_{2}$ = 40 units. Figure 2: (Color
online) Same as Fig. 1, but for various radii used in the literature. Figure
3: (Color online) The strength of the nuclear potential $\rm V_{N}~{}(MeV)$
calculated at a distance equal to $C_{1}+C_{2}+1$ fm as a function of
asymmetry parameter $A_{s}$ for the reacting partners having masses $A_{1}$ =
$A_{2}$ = 40 and $A_{1}$ = $A_{2}$ = 80 units. Here $C_{i}$ denotes the
central radius id1 . The dotted lines denote the value of the potential at
$A_{s}$ = 0.0 (for $A_{1}$ = $A_{2}$ = 40 only) using proximity potentials.
Figure 4: The theoretical fusion barrier heights $V_{B}$ (MeV) and positions
$R_{B}$ (fm) are displayed as a function of experimentally extracted values.
The shaded area represents the region within which all 12 proximity potentials
are able to reproduce experimental data. Figure 5: The percentage difference
$\Delta V_{B}(\%)$ of theoretical fusion barrier heights over experimental one
as a function of asymmetry parameter $A_{s}$. Here only 60 reactions covering
the whole mass and asymmetry range are taken. The shaded area is marked only
for those potentials where the deviation is within $\pm 5\%$. Figure 6: Same
as Fig. 5, but for percentage difference $\Delta R_{B}(\%)$. Figure 7: (Color
online) The fusion cross sections $\sigma_{fus}$ (mb) as a function of center-
of-mass energy $E_{c.m.}~{}\rm(MeV)$. For the clarity, only latest versions of
different proximity potentials are shown. The experimental data are taken from
Stefanini 2006 Stefanini06 , Newton 2001 newton01 , and Morton 1999 Morton01 .
Figure 8: (Color online) Same as Fig. 7, but for different systems explained
in the text. The experimental data are taken from Neto 1990 Neto90 , Tripathi
2001 Vandana01 , and Baby 2000 Baby2000 .
Table 1: The fusion barrier heights VB (in MeV) and positions RB (in fm) using different proximity potentials for 60 asymmetric systems. The corresponding experimental values are also listed. Reaction | Prox 77 | Prox 88 | Prox 00 | Prox 00DP | Empirical |
---|---|---|---|---|---|---
| VB | RB | VB | RB | VB | RB | VB | RB | VB | RB | $Ref.$
7Li + 27Al | 6.52 | 7.78 | 6.34 | 8.03 | 6.80 | 7.45 | 6.34 | 8.08 | 7.38 | 7.36 | Padron02
12C + 17O | 8.22 | 7.56 | 7.98 | 7.81 | 8.46 | 7.39 | 7.93 | 7.92 | 8.20 | 7.76 | tighe93
11B + 27Al | 10.68 | 7.94 | 10.39 | 8.19 | 11.09 | 7.64 | 10.62 | 8.05 | 11.20 | 7.69 | Padron02
6Li + 59Co | 12.64 | 8.41 | 12.31 | 8.66 | 12.58 | 8.49 | 11.78 | 9.14 | 12.00 | 7.60 | beck03
4He + 164Dy | 17.71 | 9.90 | 17.36 | 10.15 | 17.36 | 10.20 | 16.01 | 11.09 | 17.14 | 10.32 | vaz81
4He + 209Bi | 21.30 | 10.44 | 20.89 | 10.64 | 20.63 | 10.81 | 19.20 | 11.70 | 20.98 | 10.04 |
| | | | | | | | | $\pm$0.05 | $\pm$0.01 | kolata98
26Mg + 30Si | 25.61 | 8.64 | 24.97 | 8.89 | 25.05 | 8.86 | 24.71 | 9.01 | 24.80 | 9.05 | Morsad90
6He + 238U | 22.06 | 11.22 | 21.69 | 11.42 | 22.56 | 10.97 | 21.21 | 11.74 | 20.28 | 12.50 | trotta2000
6Li + 144Sm | 25.26 | 9.80 | 24.72 | 10.05 | 25.18 | 9.85 | 23.69 | 10.53 | 24.65 | 10.20 | rath09
14N + 59Co | 28.19 | 8.83 | 27.50 | 9.08 | 28.13 | 8.87 | 27.37 | 9.16 | 26.13 | 9.60 | gomes91
7Li + 159Tb | 25.50 | 10.20 | 25.00 | 10.45 | 26.76 | 10.15 | 24.32 | 10.77 | 23.81 | 11.03 | vaz81
24Mg + 35Cl | 31.18 | 8.60 | 30.39 | 8.85 | 30.36 | 8.90 | 30.04 | 8.98 | 30.70 | 8.84 | Cavallera90
16O + 58Ni | 33.32 | 8.85 | 32.51 | 9.10 | 33.52 | 8.82 | 32.72 | 9.09 | 31.67 | 9.30 | newton04
18O + 64Ni | 32.08 | 9.25 | 31.35 | 9.50 | 32.32 | 9.20 | 31.58 | 9.42 | 32.50 | 9.04 | Silva97
12C + 92Zr | 33.88 | 9.38 | 33.12 | 9.63 | 33.98 | 9.37 | 32.78 | 9.79 | 32.31 | 9.68 | newton04
6Li + 208Pb | 31.17 | 10.57 | 30.59 | 10.77 | 31.11 | 10.60 | 29.49 | 11.25 | 30.10 | 11.00 | Liu05
16O + 72Ge | 36.79 | 9.22 | 35.94 | 9.42 | 36.80 | 9.23 | 35.96 | 9.45 | 35.40 | 9.70 | Aguilera95
36S + 48Ca | 44.63 | 9.51 | 43.65 | 9.76 | 44.67 | 9.55 | 43.70 | 9.78 | 43.30 | | stefanini08
10Be + 209Bi | 40.50 | 11.02 | 39.78 | 11.22 | 40.59 | 10.99 | 39.11 | 11.44 | 37.60 | 13.50 | Kolata04
19F + 93Nb | 50.34 | 9.74 | 49.24 | 9.99 | 49.27 | 10.02 | 49.27 | 10.02 | 46.60 | 9.20 |
| | | | | | | | | $\pm$0.10 | $\pm$0.10 | Prasad96
12C + 152Sm | 48.37 | 10.28 | 47.41 | 10.48 | 48.98 | 10.17 | 47.60 | 10.49 | 46.39 | 10.77 | vaz81
16O + 116Sn | 53.56 | 9.94 | 52.43 | 10.19 | 53.48 | 10.01 | 52.35 | 10.23 | 50.94 | 10.36 | Vandana01
18O + 124Sn | 51.99 | 10.27 | 50.97 | 10.52 | 51.89 | 10.33 | 50.81 | 10.55 | 49.30 | 10.98 | Sinha01
48Ca + 48Ca | 53.96 | 9.89 | 52.84 | 10.09 | 53.93 | 9.89 | 52.86 | 10.11 | 51.70 | 10.38 | Trotta01
27Al + 70Ge | 57.62 | 9.59 | 56.34 | 9.84 | 57.74 | 9.58 | 57.74 | 9.58 | 55.10 | 10.20 | Aguilera90
40Ca + 48Ti | 61.67 | 9.46 | 60.27 | 9.71 | 60.71 | 9.64 | 60.71 | 9.64 | 58.17 | 9.97 |
| | | | | | | | | $\pm$0.62 | $\pm$0.07 | Sonz98
35Cl + 54Fe | 62.04 | 9.46 | 60.62 | 9.71 | 60.85 | 9.66 | 60.27 | 9.79 | 58.59 | 10.14 | Szanto90
37Cl + 64Ni | 64.41 | 9.82 | 63.03 | 10.07 | 64.02 | 9.91 | 63.37 | 10.05 | 60.60 | 10.59 | Vega90
46Ti + 46Ti | 67.15 | 9.56 | 65.64 | 9.81 | 66.34 | 9.70 | 65.38 | 9.87 | 63.30 | 10.27 | Stefanini02
12C + 204Pb | 60.73 | 10.84 | 59.61 | 11.09 | 60.96 | 10.85 | 59.08 | 11.22 | 57.55 | 11.34 | newton04
16O + 144Sm | 64.16 | 10.31 | 62.86 | 10.56 | 64.01 | 10.38 | 62.47 | 10.63 | 61.03 | 10.85 | newton04
40Ar + 58Ni | 68.84 | 9.72 | 67.33 | 9.97 | 67.93 | 9.92 | 67.93 | 9.92 | 66.32 | 10.16 | vaz81
37Cl + 73Ge | 72.43 | 10.00 | 70.91 | 10.25 | 71.88 | 10.11 | 70.74 | 10.30 | 69.20 | 10.60 | quirz01
28Si + 92Zr | 74.52 | 10.00 | 72.95 | 10.25 | 72.72 | 10.30 | 72.35 | 10.34 | 70.93 | 10.19 | newton04
16O + 186W | 73.09 | 10.86 | 71.74 | 11.06 | 71.39 | 11.18 | 70.03 | 11.40 | 68.87 | 11.12 | newton04
48Ti + 58Ni | 82.70 | 9.89 | 80.91 | 10.14 | 81.34 | 10.13 | 81.34 | 10.13 | 78.80 | 9.80 |
| | | | | | | | | $\pm$0.30 | $\pm$0.30 | Vinod96
32S + 89Y | 82.52 | 10.06 | 80.78 | 10.31 | 81.38 | 10.23 | 80.62 | 10.36 | 77.77 | 10.30 | newton04
36S + 90Zr | 82.99 | 10.30 | 81.30 | 10.55 | 82.35 | 10.41 | 81.10 | 10.60 | 79.00 | 10.64 | Stefanini2000
16O + 208Pb | 79.38 | 11.09 | 77.96 | 11.29 | 79.30 | 11.13 | 77.78 | 11.35 | 74.90 | 11.76 | Liu05
35Cl + 92Zr | 88.58 | 10.25 | 86.75 | 10.50 | 87.64 | 10.39 | 86.41 | 10.56 | 82.94 | 10.20 | newton04
28Si + 120Sn | 89.43 | 10.49 | 87.65 | 10.69 | 88.12 | 10.65 | 88.12 | 10.65 | 85.89 | 11.04 | Baby2000
19F + 197Au | 85.70 | 11.15 | 84.16 | 11.35 | 85.33 | 11.20 | 85.33 | 11.20 | 81.61 | 11.32 | newton04
16O + 238U | 86.86 | 11.39 | 85.37 | 11.59 | 87.46 | 11.30 | 85.81 | 11.56 | 80.81 | 11.45 | newton04
35Cl + 106Pd | 99.86 | 10.48 | 97.85 | 10.68 | 98.75 | 10.62 | 97.45 | 10.74 | 94.30 | 11.27 | capurro02
58Ni + 60Ni | 102.83 | 10.16 | 100.67 | 10.41 | 102.07 | 10.26 | 102.07 | 10.26 | 96.00 | 10.26 | newton04
32S + 116Sn | 101.78 | 10.49 | 99.75 | 10.74 | 100.65 | 10.64 | 99.73 | 10.76 | 97.36 | 10.80 | Vandana01
40Ca + 90Zr | 103.60 | 10.30 | 101.46 | 10.55 | 102.57 | 10.43 | 102.10 | 10.48 | 96.88 | 10.53 | newton04
48Ca + 96Zr | 99.33 | 10.80 | 97.46 | 11.00 | 98.73 | 10.90 | 97.28 | 11.04 | 95.90 | 11.21 | Stefanini06
28Si + 144Sm | 108.00 | 10.78 | 105.90 | 10.98 | 105.40 | 11.04 | 105.03 | 11.13 | 103.89 | 10.93 | newton04
50Ti + 93Nb | 112.74 | 10.71 | 110.54 | 10.96 | 111.25 | 10.87 | 110.38 | 10.99 | 106.90 | | Stelson90
40Ca + 124Sn | 123.11 | 10.90 | 120.78 | 11.10 | 121.55 | 11.01 | 121.55 | 11.01 | 112.93 | 10.08 | newton04
28Si + 208Pb | 133.90 | 11.56 | 131.59 | 11.76 | 131.10 | 11.79 | 131.10 | 11.79 | 128.07 | 11.45 | newton04
TABLE 1:-(continued).
Reaction Prox 77 Prox 88 Prox 00 Prox 00DP Empirical VB RB VB RB VB RB VB RB
VB RB $Ref.$ 40Ar + 165Ho 141.27 11.49 138.78 11.69 138.61 11.71 138.61 11.71
141.38 11.48 vaz81 32S + 232Th 163.08 11.92 160.39 12.12 162.32 11.94 160.97
12.02 155.73 11.18 newton04 40Ca + 192Os 174.70 11.71 171.71 11.96 173.90
11.74 173.07 11.79 168.07 11.05 newton04 48Ti + 208Pb 200.34 12.18 197.08
12.38 197.08 12.34 197.08 12.34 190.10 Mitsuoka07 56Fe + 208Pb 233.61 12.33
229.84 12.58 229.74 12.45 229.74 12.45 223.00 Mitsuoka07 64Ni + 208Pb 247.56
12.56 243.66 12.76 245.68 12.53 245.68 12.53 236.00 Mitsuoka07 70Zn + 208Pb
262.60 12.71 258.53 12.91 259.01 12.76 259.01 12.76 250.60 Mitsuoka07 86Kr +
208Pb 308.05 12.99 303.40 13.24 306.16 12.92 304.56 12.98 299.20 Mitsuoka07
Table 2: Fusion barrier heights VB (in MeV) and positions RB (in fm) are displayed using other different proximity potentials for 60 asymmetric systems. The limited numbers of reactions in certain cases are due to the restriction posed in different potentials. Reaction | Bass 80 | Ngo 8̂0 | AW 95 | Denisov DP
---|---|---|---|---
| VB | RB | VB | RB | VB | RB | VB | RB
7Li + 27Al | 6.20 | 8.35 | - | - | 6.31 | 8.27 | - | -
12C + 17O | 7.79 | 8.13 | - | - | 7.89 | 8.10 | - | -
11B + 27Al | 10.13 | 8.50 | - | - | 10.24 | 8.49 | - | -
6Li + 59Co | 12.00 | 8.97 | - | - | 12.14 | 8.97 | - | -
4He + 164Dy | 16.87 | 10.51 | - | - | 17.12 | 10.44 | - | -
4He + 209Bi | 20.30 | 11.00 | - | - | 20.62 | 10.95 | - | -
26Mg + 30Si | 24.33 | 9.20 | 25.65 | 8.76 | 24.42 | 9.20 | 23.84 | 9.29
6He + 238U | 21.10 | 11.83 | - | - | 21.60 | 11.59 | - | -
6Li + 144Sm | 24.08 | 10.36 | - | - | 24.34 | 10.34 | - | -
14N + 59Co | 26.79 | 9.40 | - | - | 26.90 | 9.43 | - | -
7Li + 159Tb | 24.33 | 10.76 | - | - | 24.67 | 10.72 | - | -
24Mg + 35Cl | 29.61 | 9.16 | 31.19 | 8.72 | 29.67 | 9.21 | 29.21 | 9.23
16O + 58Ni | 31.69 | 9.41 | 33.42 | 8.94 | 31.78 | 9.44 | 31.14 | 9.50
18O + 64Ni | 30.53 | 9.81 | 32.18 | 9.33 | 30.70 | 9.76 | 29.91 | 9.93
12C + 92Zr | 32.26 | 9.94 | - | - | 32.43 | 9.93 | - | -
6Li + 208Pb | 29.72 | 11.14 | - | - | 30.08 | 11.11 | - | -
16O + 72Ge | 35.02 | 9.73 | 36.92 | 9.29 | 35.14 | 9.79 | 34.46 | 9.83
36S + 48Ca | 42.48 | 10.07 | 44.69 | 9.59 | 42.69 | 10.04 | 42.11 | 10.09
10Be + 209Bi | 38.70 | 11.59 | - | - | 39.29 | 11.48 | - | -
19F + 93Nb | 48.01 | 10.25 | 50.57 | 9.78 | 48.24 | 10.26 | 47.56 | 10.32
12C + 152Sm | 46.13 | 10.79 | - | - | 46.45 | 10.82 | - | -
16O + 116Sn | 51.11 | 10.45 | 53.85 | 9.97 | 51.36 | 10.50 | 50.61 | 10.55
18O + 124Sn | 49.57 | 10.83 | 52.18 | 10.34 | 49.98 | 10.80 | 49.04 | 10.93
48Ca + 48Ca | 51.39 | 10.40 | 54.06 | 9.94 | 51.74 | 10.39 | 51.13 | 10.42
27Al + 70Ge | 54.97 | 10.11 | 57.86 | 9.60 | 55.13 | 10.12 | 54.77 | 10.09
40Ca + 48Ti | 58.83 | 9.97 | 61.90 | 9.47 | 58.91 | 9.99 | 58.76 | 9.93
35Cl + 54Fe | 59.18 | 9.92 | 62.28 | 9.47 | 59.28 | 9.98 | 59.11 | 9.92
37Cl + 64Ni | 61.47 | 10.33 | 64.67 | 9.87 | 61.71 | 10.37 | 61.37 | 10.33
46Ti + 46Ti | 64.10 | 10.07 | 67.45 | 9.56 | 64.21 | 10.07 | 64.09 | 10.02
12C + 204Pb | 58.04 | 11.40 | 57.86 | 9.60 | 58.53 | 11.38 | 55.13 | 10.12
16O + 144Sm | 61.34 | 10.82 | 64.59 | 10.31 | 61.68 | 10.83 | 60.85 | 10.88
40Ar + 58Ni | 65.75 | 10.23 | 69.19 | 9.71 | 65.91 | 10.22 | 65.71 | 10.20
37Cl + 73Ge | 69.19 | 10.51 | 72.81 | 9.98 | 69.48 | 10.48 | 69.21 | 10.49
28Si + 92Zr | 71.21 | 10.51 | 74.96 | 9.97 | 71.44 | 10.53 | 71.32 | 10.45
16O + 186W | 69.86 | 11.37 | 73.44 | 10.85 | 70.36 | 11.34 | 69.47 | 11.45
48Ti + 58Ni | 79.08 | 10.35 | 83.24 | 9.87 | 79.28 | 10.43 | 79.28 | 10.35
32S + 89Y | 78.91 | 10.52 | 83.07 | 10.03 | 79.15 | 10.59 | 79.18 | 10.50
36S + 90Zr | 79.40 | 10.76 | 83.56 | 10.26 | 79.82 | 10.76 | 79.54 | 10.73
16O + 208Pb | 75.92 | 11.60 | 79.76 | 11.07 | 76.52 | 11.60 | 75.55 | 11.66
35Cl + 92Zr | 84.76 | 10.71 | 89.22 | 10.16 | 85.09 | 10.71 | 85.11 | 10.65
28Si + 120Sn | 85.56 | 10.95 | 90.04 | 10.39 | 85.94 | 10.93 | 85.98 | 10.86
19F + 197Au | 82.04 | 11.66 | 86.19 | 11.07 | 82.76 | 11.60 | 81.92 | 11.67
16O + 238U | 83.12 | 11.90 | 87.20 | 11.37 | 83.85 | 11.88 | 82.77 | 11.98
35Cl + 106Pd | 95.67 | 10.89 | 100.71 | 10.38 | 96.09 | 10.92 | 96.24 | 10.81
58Ni + 60Ni | 98.53 | 10.56 | 103.77 | 10.02 | 98.80 | 10.61 | 99.09 | 10.53
32S + 116Sn | 97.53 | 10.95 | 102.68 | 10.39 | 97.93 | 10.98 | 98.18 | 10.85
40Ca + 90Zr | 99.28 | 10.71 | 104.55 | 10.15 | 99.58 | 10.75 | 99.93 | 10.66
48Ca + 96Zr | 95.05 | 11.26 | 100.03 | 10.74 | 95.87 | 11.23 | 95.61 | 11.19
28Si + 144Sm | 103.60 | 11.19 | 109.04 | 10.61 | 104.12 | 11.20 | 104.31 | 11.12
50Ti + 93Nb | 108.10 | 11.17 | 113.83 | 10.59 | 108.77 | 11.13 | 108.87 | 11.06
40Ca + 124Sn | 118.07 | 11.31 | 124.29 | 10.73 | 118.66 | 11.31 | 119.36 | 11.14
28Si + 208Pb | 128.56 | 11.97 | 135.03 | 11.37 | 129.53 | 11.92 | 130.06 | 11.79
40Ar + 165Ho | 135.70 | 11.90 | 142.75 | 11.29 | 136.97 | 11.86 | 137.38 | 11.73
32S + 232Th | 156.86 | 12.28 | 164.65 | 11.67 | 158.17 | 12.25 | - | -
40Ca + 192Os | 168.22 | 12.07 | 176.93 | 11.45 | 169.44 | 12.05 | 171.15 | 11.82
48Ti + 208Pb | 193.15 | 12.49 | 203.09 | 11.86 | 195.26 | 12.44 | 196.99 | 12.15
56Fe + 208Pb | 225.72 | 12.59 | 237.53 | 11.89 | 228.16 | 12.52 | 230.95 | 12.18
64Ni + 208Pb | 239.28 | 12.81 | 251.83 | 12.10 | 242.40 | 12.68 | 245.24 | 12.31
70Zn + 208Pb | 253.99 | 12.92 | 267.37 | 12.20 | 257.54 | 12.78 | 260.75 | 12.37
86Kr + 208Pb | 298.65 | 13.15 | - | - | 303.25 | 12.98 | 308.13 | 12.32
|
arxiv-papers
| 2010-05-28T04:32:55 |
2024-09-04T02:49:10.684703
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ishwar Dutt and Rajeev K. Puri",
"submitter": "Rajeev Kumar Puri",
"url": "https://arxiv.org/abs/1005.5213"
}
|
1005.5214
|
# Analytical parametrization of fusion barriers using proximity potentials
Ishwar Dutt Rajeev K. Puri rkpuri@pu.ac.in;drrkpuri@gmail.com Department of
Physics, Panjab University, Chandigarh 160014, India
###### Abstract
Using the three versions of proximity potentials, namely proximity 1977,
proximity 1988, and proximity 2000, we present a pocket formula for fusion
barrier heights and positions. This was achieved by analyzing as many as 400
reactions with mass between 15 and 296. Our parametrized formula can
reproduced the exact barrier heights and positions within an accuracy of $\pm
1\%$. A comparison with the experimental data is also in good agreement.
###### pacs:
24.10.-i, 25.70.Jj, 25.70.-z.
## I Introduction
In the low energy heavy-ion collisions, fusion of colliding nuclei and related
phenomena has always been of central interestrkp1 . Depending upon the
incident energy of the projectile as well as angular momentum and impact
parameter, the collision of nuclei can lead to several interesting phenomena
such as incomplete fusion rkp1 , multifragmentation rkp2 ; rkp3 , subthreshold
particle production rkp4 , nuclear flow rkp5 as well as formation of the
superheavy elements rkp6 . Since fusion is a low density phenomenon, several
mean field models rkp1 ; rkp6 ; blocki77 ; wr94 ; ms2000 ; wang06 ; deni02
have been developed in the recent past at microscopic/macroscopic level and
have been robust against the vast experimental data ms2000 ; wang06 ; expt
that range from symmetric to highly asymmetric colliding nuclei. The study of
mass dependence has always guided the validity of various models irrespective
of the energy range. The essential idea of developing a model is to understand
the physical mechanism behind a process or phenomenon. Extension of the
physics is also reported toward isospin degree of freedom. At the same time,
accumulation of huge experimental data ms2000 ; wang06 ; expt (that include
all kinds of masses and asymmetry of colliding nuclei) puts stringent test for
any theoretical model.
As fusion process occurs at the surface of colliding nuclei, any difference
occurring in the interior part of the potential does not make any difference
toward the fusion. One always tries to parametrize the potential in terms of
some known quantities such as the masses and charges of colliding nuclei rkp1
; deni02 ; bass73 ; ngo75 . At intermediate energies, several forms of density
dependent potentials are also available rkp2 ; rkp3 ; rkp4 ; rkp5 . Generally,
the benchmark is to parameterized the outcome in proximity fashion blocki77 .
By adding the Coulomb potential to the parameterized form of the nuclear ion-
ion potential, one obtains total ion-ion potential and ultimately, the fusion
barriers and cross sections.
Alternatively, one calculates the barrier heights as well as positions of
large number of reactions and then tries to parametrize these in terms of some
known quantities like the charges and masses of the colliding nuclei rkp1 ;
skg82 . Recently, even neutron excess dependence has also been incorporated in
some attempts ng04 . Similarly, an analytical expression to determine the
barrier heights and positions are also presented in Ref. rm01 . The cost of
such attempts was in the form of more complicated parametrized form. The
utility of such direct parametrization is that one can use these pocket
formula to find out the fusion barriers instantaneously.
As is evident from the literature, several modifications over the original
proximity potential have also been suggested in the recent years wr94 ; ms2000
. We shall here attempt to present a direct parametrization of the fusion
barrier positions as well as heights using different proximity potentials.
This attempt will introduce great simplification in obtaining the fusion
barrier positions and heights. Section II describes the models in brief, Sec.
III depicts the results, and a summary is presented in Sec. IV.
## II The Model
All proximity potentials are based on the proximity force theorem. According
to which, _“the force between two gently curved surfaces in close proximity is
proportional to the interaction potential per unit area between the two flat
surfaces”_. The nuclear part of the interaction potential in different
proximity potentials is described as a product of geometrical factor
representing the mean curvature of the interacting surfaces and an universal
function depending on the separation distance.
### II.1 $\rm Proximity~{}1977~{}(Prox~{}77)$
According to the original version of proximity blocki77 , the interaction
potential $V_{N}(r)$ between two surfaces can be written as
$V_{N}^{Prox~{}77}(r)=4\pi\gamma
b\overline{R}\Phi\left(\frac{{r}-C_{1}-C_{2}}{b}\right){~{}\rm MeV},$ (1)
where the surface energy coefficient $\gamma$ taken from the Lysekil mass
formula $(~{}\rm in~{}MeV/fm^{2})$ is written as
$\gamma=\gamma_{0}\left[1-k_{s}I^{2}\right],$ (2)
with $I=\left(\frac{N-Z}{A}\right)$; $N$, $Z$, and $A$ refer to the neutron,
proton and total mass of two interacting nuclei. Though the proximity
potential Prox 77, in principle, is for zero-neutron excess, the factor
$\gamma$ takes care of some neutron excess content. In the above formula,
$\gamma_{0}$ is the surface energy constant and $k_{s}$ is the surface-
asymmetry constant. Both constants were first parametrized by Myers and
Świa̧tecki ms66 by fitting the experimental binding energies. The first set
of these constants yielded values $\gamma_{0}$ and
$k_{s}=1.01734~{}\rm~{}MeV/fm^{2}$ and 1.79, respectively. Later on, these
values were revised to ${~{}\rm\gamma_{0}}$ = 0.9517 $~{}\rm MeV/fm^{2}$ and
$k_{s}=1.7826$ ms67 . Interestingly, most of the modified proximity type
potentials use different values of the parameter $\gamma$ wr94 ; ms2000 . The
mean curvature radius, $\overline{R}$ in Eq. (1) has the form
$\overline{R}=\frac{C_{1}C_{2}}{C_{1}+C_{2}},$ (3)
quite similar to the one used for reduced mass. Here
$C_{i}=R_{i}\left[1-\left(\frac{b}{R_{i}}\right)^{2}+\cdots\cdots\right],$ (4)
${\rm R_{i}}$, the effective sharp radius, reads as
$R_{i}=1.28A^{1/3}_{i}-0.76+0.8A^{-1/3}_{i}{~{}\rm fm}~{}~{}(i=1,2).$ (5)
The universal function $\Phi\left(\xi\right)$ was parametrized with the
following form:
$\Phi\left(\xi\right)=\left\\{\begin{array}[]{l}-\frac{1}{2}\left(\xi-2.54\right)^{2}-0.0852\left(\xi-2.54\right)^{3},\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{
for $\xi\leq 1.2511$ },\\\ -3.437\exp\left(-\frac{\xi}{0.75}\right),\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{
for $\xi\geq 1.2511$ },\end{array}\right.$ (6)
with $\xi$ = $(r-C_{1}-C_{2}$)/$b$. The width $b$ has been evaluated close to
unity. Using the above form, one can calculate the nuclear part of the
interaction potential ${V_{N}(r)}$. This model is referred as Prox 77 and
corresponding potential as $V_{N}^{Prox~{}77}(r)$.
### II.2 $\rm Proximity~{}1988~{}(Prox~{}88)$
Later on, using the more refined mass formula of Möller and Nix mn81 , the
value of coefficients $\gamma_{0}$ and $k_{s}$ were modified yielding their
values =1.2496 $\rm MeV/fm^{2}$ and 2.3, respectively. Reisdorf wr94 labeled
this modified version as ‘Proximity 1988’. Note that this set of coefficients
give stronger attraction compared to the above sets. Even a more recent
compilation by Möller and Nix mn95 yields similar values. We marked this
potential as Prox 88.
### II.3 $\rm Proximity~{}2000~{}(Prox~{}00)$
Recently, Myers and Świa̧tecki ms2000 modified Eq. (1) by using up-to-date
knowledge of nuclear radii and surface tension coefficients using their
droplet model concept. The prime aim behind this attempt was to remove
discrepancy of the order of $4\%$ reported between the results of Prox 77 and
experimental data ms2000 . Using the droplet model ms80 , matter radius
$C_{i}$ was calculated as
$C_{i}=c_{i}+\frac{N_{i}}{A_{i}}t_{i}~{}~{}~{}~{}(i=1,2),$ (7)
where $c_{i}$ denotes the half-density radii of the charge distribution and
$t_{i}$ is the neutron skin of the nucleus. To calculate $c_{i}$, these
authors ms2000 used two-parameter Fermi function values given in Ref. dv87
and remaining cases were handled with the help of parametrization of charge
distribution described below. The nuclear charge radius (denoted as $R_{00}$
in Ref. bn94 ), is given by the relation:
$R_{00i}=\sqrt{\frac{5}{3}}\left<r^{2}\right>^{1/2}$
$\displaystyle=1.240A_{i}^{1/3}\left\\{1+\frac{1.646}{A_{i}}-0.191\left(\frac{A_{i}-2Z_{i}}{A_{i}}\right)\right\\}{~{}\rm
fm}~{}~{}$ $\displaystyle(i=1,2),$ (8)
where $<r^{2}>$ represents the mean square nuclear charge radius. According to
Ref. bn94 , Eq. (8) was valid for the even-even nuclei with $8\leq Z<38$ only.
For nuclei with $Z\geq 38$, the above equation was modified by Pomorski _et
al_. bn94 as
$R_{00i}=1.256A_{i}^{1/3}\left\\{1-0.202\left(\frac{A_{i}-2Z_{i}}{A_{i}}\right)\right\\}{~{}\rm
fm}.$ (9)
These expressions give good estimate of the measured mean square nuclear
charge radius $<r^{2}>$. In the present model, authors used only Eq. (8). The
half-density radius, $c_{i}$ was obtained from the relation:
$c_{i}=R_{00i}\left(1-\frac{7}{2}\frac{b^{2}}{R_{00i}^{2}}-\frac{49}{8}\frac{b^{4}}{R_{00i}^{4}}+\cdots\right)~{}~{}~{}~{}~{}~{}~{}(i=1,2).$
(10)
Using the droplet model ms80 , neutron skin $t_{i}$ reads as
$t_{i}=\frac{3}{2}r_{0}\left[\frac{JI_{i}-\frac{1}{12}c_{1}Z_{i}A^{-1/3}_{i}}{Q+\frac{9}{4}JA^{-1/3}_{i}}\right](i=1,2).$
(11)
Here $r_{0}$ is $1.14$ fm, the value of nuclear symmetric energy coefficient
$J=32.65$ MeV and $c_{1}=3e^{2}/5r_{0}=0.757895$ MeV. The neutron skin
stiffness coefficient $Q$ was taken to be 35.4 MeV. The nuclear surface energy
coefficient $\gamma$ in terms of neutron skin was given as;
$\gamma=\frac{1}{4\pi
r^{2}_{0}}\left[18.63{\rm(MeV)}-Q\frac{\left(t^{2}_{1}+t^{2}_{2}\right)}{2r^{2}_{0}}\right],$
(12)
where $t_{1}$ and $t_{2}$ were calculated using Eq. (11). The universal
function $\Phi(\xi)$ is reported as
$\Phi\left(\xi\right)=\left\\{\begin{array}[]{ll}-0.1353+\sum\limits_{n=0}^{5}\left[c_{n}/\left(n+1\right)\right]\left(2.5-\xi\right)^{n+1},\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{
for \quad$0<\xi\leq 2.5$},\\\
-0.09551\exp\left[\left(2.75-\xi\right)/0.7176\right],\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{
for $\quad\xi\geq 2.5$}.\end{array}\right.$ (13)
The values of different constants $c_{n}$ were: $c_{0}=-0.1886$,
$c_{1}=-0.2628$, $c_{2}=-0.15216$, $c_{3}=-0.04562$, $c_{4}=0.069136$, and
$c_{5}=-0.011454$. For $\xi>2.74$, the above exponential expression is the
exact representation of the Thomas-Fermi extension of the proximity potential.
This potential is marked as Prox 00.
## III Results and Discussion
As a first step, we calculated the nuclear part of the ion-ion potential using
Prox 77, Prox 88, and Prox 00 potentials and then by adding the Coulomb
potential (= $\frac{Z_{1}Z_{2}e^{2}}{r}$), total ion-ion potential $V_{T}(r)$
for spherical colliding pair is obtained. The fusion barrier is then extracted
using conditions
$\frac{dV_{T}(r)}{dr}|_{r=R_{B}}=0,~{}~{}{\rm{and}}~{}~{}\frac{d^{2}V_{T}(r)}{dr^{2}}|_{r=R_{B}}\leq
0.$ (14)
The height of the barrier and position is marked, respectively, as $V_{B}$ and
$R_{B}$. For the present analysis, all kind of the reactions involving
symmetric $(N=Z,~{}A_{1}=A_{2})$ as well as asymmetric $(N\neq Z,~{}A_{1}\neq
A_{2})$ nuclei are considered. In all, 400 reactions covering almost whole of
the periodic table are taken into account. All nuclei considered here are
assumed to be spherical in nature, however, deformation as well as orientation
of the nuclei also affect the fusion barriers deni07 . The lightest reaction
considered here is ${}^{6}Li+^{9}Be$ whereas the heaviest one is
${}^{48}Ca+^{248}Cm$. As reported in Ref. ms2000 , proximity Prox 77
overestimate experimental data by $4\%$. It was reported to be better for
newer versions.
Once fusion barrier heights and positions were calculated, a search was made
for their parametrization. Since it is evident that barrier positions depend
on the size of the colliding systems, the best way is to parametrize them in
terms of the radius dependence i.e. in terms of $A^{1/3}$. In the literature,
several attempts exist that parametrize $R_{B}$ directly either as
$A^{{}^{\prime}}+B^{{}^{\prime}}(A_{1}^{1/3}+A_{2}^{1/3})$ anjos02 ; broglia81
; kovar79 ; cw76 or as $r_{B}~{}(=\frac{R_{B}}{A_{1}^{1/3}+A_{2}^{1/3}}$)
cngo75 ; ngo80 . We have also tried similar fits. Unfortunately, the
scattering around the mean curve was quite significant in both the cases,
therefore, we discard this kind of parametrizations. Alternatively, we plotted
the reduced fusion barrier positions $s_{B}=R_{B}-C_{1}-C_{2}$, as a function
of $\frac{Z_{1}Z_{2}}{A_{1}^{1/3}+A_{2}^{1/3}}$ for all three versions of
proximity potentials (see Fig. 1). Very encouragingly, the reduced barrier
positions $s_{B}$ of all the reactions fall on the mean curve that can be
parametrized in terms of exponential function. We noted that the scattering
around the mean positions is very small. Due to the weak Coulomb force in
lighter colliding nuclei, lesser attractive potential is needed to
counterbalance it. As a result, separation distance increases in lighter
colliding nuclei. As we go to heavier nuclei, stronger Coulomb contribution
demands more and more penetration, therefore, decreasing the value of $s_{B}$.
In other words, the fusion in lighter nuclei occurs at the outer region
compared to the heavier nuclei where $s_{B}$ is much smaller.
If we compare (a) and (b) parts of the Fig. 1, we notice that $s_{B}$, the
separation distance between nuclei is slightly more in Prox 88 compared to
Prox 77. This is due to the fact that Prox 88 has stronger surface energy
coefficient $\gamma$ [see Eq. (2) with $\gamma_{0}$ = $1.2496~{}\rm
MeV/fm^{2}$ and $k_{s}=2.3$ respectively]. This results in more attractive
nuclear potential compared to Prox 77 and therefore, counterbalancing happens
at larger distances. From the figure, it is also evident that latest proximity
potential has shallow nuclear potential compared to the other two versions.
All three proximity potentials follow similar mass/ charge dependence and can
be parametrized in terms of following function:
${s_{B}^{par}}=\alpha\exp\left[-\beta\left(x-2\right)^{1/4}\right].$ (15)
Here, $x$ = $\frac{Z_{1}Z_{2}}{A_{1}^{1/3}+A_{2}^{1/3}}$ and $\alpha$, $\beta$
are the constants whose values depend on the model one is using. The values of
$\alpha$, are 5.184 19, 5.374 57, and 5.087 58, whereas the values of $\beta$
are 0.339 79, 0.313 26, and 0.295 18 for Prox 77, Prox 88, and Prox 00,
potentials, respectively. The analytical parametrized fusion barrier positions
therefore, read as
${R_{B}^{par}}={s_{B}^{par}}+C_{1}+C_{2}.$ (16)
The quality of our parametrized fusion positions can be judged by analyzing
the percentage deviation defined as
$\Delta R_{B}~{}(\%)=\frac{R_{B}^{par}-R_{B}^{exact}}{R_{B}^{exact}}\times
100.$ (17)
We plot in Fig. 2, the percentage deviation $\Delta R_{B}~{}(\%)$ as a
function of the product of charges $Z_{1}Z_{2}$. Very encouragingly, we see
that in all three cases, our analytical parametrized form gives very good
results within $\pm 1\%$ of the actual exact barriers positions. The average
deviations calculated over 400 reactions are -0.01%, -0.02%, and 0% for Prox
77, Prox 88, and Prox 00, respectively. This is very encouraging since it is
for the first time that such accurate parametrization has been obtained. Note
that our parametrizations depend on the charges and masses of the colliding
nuclei only. This definitely introduces great simplification in the
calculation of fusion barrier positions within proximity concept.
In Fig. 3, we parametrize the fusion barrier heights $V_{B}$ as a function of
$\frac{1.44Z_{1}Z_{2}}{R_{B}^{par}}(1-\frac{0.75}{R_{B}^{par}})$, similar to
the one reported in Refs. ng04 ; broglia81 . The first part is the Coulomb
contribution whereas the second part is the reduction due to the nuclear
potential. We see that the fusion barrier heights in all three proximity
potentials can be parametrized using the following relation:
$V^{par}_{B}=\delta[\frac{1.44Z_{1}Z_{2}}{R_{B}^{par}}(1-\frac{0.75}{R_{B}^{par}})].$
(18)
Where $\delta$ is a constant having values 0.99903, 0.99868, and 1.002 for
Prox 77, Prox 88, and Prox 00, respectively. Here second term in the above
relation is introduced to take care of the deviations that happen in the lower
tail of the fusion barrier heights. We see that one can parametrize the
barrier heights very closely. The quality of our analytical parametrization is
tested in the Fig. 4, where again percentage difference between parametrized
and exact values are shown. Mathematically,
$\Delta V_{B}~{}(\%)=\frac{V_{B}^{par}-V_{B}^{exact}}{V_{B}^{exact}}\times
100.$ (19)
Very encouragingly, we see that our fits are within $\pm 1\%$ of the actual
values. Some slight deviations can be seen for lighter masses. This may also
be due to the limitations of proximity potentials in handling the lighter
masses where surface is of the order of nuclear radius. It is very encouraging
to note that our parametrized form give barrier heights and positions within
$\pm 1\%$ of the actual values. The average deviations are -0.10%, -0.12%, and
0.07% for Prox 77, Prox 88, and Prox 00, respectively. In Table 1, we display
the actual and analytical parametrized values of some selected collisions for
all three versions of proximity potentials. We note that our results are in
very close agreement with the actual value and therefore, introduces great
simplification in the calculation of fusion barriers. Finally, we compare our
outcome with experimental data in Fig. 5. Here we display our analytically
parametrized calculated fusion barrier heights $V_{B}^{par}$ [Eq. (18)] with
experimentally extracted fusion barrier heights $V_{B}^{expt}$. The
experimentally extracted fusion barrier heights displayed in this figure are
obtained in the approach, when shapes of both colliding nuclei are spherical.
The experimental data are taken from Refs ms2000 ; wang06 ; expt . It is clear
from the figure that our results are in good agreement with experimental data.
In a recent attempt id , we presented comparison of 16 different proximity
based potentials and found that potentials due to Bass wr94 , Aage Winther
aw95 , and Denisov id (marked as Bass 80, AW 95, and Denisov DP in Ref. id )
were performing better than other proximity based potentials. The analytical
parametrizations of such potentials will be presented elsewhere idd .
## IV Summary
Using three versions of proximity potentials, we obtained analytical relations
for the fusion barrier heights and positions. Our analysis is based on the
calculations of 400 reactions. Our analytical parametrized values are in very
close agreement with actual as well as experimental values. Therefore,
introducing great simplifications in the calculation of fusion barrier heights
and positions. These results can be used as a guide line for estimating the
fusion barriers in those cases where measurements do not exist and also for
the study of new nuclei yet unexplored.
## V Acknowledgments
This work was supported by a research grant from the Department of Atomic
Energy, Government of India.
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Figure 1: Reduced fusion barrier positions $s_{B}~{}\rm(fm)$ (defined as
$s_{B}=R_{B}-C_{1}-C_{2}$) as a function of the
$\frac{Z_{1}Z_{2}}{A_{1}^{1/3}+A_{2}^{1/3}}$. Parts (a), (b), and (c) show the
results with Prox 77, Prox 88, and Prox 00 versions of the proximity
potential. Our parametrized fits are shown as solid curves. The values of
constants $\alpha$ and $\beta$ are given in the text. Figure 2: The percentage
difference $\Delta R_{B}~{}(\%)$ [defined in Eq. (17)] as a function of the
product of charges of colliding pair $Z_{1}Z_{1}$. Parts (a), (b), and (c)
show the results with Prox 77, Prox 88, and Prox 00 versions of the proximity
potential. Figure 3: The fusion barrier heights $V_{B}$ (MeV), as a function
of $\frac{1.44Z_{1}Z_{2}}{R_{B}^{par}}(1-\frac{0.75}{R_{B}^{par}})$. Parts
(a), (b), and (c) show the results with Prox 77, Prox 88, and Prox 00 versions
of the proximity potential. Our parametrized fits are shown as solid curves.
The value of the constant $\delta$ is given in the text. Figure 4: Same as
Fig. 2, but for $\Delta V_{B}~{}(\%$). Figure 5: The variation of the
parametrized fusion barrier heights $V_{B}^{par}~{}\rm(MeV)$ as a function of
experimental fusion barrier heights $V_{B}^{expt}~{}\rm(MeV)$. Parts (a), (b),
and (c) show the results with Prox 77, Prox 88, and Prox 00 versions of the
proximity potential. The experimental values are taken from Refs. ms2000 ;
wang06 ; expt .
Table 1: Fusion barrier heights VB (in MeV) and positions RB (in fm), calculated using different proximity potentials along with their corresponding parametrized values are displayed for few cases. Reaction | Prox 77 | Prox 88 | Prox 00 | Prox 77 | Prox 88 | Prox 00
---|---|---|---|---|---|---
| R${}_{B}^{exact}$ | R${}_{B}^{par}$ | R${}_{B}^{exact}$ | R${}_{B}^{par}$ | R${}_{B}^{exact}$ | R${}_{B}^{par}$ | V${}_{B}^{exact}$ | V${}_{B}^{par}$ | V${}_{B}^{exact}$ | V${}_{B}^{par}$ | V${}_{B}^{exact}$ | V${}_{B}^{par}$
6Li + 9Be | 7.01 | 7.03 | 7.26 | 7.27 | 6.74 | 6.81 | 2.21 | 2.20 | 2.14 | 2.13 | 2.29 | 2.26
10B + 12C | 7.22 | 7.21 | 7.47 | 7.45 | 6.99 | 7.03 | 5.36 | 5.36 | 5.19 | 5.20 | 5.54 | 5.50
16O + 16O | 7.65 | 7.65 | 7.90 | 7.90 | 7.51 | 7.54 | 10.86 | 10.86 | 10.55 | 10.55 | 11.10 | 11.03
20Ne + 20Ne | 7.95 | 7.97 | 8.20 | 8.21 | 8.42 | 8.28 | 16.39 | 16.35 | 15.94 | 15.92 | 15.68 | 15.85
24Mg + 26Mg | 8.40 | 8.37 | 8.65 | 8.61 | 8.86 | 8.73 | 22.54 | 22.53 | 21.95 | 21.96 | 21.47 | 21.75
24Mg + 34S | 8.61 | 8.61 | 8.86 | 8.85 | 8.89 | 8.80 | 29.34 | 29.28 | 28.60 | 28.55 | 28.64 | 28.80
16O + 64Ni | 9.01 | 9.03 | 9.26 | 9.27 | 9.05 | 9.08 | 35.17 | 35.06 | 34.33 | 34.22 | 35.08 | 34.99
6Li + 238U | 10.87 | 10.97 | 11.07 | 11.21 | 10.81 | 10.93 | 34.07 | 33.72 | 33.46 | 33.04 | 34.28 | 33.94
12C + 124Sn | 9.88 | 9.94 | 10.13 | 10.18 | 9.97 | 10.00 | 40.31 | 40.14 | 39.49 | 39.26 | 40.20 | 40.04
16O + 110Pd | 9.88 | 9.90 | 10.08 | 10.13 | 10.02 | 10.01 | 49.60 | 49.42 | 48.56 | 48.38 | 49.12 | 49.07
30Si + 64Ni | 9.63 | 9.60 | 9.83 | 9.84 | 9.71 | 9.65 | 54.13 | 54.16 | 52.94 | 52.92 | 53.93 | 54.06
48Ca + 48Ca | 9.89 | 9.81 | 10.09 | 10.05 | 9.89 | 9.83 | 53.96 | 54.18 | 52.84 | 52.97 | 53.93 | 54.24
32S + 58Ni | 9.40 | 9.45 | 9.65 | 9.68 | 9.50 | 9.53 | 63.04 | 62.79 | 61.60 | 61.40 | 62.64 | 62.49
40Ar + 60Ni | 9.82 | 9.78 | 10.02 | 10.02 | 10.00 | 9.94 | 68.40 | 68.45 | 66.91 | 66.92 | 67.37 | 67.64
16O + 166Er | 10.64 | 10.66 | 10.84 | 10.89 | 10.77 | 10.76 | 68.56 | 68.25 | 67.25 | 66.89 | 67.93 | 67.87
16O + 186W | 10.86 | 10.90 | 11.06 | 11.13 | 11.18 | 11.15 | 73.09 | 72.76 | 71.74 | 71.34 | 71.39 | 71.45
36S + 90Zr | 10.30 | 10.28 | 10.55 | 10.50 | 10.41 | 10.36 | 82.99 | 83.03 | 81.30 | 81.39 | 82.35 | 82.69
35Cl + 92Zr | 10.25 | 10.25 | 10.50 | 10.47 | 10.39 | 10.36 | 88.58 | 88.45 | 86.75 | 86.71 | 87.64 | 87.85
32S + 110Pd | 10.43 | 10.45 | 10.68 | 10.68 | 10.65 | 10.65 | 94.21 | 94.05 | 92.33 | 92.15 | 92.43 | 92.70
64Ni + 64Ni | 10.48 | 10.47 | 10.73 | 10.70 | 10.60 | 10.57 | 99.84 | 100.00 | 97.86 | 97.98 | 98.90 | 99.43
40Ar + 110Pd | 10.75 | 10.73 | 10.95 | 10.95 | 11.07 | 10.98 | 103.19 | 103.25 | 101.21 | 101.30 | 100.61 | 101.37
32S + 138Ba | 10.87 | 10.87 | 11.07 | 11.09 | 10.93 | 10.96 | 110.71 | 110.40 | 108.62 | 108.33 | 109.73 | 109.89
40Ar + 130Te | 11.05 | 11.03 | 11.25 | 11.26 | 11.22 | 11.18 | 113.63 | 113.78 | 111.56 | 111.58 | 111.96 | 112.69
24Mg + 208Pb | 11.41 | 11.44 | 11.61 | 11.66 | 11.73 | 11.69 | 116.04 | 115.63 | 114.02 | 113.56 | 113.09 | 113.66
29Si + 178Hf | 11.27 | 11.28 | 11.47 | 11.50 | 11.55 | 11.49 | 120.24 | 120.00 | 118.08 | 117.83 | 117.75 | 118.32
34S + 168Er | 11.35 | 11.32 | 11.55 | 11.55 | 11.39 | 11.40 | 129.16 | 129.10 | 126.86 | 126.67 | 128.04 | 128.65
64Ni + 96Zr | 11.13 | 11.08 | 11.33 | 11.30 | 11.21 | 11.19 | 135.37 | 135.58 | 132.87 | 133.07 | 134.04 | 134.74
38S + 181Ta | 11.69 | 11.64 | 11.89 | 11.87 | 11.79 | 11.78 | 134.80 | 135.05 | 132.51 | 132.56 | 133.21 | 133.96
48Ca + 154Sm | 11.61 | 11.59 | 11.86 | 11.80 | 11.72 | 11.68 | 143.72 | 143.95 | 141.26 | 141.51 | 142.55 | 143.35
40Ar + 180Hf | 11.65 | 11.66 | 11.90 | 11.88 | 11.81 | 11.80 | 149.63 | 149.61 | 147.07 | 146.98 | 147.58 | 148.40
38S + 208Pb | 11.98 | 11.94 | 12.18 | 12.16 | 12.00 | 12.00 | 147.89 | 148.15 | 145.47 | 145.60 | 147.31 | 147.90
64Ni + 124Sn | 11.55 | 11.52 | 11.75 | 11.73 | 11.68 | 11.68 | 163.23 | 163.45 | 160.37 | 160.67 | 160.85 | 161.84
40Ar + 206Pb | 11.93 | 11.94 | 12.18 | 12.16 | 12.11 | 12.10 | 166.66 | 166.67 | 163.89 | 163.79 | 164.19 | 165.10
86Kr + 100Mo | 11.59 | 11.57 | 11.84 | 11.79 | 11.68 | 11.70 | 175.40 | 175.81 | 172.33 | 172.69 | 173.67 | 174.51
90Zr + 90Zr | 11.42 | 11.42 | 11.67 | 11.64 | 11.56 | 11.59 | 188.23 | 188.32 | 184.79 | 184.94 | 185.53 | 186.30
40Ar + 238U | 12.31 | 12.28 | 12.51 | 12.49 | 12.30 | 12.35 | 182.29 | 182.15 | 179.41 | 179.22 | 181.07 | 181.72
96Mo + 100Mo | 11.75 | 11.72 | 11.95 | 11.93 | 11.81 | 11.86 | 202.39 | 202.67 | 198.85 | 199.28 | 200.05 | 201.03
54Cr + 196Os | 12.22 | 12.19 | 12.42 | 12.40 | 12.34 | 12.34 | 201.86 | 202.01 | 198.62 | 198.75 | 199.21 | 200.31
51V + 208Pb | 12.23 | 12.24 | 12.48 | 12.45 | 12.36 | 12.40 | 208.11 | 208.09 | 204.75 | 204.73 | 205.18 | 206.18
54Cr + 209Bi | 12.33 | 12.32 | 12.53 | 12.53 | 12.59 | 12.61 | 218.37 | 218.45 | 214.85 | 214.95 | 212.95 | 214.38
96Zr + 124Sn | 12.15 | 12.13 | 12.40 | 12.34 | 12.28 | 12.29 | 222.18 | 222.53 | 218.53 | 218.91 | 219.15 | 220.48
55Mn + 208Pb | 12.35 | 12.32 | 12.55 | 12.53 | 12.24 | 12.35 | 224.74 | 224.80 | 221.13 | 221.20 | 224.89 | 224.96
70Zn + 176Yb | 12.35 | 12.31 | 12.55 | 12.52 | 12.36 | 12.41 | 230.12 | 230.47 | 226.42 | 226.76 | 228.67 | 229.41
58Fe + 208Pb | 12.39 | 12.40 | 12.64 | 12.61 | 12.38 | 12.47 | 232.38 | 232.38 | 228.67 | 228.68 | 231.26 | 231.85
59Co + 208Pb | 12.42 | 12.41 | 12.62 | 12.62 | 12.50 | 12.57 | 241.20 | 241.15 | 237.34 | 237.30 | 237.99 | 238.98
59Co + 209Bi | 12.43 | 12.42 | 12.63 | 12.63 | 12.62 | 12.69 | 244.02 | 243.90 | 240.10 | 240.01 | 238.47 | 239.75
63Cu + 197Au | 12.39 | 12.37 | 12.59 | 12.57 | 12.20 | 12.36 | 250.40 | 250.29 | 246.33 | 246.46 | 251.22 | 251.22
64Ni + 208Pb | 12.56 | 12.54 | 12.76 | 12.75 | 12.53 | 12.64 | 247.56 | 247.65 | 243.66 | 243.74 | 245.68 | 246.54
70Zn + 208Pb | 12.71 | 12.67 | 12.91 | 12.87 | 12.76 | 12.85 | 262.60 | 262.78 | 258.53 | 258.86 | 259.01 | 260.10
86Kr + 208Pb | 12.99 | 12.98 | 13.24 | 13.18 | 12.92 | 13.09 | 308.05 | 308.27 | 303.40 | 303.77 | 306.16 | 306.75
|
arxiv-papers
| 2010-05-28T04:40:18 |
2024-09-04T02:49:10.693757
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ishwar Dutt and Rajeev K. Puri",
"submitter": "Rajeev Kumar Puri",
"url": "https://arxiv.org/abs/1005.5214"
}
|
1006.0125
|
# Nonsequential Two-Photon Double Ionization of Atoms: Identifying the
Mechanism
Morten Førre morten.forre@ift.uib.no Department of Physics and Technology,
University of Bergen, N-5007 Bergen, Norway Sølve Selstø Centre of
Mathematics for Applications, University of Oslo, N-0316 Oslo, Norway Raymond
Nepstad Department of Physics and Technology, University of Bergen, N-5007
Bergen, Norway
###### Abstract
We develop an approximate model for the process of direct (nonsequential) two-
photon double ionization of atoms. Employing the model, we calculate
(generalized) total cross sections as well as energy-resolved differential
cross sections of helium for photon energies ranging from 39 to 54 eV. A
comparison with results of ab initio calculations reveals that the agreement
is at a quantitative level. We thus demonstrate that this complex ionization
process is fully described by the simple model, providing insight into the
underlying physical mechanism. Finally, we use the model to calculate
generalized cross sections for the two-photon double ionization of neon in the
nonsequential regime.
###### pacs:
32.80.Rm, 32.80.Fb, 42.50.Hz
Correlated dynamical processes in nature poses unique challenges to
experiments and theory. A prime example of this is the double ionization of
helium by one-photon impact, which has been studied for more than 40 years.
However, it is only during the last 15 years or so, that advances in theory,
modeling and experiment have enabled scientists to gain a deeper insight into
the role of electron correlations in this ionization process Briggs and
Schmidt (2000); Avaldi and Huetz (2005); Samson et al. (1998); Schneider et
al. (2002); Foumouo et al. (2006). The corresponding problem of two-photon
double ionization of helium, in the photon energy interval between 39.4 and
54.4 eV, is an outstanding quantum mechanical problem that has been, and still
is, subject to intense research worldwide, both theoretically Colgan and
Pindzola (2002); Feng and van der Hart (2003); Laulan and Bachau (2003);
Piraux et al. (2003); Hu et al. (2005); Foumouo et al. (2006); Shakeshaft
(2007); Ivanov and Kheifets (2007); Horner et al. (2007); Nikolopoulos and
Lambropoulos (2007); Feist et al. (2008); Guan et al. (2008); Foumouo et al.
(2008); Palacios et al. (2009); Nepstad et al. and experimentally, employing
state-of-the-art high-order harmonic Hasegawa et al. (2005); Nabekawa et al.
(2005); Antoine et al. (2008) and free-electron (FEL) light sources Sorokin et
al. (2007); Rudenko et al. (2008). Despite all the interest and efforts that
have been put into this research, major fundamental issues remain unresolved.
What characterizes this particular three-body breakup process is that the
electron correlation is a prerequisite for the process to occur, i.e., it
depends upon the exchange of energy between the outgoing electrons, and as
such it represents a clear departure from an independent-particle picture.
In this Letter, we present a novel approximate model for the direct or
nonsequential two-photon double ionization process in helium, sketched in Fig.
1 (a). We show that the simple model predicts the essential features of the
process, even at a quantitative level, which is quite surprising given the
very high complexity of the problem. In particular, we find very good
agreement between the model predictions and the results obtained by solving
the time-dependent Schrödinger equation from first principles, regarding
(generalized) total cross sections as well as energy-resolved differential
cross sections for the process. The proposed model may be generalized to
account for direct double ionization processes in multi-electron atoms. We
demonstrate this by calculating the generalized cross section for
nonsequential two-photon double ionization of neon.
Few-photon multiple ionization of noble gases beyond helium have been studied
experimentally in some detail Moshammer et al. (2007); Sorokin et al. (2007);
Benis et al. (2006); Rudenko et al. (2008), but to the best of our knowledge,
the cross section for the nonsequential two-photon double ionization process
has not yet been obtained. Therefore, we hope that our results will encourage
further investigation of nonsequential double ionization processes in various
noble gases.
Figure 1: (color online). a) Sketch of the direct two-photon double ionization
process in helium. The abbreviation SI and DI stands for single and double
ionization continuum, respectively, whereas the arrows illustrate the photons
that are absorbed by the system. b) Sketch of the model process for two-photon
double ionization (see text for details). c) Matrix representation of the
model Hamiltonian, for the case where the outer electron is emitted before the
inner electron (see text for more details). Atomic units (a.u.) are used in
the figure (1 a.u. of energy corresponds to 27.2 eV).
Reducing a complex quantum mechanical problem to a simple and transparent
model problem, while retaining the essential physics, is very useful in order
to access the underlying physics Lein et al. (2000); Schneider et al. (2002);
Watson et al. (1997). With such a goal in mind, we will now outline a possible
physical mechanism for the nonsequential two-photon double ionization process
in an atom, and then proceed to construct a simple quantum mechanical model
which implements these ideas. The idea behind the model is that the electrons
are considered to be distinguishable particles that can absorb one photon
each. However, in order to include the effect of the first emitted electron on
the second one, we impose the additional but important constraint that the
absorption of the second photon, by the second electron, can only occur after
the first photon absorption. In this way, and according to the principle of
conservation of energy, the first electron may transfer energy to the second
electron as it is emitted, allowing for the nonsequential ionization process
to take place.
The starting point of our model is the single-active electron approximation
(SAE) where both electrons are considered to be independent particles and
treated differently in that they are both assumed to move in their respective
ionization potentials. That is, the ’outer’ electron moves in an effective
potential set up by the nucleus of charge $Ze$ ($e$ is the elementary charge),
the ’inner’ electron and the $Z-2$ other electrons. The inner electron sees a
corresponding screened potential given by the nucleus and the $Z-2$ remaining
electrons. We will label these two different cases simply by ’$A$’ and ’$B$’,
respectively. Following this procedure, the wave function of the ground state
may be approximated by the product ansatz
$\Psi(\mathbf{r}_{A},\mathbf{r}_{B})=\psi_{A}(\mathbf{r}_{A})\psi_{B}(\mathbf{r}_{B}),$
(1)
where $\psi_{A}$ and $\psi_{B}$ refer to the one-electron wave function of
electron $A$ and $B$, respectively.
Now, the first ionization event in the direct two-photon double ionization
process can be represented by the one-electron dipole coupling between the
ground state wave function of either $A$ or $B$, i.e., the state
$\left|A,E_{A}^{0}\right>$ or $\left|B,E_{B}^{0}\right>$, and their respective
continuum states, $\left|A,E_{A}\right>$ and $\left|B,E_{B}\right>$, where
$E_{A}^{0}$ and $E_{B}^{0}$, and $E_{A}$ and $E_{B}$ represent the energies of
the ground and continuum states, respectively. In the product basis
representation (1), with the length gauge formulation of the light-matter
interaction, the dipole coupling matrix elements may be written on the
following simple form (for the case where electron $A$ is assumed to be
emitted first),
$\left<A,E_{A}^{0}\right|-e\mathbf{E}(t)\cdot\mathbf{r}_{A}\left|A,E_{A}\right>\left<\left.B,E_{B}^{0}\right|B,E_{B}^{0}\right>,$
(2)
where $\mathbf{E}(t)$ is the time-dependent electric field that defines the
laser pulse, which is assumed to be linearly polarized along the z-axis. These
coupling elements are related to the one-photon (one-electron) photoionization
cross section via the relation Cormier and Lambropoulos (1995)
$\sigma_{A}=4\pi^{2}\alpha\left(E_{A}-E_{A}^{0}\right)\left|\langle
A,E_{A}^{0}|z_{A}|A,E_{A}\rangle\right|^{2},$ (3)
where $\alpha$ is the fine structure constant.
At the instant of ionization of electron $A$, electron $B$ remains unaffected.
However, once electron $A$ has absorbed its photon, we allow for the
possibility that electron $B$ (but not $A$) can be hit by a second photon.
This secondary process is included into the model by introducing additional
dipole couplings between the $B$ ground state and its corresponding one-
electron continuum states in the following way:
$\left<B,E_{B}^{0}\right|-e\mathbf{E}(t)\cdot\mathbf{r}_{B}\left|B,E_{B}\right>\delta(E_{A},E^{{}^{\prime}}_{A})$
(4)
Note here that there are only non-vanishing couplings between SAE states
(system $A$) of the same energy, i.e., the resulting coupling matrix attains a
very simple structure, as shown in Fig. 1, with typically only a few hundred
different couplings. The same procedure may also be followed with $A$ and $B$
interchanged, however, this will neccessarily yield the same result, and
therefore need not explicitly be considered.
Figure 2: (color online). Total integrated (generalized) cross section for the
nonsequential two-photon double ionization of helium. Black line: present
model result obtained with Eq. (5); open (blue) circles: ab initio result of
Feist et al. Feist et al. (2008) obtained with a 4 fs pulse; and open (red)
squares: corresponding ab initio result of Nepstad et al. Nepstad et al. . The
vertical lines define the two-photon direct double ionization region.
The couplings (2) and (4) and the mentioned constraints, along with the
corresponding diagonal energies, constitute the entire model that we propose.
To this end, we would like to add that all excited, bound states have been
left out of the model, as they play no role in the present context. As a
matter of fact, despite the extremely simple form of the model matrix
elements, with no explicit presence of the correlation potential, it actually
allows for the possibility that the two electrons exchange energy in the
excitation process. Thus, both electrons may be emitted into the continuum
even though the energy of the secondary photon may not itself be sufficient to
eject the inner electron into the continuum.
Applying second order perturbation theory to the resulting model Hamiltonian,
one can show that the single-differential cross section for the direct two-
photon double ionization of an atom is simply given by
$\displaystyle\frac{d\sigma}{dE}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[f(E)+f\left(2\hbar\omega+E^{0}_{A}+E^{0}_{B}-E\right)\right]$
(5) $\displaystyle f(E)$ $\displaystyle\equiv$
$\displaystyle\frac{\hbar^{3}\omega^{2}}{\pi}\frac{\sigma_{A}\\!\left(E-E^{0}_{A}\right)\;\sigma_{B}\\!\left(2\hbar\omega-E+E^{0}_{A}\right)}{\left(E-E^{0}_{A}\right)\left(2\hbar\omega-E+E^{0}_{A}\right)\left(E-E^{0}_{A}-\hbar\omega\right)^{2}},$
where $\sigma_{A}$ and $\sigma_{B}$ now refer to the total one-photon single
ionization cross sections of $A$ and $B$, respectively, $E$ is the excess
energy, and where we have explicitly accounted for the exchange symmetry of
identical particles and the possibility that either the inner or the outer
electron is emitted first. At this point we would like to emphasize that the
only parameters needed in order to calculate the nonsequential two-photon
double ionization cross section within the model framework, is the effective
binding energies of electron $A$ and $B$, as well as their respective one-
photon single ionization cross sections. For instance, for helium all these
parameters are well known. The model may straightforwardly be generalized to
account for e.g. nonsequential three-photon triple ionization processes in
atoms. A more detailed exposition of the model and a derivation of the
perturbation theory expression for the cross section, will be outlined in a
forthcoming communication.
Figure 3: (color online). Electron energy distribution for two-photon double
ionization of helium at photon energies of 44.9 and 51.7 eV. Solid (black)
line: model result; and dashed (red) line: ab initio result.
In Fig. 2 we compare the total cross section obtained using the approximate
model, Eq. (5), (black line in the figure), with the ab initio result of Feist
et al. Feist et al. (2008) (blue circles) and Nepstad et al. Nepstad et al.
(red squares), both of which were obtained by solving the time-dependent
Schrödinger equation of helium from first principles. The model result is
obtained using tabulated values for the absolute one-photon photoionization
cross section of helium, as obtained experimentally by West and Marr West and
Marr (1976). Figure 3 shows corresponding energy-resolved single-differential
cross sections at two selected photon energies, 44.9 and 51.7 eV. As a matter
of fact, the agreement between the model result and the ab initio results is
almost perfect in Figs. 2 and 3, in particular for the lower photon energies,
demonstrating the strength of this extremely simple model in predicting
accurate values for the generalized cross section in direct two-photon double
ionization processes. Formula (5) predicts a sharp rise of the total cross
section in the vicinity of the threshold at 54.4 eV, which is in agreement
with recent ab initio calculations Horner et al. (2007); Shakeshaft (2007);
Horner et al. (2008); Feist et al. (2008); Palacios et al. (2009); Nepstad et
al. .
As mentioned in the introduction, the problem of nonsequential two-photon
double ionization of helium has been subject of intense research in recent
years, and accurate predictions for the generalized cross section remain
elusive Colgan and Pindzola (2002); Feng and van der Hart (2003); Laulan and
Bachau (2003); Piraux et al. (2003); Hu et al. (2005); Foumouo et al. (2006);
Shakeshaft (2007); Ivanov and Kheifets (2007); Horner et al. (2007);
Nikolopoulos and Lambropoulos (2007); Feist et al. (2008); Guan et al. (2008);
Foumouo et al. (2008); Palacios et al. (2009); Nepstad et al. ; Hasegawa et
al. (2005); Nabekawa et al. (2005); Antoine et al. (2008); Sorokin et al.
(2007); Rudenko et al. (2008) as the values obtained for the cross section for
the reaction may differ by as much as an order of magnitude. On the
theoretical side, the great discrepancies that remain between different
approaches are usually ascribed to the different ways electron correlations
are handled in the final state. To this end, we hope that the predictions of
the present model study may shed new light on this controversy.
Having justified the validity of our simple approach, we now turn to a more
complex problem, namely the process of nonsequential two-photon double
ionization of neon. Inserting, in Eq. (5), the correct first and second
ionization energies of neon, i.e., 21.6 and 40.9 eV, as well as experimental
values for the photoionization cross sections of Ne West and Marr (1976) and
Ne+ Covington et al. (2002), obtained using synchrotron radiation, the
resulting model prediction for the double ionization cross section is shown in
Fig. 4 (upper panel). The lower panel shows the corresponding electron energy
distribution at three selected photon energies. Interestingly, at lower photon
energies, the energy distribution exhibits a maximum (negative concavity) when
both electrons are emitted with the same energy, while at higher photon
energies the distribution is U-shaped. In sharp contrast to this trend, for
helium, the model yields a U-shaped energy distribution for all photon
energies (see Fig. 3).
Figure 4: Upper panel: generalized total cross section for the process of
two-photon double ionization of neon in the direct regime. The model result is
obtained using Eq. (5), inserting available experimental values for the total
one-photon single ionization cross sections of neon West and Marr (1976) and
Ne+ Covington et al. (2002), respectively. The vertical lines define the two-
photon direct double ionization region. Lower panel: normalized energy
distributions at various photon energies.
In conclusion, we have implemented an approximate and very simple model to
study the two-photon double ionization process of helium in the direct regime,
i.e., at photon energies below 54.4 eV where the sequential ionization process
is energetically inaccessible. We have investigated the validity of the model
by calculating generalized total cross sections and energy-resolved
differential cross sections and compared the model results with corresponding
results obtained by accurate ab initio calculations. Quantitative agreement
between model results and the full results was achieved in all considered
cases, demonstrating the general validity of the model for the two-photon
double ionization process. Finally, we have obtained the cross section for
nonsequential two-photon double ionization of neon, demonstrating that the
model has a great potential to be used in studies of nonsequential multiphoton
multiple ionization processes in more complex atomic systems. This is an
avenue of research we plan to pursue in the future.
###### Acknowledgements.
This work was supported by the Bergen Research Foundation (Norway). The ab
initio calculations were performed on the Cray XT4 (Hexagon) supercomputer
installation at Parallab, University of Bergen (Norway).
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|
arxiv-papers
| 2010-06-01T12:57:29 |
2024-09-04T02:49:10.740702
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Morten F{\\o}rre, S{\\o}lve Selst\\o, Raymond Nepstad",
"submitter": "Raymond Nepstad",
"url": "https://arxiv.org/abs/1006.0125"
}
|
1006.0143
|
2010 Vol. X No. XX, 000–000
11institutetext: Department of Astronomy, Peking University, Beijing 100871,
China; wuxb@bac.pku.edu.cn
22institutetext: National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100012, China
33institutetext: National Institute of Astronomical Optics & Technology,
Chinese Academy of Science, Nanjing 210042, China 44institutetext: Center for
Astrophysis, University of Science & Technology of China, Hefei 230026, China
55institutetext: Shanghai Astronomical Observatory, Chinese Academy of
Sciences, Shanghai 200030, China
Received [year] [month] [day]; accepted [year] [month] [day]
# Eight new quasars discovered by LAMOST in one extragalactic field
Xue-Bing Wu 11 Zhendong Jia 11 Zhaoyu Chen 11 Wenwen Zuo 11 Yongheng Zhao 22
Ali Luo 22 Zhongrui Bai 22 Jianjun Chen 22 Haotong Zhang 22 Hongliang Yan 22
Juanjuan Ren 22 Shiwei Sun 22 Hong Wu 22 Yong Zhang 33 Yeping Li 33 Qishuai Lu
33 You Wang 33 Jijun Ni 33 Hai Wang 33 Xu Kong 44 Shiyin Shen 55
###### Abstract
We report the discovery of eight new quasars in one extragalactic field (five
degree centered at RA=$08^{h}58^{m}08.2^{s}$,
Dec=$01^{o}32^{\prime}29.7^{\prime\prime}$) with the LAMOST commissioning
observations on December 18, 2009. These quasars, with $i$ magnitudes from
16.44 to 19.34 and redshifts from 0.898 to 2.773, were not identified in the
SDSS spectroscopic survey, though six of them with redshifts less than 2.5
were selected as quasar targets in SDSS. Except one source without near-IR
$Y$-band data, seven of these eight new quasars meet a newly proposed quasar
selection criterion involving both near-IR and optical colors. Two of them
were found in the ’redshift desert’ for quasars ($z$ from 2.2 to 3) ,
indicating that the new criterion is efficient for recovering the missing
quasars with similar optical colors as stars. Although LAMOST met some
problems during the commissioning observations, we were still able to identify
other 38 known SDSS quasars in this field, with $i$ magnitudes from 16.24 to
19.10 and redshifts from 0.297 to 4.512. Our identifications imply that a
substantial fraction of quasars may be missing in the previous quasar surveys.
The implication of our results to the future LAMOST quasar survey is
discussed.
###### keywords:
quasars: general — quasars: emission lines — galaxies: active
## 1 Introduction
Quasars are interesting objects in the universe since they can be used as
important tools to probe the accretion power around supermassive black holes,
the intergalactic medium, the large scale structure and the cosmic
reionization. The number of quasars has increased steadily in the past four
decades (Richards et al. 2009). Especially, A large number of them have been
discovered in two spectroscopic surveys, namely, the Two-Degree Fields (2dF)
survey (Boyle et al. 2000) and Sloan Digital Sky Survey (SDSS) (York et al.
2000). 2dF has discovered more than 20,000 low redshift ($z<2.2$) quasars with
UV-excess (Croom et al. 2004, Smith et al. 2005), while SDSS has identified
more than 100,000 quasars (Schneider et al. 2010; Abazajian et al. 2009). Some
dedicated methods were proposed for finding higher redshift quasars (Fan et
al. 2001a,b; Richards et al. 2002). However, the efficiency of identifying
quasars with redshift between 2.2 and 3 is still very low in SDSS (Schneider
et al. 2010). This is mainly because quasars with such redshifts have very
similar optical colors as stars and are mostly excluded by the SDSS
spectroscopy. Therefore, the redshift range from 2.2 to 3 is regarded as the
‘redshift desert’ for quasars because of the difficulty in identifying quasars
with redshifts in this range.
However, quasars in the redshift desert are usually more luminous than normal
stars in the infrared K-band (Warren et al. 2000) becasue the spectral energy
distributions (SEDs) of quasars are flat. This provides us an important way of
finding these quasars by involving the near-IR colors. Some methods have been
suggested by using the infrared K-band excess based on the UKIRT (UK Infrared
Telescope) Infrared Deep Sky Survey (UKIDSS) (Warren et al. 2000; Hewett et
al. 2006; Maddox et al. 2008). Combining the UKIDSS YJHK and SDSS ugriz
magnitudes, some criteria to select quasars have been proposed (Maddox et al.
2008; Chiu et al. 2007). More recently, based on a large SDSS-UKIDSS quasar
sample, Wu & Jia (2010) proposed to use the $Y-K$ vs. $g-z$ diagram to select
$z<4$ quasars and use the $J-K$ vs.$i-Y$ diagram to select $z<5$ quasars.
Although the success of adopting these criteria has been demonstrated by using
the existing quasar sample, we still need to apply them to discover new
quasars and investigate how many quasars missed in the previous spectroscopic
surveys.
The Large Sky Area Multi-Object Fibre Spectroscopic Telescope (LAMOST) is a
powerful instrument for spectroscopy (Su et al. 1998) and the main
construction was finished in 2008. Since 2009 LAMOST has entered its
commissioning phase, and some test observations have been done in the winter
of 2009. Although LAMOST has not reached its full capability, these
observations already led to the discovery of new quasars, including 12 quasars
behind M31 (Huo et al. 2010) and a very bright $i=16.44$ quasar with redshift
$z=2.427$ (in the redshift desert) (Wu et al. 2010; hereafter paper I). In
this paper, we report the discovery of more quasars in the same extragalactic
field where the very bright quasar was found, including another $z=2.773$
quasar in the redshift desert.
## 2 Target selection and Observation
In the winter of 2009, we have selected several extragalactic fields for the
LAMOST commissioning observations. In order to test whether the newly proposed
quasar selection criterion in the $Y-K$ vs. $g-z$ diagram is efficient in
identifying quasars (Wu & Jia 2010), we selected quasar candidates in several
sky fields overlapped between UKIDSS and SDSS surveyed area. Some additional
quasar candidates from the catalog of Richards et al. (2009) are also
included. Besides these quasar candidates, we also included many known SDSS
quasars in these fields as targets in order to compare the LAMOST spectroscopy
with SDSS. Here we report the observational results in one of these fields,
which is a five degree field centered at RA=$08^{h}58^{m}08.2^{s}$,
Dec=$01^{o}32^{\prime}29.7^{\prime\prime}$ close to the field of GAMA-09
(Robotham et al. 2010).
On December 18, LAMOST made the spectroscopic observations on this field and
357 quasar targets mostly with $i<19.1$ together with other objects were
observed with the exposure time of 30 minutes and the spectral resolution of
$R\sim 1000$. The spectra were processed using a preliminary version of LAMOST
spectral pipeline. Due to the problems in many aspects during the LAMOST
commissioning observations, the overall quality of the spectra is not
satisfactory. Only 99 of 357 quasar targets show the obvious spectral features
of quasars or stars/galaxies, and the rest spectra show either too low S/N
(signal to noise ratio) or sky light emissions only. Among these 99 objects,
46 of them can be identified as quasars and 53 of them show the features of
either stars or galaxies. 8 of 46 idenified quasars are new and 38 of them are
known SDSS quasars. Among 8 new quasars, SDSS J085543.40-001517.7 is a very
bright one ($i=16.44$) and was identified as a $z=2.427$ quasar. This is the
first quasar in the redshift desert discovered by LAMOST and its detailed
properties have been reported in paper I. For the completeness, we also
include some of its properties in this paper.
In Fig. 1 we show the SDSS finding charts111 Obtained from
http://cas.sdss.org/dr7/en/tools/chart/chart.asp of 8 new quasars in an order
of increasing RA. Clearly they all are point sources in the optical bands. We
also checked their morphology types in the UKIDSS images and all of them are
also point sources (UKIDSS mergedclass=-1) in the near-IR bands. This is
consistent with the morphology type of SDSS-UKIDSS quasars with redshifts
larger than 0.5 (Wu & Jia 2010). In Table 1 we list the main properties of
these 8 quasars, including their coordinates, magnitudes and redshifts. The
SDSS $ugriz$ magnitudes are given in AB systems and UKIDSS $YJHK$ magnitudes
are given in Vega system. All magnitudes are corrected for the Galactic
extinction using the map of Schlegel et al. (1998). The offsets between the
SDSS and UKIDSS positions are less then $0.21^{\prime\prime}$ for these 8
quasars, indicating that the mis-identifications of their UKIDSS counterparts
of these SDSS sources are very unlikely.
Table 1: Parameters of eight new quasars
Name | RA | Dec | $u$ | $g$ | $r$ | $i$ | $z$ | $Y$ | $J$ | $H$ | $K$ | LAMOST
---|---|---|---|---|---|---|---|---|---|---|---|---
(SDSS J) | (o) | (o) | | | | | | | | | | redshift
085307.31+014523.1 | 133.28049 | 1.75643 | 17.715 | 17.718 | 17.716 | 17.566 | 17.453 | 17.018 | 16.767 | 16.435 | 15.873 | 1.952
085543.40-001517.7 | 133.93086 | -0.25493 | 17.668 | 16.866 | 16.617 | 16.444 | 16.208 | 15.573 | 15.214 | 14.585 | 13.834 | 2.427
085718.29+024017.7 | 134.32625 | 2.67160 | 18.520 | 18.373 | 18.128 | 18.194 | 18.313 | — | — | 16.985 | 16.209 | 1.154
085727.85+012802.1 | 134.36605 | 1.46728 | 18.480 | 18.489 | 18.162 | 18.152 | 18.258 | 17.484 | 17.143 | 16.344 | 16.015 | 1.363
090148.15+004225.9 | 135.45065 | 0.70722 | 19.777 | 19.513 | 19.358 | 19.345 | 19.130 | 18.255 | 17.462 | 17.181 | 16.388 | 0.898
090437.02+014055.3 | 136.15428 | 1.68203 | 18.585 | 18.570 | 18.356 | 18.072 | 18.009 | 17.488 | — | 16.542 | 15.917 | 1.765
090453.24-001426.5 | 136.22187 | -0.24069 | 19.433 | 19.261 | 19.193 | 18.917 | 18.889 | 18.438 | 18.033 | 17.390 | 16.974 | 1.670
090504.87+000800.5 | 136.27030 | 0.13348 | 20.457 | 18.863 | 18.440 | 18.154 | 18.162 | 17.446 | 16.986 | 16.515 | 16.022 | 2.773
0.86The SDSS $ugriz$ magnitudes are given in AB system and the UKIDSS $YJHK$
magnitudes are given in Vega system.
Figure 1: The finding charts of 8 new quasars are shown in an order of
increasing RA. The size of each chart is 100”$\times$100”.
In Fig. 2 we show the LAMOST spectra of eight new quasars in an order of
increasing redshift (some sky light emissions were not well subtracted). The
complicated feature around 5900$\AA$ in each spectrum is due to the problem in
combining the LAMOST blue and red spectra, which overlap with each other from
5700$\AA$ to 6100$\AA$. From the spectra of six quasars with $z>1.3$, we can
clearly identify at least two broad emission lines and derive the average
redshift for them. For two quasars with $z<1.3$, only one emission line can be
reliably observed and is identified as MgII$\lambda 2798$. From the spectrum
of SDSS J085718.29+024017.7 (z=1.154), we can actually see a line appeared
around the wavelength of 4100$\AA$ although the S/N is not good in the blue
part. This is obviously the CIII]$\lambda 1909$ line and supports our
identification of the MgII line in the red part. Another support of these
identifications is from the photometric redshift estimation. For four of these
eight quasars, Richards et al. (2004) have given the photometric redshifts as
0.875, 1.075, 1.225 and 1.975, which is consistent with our spectroscopic
redshifts of 0.898, 1.154, 1.363 and 1.952, respectively.
Figure 2: The LAMOST spectra of eight new quasars are shown in an order of
increasing reshift. The most prominent emission lines are marked in each
spectrum. Figure 3: The location of two new $z>2.2$ quasars (solid triangles)
and six new $z<2.2$ quasars (open triangles) in three optical color-color
diagrams (a,b,c) and the $Y-K$ vs. $g-z$ diagram (d), in comparing with the
8996 SDSS-UKIDSS stars (Wu & Jia 2010). Black and red dots represent the
normal and later type stars, respectively. Dashed line shows the $z<4$ quasar
selection criterion proposed by Wu & Jia (2010). In diagram (d) only seven
quasars are shown because one quasar does not have $Y$ band data.
We noticed that two of eight new quasars have redshifts larger than 2.2.
Besides SDSS J085543.40-001517.7 ($z=2.427$) (see Paper I), SDSS
J090504.87+000800.5 ($z=2.773$) is also a quasar in the ’redshift desert’.
These quasars are very difficult to be identified because of their similar
optical colors as stars. However, they can be recovered by using the near-IR
colors. In Fig. 3 we show the locations of these eight quasars in three
optical color-color diagrams and the $Y-K$ vs. $g-z$ diagram, in comparison
with the 8996 SDSS-UKIDSS stars (Wu & Jia 2010). Note that in the $Y-K$ vs.
$g-z$ diagram the magnitude of $g$ and $z$ have been converted to the
magnitudes in Vega system by using the scalings (Hewett et al. 2006):
$g=g(AB)+0.103$ and $z=z(AB)-0.533$. Obviously two quasars with redshifts
larger than 2.2 locate in the stellar locus in all three optical color-color
diagrams, but are separated from stars in the $Y-K$ vs. $g-z$ diagram and
meets the selection criterion proposed by Wu & Jia (2010). For six quasars
with redshifts less than 2.2, although they are separated from the main
stellar locus in the $u-g$ vs. $g-r$ diagram, they still locate in or close to
the stellar locus in other two optical color-color diagrams. None of these 8
new quasars has the SDSS spectrum, though 6 of them with redshifts less than
2.5 were classified as quasar targets in the item of ’PrimeTarget’ of the
SDSS/DR7 database. These unidentified quasars in the SDSS spectroscopic survey
can be successfully recovered by applying the selection criterion in the $Y-K$
vs. $g-z$ diagram.
We also searched the counterparts of these new quasars in other wavelength
bands. From the VLA/FIRST radio catalog (White et al. 1997) we did not find
radio counterparts for all eight quasars within 30′′ from their SDSS
positions. Therefore, these quasar are radio-quiet ones, which is another
reason why they are not identified by the SDSS spectroscopy. We also searched
the ROSAT X-ray source catalog (Voges et al. 1999) and did not find
counterparts for them within 1’. From GALEX catalog (Morrissey et al, 2007) we
found ultraviolet counterparts within 1′′ from their SDSS positions for five
of six quasars with $z<2$. But for a $z=1.363$ quasar, SDSS
J085727.85+012802.1, and two quasars with $z>2.2$, we failed to find their
GALEX counterparts. The high GALEX detection rate (83%) of $z<2$ quasars and
the non-detection in ultraviolet for $z>2.2$ quasars in our case is well
consistent with the previous result of the SDSS-GALEX quasar sample (Trammell
et al. 2007).
Although LAMOST met some problems during the commissioning observations, we
were still able to identify other 38 known SDSS quasars in this field, with
$i$ magnitudes from 16.24 to 19.10 and redshifts from 0.297 to 4.512. The
number of known SDSS quasars with $i<19.1$ in this five degree field is 177,
and our identified 38 SDSS quasars take a fraction of 22% of them. In the
upper and lower panels of Fig. 4 we show the histograms of redshift
distribution of 177 known SDSS quasars with $i<19.1$ and 38 SDSS quasars
identified by LAMOST in this field. The contributions of 8 new quasars to
these two histograms are also demonstrated. The ratio between 8 new quasars
and 38 known SDSS quasars identified by us in this field is 21%, impling that
a substantial fraction of the quasars may be missed by the SDSS at the
magnitude limit $i<19.1$. Especially, only 4 of 177 SDSS known quasars with
$i<19.1$ in this field have redshift larger than 2.4. Our discovery of 2 new
quasars with $z>2.4$ adds a significant fraction of them. Obviously this still
needs to be confirmed by more complete spectroscopic identifications of
quasars in this field. In addition, from the lower panel of Fig. 4 we can see
that the fraction of quasars with redshifts around 1.2 is relatively lower,
which is partly due to the lower CCD efficiency around $6000\AA$ where the
blue and red spectra overlap. If we take the spectrum of a quasar with $z\sim
1.2$ with LAMOST, the MgII$\lambda 2798$ line will appear around $6000\AA$ as
the only one prominent emission line in the optical band but will be difficult
to be identified due to the current problems in combining the LAMOST blue and
red spectra around $6000\AA$. This situation will be improved after we solve
the spectral combining problems.
Figure 4: Upper panel: The histogram of redshift distribution of 177 known
SDSS quasars with $i<19.1$ in this field. Lower panel: The histogram of
redshift distribution of 38 SDSS quasars identified by LAMOST in this field.
The contributions of 8 new quasars to these two histograms are also
demonstrated by the dashed lines.
## 3 Discussion
In this paper we presented the discovery of eight new quasars with redshifts
from 0.898 to 2.773 in one extragalactic field close to GAMA-09 by the LAMOST
commissioning observations. This discovery supports the idea that by combining
the UKIDSS near-IR colors with the SDSS optical colors we are able to
efficiently recover the unindentified quasars in the SDSS spectroscopic survey
even at the magntude limit $i<19.1$. Our results indicate that not only some
quasars in the redshift desert but also some quasars with lower redshifts are
probably missed in the SDSS survey. These missing quasars may take a
substantial fraction of the quasars at the magnitude limit of SDSS
spectroscopy. Obviously this still needs to be confirmed by more complete
identifications of quasars in this field, because our identifications during
the LAMOST commissioning observations are incomplete.
Nevertheless, the success of identifing eight new quasars (including two
quasars in the redshift desert) in one extragalactic field gives us more
confidence to discover more missing quasars in the future LAMOST observations.
In the winter of 2009, LAMOST has made test observations on several sky fields
and we are now searching for more quasars from the spectra taken in these
fields. We believe that more missing quasars will be discovered soon.
A complete quasar sample is very important to the construction of the quasar
luminosity function and study the cosmological evolution of quasars. However,
as we demonstrated in this paper, because some quasars have similar optical
colors as normal stars, it is very difficult for find them in the optical
quasar surveys. The low efficiency of finding quasars in the redshift desert
($z$ from 2.2 to 3) has led to obvious incompleteness of SDSS quasar sample in
this redshift range and serious problems in constructing the luminosity
function for quasars around the redshift peak (between 2 and 3) of quasar
activity (Richards et al. 2006; Jiang et al. 2006). Therefore, recovering
these missing quasars will become an important task in the future quasar
survey. We hope that in the next a few months great progress will be made in
improving the capability of LAMOST spectroscopy. As long as LAMOST can reach
its designed capability after the commissioning phase, we expect to obtain the
largest quasar sample in the LAMOST quasar survey. This sample will
undoubtedly play a leading role in the future quasar study.
###### Acknowledgements.
We thank Michael Strauss for clarifying the SDSS target status of these new
quasars. This work was supported by the National Natural Science Foundation of
China (10525313), the National Key Basic Research Science Foundation of China
(2007CB815405). The Large Sky Area Multi-Object Fiber Spectroscopic Telescope
(LAMOST) is a National Major Scientific Project built by the Chinese Academy
of Sciences. Funding for the project has been provided by the National
Development and Reform Commission. The LAMOST is operated and managed by the
National Astronomical Observatories, Chinese Academy of Sciences. We
acknowledge the use of LAMOST, as well as the archive data from SDSS, UKIDSS,
FIRST, ROSAT and GALEX.
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|
arxiv-papers
| 2010-06-01T14:45:24 |
2024-09-04T02:49:10.746183
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xue-Bing Wu, Zhendong Jia, Zhaoyu Chen, Wenwen Zuo, Yongheng Zhao, Ali\n Luo, Zhongrui Bai, Jianjun Chen, Haotong Zhang, Hongliang Yan, Juanjuan Ren,\n Shiwei Sun, Hong Wu, Yong Zhang, Yeping Li, Qishuai Lu, You Wang, Jijun Ni,\n Hai Wang, Xu Kong, Shiyin Shen",
"submitter": "Xue-Bing Wu",
"url": "https://arxiv.org/abs/1006.0143"
}
|
1006.0171
|
# Advances in Modeling of Scanning Charged-Particle-Microscopy Images
Petr Cizmar András E. Vladár and Michael T. Postek National Institute of
Standards and Technology (NIST) 111 Contribution of the National Institute of
Standards and Technology; not subject to copyright. Certain commercial
equipment is identified in this report to adequately describe the experimental
procedure. Such identification does not imply recommendation or endorsement by
the National Institute of Standards and Technology, nor does it imply that the
equipment identified is necessarily the best available for the purpose. 100
Bureau Drive Gaithersburg MD 20899 USA
###### Abstract
Modeling artificial scanning electron microscope (SEM) and scanning ion
microscope images has recently become important. This is because of the need
to provide repeatable images with a priori determined parameters. Modeled
artificial images are highly useful in the evaluation of new imaging and
metrological techniques, like image-sharpness calculation, or drift-corrected
image composition (DCIC). Originally, the NIST-developed artificial image
generator was designed only to produce the SEM images of gold-on-carbon
resolution sample for image-sharpness evaluation. Since then, the new improved
version of the software was written in C++ programming language and is in the
Public Domain. The current version of the software can generate arbitrary
samples, any drift function, and many other features. This work describes
scanning in charged-particle microscopes, which is applied both in the
artificial image generator and the DCIC technique. As an example, the
performance of the DCIC technique is demonstrated.
## 1 Introduction
Computational scanning electron microscopy[1] through rapid artificial image
modeling is gaining importance. It is a useful tool for evaluation of imaging
and metrology methods, since real SEMs or other charged-particle microscopes
cannot always provide repeatable images. For example, it is virtually
impossible to obtain two real SEM images that only differ in random noise.
This is usually caused by many perturbing factors like drift, sample charging,
or electro-magnetic fields. The artificial image generator is capable of
modeling all important effects in a deterministic way. One can a priori choose
the drift function, the type [2, 3] and magnitude and type of noise, the
charged-particle-beam profile, etc. That being the case, computer generated
artificial images may be input to the imaging and metrological techniques and
the results compared to the chosen parameters, hence indicating the
performance of given techniques. None of these is possible with the real
images, where these effects are present there, but all are random and often
even unknown.
An advanced version of the artificial SEM image generator [4, 5] has been
released as a public-domain software. It is implemented as a library written
in C++. This also allows for linking with programs written in many other
programming languages. The software works in Linux, Mac OSX, Windows, and very
probably in other UN*X systems as well, however, the latter has not yet been
tested. For faster and easier designing of calculations, Lua[6] scripting was
implemented. Lua is a scripting language originally designed for data-entry
applications. These days it is mostly employed in computer games. It is one of
the simplest and fastest scripting languages available. A simple graphical
user interface (GUI) has been written mainly for demonstration. One can very
easily generate images of two types; gold-on-carbon resolution sample and
periodic semiconductor cross structures. The GUI depends on wxWidgets[7]
library which is multiplatform as well.
One of the techniques that have been tested with modeled images is the drift-
corrected image composition (DCIC)[8], which outputs significantly more
accurate images than the traditional imaging techniques. This is necessary for
sub-nanometer-scale metrology, since the conventional “slow-scan” and “fast-
scan” techniques provide images that are often distorted or blurry. The DCIC
works with frames that are taken as quickly as the capabilities of the
instrument permit. Physical drift causes displacement between each couple of
frames. This displacement is searched for with cross-correlation. Since the
quickly acquired frames are usually extensively noisy, a noise reduction is a
part of the DCIC technique.
## 2 Drift Distortion
In the scanning microscopes, the image is formed by scanning across the sample
in a raster pattern. Intensity value is acquired at each location on the
sample. In digital scanning microscopes, that corresponds with a pixel in the
image. The intensity value $\xi(\vec{r})$ depends on the landing position of
the electron beam $\vec{r}$. Most SEMs use the raster pattern. Let the raster
pattern be defined by the time-dependent vector function:
$\displaystyle\vec{r}_{r}(t)$ $\displaystyle=$ $\displaystyle
M\left(x(t)\vec{e}_{x}+y(t)\vec{e}_{y}\right),$ (1) $\displaystyle t_{p}$
$\displaystyle=$ $\displaystyle t_{D}+t_{d},$ $\displaystyle y(t)$
$\displaystyle=$
$\displaystyle\left\lfloor\frac{t}{Xt_{p}+t_{j}}\right\rfloor,$ (2)
$\displaystyle x(t)$ $\displaystyle=$
$\displaystyle\left\lfloor\frac{t}{t_{p}}\right\rfloor-Xy(t),$ (3)
$\displaystyle 0\leq$ $\displaystyle t$ $\displaystyle\leq Y(Xt_{p}+t_{j}),$
where $t$ is time, $M$ is a single-pixel step length. $x$ and $y$ are column
and row indexes in the SEM image. $\vec{e_{x}}$ and $\vec{e_{y}}$ are the unit
vectors in x- and y-direction, $t_{D}$ is the pixel-dwell time, $t_{d}$ is the
dead time between acquisition of two pixels, $t_{j}$ is the time needed to
move the beam to the beginning of the new line. $\lfloor q\rfloor$ is a symbol
for the ${\rm floor}(q)$ function as used in programming languages. $X$ and
$Y$ are the pixel-width and pixel-height of the SEM image.
Let the SEM imaging be defined as a relation between the intensity map of the
sample $\xi(\vec{r})$ and the SEM image $I(x,y)$:
$I(x(t),y(t))=K\xi(\vec{r}(t)).$ (4)
The relation between $I$ and $\xi$ may in practice be very general. For
simplicity, let $K$ be a constant in this manuscript, since this does not
affect generality of the DCIC technique. In the ideal case:
$\vec{r}(t)=\vec{r}_{r}(t)$; however, drift and space distortions are always
present in scanning microscopes and they can significantly affect the position
$\vec{r}$:
$\vec{r}(t)=\vec{r}_{r}(t)+\vec{D}_{d}(t)+\vec{D}_{s}(\vec{r}_{r}).$ (5)
The space distortion $\vec{D}_{s}$ is constant in time and may be compensated
for, when its function is known. This distortion may be caused by non-
linearities in deflection amplifiers and is significant mostly at low
magnifications. On the other hand, the drift distortion $\vec{D}_{d}$ is
changing in time, its function is usually unknown, and it may extensively
affect the high-magnification images. The drift distortion may arise from
several sources; e.g. translational motion of the sample, tilt or deformation
of the electron-optical column, outer forces and vibrations, or temperature
expansion. High-magnification images are very sensitive to drift distortion,
since microscopic displacements, tilts, or temperature changes can easily
cause nanometer distortions and displacements, which can significantly impair
the SEM image and its usability for nanometer-scale measurements.
Figure 1: Series of artificial images of a semiconductor structure composed
using the traditional “fast-scan” technique. Compositions of 2, 4, 8, 16, 32,
64, 128, 256, and 512 frames (from the top left). Images are normalized.
Figure 2: Series of artificial images of a semiconductor structure composed
using the DCIC method. Compositions of 2, 4, 8, 16, 32, 64, 128, 256, and 512
frames (from the top left).
The drift-distortion function is generally unknown, however, since it
characterizes motion of physical bodies, it must be continuous and thus
square-integrable. Therefore, drift-distortion function may be Fourier-series
expanded:
$\displaystyle D_{cd}(t)$ $\displaystyle=$
$\displaystyle\sum\limits_{n=-\infty}^{\infty}c_{n}{\rm e}^{-{\rm i}nt},$ (6)
$\displaystyle\vec{D}_{d}$ $\displaystyle=$
$\displaystyle\Re(D_{cd})\vec{e}_{x}+\Im(D_{cd})\vec{e}_{y},$ (7)
$\displaystyle U$ $\displaystyle\propto$
$\displaystyle\sum_{n=-\infty}^{\infty}c_{n}^{2}n^{2},$ (8)
where $c_{n}$ are the (complex) Fourier coefficients, $U$ is the overall
energy of the drifting system. Since $U$ is limited, for high $n$ the
coefficients $c_{n}$ must be nearing zero. In practice, for frequencies higher
than 200 Hz, $c_{n}$ correspond to noise only and are negligible. Therefore,
the $D_{cd}(t)$ can be written:
$D_{cd}(t)\approx\sum\limits_{n=-N}^{N}c_{n}{\rm e}^{-{\rm i}nt},\\\ $ (9)
where $N$ represents the highest significant angular frequency.
## 3 “Fast-scan” Imaging
The imaged intensity signal in the SEM always contains noise. The intensity
function is a superposition of a real signal and noise:
$\xi(\vec{r},t)=\xi_{s}(\vec{r})+\xi_{n}(t),$ (10)
where $\xi_{s}$ is the position-dependent real signal and $\xi_{n}$ is the
time-dependent noise. $\xi_{n}$ is a superposition of all noise contributions
present in the SEM: Poisson noise originating from the electron source and the
secondary emission, the noise originating from the amplifier and electronics,
quantization-error noise, etc. Due to the central limit theorem, it is
legitimate to suppose that the mean value of this noise is zero:
$<\xi_{n}(t)>=0.$ (11)
In order to obtain a SEM image with a desired level of noise, the overall
pixel dwell-time $t_{D}$ must be sufficiently high. Unfortunately, the
electron yield is usually low and the overall pixel-dwell time must often be
set to times ranging from tens to several hundreds of $\mu$s.
In the SEM, there are two common methods to achieve this, i.e “slow-scan” and
“fast scan”, while the latter is useful for metrological application.
“Fast-scan” is one of the common imaging methods in SEMs. The image is
composed from multiple ($N$) frames, for which averaging is the mostly applied
technique. The frames are acquired with the lowest possible pixel-dwell time
$t_{D}$. The image pixel value is an average of corresponding frame-pixel
values:
$\displaystyle I_{k}(x(t_{0}),y(t_{0}))$ $\displaystyle=$ $\displaystyle
K\xi_{s}(\vec{r}(t_{0}+kt_{f}))+$ (12) $\displaystyle+$ $\displaystyle
K\xi_{n}(t_{0}+kt_{f}),$ $\displaystyle I(x,y)$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{k=0}^{N}I_{k}(x,y).$ (13) $\displaystyle t_{f}$
$\displaystyle=$ $\displaystyle Y(Xt_{p}+t_{j})+t_{jj},$ (14)
$t_{f}$ is a time period between beginnings of acquisition of two following
frames, $t_{jj}$ is the dead time between the end of acquisition of one frame
and beginning of the next one. Considering Eq (11), the higher $N$, the lower
noise level is present in the image. The required noise-level thus determines
the number of composed images $N$. For high $N$:
$\sum_{k=0}^{N-1}\xi_{n}(t_{0}+kt_{f})\approx 0.\\\ $ (15)
Because the scanning raster pattern is constant for all frames,
$\vec{r}_{r}(t_{0}+kt_{f})=\vec{r}(t_{0}).\\\ $ (16)
Eq (12) may be expanded:
$\displaystyle I(x(t_{0}),y(t_{0}))$ $\displaystyle=$
$\displaystyle\frac{K}{N}\sum_{k=0}^{N-1}\xi_{s}[\vec{r}_{r}(t_{0})+$ (17)
$\displaystyle+$
$\displaystyle\vec{D}_{s}(\vec{r}_{r}(t_{0}))+\vec{D}_{d}(t_{0}+kt_{f})].$
With current SEMs, the frame-acquisition time $t_{f}$ can be much lower than
the period of even the highest drift-distortion frequencies. The drift-
distortion within the single-frame acquisition time is then minimal. However,
it becomes significant during acquisition of the whole image, especially, when
the dead times $t_{jj}$ are prohibitively high, which is the case even with
some current instruments.
## 4 Drift-Corrected Image Composition (DCIC)
The “fast-scan” method may be significantly improved using drift-distortion
correction, when the images are acquired quickly enough. Since the space-
distortion $\vec{D}_{s}$ is much less pronounced and much smaller that the
drift-distortion $\vec{D}_{d}$ at very high magnifications, it will be
neglected from now on. The Eq (17) then becomes:
$\displaystyle I(x,y)$ $\displaystyle=$
$\displaystyle\frac{K}{N}\sum_{k=0}^{N-1}\xi_{s}[\vec{r}_{r}(r)+\vec{D}_{dk}],$
(18) $\displaystyle\vec{D}_{dk}$ $\displaystyle=$
$\displaystyle\vec{D}_{d}(t_{0}+kt_{f}).$ (19)
The image is in this case the mean value of $N$ displaced images.
Under certain conditions, it is possible to find the displacement vectors of
the images, which are equal to the drift-distortion values $\vec{D}_{dk}$. The
drift-distortion then may be compensated for, which allows for acquisition of
a corrected, more accurate image. One possible approach is a cross-
correlation-based displacement detection, which is used in the DCIC technique.
The maximum of the cross-correlation function is searched for. Its position is
equal to the searched displacement vector $\vec{D}_{dk}$.
In the DCIC technique, the cross-correlation with noise reduction is applied.
This is necessary, because the quickest-acquired images are usually very noisy
and the peak in the cross-correlation function becomes overridden by numerous
other peaks, corresponding to random correlation of noise. This often makes
finding the displacement vector impossible. This issue can be tackled by low-
pass frequency filtering performed in the frequency domain. The cut-off
frequency is determined by the filter-radius $R$.
Figure 3: Error distribution of the displacement detection. Artificial SEM
image of a periodic semiconductor sample was used.
Plain maximum search in a discrete function limits the accuracy to a minimum
of one pixel. However, in the DCIC, the detection of the displacement vector
$\vec{D}_{dk}$ is performed with sub-pixel resolution. The peak in the two-
dimensional cross-correlation function is interpolated with a polynomial
third-order two-dimensional polynomial function and the algorithm then
searches for its maximum.
The technique is very powerful, since it can correct for the drift-related
distortions and blur in extremely noisy images. (See Figs 1 and 2)
## 5 Accuracy of the DCIC technique
The accuracy of the detected displacement vector $\vec{D}_{dk}$ characterizes
the accuracy of the DCIC imaging technique. Errors in the displacement vector
can cause blur. Such blur can under certain circumstances be larger than with
application of the original “fast-scan” technique ($\vec{D}_{dk}=\vec{0}$). In
metrological applications, where dimensions are measured from the images, the
drift-related displacement is the main source of errors.
The artificial SEM images have been successfuly used to evaluate accuracy of
the DCIC technique. The artificial-image generator is, unlike any other source
of SEM images, capable of modeling all necessary characteristics for this
application, e.g. arbitrary drift functions, dead times, arbitrary types of
samples, etc.
A performance characteristics must be chosen to investigate the limits of an
imaging technique. Application of the standard deviation of the displacement
vector would be a good candidate, if the distribution of errors was Gaussian.
In order to find this out, a large set of artificial images (500 000) randomly
differing in displacement and noise has been applied to find the error
distribution of the displacement detection. The DCIC technique has processed
all generated frames and has output corresponding displacement values. The
two-dimensional histogram of these values forms the resulting distribution,
which is shown in the Fig 3. These data have clearly indicated that the error
distribution is not (always) Gaussian. Using standard (Gaussian) error
processing has therefore been unsuitable and thus we have chosen the mean
error $\delta_{D}$ as the performance characteristics.
$\bar{\delta}_{D}=\frac{1}{N-1}\sum_{k=1}^{N-1}\delta_{Dk},$ (20)
where $\delta_{Dk}$ is the error of the displacement vector $\vec{D}_{dk}$ and
$N$ is the number of frames. Since the correct displacement vector
$\vec{D}_{ci}$ is known (it is determined by the artificial-image generator),
$\delta_{Dk}=|\vec{D}_{dk}-\vec{D}_{ck}|.$ (21)
The performance of the DCIC technique is obviously limited, because noise,
blur, contrast, and other parameters affect it significantly. For instance, if
the frames were extremely blurred and the cross-correlation maximum would be
overly wide and the mean error of the displacement vector would be excessively
high. It is therefore useful to find the dependences of $\delta_{D}$ on noise
and blur and provide a set of limiting parameters.
Figure 4: Evaluation of the DCIC technique. Dependence of the mean displacement detection error on the magnitude ($\sigma_{g}$) of Gaussian noise. Each represents 5000 artificial images of the gold-on-carbon resolution sample sized 512x512 pixels. The error-bars denote the standard deviation of the displacement vector detection error. $\sigma_{g}=10^{-2}$ | $\sigma_{g}=10^{-1}$ | $\sigma_{g}=1$ | $\sigma_{g}=10$
---|---|---|---
$SNR=60$ | $SNR=6$ | $SNR=0.6$ | $SNR=0.06$
$SNR_{dB}=17.8{\rm~{}dB}$ | $SNR_{dB}=7.78{\rm~{}dB}$ | $SNR_{dB}=-2.22{\rm~{}dB}$ | $SNR_{dB}=-12.2{\rm~{}dB}$
| | |
Figure 5: Gaussian-noise scale. Artificial images of the gold-on-carbon
resolution sample with Gaussian noise of different magnitudes. Figure 6:
Evaluation of the DCIC technique. Dependence of the mean displacement
detection error on Gaussian blur ($\sigma_{b}$). This blur simulates the
effect of the charged-particle-beam profile. Each represents 5000 artificial
images of the gold-on-carbon resolution sample sized 512x512 pixels. The
error-bars denote the standard deviation of the displacement vector detection
error.
The dependence of the mean error of the detected displacement
$\bar{\delta}_{D}$ on noise and blur have been both investigated with
application of artificial images. Gaussian noise and Gaussian blur have been
chosen for simplicity, although the type of noise and the blur profile may be
arbitrary. For every step in noise and blur, 5000 artificial images of the
gold-on-carbon resolution sample have been generated and processed by the DCIC
algorithm. The results of these tests are shown in Figs 4 and 6. For reference
images showing different magnitudes of Gaussian noise see Fig 5. These tests
demonstrate the capability of the DCIC technique to find the displacements
with sub-pixel accuracy. In the noise test, this is maintained up to the
$\sigma_{g}=8$, which roughly corresponds to signal to noise ratio (SNR)
around 0.1 and the dependence is almost linear. The dependence on blur
indicates that the sub-pixel accuracy is sustained up to $\sigma_{b}=14$.
## 6 Conclusion
Modeled artificial SEM images were first employed in assessment of the image-
sharpness calculation techniques[1] and have been adopted as a part of the
developed international standard for image sharpness. Since then, a new highly
improved version of the software was written. This version supports arbitrary
non-overlapping two-dimensional samples, rigorous generation of Poisson and
Gaussian noise, arbitrary drift functions, dead times and other features.
Scripting in Lua scripting language was implemented to make the calculations
easier to design. This new tool was then used in evaluation of the new imaging
technique of DCIC. By finding dependence of the error in detection of the
displacement on noise and blur, the sub-pixel accuracy was demonstrated even
for high magnitudes of noise or blur. This makes the DCIC and modeling of
microscope images useful and important tools for nanoscale metrology and
nanotechnology.
## References
* [1] M. T. Postek, A. E. Vladar, J. R. Lowney, and W. J. Keery, “Two-dimensional simulation and modeling in scanning electron microscope imaging and metrology research,” Scanning 24, pp. 179–185, JUL-AUG 2002.
* [2] G. E. P. Box and M. E. Muller, “A Note on the Generation of Random Normal Deviates,” Annals of Mathematical Statistics 29(2), pp. 610–611, 1958.
* [3] D. E. Knuth, Art of Computer Programming, Volume 2: Seminumerical Algorithms, Addison-Wesley Professional, third ed., November 1997.
* [4] P. Cizmar, A. E. Vladar, B. Ming, and M. T. Postek, “Simulated SEM Images for Resolution Measurement,” Scanning 30, pp. 381–391, Sep-Oct 2008.
* [5] P. Cizmar, A. E. Vladar, and M. T. Postek, “Optimization of accurate sem imaging by use of artificial images,” Scanning Microscopy 2009 7378(1), p. 737815, SPIE, 2009.
* [6] R. Ierusalimschy, L. H. de Figueiredo, and W. Celes, Lua 5.1 Reference Manual, Lua.org, 2006.
* [7] J. Smart, K. Hock, and S. Csomor, Cross-Platform GUI Programming with wxWidgets, Prentice Hall, 2005.
* [8] P. Cizmar, A. E. Vladar, and M. T. Postek, “Real-Time Image Composition with Correction of Drift Distortion,” ArXiv e-prints , Oct. 2009.
|
arxiv-papers
| 2010-05-14T14:48:22 |
2024-09-04T02:49:10.752097
|
{
"license": "Public Domain",
"authors": "Petr Cizmar, Andras E. Vladar, and Michael T. Postek",
"submitter": "Petr Cizmar",
"url": "https://arxiv.org/abs/1006.0171"
}
|
1006.0261
|
# Short-range force detection using optically-cooled levitated microspheres
Andrew A. Geraci aageraci@boulder.nist.gov Scott B. Papp John Kitching Time
and Frequency Division, National Institute of Standards and Technology,
Boulder, CO 80305 USA
###### Abstract
We propose an experiment using optically trapped and cooled dielectric
microspheres for the detection of short-range forces. The center-of-mass
motion of a microsphere trapped in vacuum can experience extremely low
dissipation and quality factors of $10^{12}$, leading to yoctonewton force
sensitivity. Trapping the sphere in an optical field enables positioning at
less than $1$ $\mu$m from a surface, a regime where exotic new forces may
exist. We expect that the proposed system could advance the search for non-
Newtonian gravity forces via an enhanced sensitivity of $10^{5}-10^{7}$ over
current experiments at the $1$ $\mu$m length scale. Moreover, our system may
be useful for characterizing other short-range physics such as Casimir forces.
###### pacs:
04.80.Cc,07.10.Pz,42.50.Pq
Since the pioneering work of Ashkin and coworkers in the 1970s ashkin1 ,
optical trapping of dielectric objects has become an extraordinarily rich area
of research. Optical tweezers are used extensively in biophysics to study and
manipulate the dynamics of single molecules, and in soft condensed-matter
physics to study macromolecular interactions grier ; block . Much recent work
has focused on trapping in solution where strong viscous damping dominates
particle motion. There has also been interest in extending the regime that
Ashkin and coworkers opened, namely, trapping sub-wavelength particles in
vacuum where particle motion is strongly decoupled from a room-temperature
environment ashkin1 ; beadexpts .
Recent theoretical studies have suggested that nanoscale dielectric objects
trapped in ultrahigh vacuum might be cooled to their ground state of (center-
of-mass) motion via radiation pressure forces of an optical cavity kimble ;
cirac . This remarkable result is made possible by isolation from the thermal
bath, robust decoupling from internal vibrations, and lack of a clamping
mechanism. In fact, a trapped dielectric nanosphere has been predicted to
attain an ultrahigh mechanical quality factor $Q$ exceeding $10^{12}$ for the
center-of-mass mode, limited by background gas collisions. Such large $Q$
factors enable cavity cooling, for which the lowest possible phonon occupation
of the mechanical oscillator is $n_{T}/Q$, where $n_{T}$ is the number of
room-temperature thermal phonons. Although such $Q$ factors have yet to be
observed in experiment, optically levitated microspheres have been trapped in
vacuum for lifetimes exceeding $1000$ s ashkin1 and electrically levitated
microspheres have exhibited pressure-limited damping that is consistent with
theoretical predictions down to $\sim 10^{-6}$ Torr kendall .
In addition to being beneficial for ground-state cooling and studies of
quantum coherence in mesoscopic systems, mechanical oscillators with high
quality factors also enable sensitive resonant force detection rugar2 ; yocto
. Optically levitated microspheres in vacuum have been studied theoretically
in the context of reaching and exceeding the standard quantum limit of
position measurement libbrecht . In this paper, we discuss the force sensing
capability of a microsphere trapped inside a medium-finesse optical cavity at
ultra-high vacuum, and propose an experiment that could extend the search for
non-Newtonian gravity-like forces that may occur at micron scale distances.
Such forces could be mediated by particles residing in sub-millimeter scale
extra spatial dimensions add or by moduli in the case of gauge-mediated
supersymmetry breaking sg . The apparatus we propose is also well suited to
studying Casimir forces Casimir , and may be useful for studying radiative
heat transfer at the nano-scale heatxfer .
Corrections to Newtonian gravity at short range are generally parameterized
according to a Yukawa-type potential
$V=-\frac{G_{N}m_{1}m_{2}}{r}\left[1+\alpha e^{-r/\lambda}\right],$ (1)
where $m_{1}$ and $m_{2}$ are two masses interacting at distance $r$, $\alpha$
is the strength of the potential relative to gravity, and $\lambda$ is the
range of the interaction. For two objects of mass density $\rho$ and linear
dimesion $\lambda$ with separation $r\approx\lambda$, a Yukawa-force scales
roughly as $F_{Y}\sim G_{N}\rho^{2}\alpha\lambda^{4}$, decreasing rapidly with
smaller $\lambda$. For example, taking gold masses, for an interaction
potential with $\alpha=10^{5}$ and $\lambda=1$ $\mu$m, $F_{Y}\sim 10^{-21}$ N.
As we will discuss, the thermal-noise-limited force sensitivity of micron
scale, optically levitated silica micro-spheres at $300$ K with $Q=10^{12}$
can be of order $\sim 10^{-21}$ N$/\sqrt{{\rm{Hz}}}$, and therefore allows
probing deep into unexplored regimes. For instance, current experimental
limits at $\lambda=1$ $\mu$m have ruled out interactions with $|\alpha|$
exceeding $10^{10}$.
Figure 1: (color online) a) Proposed experimental geometry. A sub-wavelength
dielectric microsphere of radius $a$ is trapped with light in an optical
cavity. The sphere is positioned at an anti-node occurring at distance $z$
from a gold-coated SiN membrane. Light of a second wavelength
$\lambda_{\rm{cool}}=2\lambda_{\rm{trap}}/3$ is used to simultaneously cool
and measure the center of mass motion of the sphere. The sphere displacement
$\delta z$ results in a phase shift $\delta\phi$ in the cooling light
reflected from the cavity. For the short-range gravity measurement, a source
mass of thickness $t$ with varying density sections is positioned on a
moveable element behind the mirror surface that oscillates harmonically with
an amplitude $\delta y$. The source mass is coated with a thin layer of gold
to provide an equipotential. (b) Displacement spectral density (blue) due to
thermal noise and shot-noise limited displacement sensitivity (flat line, red)
for parameters discussed in the text.
The proposed experimental setup is shown schematically in Fig. 1. A dielectric
microsphere of radius $a=150$ nm is optically levitated and cooled in an
optical cavity of length $L$ by use of two light fields of wavenumbers
$k_{t}=2\pi/\lambda_{\rm{trap}}$ and $k_{c}=2\pi/\lambda_{\rm{cool}}$,
respectively. The silica microsphere has density $\rho=2300$ kg/m3, dielectric
constant $\epsilon=2$, and is trapped near the position of the closest anti-
node of the cavity trapping field to a gold mirror surface. The mirror is a
$200$ nm thick SiN membrane coated with $200$ nm of gold. A source mass of
thickness $t=5$ $\mu$m and length $20$ $\mu$m with varying density sections of
width $2$ $\mu$m (e.g., Au and Si) is positioned at edge-to-edge separation
$d=1$ $\mu$m from the sphere. Below we describe trapping and cooling of the
microsphere’s center-of-mass motion, detection of Casimir forces between the
microsphere and gold mirror, and the search for gravity-like forces on the
microsphere due to the source mass.
Following Ref. kimble , the sub-wavelength dielectric particle has a center-
of-mass resonance frequency $\omega_{0}=\left[\frac{6k_{t}^{2}I_{t}}{\rho
c}{\mathcal{R}}e\frac{\epsilon-1}{\epsilon+2}\right]^{1/2}$, where $I_{t}$ is
the intracavity intensity of the trapping light. The trap depth is
$U=\frac{3I_{t}V}{c}\frac{\epsilon-1}{\epsilon+2}$, where $V$ is the volume of
the microsphere. For concreteness, we consider a cavity of length $L=0.15$ m,
finesse ${\mathcal{F}}=200$, driven with a trapping laser of power $P_{t}=2$
mW and wavelength $\lambda_{\rm{trap}}=1.5$ $\mu$m. We choose a cavity mode
waist $w=15$ $\mu$m. The Gaussian profile of the trapping beam near the mode
waist provides transverse confinement, with an oscillation frequency of $\sim
1$ kHz. Tighter transverse confinement could be established by use of a
transverse standing wave potential. The cooling light has input power
$P_{c}=48$ $\mu$W, and an optimized red detuning of $\delta=-0.23\kappa$,
where the cavity decay rate is $\kappa=\pi c/L{\mathcal{F}}$. The cooling
light causes a slight shift $z_{0}$ in the axial equilibrium position of the
microsphere, given by
$z_{0}=\frac{1}{2}\frac{k_{c}}{k_{t}^{2}}\frac{I_{c}}{I_{t}}\approx 2$ nm,
where $I_{c}$ is the intracavity intensity of the cooling mode. The
optomechanical coupling of the cooling mode is
$g=\frac{3V}{4V_{c}}\frac{\epsilon-1}{\epsilon+2}\omega_{c},$ where $V_{c}=\pi
w^{2}L/4$ is the cavity mode volume kimble , and $\omega_{c}=k_{c}c$. The
optimum detuning is determined by minimizing the final phonon occupancy
$n_{f}$, which depends on the laser-cooling rate and heating due to photon
recoil from light scattered by the sphere. Additional cavity loss due to
photon scattering is negligible: $\sim 10^{-3}\kappa$ for our parameters.
Values of the trapping and cooling parameters appear in Table 1.
A mechanical oscillator with frequency $\sim 37$ kHz and $Q\sim 10^{12}$ will
respond to perturbations with a characteristic time scale of
$2Q/\omega_{0}\sim 10^{7}$ s. The cooling serves both to damp the $Q$ factor
so that perturbations to the system ring down within reasonably short periods
of time, and to localize the sphere by reducing the amplitude of the thermal
motion. Because of the low cavity finesse, the cooling is not in the resolved
sideband regime. Still, for the parameters discussed above the phonon
occupation of the microsphere oscillation is reduced by a factor of over
$10^{5}$. This corresponds to operating with an effective $Q_{\rm{eff}}\approx
10^{5}$ and ring down time of $\approx 1$ s. Cooling of the transverse motion
is also required, as the rms position spread must be maintained to be less
than $\sim 0.1$ $\mu$m. We imagine this can be done with active feedback to
modulate the power of a transverse trapping laser using the signal from a
transverse position measurement, for example generated by measuring scattered
light incident on a quadrant photodiode. A modest cooling factor of $\approx
1000$ in the transverse directions is sufficient to yield the required
localization.
The cooling light is also used to detect the position of the sphere. The phase
of the cooling light reflected from the cavity is modulated by microsphere
motion through the optomechanical coupling
$\partial{\omega_{c}}/\partial{z}=2k_{c}g$. Photon shot-noise limits the
minimum detectable phase shift to $\delta\phi\approx 1/(2\sqrt{I})$ where
$I\equiv P_{c}/(\hbar\omega_{c})$ hadjar . The corresponding photon shot-noise
limited displacement sensitivity is
$\sqrt{S_{z}(\omega)}=\frac{\kappa}{4k_{c}g}\frac{1}{\sqrt{I}}\sqrt{1+\frac{4\omega^{2}}{\kappa^{2}}}$
hadjar , for an impedance matched cavity. This displacement sensitivity is
generally well below the thermal noise limited sensitivity, as shown in Fig.
1(b). We assume that substrate vibrational noise, electronics noise and laser
noise can be controlled at a level comparable to the photon shot noise. The
minimum detectable force due to thermal noise at temperature $T_{\rm{eff}}$ is
$F_{\rm{min}}=\sqrt{\frac{4kk_{B}T_{\rm{eff}}b}{\omega_{0}Q_{\rm{eff}}}}$,
where $k$ is the center-of-mass mode spring constant, and $b$ is the bandwidth
of the measurement. We assume an initial center-of-mass temperature
$T_{\rm{CM}}=300$ K, and that $Q\approx\omega_{0}/\gamma_{g}$ is limited by
background gas collisions, with loss rate
$\gamma_{g}=16P_{\rm{gas}}/(\pi\bar{v}\rho a)$ epstein , for a background air
pressure of $P_{\rm{gas}}=10^{-10}$ Torr and rms gas velocity $\bar{v}$.
Cooling the center-of-mass mode to $T_{\rm{eff}}=0.9$ mK results in
$F_{\rm{min}}\sim 10^{-21}$ N$/\sqrt{\rm{Hz}}$ as shown in Table 1. In this
regime $F_{\rm{min}}$ scales linearly with the sphere radius $a$.
The microsphere absorbs optical power from both the trapping and cooling light
in the cavity, which results in an increased internal temperature
$T_{\rm{int}}$. Assuming negligible cooling due to gas collisions, the
absorbed power is re-radiated as blackbody radiation. $T_{\rm{int}}$ is listed
in Table 1 for fused silica with dielectric response
$\epsilon=\epsilon_{1}+i\epsilon_{2}$, with $\epsilon_{1}=2$ and
$\epsilon_{2}=1.0\times 10^{-5}$ as in Ref. fusedsilicaloss , and
$\epsilon_{\rm{bb}}=0.1$ as in Ref. kimble , for an environmental temperature
$T_{\rm{ext}}=300$ K. We assume $T_{\rm{int}}$ and $T_{\rm{CM}}$ are not
significantly coupled over the time scale of the experimental measurements at
$P_{\rm{gas}}\sim 10^{-10}$ Torr.
Parameter | Units | Value
---|---|---
$\lambda_{\rm{trap}}$ | $\mu$m | $1.5$
$U/k_{B}$ | K | $3.7\times 10^{3}$
$\omega_{0}/2\pi$ | Hz | $3.7\times 10^{4}$
$T_{\rm{int}}$ | K | $900$
$Q,(Q_{\rm{eff}})$ | - | $6.1\times 10^{11},(1.0\times 10^{5})$
$\delta/\kappa$ | - | $-0.23$
$n_{T},(n_{f})$ | - | $1.7\times 10^{8}$,$(510)$
$\sqrt{S_{z}}$ | m$/\sqrt{\rm{Hz}}$ | $4.7\times 10^{-13}$
$F_{\rm{min}}$ | N$/\sqrt{\rm{Hz}}$ | $1.9\times 10^{-21}$
$z_{\rm{th}}$ | m$/\sqrt{\rm{Hz}}$ | $2.6\times 10^{-11}$
Table 1: Parameters for trapping and cooling a silica sphere with radius
$a=150$ nm.
Casimir Force. The Casimir force between a dielectric sphere and metal plane
can be written using the proximity force approximation (PFA) as Casimir
$F_{\rm{c}}=-\eta\frac{\pi^{3}a\hbar c}{360(z-a)^{3}}$ in the limit that
$(z-a)\ll a$. The prefactor $\eta$ characterizes the reduction in the force
compared with that between two perfect conductors lambrecht . For $z\gg a$,
the force takes the Casimir-Polder casimirpolder form
$F_{\rm{cp}}=-\frac{3\hbar c\alpha_{V}}{8\pi^{2}\epsilon_{0}}\frac{1}{z^{5}}$,
where $\alpha_{V}=3\epsilon_{0}V\frac{\epsilon-1}{\epsilon+2}$ is the electric
polarizability. Our setup is capable of probing the unexplored transition
between these two regimes, and of testing the PFA, which is expected to be
valid for $z-a\lesssim a$ jaffe . To estimate $\eta$, we adopt a similar
approach to that taken in Ref. lambrecht to determine the force between a
metal and dielectric plate. We assume dielectric spheres separated from a
metal mirror will have a similar pre-factor. Taking an infinite plate with
$\epsilon=2$ and thickness $2a$, and another with gold of thickness $200$ nm,
we find $\eta\approx 0.13$ at $(z-a)=225$ nm.
For a sphere located at the position of the first anti-node of the trapping
field, the Casimir force displaces the equilibrium position by approximately
$-3$ nm. The gradient of the Casimir force near the static mirror surface
produces a fractional shift in the resonance frequency of the sphere given by
$|\delta\omega_{0}/\omega_{0}|=\frac{|\partial{F_{\rm{c}}}/\partial{z}|}{2k}$.
A similar frequency shift has been measured for an atomic sample harber . The
shift is shown in Fig. 2 as a function of mirror separation $(z-a)$ for
$\eta=0.13$. The minimum detectable frequency shift due to thermal noise is
given by
$|\delta\omega_{0}/\omega_{0}|_{\rm{min}}=\sqrt{\frac{k_{B}T^{\rm{f}}_{\rm{CM}}b}{k\omega_{0}Q_{\rm{eff}}z_{\rm{rms}}^{2}}}$.
For $z_{\rm{rms}}=5$ nm, $|\delta\omega_{0}/\omega_{0}|\approx 10^{-7}$ can be
detected in $\sim 1$ s. Other sources of systematic frequency shifts near the
surface, for example from variation of the cavity finesse with bead position
or from diffuse scattered light on the gold surface, would need to be
experimentally characterized. Also, surface roughness of the microsphere can
modify the Casimir force neto . Rotation of the microsphere may lead to an
effective averaging over surface inhomogeneities.
Figure 2: (color online) Fractional frequency shift due to Casimir interaction
of microspheres of radius $a=150$ nm at distance $z$ from the gold surface.
The PFA is expected to be valid at short distances, while the Casimir-Polder
form is expected at large distances, and the transition region is shown as a
shaded area. (inset) Optical and Casimir contributions to the cavity trapping
potential.
Search for non-Newtonian gravity. To generate a modulation of any Yukawa-type
force at the resonance frequency of the center-of-mass mode along $z$, the
source mass is mounted on a cantilever beam that undergoes a lateral tip
displacement of 2.6 $\mu$m at a frequency of $\omega_{0}/3$. The mechanical
motion occurs at a sub-harmonic of the microsphere resonance to avoid direct
vibrational coupling. To estimate the force on the sphere a numerical
integration over the geometry of the masses is performed. For $b=10^{-5}$ Hz,
the estimated search reach is shown in Fig. 3 (a). Several orders of magnitude
of improvement are possible between $0.1$ nm and a few microns, due to the
proximity of the masses and high force sensitivity.
Figure 3: (color online) Experimental constraints decca07 ; decca05 ; masuda08
; lamoreaux97 ; chiaverini03 ; geraci08 ; kapner07 and theoretical
predictions bec for short-range forces due to an interaction potential of
Yukawa form $V=-\frac{G_{N}m_{1}m_{2}}{r}\left[1+\alpha
e^{-r/\lambda}\right]$. Lines (a) and (b) denote the projected improved search
reach for microspheres of radius $a=150$ nm and $a=1500$ nm, respectively.
The source mass surface is coated with $200$ nm of gold in order to screen the
differential Casimir force, depending on which material is directly beneath
the microsphere. Following the method of Ref. lambrecht , the differential
Casimir force is $9\times 10^{-24}$ N, which is comparable to the sensitivity
of the experiment at $10^{-5}$ Hz bandwidth. The gold coating on the cavity
mirror membrane attenuates this Casimir interaction even further. Patch
potentials on the mirror surface and any electric charge on the sphere can
produce spurious forces on the sphere. By translating the position of the
optical trap along the surface, these and other backgrounds, e.g. vibration,
can be distinguished from a Yukawa-type signal, as any Yukawa-type signal
should exhibit a spatial periodicity associated with the alternating density
pattern, similar to the system discussed in Ref. geraci08 .
Increasing the radius of the sphere can significantly enhance the search for
non-Newtonian effects at longer range. Curve (b) in Fig. 3 shows the estimated
search reach that would be obtained by scaling the sphere size by a factor of
10 and positioning it at edge-to-edge separation of $3.8$ $\mu$m from a source
mass with thickness $t=10$ $\mu$m consisting of sections of width $10$ $\mu$m
driven at an amplitude of $13$ $\mu$m. Such a larger sphere could be trapped
in an optical lattice potential with the incident beams at a shallow angle,
instead of in an optical cavity, to enable sub-wavelength confinement. In this
case cooling could be performed by use of active feedback. Alternatively it
may be possible to trap the larger $1.5$ $\mu$m sphere in a cavity by use of
longer wavelength light (e.g., $\lambda_{\rm{trap}}=10.6$ $\mu$m) by choosing
a sphere material such as ZnSe with lower optical loss at this wavelength.
The experiment we have proposed may allow improvement by several orders of
magnitude in the search for non-Newtonian gravity below the $10$ $\mu$m length
scale. An experimental challenge will be to capture and cool individual
microspheres and precisely control their position near a surface. Previous
experimental work has been successful at optically trapping $1.5$ $\mu$m
radius spheres beadexpts , and similar techniques may work for the setup
proposed here. Extrapolating the results of Ref. kendall at $10^{-6}$ Torr
for the system we consider would yield a pressure-limited $Q\sim 10^{9}$. In
the absence of additional damping mechanisms, we expect that $Q\approx
10^{12}$ could be achieved at lower pressure. Further improvements in force
sensitivity may be possible in a cryogenic environment.
We thank John Bollinger and Jeff Sherman for a careful reading of this
manuscript. AG and SP acknowledge support from the NRC.
## References
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* (2) D. G. Grier, Nature 424, 810 (2003).
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* (6) O. Romero-Isart et. al., New J. Phys. 12, 033015 (2010).
* (7) L. D. Hinkle and B. R. F. Kendall, J. Vac. Sci. Technol. A 8, 2802 (1990).
* (8) D. Rugar et. al., Nature 430, 329 (2004).
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* (10) K. G. Libbrecht and E. D. Black, Phys. Lett. A 321, 99 (2004).
* (11) N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Lett. B 429, 263 (1998).
* (12) S. Dimopoulos and G. F. Guidice, Phys.Lett.B 379, 105 (1996).
* (13) H. B. G. Casimir, Proc. Kon. Nederland. Akad. Wetensch. B51, 793 (1948).
* (14) E. Rousseau et. al., Nature Photonics 3, 514 (2009).
* (15) Y. Hadjar et. al., Europhys. Lett. 47, 545 (1999).
* (16) P. S. Epstein, Phys. Rev. 23, 710 (1924).
* (17) R. Kitamura, L. Pilon, and M. Jonasz, Appl. Opt. 46, 8118 (2007).
* (18) A. Lambrecht and S. Reynaud, Eur. Phys. J. D 8, 309 (2000).
* (19) H. B. G. Casimir and P. Polder, Phys. Rev. 73, 360 (1948).
* (20) A. Scardicchio, R. L. Jaffe, Nucl.Phys. B 704, 552 (2005).
* (21) D. M. Harber et. al., Phys. Rev. A 72, 033610 (2005).
* (22) P. A. Maia Neto, A. Lambrecht, and S. Reynaud, Europhys. Lett. 69, 924 (2005).
* (23) R. S. Decca et. al., Phys. Rev. D 75, 077101 (2007).
* (24) R. S. Decca et. al., Phys. Rev. Lett. 94, 240401 (2005).
* (25) M. Masuda and M. Sasaki, Phys. Rev. Lett. 102, 171101 (2009).
* (26) S. K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997).
* (27) J. Chiaverini et. al., Phys. Rev. Lett. 90, 151101 (2003).
* (28) A. A. Geraci et. al., Phys. Rev. D 78, 022002 (2008).
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|
arxiv-papers
| 2010-06-01T22:01:20 |
2024-09-04T02:49:10.759581
|
{
"license": "Public Domain",
"authors": "Andrew A. Geraci, Scott B. Papp, and John Kitching",
"submitter": "Andrew A. Geraci",
"url": "https://arxiv.org/abs/1006.0261"
}
|
1006.0386
|
# A Smart Approach for GPT Cryptosystem Based on Rank Codes
Haitham Rashwan Department of Communications
InfoLab21, South Drive
Lancaster University
Lancaster UK LA1 4WA
Email: h.rashwan@lancaster.ac.uk Ernst M. Gabidulin Department of Radio
Engineering
Moscow Institute
of Physics and Technology
(State University)
141700 Dolgoprudny, Russia
Email: gab@mail.mipt.ru Bahram Honary Department of Communications
InfoLab21, South Drive
Lancaster University
Lancaster UK LA1 4WA
Email: b.honary@lancaster.ac.uk
###### Abstract
The concept of Public- key cryptosystem was innovated by McEliece’s
cryptosystem. The public key cryptosystem based on rank codes was presented in
1991 by Gabidulin –Paramonov–Trejtakov (GPT). The use of rank codes in
cryptographic applications is advantageous since it is practically impossible
to utilize combinatoric decoding. This has enabled using public keys of a
smaller size. Respective structural attacks against this system were proposed
by Gibson and recently by Overbeck. Overbeck’s attacks break many versions of
the GPT cryptosystem and are turned out to be either polynomial or exponential
depending on parameters of the cryptosystem. In this paper, we introduce a new
approach, called the Smart approach, which is based on a proper choice of the
distortion matrix $\mathbf{X}$. The Smart approach allows for withstanding all
known attacks even if the column scrambler matrix $\mathbf{P}$ over the base
field $\mathbb{F}_{q}$.
## I Introduction
McEliece [1] has introduced the first code-based public-key cryptosystem
(PKC). The system is based on Goppa codes in the Hamming metric, which is
connected to the hardness of the general decoding problem. It is a strong
cryptosystem but the size of a public key is too large (500 000 bits) for
practical implementations to be efficient.
Neiderreiter [2] has introduced a new PKC based on a family of Generalized
Reed-Solomon codes; its public key size is less than the McEliece
cryptosystem, but still large for practical application.
Also, Gabidulin Paramonov and Trietakov have proposed a new public key
cryptosystem, which is now called the GPT cryptosystem, based on _rank_ error
correcting codes in [3, 4]. The GPT cryptosystem has two advantages over
McEliece’s Cryptosystem. Firstly, it is more robust against decoding attacks
than McEliece’s Cryptosystem; secondly, the key size of the GPT is much
smaller and more useful in terms of practical applications than McEliece’s
cryptosystem.
Rank codes are well structured. Subsequently in a series of works, Gibson [5,
6] developed attacks that break the GPT system for public keys of about $5$
Kbits. The Gibson’s attacks are efficient for practical values of parameters
$n\leq 30$, where $n$ is the length of rank code with the field
$\mathbb{F}_{2^{N}}$ as an alphabet.
Several proposals of the GPT PKC were introduced to withstand Gibson’s attacks
[7, 8]. One proposal is to use a rectangular row scramble matrix instead of a
square matrix. The proposal allows working with subcodes of the rank codes
which have much more complicated structure. Another proposal exploits a
modification of Maximum Rank Distance (MRD) codes where the concept of a
_column_ scramble matrix was also introduced. A new variant, called reducible
rank codes, is also implemented to modify the GPT cryptosystem [9, 10]. All
these variants withstand Gibson’s attack.
Recently, R. Overbeck [11, 12], and [13] has proposed new attacks, which are
more effective than any of Gibson’s attacks. His method is based on two
factors : a) a column scrambler _P_ that is defined over the _base field_ ,
and b) the unsuitable choice of a distortion matrix _X_. However, Overbeck
managed to break many instances of the GPT cryptosystem based on the general
and developed ideas of Gibson.
Kshevetskiy in [19] suggested a secure approach towards the choice of
parameters for avoiding Overbeck’s attacks based on suitable choice of the
distortion matrix X. Independently, Loidreau in [20] proposed similar method.
Gabidulin [14] has offered a new approach called the Advanced approach, which
makes the cryptographer define a proper column scrambler matrix over the
extension field without violating the standard mode of the PKC. The Advanced
approach allows the decryption of the authorised party, and prevents an
unauthorized party from breaking the system by means of any known attacks.The
two approaches withstand Overbeck and Gibson’s attacks.
Recently, we have presented another variant of the GPT public key cryptosystem
[21], based on a proper choice of column scrambler matrix over the extension
field. This variant, which we call the Instrumental approach, is secure
against all known attacks.
In this paper, we introduce a new approach called the Smart approach, which is
based on a proper choice of the distortion matrix _X_. The Smart approach
allows for withstanding all known attacks even if the column scrambler matrix
$\mathbf{P}$ over the base field $\mathbb{F}_{q}$.
The rest of this paper is structured as follows. Section 2 gives a short
introduction to rank codes. Section 3 describes the GPT cryptosystems. Section
4 discusses the Overbeck’s attacks. Section 5 presents the Smart approach of
GPT PKC cryptosystem with two examples. Finally, section 6 concludes the paper
with some remarks.
## II Rank codes
Let us introduce the basic notion of rank codes [3], [15]. Let
$\mathbb{F}_{q}$ be a finite field of $q$ elements and let
$\mathbb{F}_{q^{N}}$ be an extension field of degree $N$. Let
$\mathbf{x}=(x_{1},x_{2},\dots,x_{n})$ be a vector with coordinates in
$\mathbb{F}_{q^{N}}$.
The Rank norm of x is defined as the maximal number of $\emph{x}_{i}$, which
are linearly independent over the base field $\mathbb{F}_{q}$ and is denoted
$\mathrm{Rk}(\mathbf{x}\mid\mathbb{F}_{q})$.
Similarly, for a matrix M with entries in $\mathbb{F}_{q^{N}}$, the columns
rank is defined as the maximal number of columns, which are linearly
independent over the base field $\mathbb{F}_{q}$, and is denoted
$\mathrm{Rk_{col}}(M|\mathbb{F}_{q})$.
We distinguish two ranks of the matrix:
1. 1.
The usual rank of matrix $M$ over $\mathbb{F}_{q^{N}}$ –
$\mathrm{Rk}(M\mid\mathbb{F}_{q^{N}})$.
2. 2.
The column rank of a matrix $M$ over the base field $\mathbb{F}_{q}$ –
$\mathrm{Rk_{col}}(M\mid\mathbb{F}_{q})$.
The column rank of the matrix M depends on the field. In particular,
$\mathrm{Rk_{col}}(M\mid\mathbb{F}_{q})\geq\mathrm{Rk_{col}}(M|\mathbb{F}_{q^{N}})$
The Rank distance between $\mathbf{x}$ and $\mathbf{y}$ is defined as the rank
norm of the difference $\mathbf{x-y}$:
$d(\mathbf{x,y})=\mathrm{Rk_{col}}(\mathbf{x-y}\leavevmode\nobreak\
\mid\leavevmode\nobreak\ \mathbb{F}_{q})$.
Any linear $(n,k,d)$ code $\mathcal{C}\subset\mathbb{F}^{n}_{q^{N}}$ fulfils
the Singleton-style bound [15] for the rank distance:
$Nk\leq Nn-(d-1)\max\\{N,n\\}.$ (1)
A code $\mathcal{C}$ reaching that bound is called a Maximal Rank Distance
(MRD) code.
The theory of optimal MRD (Maximal Rank Distance) codes is given in [15].
The notation $g[i]:=g^{q^{i\leavevmode\nobreak\
\mathrm{mod}\leavevmode\nobreak\ n}}$ means the ${i}$-th Frobenius power of
$g$. It allows to consider both positive and negative Frobenius powers $i$.
For $n\leq N$, a generator matrix $\mathbf{G}_{k}$ of a $(n,k,d)$ MRD code is
defined by a matrix of the following form:
$\mathbf{G}_{k}=\begin{bmatrix}g_{1}&g_{2}&\dots&g_{n}\\\
g_{1}^{[1]}&g_{2}^{[1]}&\dots&g_{n}^{[1]}\\\ \vdots&\vdots&\ddots&\vdots\\\
g_{1}^{[k-1]}&g_{2}^{[k-1]}&\dots&g_{n}^{[k-1]}\end{bmatrix}$ (2)
where $g_{1},g_{2},\ldots,g_{n}$ are any set of elements of the extension
field $\mathbb{F}_{q^{N}}$ which are linearly independent over the base field
$\mathbb{F}_{q}$.
A code with the generator matrix (2) is referred to as $(n,k,d)$ code, where
$n$ is code length, $k$ is the number of information symbols, $d$ is code
distance. For MRD codes, $d=n-k+1$. Let $\mathbf{m}=(m_{1},m_{2},\dots,m_{k})$
be an information vector of dimension $k$. The corresponding code vector is
the $n$-vector
$\mathbf{g}(\mathbf{m})=\mathbf{mG}_{k}.$
If $\mathbf{y}=\mathbf{g}(\mathbf{m})+\mathbf{e}$ and
$\mathrm{Rk}(\mathbf{e})=s\leq t=\frac{d-1}{2}$ , then the information vector
$\mathbf{m}$ can be recovered uniquely from $\mathbf{y}$ by some decoding
algorithm. There exist fast decoding algorithms for MRD codes [15], [16]. A
decoding procedure requires elements of the $(n-k)\times n$ parity check
matrix $\mathbf{H}$ such that $\mathbf{G}_{k}\mathbf{H}^{T}=0$. For decoding,
the matrix $\mathbf{H}$ should be of the form
$\mathbf{H}=\begin{bmatrix}h_{1}&h_{2}&\dots&h_{n}\\\
h_{1}^{[1]}&h_{2}^{[1]}&\dots&h_{n}^{[1]}\\\ \vdots&\vdots&\ddots&\vdots\\\
h_{1}^{[d-2]}&h_{2}^{[d-2]}&\dots&h_{n}^{[d-2]}\end{bmatrix},$ (3)
where elements $h_{1},h_{2},\dots,h_{n}$ are in the extension field
$\mathbb{F}_{q^{N}}$ and are linearly independent over the base field
$\mathbb{F}_{q}$.
The optimal code has the following design parameters: code length $n\leq N$;
dimension $k=n-d+1$, rank code distance $d=n-k+1$.
## III The GPT Cryptosystem
Description of the standard GPT cryptosystem.
The GPT cryptosystem is described as follows:
Plaintext: A Plaintext is any $k$-vector
$\mathbf{m}=(m_{1},m_{2},\dots,m_{k})$,
$m_{s}\in\mathbb{F}_{q^{N}},\,\leavevmode\nobreak\ s=1,2,\ldots,k$.
In previous works, different representations of the public key are given. All
of them can be reduced to the following form.
The Public key is a $k\times(n+t_{1})$ generator matrix
$\mathbf{G}_{pub}=\mathbf{S}\begin{bmatrix}\mathbf{X}&\mathbf{G}_{k}\\\
\end{bmatrix}\mathbf{P}.$ (4)
Let us explain roles of the factors.
* •
The main matrix $\mathbf{G}_{k}$ is given by 2. It is used to correct rank
errors. Errors of rank not greater than $\frac{n-k}{2}$ can be corrected.
* •
A matrix $\mathbf{S}$ is a row scrambler. This matrix is a non singular square
matrix of order $k$ over $\mathbb{F}_{q^{N}}$.
* •
A matrix $\mathbf{X}$ is a distortion $(k\times t_{1})$ matrix over
$\mathbb{F}_{q^{N}}$ with full column rank
$\mathrm{Rk_{col}}(X\mid\mathbb{F}_{q})=t_{1}$ and rank
$\mathrm{Rk}(\mathbf{X}\mid\mathbb{F}_{q^{N}})=t_{X},\leavevmode\nobreak\
t_{X}\leq t_{1}$. The matrix
$\begin{bmatrix}\mathbf{X}&\mathbf{G}_{k}\end{bmatrix}$ has full column rank
$\mathrm{Rk_{col}}(\begin{bmatrix}\mathbf{X}&\mathbf{G}_{k}\\\
\end{bmatrix}\mid\mathbb{F}_{q})=n+t_{1}$.
* •
A matrix $\mathbf{P}$ is a square _column scramble_ matrix of order
$(t_{1}+n)$ over $\mathbb{F}_{q}$.
* •
$t_{1}+n$ may be greater than $N$, but $n\leq N$.
The Private keys are matrices $\mathbf{S},\leavevmode\nobreak\
\mathbf{G}_{k},\leavevmode\nobreak\ \mathbf{X},\leavevmode\nobreak\
\mathbf{P}$ separately and (explicitly) a fast decoding algorithm of an MRD
code. Note also, that the matrix $\mathbf{X}$ is not used to decrypt a
ciphertext and can be deleted after calculating the Public key.
Encryption: Let $\mathbf{m}=(m_{1},m_{2},\dots,m_{k})$ be a plaintext. The
corresponding ciphertext is given by
$\mathbf{c}=\mathbf{mG}_{\mathrm{pub}}+\mathbf{e}=\mathbf{mS}\begin{bmatrix}\mathbf{X}&\mathbf{G}_{k}\end{bmatrix}\mathbf{P}+\mathbf{e},$
(5)
where $\mathbf{e}$ is an artificial vector of errors of rank $t_{2}$ or less.
It is assumed that $t_{1}+t_{2}\leq t=\lfloor\frac{n-k}{2}\rfloor$
Decryption: The legitimate receiver upon receiving $\mathbf{c}$ calculates
$\mathbf{c}^{{}^{\prime}}=(c_{1}^{{}^{\prime}},c_{2}^{{}^{\prime}},\ldots,c_{t_{1}+n}^{{}^{\prime}})=$
$\mathbf{c}\mathbf{P}^{-1}=\mathbf{mS}\begin{bmatrix}\mathbf{X}&\mathbf{G}_{k}\\\
\end{bmatrix}+\mathbf{e}\mathbf{P}^{-1}$
Then from $\mathbf{c}^{{}^{\prime}}$ he extracts the subvector
$\mathbf{c}^{{}^{\prime\prime}}=(c_{t_{1}+1}^{{}^{\prime}},c_{t_{1}+2}^{{}^{\prime}},\ldots,c_{t_{1}+n}^{{}^{\prime}})=\mathbf{mSG}_{k}+\mathbf{e}^{{}^{\prime\prime}},$
(6)
where $e^{{}^{\prime\prime}}$ is the subvector of $\mathbf{eP}^{-1}$. Then the
legitimate receiver applies the fast decoding algorithm to correct the error
$\mathbf{e}^{{}^{\prime\prime}}$, extracts $\mathbf{mS}$ and recovers $m$ as
$\mathbf{m}=(\mathbf{mS})\mathbf{S}^{-1}$.
In this system, the size of the public key is $V=k(t_{1}+n)N$ bits, and the
information rate is $R=\frac{k}{t_{1}+n}$.
## IV Overbeck’s Attack
In [11, 12], and [13], new attacks are proposed on the GPT PKC described in
the form of 4. It is claimed, that similar attacks can be proposed on all the
variants of GPT PKC.
We recall briefly this attack.
We need some notations.
For $x\in\mathbb{F}_{q^{N}}$ let $\sigma(x)=x^{q}$ be the Frobenius
automorphism.
For the matrix $\mathbf{T}=(t_{ij})$ over $\mathbb{F}_{q^{N}}$, let
$\sigma(\mathbf{T})=(\sigma(t_{ij}))=(t_{ij}^{q})$.
For any integer $s$, let
$\sigma^{s}(\mathbf{T})=\sigma(\sigma^{s-1}(\mathbf{T}))$.
It is clear that $\sigma^{N}=\sigma$. Thus the inverse exists
$\sigma^{-1}=\sigma^{N-1}$.
The following simple properties if $\sigma$ are useful:
* •
$\sigma(a+b)=\sigma(a)+\sigma(b)$.
* •
$\sigma(ab)=\sigma(a)\sigma(b)$.
* •
In general, for matrices $\sigma(\mathbf{T})\neq\mathbf{T}$.
* •
If $\mathbf{P}$ is a matrix over the _base_ field $\mathbb{F}_{q}$, then
$\sigma(\mathbf{P})=\mathbf{P}$.
Description of Overbeck’s attack: To break a system, a cryptanalyst constructs
from the public key
$\mathbf{G}_{\mathrm{pub}}=\mathbf{S}\begin{bmatrix}\mathbf{X}&\mathbf{G}_{k}\end{bmatrix}\mathbf{P}$
the _extended_ public key $\mathbf{G}_{\mathrm{ext,pub}}$ as follows:
$\mathbf{G}_{\mathrm{ext,pub}}=\left\|\begin{matrix}\mathbf{G}_{\mathrm{pub}}\\\
\sigma(\mathbf{G}_{\mathrm{pub}})\\\ \dots\\\
\sigma^{u}(\mathbf{G}_{\mathrm{pub}})\\\ \end{matrix}\right\|=$
$\left\|\begin{matrix}\mathbf{S}&\begin{bmatrix}\mathbf{X}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ &\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathbf{G}_{k}\end{bmatrix}&\mathbf{P}\\\
\sigma(\mathbf{S})&\begin{bmatrix}\sigma(\mathbf{X})\leavevmode\nobreak\
&\leavevmode\nobreak\ \sigma(\mathbf{G}_{k})\end{bmatrix}&\mathbf{P}\\\
\dots&\dots\dots\leavevmode\nobreak\ \dots\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ &\dots\\\
\sigma^{u}(\mathbf{S})&\begin{bmatrix}\sigma^{u}(\mathbf{X})&\sigma^{u}(\mathbf{G}_{k})\end{bmatrix}&\mathbf{P}\\\
\end{matrix}\right\|.$ (7)
The property that $\sigma(\mathbf{P})=\mathbf{P}$, if $\mathbf{P}$ is a matrix
over the _base_ field $\mathbb{F}_{q}$, is used in (7).
Rewrite this matrix as
$\mathbf{G}_{\mathrm{ext,pub}}=\mathbf{S}_{\mathrm{ext}}\begin{bmatrix}\mathbf{X}_{\mathrm{ext}}&\mathbf{G}_{\mathrm{ext}}\end{bmatrix}\mathbf{P},$
(8)
where
$\begin{array}[]{c}\mathbf{S}_{\mathrm{ext}}=\mathrm{Diag}\begin{bmatrix}\mathbf{S}&\sigma(\mathbf{S})&\dots&\sigma^{u}(\mathbf{S})\end{bmatrix}\\\\[8.53581pt]
\mathbf{X}_{\mathrm{ext}}=\begin{bmatrix}\mathbf{X}\\\ \sigma(\mathbf{X})\\\
\vdots\\\ \sigma^{u}(\mathbf{X})\\\
\end{bmatrix},\quad\mathbf{G}_{\mathrm{ext}}=\begin{bmatrix}\mathbf{G}_{k}\\\
\sigma(\mathbf{G}_{k})\\\ \vdots\\\ \sigma^{u}(\mathbf{G}_{k})\\\
\end{bmatrix}.\par\end{array}$ (9)
Choose
$u=n-k-1.$ (10)
For a $k\times t_{1}$ matrix $\mathbf{X}$, let $\mathbf{X}_{1}$ be the
$(k-1)\times t_{1}$ matrix, obtained from $\mathbf{X}$ by deleting the _last_
row. Similarly, let $\mathbf{X}_{2}$ be the $(k-1)\times t_{1}$ matrix,
obtained from $\mathbf{X}$ by deleting the _first_ row.
Define a linear mapping $T:\mathbb{F}_{q^{N}}^{k\times
t_{1}}\rightarrow\mathbb{F}_{q^{N}}^{(k-1)\times t_{1}}$ by the rule: if
$\mathbf{X}\in\mathbb{F}_{q^{N}}^{k\times t_{1}}$, then
$T(\mathbf{X})=\mathbf{Y}=\sigma(\mathbf{X}_{1})-\mathbf{X}_{2}.$ Let
$\mathbf{Y}_{\mathrm{ext}}=\begin{bmatrix}\mathbf{Y}&\sigma(\mathbf{Y})&\sigma^{2}(\mathbf{Y})&\dots&\sigma^{u-1}(\mathbf{Y})\end{bmatrix}^{\top}$
(11)
Using this and other suitable transformations of rows, one can rewrite for
analysis (8) and (9) in the form
$\tilde{\mathbf{G}}_{\mathrm{pub,ext}}=\tilde{\mathbf{S}}_{\mathrm{ext}}\begin{bmatrix}\mathbf{Z}&|&\mathbf{G}_{n-1}\\\
\mathbf{Y}_{\mathrm{ext}}&|&0\\\ \end{bmatrix}\mathbf{P}$ (12)
where $\mathbf{G}_{n-1}$ is the generator matrix of the $(n,n-1,2)$ MRD code.
Let us try to find a solution $\mathbf{u}$ of the system
$\tilde{\mathbf{S}}_{ext}\begin{bmatrix}\mathbf{Z}&|&\mathbf{G}_{n-1}\\\
\mathbf{Y}_{ext}&|&0\\\ \end{bmatrix}\mathbf{P}\mathbf{u}^{T}=\mathbf{0},$
(13)
where $\mathbf{u}$ is a vector-row over the extension field
$\mathbb{F}_{q^{N}}$ of length $t_{1}+n$. Represent the vector
$\mathbf{P}\mathbf{u}^{T}$ as
$\mathbf{P}\mathbf{u}^{T}=\begin{bmatrix}\mathbf{y}&\mathbf{h}\end{bmatrix}^{T},$
where the subvector $\mathbf{y}$ has length $t_{1}$ and $\mathbf{h}$ has
length $n$. Then the system (13) is equivalent to the following system:
$\displaystyle\mathbf{Z}\mathbf{y}^{T}+\mathbf{G}_{n-1}\mathbf{h}^{T}=\mathbf{0},$
(14) $\displaystyle\mathbf{Y}_{ext}\mathbf{y}^{T}=\mathbf{0}.$ (15)
Assume that the next condition is valid:
$\mathrm{Rk}(\mathbf{Y}_{ext}|\mathbb{F}_{q^{N}})=t_{1}.$ (16)
Then the equation (15) has only the trivial solution
$\mathbf{y}^{T}=\mathbf{0}$. The equation (14) becomes
$\mathbf{G}_{n-1}\mathbf{h}^{T}=\mathbf{0}.$ (17)
It allows to find the first row of the parity check matrix for the code with
the generator matrix (12) (see,[11, 12], and [13], for details). Hence this
solution breaks a GPT cryptosystem in polynomial time. The Overbeck’s attack
requires $O((n+t_{1})^{3})$ operation over $\mathbb{F}_{q^{N}}$ since all the
steps of the attack have at most cubic complexity on $n+t_{1}$.
## V Smart approach
To withstand Overbeck’s attack, the cryptographer should choose the matrix
$\mathbf{X}$ in such a manner that
$\mathrm{Rk}(\mathbf{Y}_{ext}\mid\mathbb{F}_{q^{N}})=t_{1}-a,$ (18)
where $a\geq 2$. In this case, the system (15) has $q^{aN}$ solutions
$\mathbf{y}^{T}$. Hence the exhaustive search over $\mathbf{y}^{T}$ is needed.
The work function has order $O(q^{aN}(n+t_{1})^{3})$ and Overback’s attack
fails.
One method to provide the condition (18) is proposed in [19, 20]. Choose the
matrix $\mathbf{X}$ over the extension field $\mathbb{F}_{q^{N}}$ in such a
manner that the following conditions are satisfied:
$\begin{array}[]{lclcl}t_{1}&=&\mathrm{Rk_{col}}(\mathbf{X}\mid\mathbb{F}_{q})&>&n-k.\\\
r_{X}&=&\mathrm{Rk}(\mathbf{X}\mid\mathbb{F}_{q^{N}})&=&\left\lfloor\frac{t_{1}-a}{n-k}\right\rfloor\leq
k.\end{array}$ (19)
Overbeck’s attack is exponential on $a$ and has the minimum complexity at
least $O\left(q^{aN}(n+t_{1})^{3}\right)$.
We propose an alternative Smart approach. The point is to choose the matrix
$\mathbf{X}$ in such a manner that the corresponding matrix
$\mathbf{Y}=T(\mathbf{X})$ has column rank
$\mathrm{Rk}(\mathbf{Y}\mid\mathbb{F}_{q})$ not greater than $t_{1}-a,\,a\geq
2$.
The following result is evident.
###### Lemma 1
If $\mathrm{Rk}(\mathbf{Y}\mid\mathbb{F}_{q})=s$, then
$\mathrm{Rk}(\mathbf{Y}_{\mathrm{ext}}\mid\mathbb{F}_{q})=s$.
###### Corollary 1
$\mathrm{Rk}(\mathbf{Y}_{\mathrm{ext}}\mid\mathbb{F}_{q^{N}})\leq\mathrm{Rk}(\mathbf{Y}_{\mathrm{ext}}\mid\mathbb{F}_{q})=s=\mathrm{Rk}(\mathbf{Y}\mid\mathbb{F}_{q})$.
### The simple case
Let a matrix $\mathbf{X}$ be of the following form:
$\mathbf{X}=\begin{bmatrix}\mathbf{m}\\\ \mathbf{m}^{[1]}\\\ \vdots\\\
\mathbf{m}^{[k-1]}\end{bmatrix}+\begin{bmatrix}\mathbf{0}\\\ \mathbf{s}_{1}\\\
\vdots\\\ \mathbf{s}_{k-1}\end{bmatrix}.$ (20)
Here $\mathbf{m}$ is a random vector over the extension field
$\mathbb{F}_{q^{N}}$ with full column rank $t_{1}$ and vectors
$\mathbf{s}_{i},\;i=1,\dots,k-1,$ are random vectors over the _base_ field
$\mathbb{F}_{q}$ such that the matrix
$\begin{bmatrix}\mathbf{0}&\mathbf{s}_{1}&\dots&\mathbf{s}_{k-1}\end{bmatrix}^{\top}$
has rank $t_{1}-a$. Then the matrix $\mathbf{Y}=T(\mathbf{X})$ has the form
$\mathbf{Y}=\begin{bmatrix}-\mathbf{s}_{1}&\mathbf{s}_{1}-\mathbf{s}_{2}&\dots&\mathbf{s}_{k-1}-\mathbf{s}_{k}\end{bmatrix}^{\top}.$
(21)
This matrix is a matrix over the _base_ field $\mathbb{F}_{q}$ and has rank
$t_{1}-a$ too. It follows that
$\sigma(\mathbf{Y})=\begin{bmatrix}\sigma(-\mathbf{s}_{1})\\\
\sigma(\mathbf{s}_{1}-\mathbf{s}_{2})\\\ \vdots\\\
\sigma(\mathbf{s}_{k-1}-\mathbf{s}_{k})\end{bmatrix}=\begin{bmatrix}-\mathbf{s}_{1}\\\
\mathbf{s}_{1}-\mathbf{s}_{2}\\\ \vdots\\\
\mathbf{s}_{k-1}-\mathbf{s}_{k}\end{bmatrix}=\mathbf{Y}.$ (22)
Hence
$\mathbf{Y}_{ext}=\begin{bmatrix}\mathbf{Y}\\\ \sigma(\mathbf{Y})\\\ \dots\\\
\sigma^{u-1}(\mathbf{Y})\\\ \end{bmatrix}=\begin{bmatrix}\mathbf{Y}\\\
\mathbf{Y}\\\ \dots\\\ \mathbf{Y}\\\ \end{bmatrix}.$ (23)
Therefore
$\mathrm{Rk}(\mathbf{Y}_{ext}\mid\mathbb{F}_{q^{N}})=\mathrm{Rk}(\mathbf{Y}\mid\mathbb{F}_{q^{N}})=t_{1}-a,$
and the condition (18) is satisfied.
As in the previous case, the proposed Smart approach shows that Overbeck’s
attack is exponential on $a$ and has the bit complexity at least
$O\left(q^{aN}(n+t_{1})^{3}\right)$.
It has been shown that the Smart approach presented above is secure against
all known attacks including the recent attack presented by Overbeck in [13].
###### Example 1
Let $n=8,\leavevmode\nobreak\ k=4,\leavevmode\nobreak\
N=8,\leavevmode\nobreak\ t=5,\leavevmode\nobreak\ t_{1}=4,\leavevmode\nobreak\
q=2,\leavevmode\nobreak\ a=2$
Let the extension field $\mathbb{F}_{2^{8}}$ be defined by the primitive
polynomial $r(x)=1+x^{2}+x^{3}+x^{4}+x^{8},$ and let $\alpha$ be a primitive
element of the field. Choose the matrix $\mathbf{X}$ as in (20). A vector
$\mathbf{m}$ of full column rank $t_{1}=4$ is defined as
$\mathbf{m}=\begin{bmatrix}\alpha^{3}&\alpha^{5}&\alpha^{6}&\alpha^{2}\end{bmatrix}.$
Choose vectors $\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{s}_{3}$ as
$\mathbf{s}_{1}=\begin{bmatrix}1&1&0&0\end{bmatrix}$,
$\mathbf{s}_{2}=\begin{bmatrix}1&1&1&1\end{bmatrix}$,
$\mathbf{s}_{3}=\begin{bmatrix}0&0&1&1\end{bmatrix}.$ Then we obtain
$\begin{array}[]{rl}\mathbf{X}=&\begin{bmatrix}\alpha^{3}&\alpha^{5}&\alpha^{6}&\alpha^{2}\\\
\alpha^{6}&\alpha^{10}&\alpha^{12}&\alpha^{4}\\\
\alpha^{12}&\alpha^{20}&\alpha^{24}&\alpha^{8}\\\
\alpha^{24}&\alpha^{40}&\alpha^{48}&\alpha^{16}\\\
\end{bmatrix}+\begin{bmatrix}0&0&0&0\\\ 1&1&0&0\\\ 1&1&1&1\\\ 0&0&1&1\\\
\end{bmatrix}=\\\\[8.53581pt]
&\begin{bmatrix}\alpha^{3}&\alpha^{5}&\alpha^{6}&\alpha^{2}\\\
\alpha^{6}+1&\alpha^{10}+1&\alpha^{12}&\alpha^{4}\\\
\alpha^{12}+1&\alpha^{20}+1&\alpha^{24}+1&\alpha^{8}+1\\\
\alpha^{24}&\alpha^{40}&\alpha^{48}+1&\alpha^{16}+1\\\
\end{bmatrix}\end{array}.$ (24)
The corresponding matrix $\mathbf{Y}$ is as follows:
$\mathbf{Y}=\begin{bmatrix}1&1&0&0\\\ 0&0&1&1\\\ 1&1&0&0\\\ \end{bmatrix}.$
(25)
It has rank $t_{1}-a=2$. The attack is exponential on $a$ and has the bit
complexity at least $O(q^{aN}(n+t_{1})^{3})=O(2^{37}$ bite operations.
### The general case
Let $\mathbf{X}$ be a matrix consisting of $a$ Frobenius-type columns and
$t_{1}-a$ non-Frobenius columns. A column $\mathbf{w}$ is called Frobenius-
type if it has the form
$\mathbf{w}=\begin{pmatrix}w&w^{[1]}&\dots&w^{[k-1]}\end{pmatrix}^{\top}$. It
is clear that $T(\mathbf{w})=\mathbf{0}$. Hence the matrix
$\mathbf{Y}=T(\mathbf{X})$ will have $a$ all zero columns and column rank
$t_{1}-a$ and by Corollary 1 the matrix $\mathbf{Y}_{\mathrm{ext}}$ has rank
not greater than $t_{1}-a$. The result is valid also if suitable linear
combinations of non-Frobenius columns are added to Frobenius-type columns.
###### Example 2
In conditions of the previous example, let matrix $\mathbf{X}$ be as follows:
$\mathbf{X}=\begin{bmatrix}\alpha^{3}+\alpha^{6}&\alpha^{5}+\alpha^{2}&\alpha^{6}&\alpha^{2}\\\
\alpha^{6}+\alpha^{12}&\alpha^{10}+\alpha^{5}&\alpha^{12}&\alpha^{5}\\\
\alpha^{12}+\alpha^{12}&\alpha^{20}+\alpha^{5}&\alpha^{12}&\alpha^{5}\\\
\alpha^{24}+\alpha^{12}&\alpha^{40}+\alpha^{2}&\alpha^{12}&\alpha^{2}\\\
\end{bmatrix}.$
The third column is added to the first Frobenius-type, and the fourth is added
to the second Frobenius-type, so $a=2$. Column rank of $\mathbf{X}$ is
$t_{1}=4$. The corresponding matrix $\mathbf{Y}=T(\mathbf{X})$ is of the form:
$\mathbf{Y}=\begin{bmatrix}0&\alpha^{4}+\alpha^{5}&0&\alpha^{4}+\alpha^{5}\\\
\alpha^{24}+\alpha^{12}&\alpha^{4}+\alpha^{5}&\alpha^{24}+\alpha^{12}&\alpha^{4}+\alpha^{5}\\\
\alpha^{24}+\alpha^{12}&\alpha^{10}+\alpha^{5}&\alpha^{24}+\alpha^{12}&\alpha^{10}+\alpha^{5}\\\
\end{bmatrix}.$
It has rank $t_{1}-a=2$.
In general, Overbeck’s attack fails when $aN\geq 60$.
## VI Conclusion
We have introduced the Smart approach as a technique of withstanding
Overbeck’s attack on the GPT Public key cryptosystem, which is based on rank
codes.
It is shown that proper choice of the distortion matrix $\mathbf{X}$ over the
extension field $\mathbb{F}_{q^{N}}$ allows the decryption by the authorized
party and prevents the unauthorized party from breaking the system by means of
any known attacks.
## References
* [1] R.J. McEliece, “A Public Key Cryptosystem Based on Algebraic Coding Theory,” JPL DSN Progress Report 42–44, Pasadena, CA, pp. 114–116, 1978.
* [2] H. Niederreiter, (1986), Knapsack-Type Cryptosystem and Algebraic Coding Theory, Probl. Control and Inform. Theory, vol. 15, pp. 19-34,1986.
* [3] E.M. Gabidulin, A.V. Paramonov, O.V. Tretjakov, “Ideals over a Non-commutative Ring and Their Application in Cryptology”, in: Advances in Cryptology — Eurocrypt ’91, LNCS 547, 1991, pp. 482–489.
* [4] E.M. Gabidulin, “Public-Key Cryptosystems Based on Linear Codes over Large Alphabets: Efficiency and Weakness,” in: Codes and Ciphers, Editor: P.G. Farrell, pp. 17–32, Essex: Formara Limited, 1995.
* [5] J. K. Gibson, “Severely denting the Gabidulin version of the McEliece public key cryptosystem,” // _Designs, Codes and Cryptography, 6(1)_ , 1995, pp. 37–45.
* [6] J. K. Gibson, “The security of the Gabidulin public-key cryptosystem,” in: U. M. Maurer, ed. // _Advances in Cryptology – EUROCRYPT’96, LNCS 1070_ , 1996, pp. 212–223.
* [7] E.M. Gabidulin, A.V. Ourivski, “Improved GPT Public Key Cryptosystems.” // In: P. Farrell, M. Darnell, B. Honary (Ed’s), _”Coding, Communications, and Broadcasting”_ , Research Studies Press, 2000, pp. 73-102.
* [8] A. V. Ourivski, E. M. Gabidulin, “Column Scrambler for the GPT Cryptosystem.” // _Discrete Applied Mathematics._ 128(1): 207-221 (2003).
* [9] E. M. Gabidulin, A. V. Ourivski, B. Honary, B. Ammar, “Reducible Rank Codes and Their Applications to Cryptography.” // _IEEE Transactions on Information Theory._ 49(12): 3289-3293 (2003).
* [10] A. S. Kshevetskiy, E. M. Gabidulin, “High-weight errors in public-key cryptosystems based on reducible rank codes.” // In: _Proc. of ISCTA_ , 2005.
* [11] Overbeck, R.: A new structural attack for GPT and variants. In: Proc. of Mycrypt 2005, vol. 3517 of LNCS, pp. 5 63. Springer-Verlag (2005).
* [12] Overbeck R.: Extending Gibson’s attacks on the GPT cryptosystem. In Proc. of WCC 2005, volume 3969 of LNCS, pp. 178-188, Springer Verlag,2006.
* [13] Overbeck R : Structural Attacks for Public Key Cryptosystems based on Gabidulin Codes, Journal of Cryptology, volume 21, number 2, April 2008
* [14] E. M. Gabidulin, ”Attacks and counter-attacks on the GPT public key cryptosystem,” _Designs, Codes and Cryptography._ V. 48, No. 2/ August 2008. Pp. 171-177, Springer Netherlands, DOI 10.1007/s10623-007-9160-8.
* [15] E.M. Gabidulin, “Theory of Codes with Maximum Rank Distance,” Probl. Inform. Transm., vol. 21, No. 1, pp. 1–12, July, 1985.
* [16] E. M. Gabidulin, “A Fast Matrix Decoding Algorithm For Rank-Error-Correcting Codes.” In: (Eds G. Cohen, S. Litsyn, A. Lobstein, G. Zemor), Algebraic coding , pp. 126-132, Lecture Notes in Computer Science No. 573, Springer-Verlag, Berlin, 1992.
* [17] T. Johansson, A.V. Ourivski, “New technique for decoding codes in the rank metric and its cryptography applications,” _Problems Inform. Transm._ 38(3), 237 246 (2002).
* [18] F. Levy-dit-Vehel1, J.-Ch. Jean-Charles Faug‘ere, and L. Perret,“Cryptanalysis of MinRank.” Advances in Cryptology - CRYPTO 2008, 28th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 17-21, 2008, Proceedings. Series: Lecture Notes in Computer Science. Subseries: Security and Cryptology , Vol. 5157. Wagner, David (Ed.). 2008. Pp. 280-296.
* [19] Kshevetskiy A.S.: Security of GPT-like cryptosystems based on linear rank codes. Signal Design and Its Applications in Communications, 2007. IWSDA 2007. On page(s): 143-147.
* [20] P. Loidreau, “Designing a rank metric based McEliece cryptosystem.” PQCrypto 2010. The Third International Workshop on Post-Quantum Cryptography. Darmstadt, Germany, May 25-28, 2010.
* [21] E. M. Gabidulin, H.Rashwan and B. Honary,, “On improving security of GPT cryptosystems.“ IEEE International Symposium on Information Theory , June 2009.
|
arxiv-papers
| 2010-06-02T14:18:25 |
2024-09-04T02:49:10.769265
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Haitham Rashwan, Ernst M. Gabidulin, Bahram Honary",
"submitter": "Haitham Rashwan",
"url": "https://arxiv.org/abs/1006.0386"
}
|
1006.2095
|
# Lepton flavor violating Higgs decays induced by massive unparticle
E. O. Iltan,
Middle East Technical University, Northern Cyprus Campus,
Guzelyurt, Mersin 10, TURKEY
E-mail address: eiltan@newton.physics.metu.edu.tr
###### Abstract
We predict the branching ratios of the lepton flavor violating Higgs decays
$H^{0}\rightarrow e^{\pm}\mu^{\pm}$, $H^{0}\rightarrow e^{\pm}\tau^{\pm}$ and
$H^{0}\rightarrow\mu^{\pm}\tau^{\pm}$ with the assumption that lepton flavor
violation is due to the unparticle mediation. Here, we consider an effective
interaction which breaks the conformal invariance after the electroweak
symmetry breaking and causes that unparticle becomes massive. The new
interaction results in a modification of the mediating unparticle propagator
and brings additional contribution to the branching ratios of the lepton
flavor violating decays with the new vertex including Higgs field and two
unparticle fields. We observe that the branching ratios of the decays under
consideration lie in the range of $10^{-6}-10^{-4}$.
The standard model (SM) electroweak symmetry breaking mechanism which can
explain the production of the masses of fundamental particles will be tested
at the Large Hadron Collider (LHC) and, hopefully, the Higgs boson $H^{0}$,
which is responsible for this mechanism will be hunt soon. The possible decays
of the Higgs boson to the SM particles are worthwhile to study and, among
them, the lepton flavor violating (LFV) decays reach great interest [1, 2, 4,
3, 5] since the LF violation mechanism is sensitive to the physics beyond the
SM. The addition of the new Higgs doublet to the SM particle spectrum is one
of the possibility to switch on the LFV interactions, arising from the tree
level LFV couplings. In [1, 2, 3], $H^{0}\rightarrow\tau\mu$ decay has been
analyzed and the branching ratio (BR) at the order of magnitude of $0.001-0.1$
has been estimated. In [4], the observable BRs of LF changing $H^{0}$ decays
have been obtained in the SM with right handed neutrinos. Another possibility
to switch on the LF violation is to introduce the intermediate scalar
unparticle (U) with the effective U-lepton-lepton vertex in the loop level. In
[5], the BRs of the LFV Higgs decays $H^{0}\rightarrow e^{\pm}\mu^{\pm}$,
$H^{0}\rightarrow e^{\pm}\tau^{\pm}$ and $H^{0}\rightarrow\mu^{\pm}\tau^{\pm}$
have been estimated, by respecting the unparticle idea. Unparticles,
introduced by Georgi [6, 7], come out as new degrees of freedom due to the SM-
ultraviolet sector interaction; they are massless and have non integral
scaling dimension $d_{u}$, around, $\Lambda_{U}\sim 1\,TeV$.
In the present work we study the LFV SM Higgs decays by considering that the
LF violation exists in the one loop level and it is carried by the effective
U-lepton-lepton vertex. The effective interaction lagrangian, which is
responsible for the LFV decays, is
$\displaystyle{\cal{L}}_{FV}=\frac{1}{\Lambda_{U}^{du-1}}\Big{(}\lambda_{ij}^{S}\,\bar{l}_{i}\,l_{j}+\lambda_{ij}^{P}\,\bar{l}_{i}\,i\gamma_{5}\,l_{j}\Big{)}\,U\,,$
(1)
with the lepton field $l$ and scalar (pseudoscalar) coupling
$\lambda_{ij}^{S}$ ($\lambda_{ij}^{P}$). Here we consider the operators with
the lowest possible dimension since their contributions are dominant in the
low energy effective theory (see [8]). Furthermore, we consider that there
exists an additional interaction which ensures a non-zero mass to unparticle
after the electroweak symmetry breaking [9] as
$\displaystyle{\cal{L}}_{U}=-\frac{\lambda}{\Lambda_{U}^{2\,du-2}}\,U^{2}\,H^{\dagger}\,H\,,$
(2)
and we get
$\displaystyle{\cal{L}}_{U}=-\frac{1}{2}\,\frac{\lambda}{\Lambda_{U}^{2\,du-2}}\,U^{2}\,\Bigg{(}H^{0\,2}+2\,v\,H^{0}+v^{2}\Bigg{)}\,,$
(3)
when the Higgs doublet develops the vacuum expectation value. The interaction
in eq.(3) leads to the lagrangian
$\displaystyle{\cal{L^{\prime}}}_{U}=-\frac{m_{U}^{4-2\,d_{U}}}{v}\,U^{2}\,H^{0}\,,$
(4)
with the unparticle mass
$\displaystyle
m_{U}=\Bigg{(}\frac{\sqrt{\lambda}\,v}{\Lambda_{U}^{du-1}}\Bigg{)}^{\frac{1}{2-d_{U}}}\,,$
(5)
and this term results in an additional diagram driving the LFV decays with the
help of U-lepton-lepton vertices (see Fig.1-(d)). Here, the non-zero
unparticle mass $m_{U}$ is the sign of the broken conformal invariance and one
expects that the unparticle propagator is modified. The propagator is model
dependent (see [10]) and we consider the one in the simple model [11, 12]
$\displaystyle\int\,d^{4}x\,e^{ipx}\,<0|T\Big{(}U(x)\,U(0)\Big{)}0>=i\frac{A_{d_{u}}}{2\,\pi}\,\int_{0}^{\infty}\,ds\,\frac{s^{d_{u}-2}}{p^{2}-\mu^{2}-s+i\epsilon}\,,$
(6)
with
$\displaystyle
A_{d_{u}}=\frac{16\,\pi^{5/2}}{(2\,\pi)^{2\,d_{u}}}\,\frac{\Gamma(d_{u}+\frac{1}{2})}{\Gamma(d_{u}-1)\,\Gamma(2\,d_{u})}\,,$
(7)
and the scale $\mu$ where unparticle sector becomes a particle sector. This
choice has clues about the unparticle nature of the hidden sector, it carries
the information on the effects of the broken scale invariance and ensures a
possibility to estimate the scale invariance breaking effects111Notice that
the modification in the propagator needs a further analysis in order to
understand whether it is based on a consistent quantum field theory and this
is beyond the scope of the present manuscript.. In our calculations we choose
$\mu=m_{U}$ and $d_{u}\sim 1.0$ which is the case that unparticle behaves as
if it is almost gauge singlet scalar222 This is the case that $m_{U}$ lies
near the electroweak scale [9]..
Now, we are ready to present the BR for $H^{0}\rightarrow
l_{1}^{-}\,l_{2}^{+}$ decay,
$\displaystyle BR(H^{0}\rightarrow
l_{1}^{-}\,l_{2}^{+})=\frac{1}{16\,\pi\,m_{H^{0}}}\,\frac{|M|^{2}}{\Gamma_{H^{0}}}\,,$
(8)
where $M$ is the matrix element of the LFV $H^{0}\rightarrow
l_{1}^{-}\,l_{2}^{+}$ decay (see Fig.1) and $\Gamma_{H^{0}}$ is the Higgs
total decay width. The square of the matrix element $|M|^{2}$ reads
$\displaystyle|M|^{2}=2\Big{(}m_{H^{0}}^{2}-(m_{l_{1}^{-}}+m_{l_{2}^{+}})^{2}\Big{)}\,|A|^{2}+2\Big{(}m_{H^{0}}^{2}-(m_{l_{1}^{-}}-m_{l_{2}^{+}})^{2}\Big{)}\,|A^{\prime}|^{2}\,,$
(9)
with the amplitudes
$\displaystyle A$ $\displaystyle=$
$\displaystyle\int^{1}_{0}\,dx\,f_{self}^{S}+\int^{1}_{0}\,dx\,\int^{1-x}_{0}\,dy\,f_{vert}^{S}\,,$
$\displaystyle A^{\prime}$ $\displaystyle=$
$\displaystyle\int^{1}_{0}\,dx\,f_{self}^{\prime\,S}+\int^{1}_{0}\,dx\,\int^{1-x}_{0}\,dy\,f_{vert}^{\prime\,S}\,.$
(10)
The functions333$f_{self}^{S}$, $f_{self}^{\prime\,S}$ are the same as the
functions presented in [5] except that the propagators $L_{self}$ and
$L_{self}^{\prime}$ contain the unparticle mass term $m_{U}$. On the other
hand $f_{vert}^{S}$, $f_{vert}^{\prime\,S}$ include additional part
proportional to the parameter $c_{2}$ which comes from the new interaction
(see eq.(4)) leading to the vertex given in Fig.1-d $f_{self}^{S}$,
$f_{self}^{\prime\,S}$, $f_{vert}^{S}$, $f_{vert}^{\prime\,S}$ are
$\displaystyle f_{self}^{S}$ $\displaystyle=$
$\displaystyle\frac{-i\,c_{1}\,(1-x)^{1-d_{u}}}{16\,\pi^{2}\,\Big{(}m_{l_{2}^{+}}-m_{l_{1}^{-}}\Big{)}\,(1-d_{u})}\,\sum_{i=1}^{3}\,\Big{\\{}(\lambda_{il_{1}}^{S}\,\lambda_{il_{2}}^{S}+\lambda_{il_{1}}^{P}\lambda_{il_{2}}^{P})\,m_{l_{1}^{-}}\,m_{l_{2}^{+}}\,(1-x)$
$\displaystyle\times$ $\displaystyle\Big{(}L_{self}^{d_{u}-1}-L_{self}^{\prime
d_{u}-1}\Big{)}-(\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{P}-\lambda_{il_{1}}^{S}\lambda_{il_{2}}^{S})\,m_{i}\,\Big{(}m_{l_{2}^{+}}\,L_{self}^{d_{u}-1}-m_{l_{1}^{-}}\,L_{self}^{\prime
d_{u}-1}\Big{)}\Big{\\}}\,,$ $\displaystyle f_{self}^{\prime\,S}$
$\displaystyle=$
$\displaystyle\frac{i\,c_{1}\,(1-x)^{1-d_{u}}}{16\,\pi^{2}\,\Big{(}m_{l_{2}^{+}}+m_{l_{1}^{-}}\Big{)}\,(1-d_{u})}\,\sum_{i=1}^{3}\,\Big{\\{}(\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{S}+\lambda_{il_{1}}^{S}\lambda_{il_{2}}^{P})\,m_{l_{1}^{-}}\,m_{l_{2}^{+}}\,(1-x)$
$\displaystyle\times$ $\displaystyle\Big{(}L_{self}^{d_{u}-1}-L_{self}^{\prime
d_{u}-1}\Big{)}-(\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{S}-\lambda_{il_{1}}^{S}\lambda_{il_{2}}^{P})\,m_{i}\,\Big{(}m_{l_{2}^{+}}\,L_{self}^{d_{u}-1}+m_{l_{1}^{-}}\,L_{self}^{\prime
d_{u}-1}\Big{)}\Big{\\}}\,,$ $\displaystyle f_{vert}^{S}$ $\displaystyle=$
$\displaystyle\frac{i\,c_{1}\,m_{i}\,(1-x-y)^{1-d_{u}}}{16\,\pi^{2}}\,\sum_{i=1}^{3}\,\frac{1}{\,L_{vert}^{2-d_{u}}}\,\Bigg{\\{}(\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{P}-\lambda_{il_{1}}^{S}\,\lambda_{il_{2}}^{S})\,\Big{\\{}(1-x-y)$
$\displaystyle\times$
$\displaystyle\Bigg{(}m_{l_{1}^{-}}^{2}\,x+m_{l_{2}^{+}}^{2}\,y-m_{l_{2}^{+}}\,m_{l_{1}^{-}}\Bigg{)}+x\,y\,m_{H^{0}}^{2}-\frac{2\,L_{vert}}{1-d_{u}}-m_{i}^{2}\Big{\\}}$
$\displaystyle-$
$\displaystyle(\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{P}+\lambda_{il_{1}}^{S}\lambda_{il_{2}}^{S})\,m_{i}\,\Big{(}m_{l_{1}^{-}}\,(2\,x-1)+m_{l_{2}^{+}}\,(2\,y-1)\Big{)}\Bigg{\\}}$
$\displaystyle-$
$\displaystyle\frac{i\,c_{2}\,\Gamma[3-2\,d_{u}]\,(x\,y)^{1-d_{u}}}{16\,\pi^{2}\,\Gamma[2-d_{u}]^{2}}\,\sum_{i=1}^{3}\,\frac{1}{\,L_{2vert}^{3-2\,d_{u}}}\,\Bigg{\\{}m_{i}\,(\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{P}-\lambda_{il_{1}}^{S}\,\lambda_{il_{2}}^{S})$
$\displaystyle-$
$\displaystyle(\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{P}+\lambda_{il_{1}}^{S}\lambda_{il_{2}}^{S})\,\Big{(}m_{l_{1}^{-}}\,x+m_{l_{2}^{+}}\,y\Big{)}\Bigg{\\}}\,,$
$\displaystyle f_{vert}^{\prime\,S}$ $\displaystyle=$
$\displaystyle\frac{i\,c_{1}\,m_{i}\,(1-x-y)^{1-d_{u}}}{16\,\pi^{2}}\,\sum_{i=1}^{3}\,\frac{1}{\,L_{vert}^{2-d_{u}}}\,\Bigg{\\{}(\lambda_{il_{1}}^{S}\,\lambda_{il_{2}}^{P}-\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{S})\,\Big{\\{}(1-x-y)$
(11) $\displaystyle\times$
$\displaystyle\Bigg{(}m_{l_{1}^{-}}^{2}\,x+m_{l_{2}^{+}}^{2}\,y+m_{l_{2}^{+}}\,m_{l_{1}^{-}}\Bigg{)}+x\,y\,m_{H^{0}}^{2}-\frac{2\,L_{vert}}{1-d_{u}}-m_{i}^{2}\Big{\\}}$
$\displaystyle+$
$\displaystyle(\lambda_{il_{1}}^{S}\lambda_{il_{2}}^{P}+\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{S})\,m_{i}\,\Big{(}m_{l_{1}^{-}}\,(2\,x-1)+m_{l_{2}^{+}}\,(1-2\,y)\Big{)}\Bigg{\\}}$
$\displaystyle-$
$\displaystyle\frac{i\,c_{2}\,\Gamma[3-2\,d_{u}]\,(x\,y)^{1-d_{u}}}{16\,\pi^{2}\,\Gamma[2-d_{u}]^{2}}\,\sum_{i=1}^{3}\,\frac{1}{\,L_{2vert}^{3-2\,d_{u}}}\,\Bigg{\\{}m_{i}\,(\lambda_{il_{1}}^{S}\,\lambda_{il_{2}}^{P}-\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{S})$
$\displaystyle+$
$\displaystyle(\lambda_{il_{1}}^{S}\lambda_{il_{2}}^{P}+\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{S})\,\Big{(}m_{l_{1}^{-}}\,x-m_{l_{2}^{+}}\,y\Big{)}\Bigg{\\}}\,,$
where $L_{self}$, $L_{self}^{\prime}$, $L_{vert}$, and $L_{2vert}$ are
$\displaystyle L_{self}$ $\displaystyle=$ $\displaystyle
x\,\Big{(}m_{l_{1}^{-}}^{2}\,(1-x)-m_{i}^{2}\Big{)}+m_{U}^{2}\,(x-1)\,,$
$\displaystyle L_{self}^{\prime}$ $\displaystyle=$ $\displaystyle
x\,\Big{(}m_{l_{2}^{+}}^{2}\,(1-x)-m_{i}^{2}\Big{)}+m_{U}^{2}\,(x-1)\,,$
$\displaystyle L_{vert}$ $\displaystyle=$
$\displaystyle(m_{l_{1}^{-}}^{2}\,x+m_{l_{2}^{+}}^{2}\,y)\,(1-x-y)-m_{i}^{2}\,(x+y)+m_{H^{0}}^{2}\,x\,y-m_{U}^{2}\,(1-x-y)\,,$
$\displaystyle L_{2vert}$ $\displaystyle=$
$\displaystyle(m_{l_{1}^{-}}^{2}\,x+m_{l_{2}^{+}}^{2}\,y)\,(1-x-y)-m_{i}^{2}\,(1-x-y)+m_{H^{0}}^{2}\,x\,y-m_{U}^{2}\,(x+y)\,,$
(12)
with
$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle\frac{g\,A_{d_{u}}\,e^{-i\,\pi\,d_{u}}}{4\,m_{W}\,sin\,(d_{u}\pi)\,\Lambda_{u}^{2\,(d_{u}-1)}}\,,$
$\displaystyle c_{2}$ $\displaystyle=$
$\displaystyle\frac{A^{2}_{d_{u}}\,m_{U}^{4-2\,d_{u}}\,e^{-2\,i\,\pi\,d_{u}}}{4\,v\,sin^{2}\,(d_{u}\pi)\,\Lambda_{u}^{2\,(d_{u}-1)}}\,.$
(13)
Here $\lambda_{il_{1(2)}}^{S,P}$ are the scalar and pseudoscalar couplings
related to the $U-i-l_{1}^{-}\,(l_{2}^{+})$ interaction where $i$,
($i=e,\mu,\tau$) is the internal lepton and $l_{1}^{-}\,(l_{2}^{+})$ the
outgoing lepton (anti lepton). Notice that, in the numerical calculations, we
consider the BR due to the production of sum of charged states, namely,
$\displaystyle BR(H^{0}\rightarrow
l_{1}^{\pm}\,l_{2}^{\pm})=\frac{\Gamma(H^{0}\rightarrow(\bar{l}_{1}\,l_{2}+\bar{l}_{2}\,l_{1}))}{\Gamma_{H^{0}}}\,.$
(14)
Discussion
This section is devoted to the analysis of the BRs of the LFV
$H^{0}\rightarrow l_{1}^{-}l_{2}^{+}$ decays in the case that the LF violation
is carried by the U\- lepton-lepton vertex. The LFV decays exist at least in
the loop level with the help of the internal unparticle mediation. The
interaction Lagrangian given in eq.(2) results in a nonzero mass for
unparticle after the electroweak symmetry breaking and the propagator of
unparticle existing in the loop should be modified. In the present work we
take the propagator as (see eq.(6))
$\displaystyle
P(p^{2})=\frac{i\,A_{d_{u}}}{2\,sin\,\pi\,d_{u}}\frac{e^{-i\,d_{u}\,\pi}}{(p^{2}-m_{U}^{2})^{2-d_{u}}}\,,$
(15)
which becomes a massive scalar propagator for $d_{u}=1$.
The LF violation is carried by single unparticle mediation and two unparticles
mediation in the loop (see Fig.1). The possible two unparticles mediation
brings an additional contribution to the LFV decays with the strength which is
a function of unparticle mass $m_{U}$, reaching $246\,GeV$ when $d_{u}\sim
1.0$ for the coupling $\lambda\sim 1.0$. In our numerical calculations we take
the scaling parameter $d_{u}$ not far from $1.0$, namely $1.0\leq d_{u}\leq
1.2$. On the other hand we take the coupling $\lambda$ as $\lambda\leq 1.0$ in
order to guarantee that the calculations are perturbative in case of
$d_{u}\sim 1.0$ and we choose the energy scale $\Lambda_{u}$ as
$\Lambda_{u}\sim 1.0\,(TeV)$. The FV U\- lepton-lepton couplings, the scalar
$\lambda^{S}_{ij}$ and pseudo scalar $\lambda^{P}_{ij}$, are among the free
parameters which we choose $\lambda^{S}_{ij}=\lambda^{P}_{ij}=\lambda_{ij}$.
Furthermore, we first consider that the diagonal $\lambda_{ii}=\lambda_{0}$
and off diagonal $\lambda_{ij}=\kappa\lambda_{0},i\neq j$ couplings are family
blind with $\kappa<1$. Second we assume that the the diagonal couplings
$\lambda_{ii}$ carry the lepton family hierarchy, namely,
$\lambda_{\tau\tau}>\lambda_{\mu\mu}>\lambda_{ee}$, on the other hand, the
off-diagonal couplings, $\lambda_{ij}$ are family blind, universal and
$\lambda_{ij}=\kappa\lambda_{ee}$. In our numerical calculations, we choose
$\kappa=0.5$ and we take the magnitude of the FV coupling(s) at most $1.0$ in
order to ensure that the calculations are the perturbative for $d_{u}=1.0$.
In order to estimate the BR of the LFV decays under consideration one needs
the Higgs mass and its total decay width. The theoretical upper and lower
bounds of Higgs mass read $1.0\,TeV$ and $0.1\,TeV$ [13], respectively. This
is due to the fact that one does not meet the unitarity problem and the
instability of the Higgs potential both. Furthermore, the electroweak
measurements predict the range of the Higgs mass as
$m_{H^{0}}=129^{+74}_{-49}$ [14] which is not in contradiction with the
theoretical results. The total Higgs decay width is another parameter which
should be restricted and it is estimated by using the possible decays for the
chosen Higgs mass444For the light (heavy) Higgs boson, $m_{H^{0}}\leq
130\,GeV$ ($m_{H^{0}}\sim 180\,GeV$), the leading decay mode is $b\bar{b}$
pair [15, 16, 17] ($H^{0}\rightarrow WW\rightarrow
l^{+}l^{\prime-}\nu_{l}\nu_{l^{\prime}}$ [18, 19, 20]).. Notice that
throughout our calculations we choose $m_{H^{0}}=120\,(GeV)$ and we use the
input values given in Table (1).
Parameter | Value
---|---
$m_{e}$ | $0.0005$ (GeV)
$m_{\mu}$ | $0.106$ (GeV)
$m_{\tau}$ | $1.780$ (GeV)
$\Gamma(H^{0})|_{m_{H^{0}}=120\,GeV}$ | $0.0029$ (GeV)
$G_{F}$ | $1.1663710^{-5}(GeV^{-2})$
Table 1: The values of the input parameters used in the numerical
calculations.
In Fig.2, we present the BR$(H^{0}\rightarrow\mu^{\pm}\,e^{\pm})$ with respect
to the scale parameter $d_{u}$ for the flavor blind (FB) couplings
$\lambda_{ee}=\lambda_{\mu\mu}=\lambda_{\tau\tau}=1.0$. Here, the solid (long
dashed-short dashed-dotted) line represents the BR for
$\lambda=0.0\,(0.2-0.5-1.0)$. The possible interaction of unparticle with the
Higgs scalar leads to a nonzero mass for unparticle after the spontaneous
symmetry breaking and the mass term leads to a suppression in the BR. The
additional term coming from the $U-U-H^{0}$ vertex does not result is an
enhancement in the BR. The BR reaches to the values of the order of $10^{-4}$
for $\lambda=0$ and $d_{u}\sim 1.0$. For $\lambda\sim 1.0$ and near $d_{u}\sim
1.0$ 555This is the case that unparticle mass is near the vacuum expectation
value, namely $m_{U}\sim 246\,GeV$. the BR is of the order of $10^{-6}$. Fig.3
represents the BR$(H^{0}\rightarrow\mu^{\pm}\,e^{\pm})$ with respect to
$\lambda$ for the scale parameter $d_{u}=1$. Here, the solid (long dashed-
short dashed) line represents the BR for
$\lambda_{ee}=\lambda_{\mu\mu}=\lambda_{\tau\tau}=1.0$
($\lambda_{ee}=0.1,\,\lambda_{\mu\mu}=0.5,\,\lambda_{\tau\tau}=1.0$-$\lambda_{ee}=0.01,\,\lambda_{\mu\mu}=0.1,\,\lambda_{\tau\tau}=1.0$).
This figure shows the strong sensitivity of the BR to the $U-U-H^{0}$
interaction strength $\lambda$, especially for $\lambda<0.3$.
Fig.4 (5) shows the
BR$(H^{0}\rightarrow\tau^{\pm}\,e^{\pm}\,(\tau^{\pm}\,\mu^{\pm}))$ with
respect to the scale parameter $d_{u}$, for the FB couplings
$\lambda_{ee}=\lambda_{\mu\mu}=\lambda_{\tau\tau}=1.0$. Here, the solid-long
dashed-short dashed-dotted lines represent the BR for
$\lambda=0.0-0.2-0.5-1.0$. In the case of $d_{u}\sim 1.0$, the BR is almost
$5.0\times 10^{-6}$ ($6.0\times 10^{-6}$) for $\lambda\sim 1.0$ and enhances
up to $4.0\times 10^{-4}$ for $\lambda=0$ and $d_{u}\sim 1.0$. Similar to the
previous decay the mass term leads to a suppression in the BR and the
additional term coming from the $U-U-H^{0}$ vertex is not enough to enhance
the BR over the numerical values which is obtained for the massless unparticle
case.
In Fig.6 (7) we present the
BR$(H^{0}\rightarrow\tau^{\pm}\,e^{\pm}\,(\tau^{\pm}\,\mu^{\pm}))$ with
respect to $\lambda$ for the scale parameter $d_{u}=1$. Here, the solid (long
dashed-short dashed) line represents the BR for
$\lambda_{ee}=\lambda_{\mu\mu}=\lambda_{\tau\tau}=1.0$
($\lambda_{ee}=0.1,\,\lambda_{\mu\mu}=0.5,\,\lambda_{\tau\tau}=1.0$-$\lambda_{ee}=0.01,\,\lambda_{\mu\mu}=0.1,\,\lambda_{\tau\tau}=1.0$).
It is observed that the BR is suppressed more than one order in the range
$0.0<\lambda<1.0$ and this suppression is strong for $\lambda<0.3$.
Conclusion
As a summary, the mass of unparticle which arises with unparticle Higgs scalar
interaction results in that the BRs of the LFV $H^{0}\rightarrow
l_{1}^{\pm}\,l_{2}^{\pm}$ decays are suppressed. The BRs are of the order of
$10^{-6}$ for $\lambda\sim 1.0$ and $d_{u}\sim 1.0$. If the unparticle-Higgs
scalar interaction is switched off unparticle remains massless and the BRs of
the decays studied reach to the values of the order of $10^{-4}$ for FB
U-lepton-lepton couplings. With the possible production of the Higgs boson
$H^{0}$ at the LHC the theoretical results of the BRs of the LFV Higgs decays
will be tested and the new physics which drives the flavor violation,
including the unparticle sector will be searched.
## References
* [1] U. Cotti, L. Diaz-Cruz, C. Pagliarone, E. Vataga, hep-ph/0111236 (2001).
* [2] T. Han, D. Marfatia, Phys. Rev. Lett. D86, 1442 (2001).
* [3] K. A. Assamagan, A. Deandrea, P.A. Delsart, Phys. Rev. D67 035001 (2003).
* [4] J. G. Koerner,A. Pilaftsis, K. Schilcher, Phys. Rev. D47, 1080 (1993).
* [5] E. O. Iltan, Mod. Phys. Lett. A24, 1361 (2009).
* [6] H. Georgi, Phys. Rev. Lett. 98, 221601 (2007).
* [7] H. Georgi, Phys. Lett. B650, 275 (2007).
* [8] S. L. Chen, X. G. He, Phys. Rev. D76, 091702 (2007).
* [9] T. Kikuchi, N. Okada, Phys. Lett. B665, 186 (2008).
* [10] A. Delgado, J. R. Espinosa, J. M. No and M. Quiros, JHEP 0804, 028 (2008)
* [11] P. J. Fox, A. Rajaraman, Y. Shirman, Phys. Rev. D76, 075004 (2007).
* [12] A. Rajaraman, Phys. Lett. B671, 411 (2009).
* [13] . K. G. Hagiawara, Particle Data Group Collaboration, Phys. Rev. D66, 010001 (2002).
* [14] C. Amsler et al. (Particle Data Group),Phys. Lett. B667, 1 (2008).
* [15] A. Djouadi, J. Kalinowski, M. Spira, Comput. Phys. Commun. 108, 56 (1998).
* [16] M. Spira, P. Zerwas, Lect. Notes Phys. 512, 161 (1998).
* [17] V. Drollinger, T. Muller, D. Denegri, hep-ph/0111312.
* [18] M. Carena, J. S. Conway, H. E. Haber, J. D. Hobbs, et. al., Physics at Run II: Supersymmery/Higgs workshop, hep-ph/0010338 (2000).
* [19] M. Dittmar, H. K. Dreiner, hep-ph/9703401 (1997).
* [20] M. Dittmar, H. K. Dreiner, Phys. Rev. D55, 167 (1997).
Figure 1: One loop diagrams contribute to $H^{0}\rightarrow
l_{1}^{-}\,l_{2}^{+}$ decay with scalar unparticle mediator. Solid line
represents the lepton field: $i$ represents the internal lepton, $l_{1}^{-}$
($l_{2}^{+}$) outgoing lepton (anti lepton), dashed line the Higgs field,
double dashed line unparticle field.
Figure 2: $d_{u}$ dependence of the BR $(H^{0}\rightarrow\mu^{\pm}\,e^{\pm})$
for $\lambda_{ee}=\lambda_{\mu\mu}=\lambda_{\tau\tau}=1.0$. Here, the solid
(long dashed-short dashed-dotted) line represents the BR for
$\lambda=0.0\,(0.2-0.5-1.0)$. Figure 3: $\lambda$ dependence of the BR
$(H^{0}\rightarrow\mu^{\pm}\,e^{\pm})$ for $d_{u}=1$. Here, the solid (long
dashed-short dashed) line represents the BR for
$\lambda_{ee}=\lambda_{\mu\mu}=\lambda_{\tau\tau}=1.0$
($\lambda_{ee}=0.1,\,\lambda_{\mu\mu}=0.5,\,\lambda_{\tau\tau}=1.0$-$\lambda_{ee}=0.01,\,\lambda_{\mu\mu}=0.1,\,\lambda_{\tau\tau}=1.0$).
Figure 4: The same as Fig.2 but for $H^{0}\rightarrow\tau^{\pm}\,e^{\pm}$
decay. Figure 5: The same as Fig.2 but for
$H^{0}\rightarrow\tau^{\pm}\,\mu^{\pm}$ decay. Figure 6: The same as Fig.3 but
for $H^{0}\rightarrow\tau^{\pm}\,e^{\pm}$ decay. Figure 7: The same as Fig.3
but for $H^{0}\rightarrow\tau^{\pm}\,\mu^{\pm}$ decay.
|
arxiv-papers
| 2010-06-10T17:49:18 |
2024-09-04T02:49:10.833049
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. O. Iltan",
"submitter": "Erhan Iltan",
"url": "https://arxiv.org/abs/1006.2095"
}
|
1006.2115
|
# Erlangen Program at Large: Outline
Vladimir V. Kisil School of Mathematics, University of Leeds, Leeds, LS2 9JT,
UK kisilv@maths.leeds.ac.uk http://www.maths.leeds.ac.uk/~kisilv/
###### Abstract.
This is an outline of _Erlangen Program at Large_. Study of objects and
properties, which are invariant under a group action, is very fruitful far
beyond the traditional geometry. In this paper we demonstrate this on the
example of the group $SL_{2}{}(\mathbb{R}{})$. Starting from the conformal
geometry we develop analytic functions and apply these to functional calculus.
Finally we provide an extensive description of open problems.
###### Key words and phrases:
Special linear group, Hardy space, Clifford algebra, elliptic, parabolic,
hyperbolic, complex numbers, dual numbers, double numbers, split-complex
numbers, Cauchy-Riemann-Dirac operator, Möbius transformations, functional
calculus, spectrum, quantum mechanics, non-commutative geometry.
###### 2000 Mathematics Subject Classification:
Primary 30G35; Secondary 22E46, 30F45, 32F45, 43A85, 30G30, 42C40, 46H30,
47A13, 81R30, 81R60.
On leave from the Odessa University.
###### Contents
1. 1 Introduction
1. 1.1 Make a Guess in Three Attempts
2. 1.2 Erlangen program at large
2. 2 Geometry
1. 2.1 Cycles as Invariant Objects
2. 2.2 Invariance of FSCc
3. 2.3 Invariants: algebraic and geometric
4. 2.4 Joint invariants: orthogonality
5. 2.5 Higher order joint invariants: s-orthogonality
6. 2.6 Distance, length and perpendicularity
3. 3 Analytic Functions
1. 3.1 Wavelet Transform and Cauchy Kernel
2. 3.2 The Dirac (Cauchy-Riemann) and Laplace Operators
3. 3.3 The Taylor expansion
4. 4 Functional Calculus
1. 4.1 Another Approach to Analytic Functional Calculus
2. 4.2 Representations in Banach Algebras
3. 4.3 Jet Bundles and Prolongations
4. 4.4 Spectrum and the Jordan Normal Form of a Matrix
5. 4.5 Spectral Mapping Theorem
5. 5 Open Problems
1. 5.1 Geometry
2. 5.2 Analytic Functions
3. 5.3 Functional Calculus
4. 5.4 Quantum Mechanics
## 1\. Introduction
The simplest objects with non-commutative multiplication may be $2\times 2$
matrices with real entries. Such matrices _of determinant one_ form a closed
set under multiplication (since $\det(AB)=\det A\cdot\det B$), the identity
matrix is among them and any such matrix has an inverse (since $\det A\neq
0$). In other words those matrices form a group, the $SL_{2}{}(\mathbb{R}{})$
group [Lang85]—one of the two most important Lie groups in analysis. The other
group is the Heisenberg group [Howe80a]. By contrast the “$ax+b$”-group, which
is often used to build wavelets, is only a subgroup of
$SL_{2}{}(\mathbb{R}{})$, see the numerator in (1.1).
The simplest non-linear transforms of the real line—linear-fractional or
Möbius maps—may also be associated with $2\times 2$ matrices [Beardon05a]*Ch.
13:
(1.1) $g:x\mapsto g\cdot x=\frac{ax+b}{cx+d},\text{ where
}g=\begin{pmatrix}a&b\\\ c&d\end{pmatrix},x\in\mathbb{R}{}.$
An enjoyable calculation shows that the composition of two transforms (1.1)
with different matrices $g_{1}$ and $g_{2}$ is again a Möbius transform with
matrix the product $g_{1}g_{2}$. In other words (1.1) it is a (left) action of
$SL_{2}{}(\mathbb{R}{})$.
According to F. Klein’s _Erlangen program_ (which was influenced by S. Lie)
any geometry is dealing with invariant properties under a certain group
action. For example, we may ask: _What kinds of geometry are related to the
$SL_{2}{}(\mathbb{R}{})$ action (1.1)_?
The Erlangen program has probably the highest rate of
$\frac{\text{praised}}{\text{actually used}}$ among mathematical theories not
only due to the big numerator but also due to undeserving small denominator.
As we shall see below Klein’s approach provides some surprising conclusions
even for such over-studied objects as circles.
### 1.1. Make a Guess in Three Attempts
It is easy to see that the $SL_{2}{}(\mathbb{R}{})$ action (1.1) makes sense
also as a map of complex numbers $z=x+\mathrm{i}y$, $\mathrm{i}^{2}=-1$.
Moreover, if $y>0$ then $g\cdot z$ has a positive imaginary part as well, i.e.
(1.1) defines a map from the upper half-plane to itself.
However there is no need to be restricted to the traditional route of complex
numbers only. Less-known _dual_ and _double_ numbers [Yaglom79]*Suppl. C have
also the form $z=x+\mathrm{i}y$ but different assumptions on the imaginary
unit $\mathrm{i}$: $\mathrm{i}^{2}=0$ or $\mathrm{i}^{2}=1$ correspondingly.
Although the arithmetic of dual and double numbers is different from the
complex ones, e.g. they have divisors of zero, we are still able to define
their transforms by (1.1) in most cases.
Three possible values $-1$, $0$ and $1$ of $\sigma:=\mathrm{i}^{2}$ will be
refereed to here as _elliptic_ , _parabolic_ and _hyperbolic_ cases
respectively. We repeatedly meet such a division of various mathematical
objects into three classes. They are named by the historically first
example—the classification of conic sections—however the pattern persistently
reproduces itself in many different areas: equations, quadratic forms,
metrics, manifolds, operators, etc. We will abbreviate this separation as
_EPH-classification_. The _common origin_ of this fundamental division can be
seen from the simple picture of a coordinate line split by zero into negative
and positive half-axes:
(1.2)
Connections between different objects admitting EPH-classification are not
limited to this common source. There are many deep results linking, for
example, ellipticity of quadratic forms, metrics and operators. On the other
hand there are still a lot of white spots and obscure gaps between some
subjects as well.
To understand the action (1.1) in all EPH cases we use the Iwasawa
decomposition [Lang85] of $SL_{2}{}(\mathbb{R}{})=ANK$ into _three_ one-
dimensional subgroups $A$, $N$ and $K$:
(1.3) $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}={\begin{pmatrix}\alpha&0\\\
0&\alpha^{-1}\end{pmatrix}}{\begin{pmatrix}1&\nu\\\
0&1\end{pmatrix}}{\begin{pmatrix}\cos\phi&\sin\phi\\\
-\sin\phi&\cos\phi\end{pmatrix}}.$
Subgroups $A$ and $N$ act in (1.1) irrespectively to value of $\sigma$: $A$
makes a dilation by $\alpha^{2}$, i.e. $z\mapsto\alpha^{2}z$, and $N$ shifts
points to left by $\nu$, i.e. $z\mapsto z+\nu$.
Figure 1. Action of the $K$ subgroup. The corresponding $K$-orbits are thick
circles, parabolas and hyperbolas. Thin traversal lines are images of the
vertical axis for certain values of the parameter $\phi$.
By contrast, the action of the third matrix from the subgroup $K$ sharply
depends on $\sigma$, see Fig. 1. In elliptic, parabolic and hyperbolic cases
$K$-orbits are circles, parabolas and (equilateral) hyperbolas
correspondingly. Thin traversal lines in Fig. 1 join points of orbits for the
same values of $\phi$ and grey arrows represent “local velocities”—vector
fields of derived representations.
### 1.2. Erlangen program at large
As we already mentioned the division of mathematics into areas is only
apparent. Therefore it is unnatural to limit Erlangen program only to
“geometry”. We may continue to look for $SL_{2}{}(\mathbb{R}{})$ invariant
objects in other related fields. For example, transform (1.1) generates
unitary representations on certain $L_{2}{}$ spaces, cf. (1.1):
(1.4) $g^{-1}:f(x)\mapsto\frac{1}{(cx+d)^{m}}f\left(\frac{ax+b}{cx+d}\right).$
For $m=1$, $2$, …the invariant subspaces of $L_{2}{}$ are Hardy and (weighted)
Bergman spaces of complex analytic functions. All main objects of _complex
analysis_ (Cauchy and Bergman integrals, Cauchy-Riemann and Laplace equations,
Taylor series etc.) may be obtaining in terms of invariants of the _discrete
series_ representations of $SL_{2}{}(\mathbb{R}{})$ [Kisil02c]*§ 3. Moreover
two other series (_principal_ and _complimentary_ [Lang85]) play the similar
rôles for hyperbolic and parabolic cases [Kisil02c] [Kisil05a].
Moving further we may observe that transform (1.1) is defined also for an
element $x$ in any algebra $\mathfrak{A}$ with a unit $\mathbf{1}$ as soon as
$(cx+d\mathbf{1})\in\mathfrak{A}$ has an inverse. If $\mathfrak{A}$ is
equipped with a topology, e.g. is a Banach algebra, then we may study a
_functional calculus_ for element $x$ [Kisil02a] in this way. It is defined as
an intertwining operator between the representation (1.4) in a space of
analytic functions and a similar representation in a left
$\mathfrak{A}$-module.
In the spirit of Erlangen program such functional calculus is still a
geometry, since it is dealing with invariant properties under a group action.
However even for a simplest non-normal operator, e.g. a Jordan block of the
length $k$, the obtained space is not like a space of point but is rather a
space of $k$-th _jets_ [Kisil02a]. Such non-point behaviour is oftenly
attributed to _non-commutative geometry_ and Erlangen program provides an
important input on this fashionable topic [Kisil02c].
Of course, there is no reasons to limit Erlangen program to
$SL_{2}{}(\mathbb{R}{})$ group only, other groups may be more suitable in
different situations. However $SL_{2}{}(\mathbb{R}{})$ still possesses a big
unexplored potential and is a good object to start with.
## 2\. Geometry
### 2.1. Cycles as Invariant Objects
###### Definition 2.1.
The common name _cycle_ [Yaglom79] is used to denote circles, parabolas and
hyperbolas (as well as straight lines as their limits) in the respective EPH
case.
(a) (b)
Figure 2. $K$-orbits as conic sections: circles are sections by the plane
$EE^{\prime}$; parabolas are sections by $PP^{\prime}$; hyperbolas are
sections by $HH^{\prime}$. Points on the same generator of the cone correspond
to the same value of $\phi$.
It is well known that any cycle is a _conic sections_ and an interesting
observation is that corresponding $K$-orbits are in fact sections of the same
two-sided right-angle cone, see Fig. 2. Moreover, each straight line
generating the cone, see Fig. 2(b), is crossing corresponding EPH $K$-orbits
at points with the same value of parameter $\phi$ from (1.3). In other words,
all three types of orbits are generated by the rotations of this generator
along the cone.
$K$-orbits are $K$-invariant in a trivial way. Moreover since actions of both
$A$ and $N$ for any $\sigma$ are extremely “shape-preserving” we find natural
invariant objects of the Möbius map:
###### Theorem 2.2 ([Kisil06a]).
The family of all cycles from Defn. 2.1 is invariant under the action (1.1).
According to Erlangen ideology we shall study invariant properties of cycles.
### 2.2. Invariance of FSCc
Fig. 2 suggests that we may get a unified treatment of cycles in all EPH by
consideration of a higher dimension spaces. The standard mathematical method
is to declare objects under investigations (cycles in our case, functions in
functional analysis, etc.) to be simply points of some bigger space. This
space should be equipped with an appropriate structure to hold externally
information which were previously inner properties of our objects.
A generic cycle is the set of points $(u,v)\in\mathbb{R}^{2}{}$ defined for
all values of $\sigma$ by the equation
(2.1) $k(u^{2}-\sigma v^{2})-2lu-2nv+m=0.$
This equation (and the corresponding cycle) is defined by a point $(k,l,n,m)$
from a projective space $\mathbb{P}^{3}{}$, since for a scaling factor
$\lambda\neq 0$ the point $(\lambda k,\lambda l,\lambda n,\lambda m)$ defines
the same equation (2.1). We call $\mathbb{P}^{3}{}$ the _cycle space_ and
refer to the initial $\mathbb{R}^{2}{}$ as the _point space_.
In order to get a connection with Möbius action (1.1) we arrange numbers
$(k,l,n,m)$ into the matrix
(2.2) $C_{\breve{\sigma}}^{s}=\begin{pmatrix}l+\mathrm{\breve{\i}}sn&-m\\\
k&-l+\mathrm{\breve{\i}}sn\end{pmatrix},$
with a new imaginary unit $\mathrm{\breve{\i}}$ and an additional parameter
$s$ usually equal to $\pm 1$. The values of
$\breve{\sigma}:=\mathrm{\breve{\i}}^{2}$ is $-1$, $0$ or $1$ independently
from the value of $\sigma$. The matrix (2.2) is the cornerstone of (extended)
Fillmore–Springer–Cnops construction (FSCc) [Cnops02a] and closely related to
technique recently used by A.A. Kirillov to study the Apollonian gasket
[Kirillov06].
The significance of FSCc in Erlangen framework is provided by the following
result:
###### Theorem 2.3.
The image $\tilde{C}_{\breve{\sigma}}^{s}$ of a cycle $C_{\breve{\sigma}}^{s}$
under transformation (1.1) with $g\in SL_{2}{}(\mathbb{R}{})$ is given by
similarity of the matrix (2.2):
(2.3) $\tilde{C}_{\breve{\sigma}}^{s}=gC_{\breve{\sigma}}^{s}g^{-1}.$
In other words FSCc (2.2) _intertwines_ Möbius action (1.1) on cycles with
linear map (2.3).
There are several ways to prove (2.3): either by a brute force calculation
(fortunately performed by a CAS) [Kisil05a] or through the related
orthogonality of cycles [Cnops02a], see the end of the next section 2.3.
The important observation here is that FSCc (2.2) uses an imaginary unit
$\mathrm{\breve{\i}}$ which is not related to $\mathrm{i}$ defining the
appearance of cycles on plane. In other words any EPH type of geometry in the
cycle space $\mathbb{P}^{3}{}$ admits drawing of cycles in the point space
$\mathbb{R}^{2}{}$ as circles, parabolas or hyperbolas. We may think on points
of $\mathbb{P}^{3}{}$ as ideal cycles while their depictions on
$\mathbb{R}^{2}{}$ are only their shadows on the wall of Plato’s cave.
(a) (b)
Figure 3. (a) Different EPH implementations of the same cycles defined by
quadruples of numbers.
(b) Centres and foci of two parabolas with the same focal length.
Fig. 3(a) shows the same cycles drawn in different EPH styles. Points
$c_{e,p,h}=(\frac{l}{k},-\sigma\frac{n}{k})$ are their respective
e/p/h-centres. They are related to each other through several identities:
(2.4) $c_{e}=\bar{c}_{h},\quad c_{p}=\frac{1}{2}(c_{e}+c_{h}).$
Fig. 3(b) presents two cycles drawn as parabolas, they have the same focal
length $\frac{n}{2k}$ and thus their e-centres are on the same level. In other
words _concentric_ parabolas are obtained by a vertical shift, not scaling as
an analogy with circles or hyperbolas may suggest.
Fig. 3(b) also presents points, called e/p/h-foci:
(2.5) $f_{e,p,h}=\left(\frac{l}{k},-\frac{\det
C_{\breve{\sigma}}^{s}}{2nk}\right),$
which are independent of the sign of $s$. If a cycle is depicted as a parabola
then h-focus, p-focus, e-focus are correspondingly geometrical focus of the
parabola, its vertex, and the point on the directrix nearest to the vertex.
As we will see, cf. Thms. 2.5 and 2.7, all three centres and three foci are
useful attributes of a cycle even if it is drawn as a circle.
### 2.3. Invariants: algebraic and geometric
We use known algebraic invariants of matrices to build appropriate geometric
invariants of cycles. It is yet another demonstration that any division of
mathematics into subjects is only illusive.
For $2\times 2$ matrices (and thus cycles) there are only two essentially
different invariants under similarity (2.3) (and thus under Möbius action
(1.1)): the _trace_ and the _determinant_. The latter was already used in
(2.5) to define cycle’s foci. However due to projective nature of the cycle
space $\mathbb{P}^{3}{}$ the absolute values of trace or determinant are
irrelevant, unless they are zero.
Alternatively we may have a special arrangement for normalisation of
quadruples $(k,l,n,m)$. For example, if $k\neq 0$ we may normalise the
quadruple to $(1,\frac{l}{k},\frac{n}{k},\frac{m}{k})$ with highlighted
cycle’s centre. Moreover in this case $\det{C^{s}_{\breve{\sigma}}}$ is equal
to the square of cycle’s radius, cf. Section 2.6. Another normalisation
$\det{C^{s}_{\breve{\sigma}}}=1$ is used in [Kirillov06] to get a nice
condition for touching circles.
We still get important characterisation even with non-normalised cycles, e.g.,
invariant classes (for different $\breve{\sigma}$) of cycles are defined by
the condition $\det C_{\breve{\sigma}}^{s}=0$. Such a class is parametrises
only by two real number and as such is easily attached to certain point of
$\mathbb{R}^{2}{}$. For example, the cycle $C_{\breve{\sigma}}^{s}$ with $\det
C_{\breve{\sigma}}^{s}=0$, $\breve{\sigma}=-1$ drawn elliptically represent
just a point $(\frac{l}{k},\frac{n}{k})$, i.e. (elliptic) zero-radius circle.
The same condition with $\breve{\sigma}=1$ in hyperbolic drawing produces a
null-cone originated at point $(\frac{l}{k},\frac{n}{k})$:
$(u-\frac{l}{k})^{2}-(v-\frac{n}{k})^{2}=0,$
i.e. a zero-radius cycle in hyperbolic metric.
Figure 4. Different $\mathrm{i}$-implementations of the same
$\breve{\sigma}$-zero-radius cycles and corresponding foci.
In general for every notion there is nine possibilities: three EPH cases in
the cycle space times three EPH realisations in the point space. Such nine
cases for “zero radius” cycles is shown on Fig. 4. For example, p-zero-radius
cycles in any implementation touch the real axis.
This “touching” property is a manifestation of the _boundary effect_ in the
upper-half plane geometry [Kisil05a]*Rem. 3.4. The famous question on hearing
drum’s shape has a sister:
> _Can we see/feel the boundary from inside a domain?_
Both orthogonality relations described below are “boundary aware” as well. It
is not surprising after all since $SL_{2}{}(\mathbb{R}{})$ action on the
upper-half plane was obtained as an extension of its action (1.1) on the
boundary.
According to the categorical viewpoint internal properties of objects are of
minor importance in comparison to their relations with other objects from the
same class. Thus from now on we will look for invariant relations between two
or more cycles.
### 2.4. Joint invariants: orthogonality
The most expected relation between cycles is based on the following Möbius
invariant “inner product” build from a trace of product of two cycles as
matrices:
(2.6) $\left\langle
C_{\breve{\sigma}}^{s},\tilde{C}_{\breve{\sigma}}^{s}\right\rangle=\mathop{tr}(C_{\breve{\sigma}}^{s}\tilde{C}_{\breve{\sigma}}^{s})$
By the way, an inner product of this type is used, for example, in GNS
construction to make a Hilbert space out of $C^{*}$-algebra. The next standard
move is given by the following definition.
###### Definition 2.4.
Two cycles are called $\breve{\sigma}$-orthogonal if $\left\langle
C_{\breve{\sigma}}^{s},\tilde{C}_{\breve{\sigma}}^{s}\right\rangle=0$.
For the case of $\breve{\sigma}\sigma=1$, i.e. when geometries of the cycle
and point spaces are both either elliptic or hyperbolic, such an orthogonality
is the standard one, defined in terms of angles between tangent lines in the
intersection points of two cycles. However in the remaining seven ($=9-2$)
cases the innocent-looking Defn. 2.4 brings unexpected relations.
Figure 5. Orthogonality of the first kind in the elliptic point space.
Each picture presents two groups (green and blue) of cycles which are
orthogonal to the red cycle $C^{s}_{\breve{\sigma}}$. Point $b$ belongs to
$C^{s}_{\breve{\sigma}}$ and the family of blue cycles passing through $b$ is
orthogonal to $C^{s}_{\breve{\sigma}}$. They all also intersect in the point
$d$ which is the inverse of $b$ in $C^{s}_{\breve{\sigma}}$. Any orthogonality
is reduced to the usual orthogonality with a new (“ghost”) cycle (shown by the
dashed line), which may or may not coincide with $C^{s}_{\breve{\sigma}}$. For
any point $a$ on the “ghost” cycle the orthogonality is reduced to the local
notion in the terms of tangent lines at the intersection point. Consequently
such a point $a$ is always the inverse of itself.
Elliptic (in the point space) realisations of Defn. 2.4, i.e. $\sigma=-1$ is
shown in Fig. 5. The left picture corresponds to the elliptic cycle space,
e.g. $\breve{\sigma}=-1$. The orthogonality between the red circle and any
circle from the blue or green families is given in the usual Euclidean sense.
The central (parabolic in the cycle space) and the right (hyperbolic) pictures
show non-local nature of the orthogonality. There are analogues pictures in
parabolic and hyperbolic point spaces as well [Kisil05a].
This orthogonality may still be expressed in the traditional sense if we will
associate to the red circle the corresponding “ghost” circle, which shown by
the dashed line in Fig. 5. To describe ghost cycle we need the _Heaviside
function_ $\chi(\sigma)$:
(2.7) $\chi(t)=\left\\{\begin{array}[]{ll}1,&t\geq 0;\\\
-1,&t<0.\end{array}\right.$
###### Theorem 2.5.
A cycle is $\breve{\sigma}$-orthogonal to cycle $C_{\breve{\sigma}}^{s}$ if it
is orthogonal in the usual sense to the $\sigma$-realisation of “ghost” cycle
$\hat{C}_{\breve{\sigma}}^{s}$, which is defined by the following two
conditions:
1. (i)
$\chi(\sigma)$-centre of $\hat{C}_{\breve{\sigma}}^{s}$ coincides with
$\breve{\sigma}$-centre of $C_{\breve{\sigma}}^{s}$.
2. (ii)
Cycles $\hat{C}_{\breve{\sigma}}^{s}$ and $C^{s}_{\breve{\sigma}}$ have the
same roots, moreover $\det\hat{C}_{\sigma}^{1}=\det
C^{\chi(\breve{\sigma})}_{\sigma}$.
The above connection between various centres of cycles illustrates their
meaningfulness within our approach.
One can easy check the following orthogonality properties of the zero-radius
cycles defined in the previous section:
1. (i)
Since $\left\langle
C_{\breve{\sigma}}^{s},{C}_{\breve{\sigma}}^{s}\right\rangle=\det{C}_{\breve{\sigma}}^{s}$
zero-radius cycles are self-orthogonal (isotropic) ones.
2. (ii)
A cycle ${C^{s}_{\breve{\sigma}}}$ is $\sigma$-orthogonal to a zero-radius
cycle $Z^{s}_{\breve{\sigma}}$ if and only if ${C^{s}_{\breve{\sigma}}}$
passes through the $\sigma$-centre of $Z^{s}_{\breve{\sigma}}$.
### 2.5. Higher order joint invariants: s-orthogonality
With appetite already wet one may wish to build more joint invariants. Indeed
for any homogeneous polynomial $p(x_{1},x_{2},\ldots,x_{n})$ of several non-
commuting variables one may define an invariant joint disposition of $n$
cycles ${}^{j}\\!{C^{s}_{\breve{\sigma}}}$ by the condition:
$\mathop{tr}p({}^{1}\\!{C^{s}_{\breve{\sigma}}},{}^{2}\\!{C^{s}_{\breve{\sigma}}},\ldots,{}^{n}\\!{C^{s}_{\breve{\sigma}}})=0.$
However it is preferable to keep some geometrical meaning of constructed
notions.
An interesting observation is that in the matrix similarity of cycles (2.3)
one may replace element $g\in SL_{2}{}(\mathbb{R}{})$ by an arbitrary matrix
corresponding to another cycle. More precisely the product
${C^{s}_{\breve{\sigma}}}{\tilde{C}^{s}_{\breve{\sigma}}}{C^{s}_{\breve{\sigma}}}$
is again the matrix of the form (2.2) and thus may be associated to a cycle.
This cycle may be considered as the reflection of
${\tilde{C}^{s}_{\breve{\sigma}}}$ in ${C^{s}_{\breve{\sigma}}}$.
###### Definition 2.6.
A cycle ${C^{s}_{\breve{\sigma}}}$ is s-orthogonal _to_ a cycle
${\tilde{C}^{s}_{\breve{\sigma}}}$ if the reflection of
${\tilde{C}^{s}_{\breve{\sigma}}}$ in ${C^{s}_{\breve{\sigma}}}$ is orthogonal
(in the sense of Defn. 2.4) to the real line. Analytically this is defined by:
(2.8)
$\mathop{tr}({C^{s}_{\breve{\sigma}}}{\tilde{C}^{s}_{\breve{\sigma}}}{C^{s}_{\breve{\sigma}}}R^{s}_{\breve{\sigma}})=0.$
Due to invariance of all components in the above definition s-orthogonality is
a Möbius invariant condition. Clearly this is not a symmetric relation: if
${C^{s}_{\breve{\sigma}}}$ is s-orthogonal to
${\tilde{C}^{s}_{\breve{\sigma}}}$ then ${\tilde{C}^{s}_{\breve{\sigma}}}$ is
not necessarily s-orthogonal to ${C^{s}_{\breve{\sigma}}}$.
Figure 6. Orthogonality of the second kind for circles. To highlight both
similarities and distinctions with the ordinary orthogonality we use the same
notations as that in Fig. 5.
Fig. 6 illustrates s-orthogonality in the elliptic point space. By contrast
with Fig. 5 it is not a local notion at the intersection points of cycles for
all $\breve{\sigma}$. However it may be again clarified in terms of the
appropriate s-ghost cycle, cf. Thm. 2.5.
###### Theorem 2.7.
A cycle is s-orthogonal to a cycle $C^{s}_{\breve{\sigma}}$ if its orthogonal
in the traditional sense to its _s-ghost cycle_
${\tilde{C}^{\breve{\sigma}}_{\breve{\sigma}}}={C^{\chi(\sigma)}_{\breve{\sigma}}}\mathbb{R}^{\breve{\sigma}}_{\breve{\sigma}}{}{C^{\chi(\sigma)}_{\breve{\sigma}}}$,
which is the reflection of the real line in
${C^{\chi(\sigma)}_{\breve{\sigma}}}$ and $\chi$ is the _Heaviside function_
(2.7). Moreover
1. (i)
$\chi(\sigma)$-Centre of ${\tilde{C}^{\breve{\sigma}}_{\breve{\sigma}}}$
coincides with the $\breve{\sigma}$-focus of ${C^{s}_{\breve{\sigma}}}$,
consequently all lines s-orthogonal to ${C^{s}_{\breve{\sigma}}}$ are passing
the respective focus.
2. (ii)
Cycles ${C^{s}_{\breve{\sigma}}}$ and
${\tilde{C}^{\breve{\sigma}}_{\breve{\sigma}}}$ have the same roots.
Note the above intriguing interplay between cycle’s centres and foci. Although
s-orthogonality may look exotic it will naturally appear in the end of next
Section again.
Of course, it is possible to define another interesting higher order joint
invariants of two or even more cycles.
### 2.6. Distance, length and perpendicularity
Geo _metry_ in the plain meaning of this word deals with _distances_ and
_lengths_. Can we obtain them from cycles?
(a) (b) (c)
Figure 7. (a) The square of the parabolic diameter is the square of the
distance between roots if they are real ($z_{1}$ and $z_{2}$), otherwise the
negative square of the distance between the adjoint roots ($z_{3}$ and
$z_{4}$).
(b) Distance as extremum of diameters in elliptic ($z_{1}$ and $z_{2}$) and
parabolic ($z_{3}$ and $z_{4}$) cases.
(c) Perpendicular as the shortest route to a line.
We mentioned already that for circles normalised by the condition $k=1$ the
value
$\det{C^{s}_{\breve{\sigma}}}=\left\langle{C^{s}_{\breve{\sigma}}},{C^{s}_{\breve{\sigma}}}\right\rangle$
produces the square of the traditional circle radius. Thus we may keep it as
the definition of the _radius_ for any cycle. But then we need to accept that
in the parabolic case the radius is the (Euclidean) distance between (real)
roots of the parabola, see Fig. 7(a).
Having radii of circles already defined we may use them for other measurements
in several different ways. For example, the following variational definition
may be used:
###### Definition 2.8.
The _distance_ between two points is the extremum of diameters of all cycles
passing through both points, see Fig. 7(b).
If $\breve{\sigma}=\sigma$ this definition gives in all EPH cases the distance
between endpoints of a vector $z=u+\mathrm{i}v$ as follows:
(2.9) $d_{e,p,h}(u,v)^{2}=(u+\mathrm{i}v)(u-\mathrm{i}v)=u^{2}-\sigma v^{2}.$
The parabolic distance $d_{p}^{2}=u^{2}$, see Fig. 7(b), algebraically sits
between $d_{e}$ and $d_{h}$ according to the general principle (1.2) and is
widely accepted [Yaglom79]. However one may be unsatisfied by its degeneracy.
An alternative measurement is motivated by the fact that a circle is the set
of equidistant points from its centre. However the choice of “centre” is now
rich: it may be either point from three centres (2.4) or three foci (2.5).
###### Definition 2.9.
The _length_ of a directed interval $\overrightarrow{AB}$ is the radius of the
cycle with its _centre_ (denoted by $l_{c}(\overrightarrow{AB})$) or _focus_
(denoted by $l_{f}(\overrightarrow{AB})$) at the point $A$ which passes
through $B$.
These definition is less common and have some unusual properties like non-
symmetry: $l_{f}(\overrightarrow{AB})\neq l_{f}(\overrightarrow{BA})$. However
it comfortably fits the Erlangen program due to its
$SL_{2}{}(\mathbb{R}{})$-_conformal invariance_ :
###### Theorem 2.10 ([Kisil05a]).
Let $l$ denote either the EPH distances (2.9) or any length from Defn. 2.9.
Then for fixed $y$, $y^{\prime}\in\mathbb{R}^{\sigma}{}$ the limit:
$\lim_{t\rightarrow 0}\frac{l(g\cdot
y,g\cdot(y+ty^{\prime}))}{l(y,y+ty^{\prime})},\qquad\text{ where }g\in
SL_{2}{}(\mathbb{R}{}),$
exists and its value depends only from $y$ and $g$ and is independent from
$y^{\prime}$.
We may return from distances to angles recalling that in the Euclidean space a
perpendicular provides the shortest root from a point to a line, see Fig.
7(c).
###### Definition 2.11.
Let $l$ be a length or distance. We say that a vector $\overrightarrow{AB}$ is
_$l$ -perpendicular_ to a vector $\overrightarrow{CD}$ if function
$l(\overrightarrow{AB}+\varepsilon\overrightarrow{CD})$ of a variable
$\varepsilon$ has a local extremum at $\varepsilon=0$.
A pleasant surprise is that $l_{f}$-perpendicularity obtained thought the
length from focus (Defn. 2.9) coincides with already defined in Section 2.5
s-orthogonality as follows from Thm. 2.7(i). It is also possible [Kisil08a] to
make $SL_{2}{}(\mathbb{R}{})$ action isometric in all three cases.
All these study are waiting to be generalised to high dimensions and Clifford
algebras provide a suitable language for this [Kisil05a].
## 3\. Analytic Functions
We saw in the previous section that an inspiring geometry of cycles can be
recovered from the properties of $SL_{2}{}(\mathbb{R}{})$. In this section we
consider a realisation of the function theory within Erlangen approach
[Kisil97c, Kisil97a, Kisil01a, Kisil02c].
### 3.1. Wavelet Transform and Cauchy Kernel
Elements of $SL_{2}{}(\mathbb{R}{})$ could be also represented by $2\times
2$-matrices with complex entries such that:
$g={\left(\\!\\!\begin{array}[]{cc}\alpha&\bar{\beta}\\\
\beta&\bar{\alpha}\end{array}\\!\\!\right)},\qquad
g^{-1}={\left(\\!\\!\begin{array}[]{cc}\bar{\alpha}&-\bar{\beta}\\\
-\beta&\alpha\end{array}\\!\\!\right)},\qquad\left|\alpha\right|^{2}-\left|\beta\right|^{2}=1.$
This realisations of $SL_{2}{}(\mathbb{R}{})$ (or rather $SU(2,\mathbb{C}{})$)
is more suitable for function theory in the unit disk. It is obtained from the
form, which we used before for the upper half-plane, by means of the Cayley
transform [Kisil05a, § 8.1].
We may identify the unit disk $\mathbb{D}{}$ with the homogeneous space
$SL_{2}{}(\mathbb{R}{})/\mathbb{T}{}$ for the unit circle $\mathbb{T}{}$
through the important decomposition
$SL_{2}{}(\mathbb{R}{})\sim\mathbb{D}{}\times\mathbb{T}{}$ with
$K=\mathbb{T}{}$—the only compact subgroup of $SL_{2}{}(\mathbb{R}{})$:
(3.7) $\displaystyle{\left(\\!\\!\begin{array}[]{cc}\alpha&\bar{\beta}\\\
\beta&\bar{\alpha}\end{array}\\!\\!\right)}$ $\displaystyle=$
$\displaystyle\left|\alpha\right|{\left(\\!\\!\begin{array}[]{cc}1&\bar{\beta}\bar{\alpha}^{-1}\\\
{\beta}{\alpha}^{-1}&1\end{array}\\!\\!\right)}{\left(\\!\\!\begin{array}[]{cc}\frac{{\alpha}}{\left|\alpha\right|}&0\\\
0&\frac{\bar{\alpha}}{\left|\alpha\right|}\end{array}\\!\\!\right)}$ (3.12)
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1-\left|u\right|^{2}}}{\left(\\!\\!\begin{array}[]{cc}1&u\\\
\bar{u}&1\end{array}\\!\\!\right)},{\left(\\!\\!\begin{array}[]{cc}e^{i\omega}&0\\\
0&e^{-i\omega}\end{array}\\!\\!\right)}$
where
$\omega=\arg\alpha,\qquad
u=\bar{\beta}\bar{\alpha}^{-1},\qquad\left|u\right|<1.$
Each element $g\in SL_{2}{}(\mathbb{R}{})$ acts by the linear-fractional
transformation (the Möbius map) on $\mathbb{D}{}$ and $\mathbb{T}{}$
$H_{2}{}(\mathbb{T}{})$ as follows:
(3.13)
$g^{-1}:z\mapsto\frac{\bar{\alpha}z-\bar{\beta}}{\alpha-{\beta}z},\qquad\textrm{
where }\quad
g^{-1}={\left(\\!\\!\begin{array}[]{cc}\bar{\alpha}&-\bar{\beta}\\\
-\beta&\alpha\end{array}\\!\\!\right)}.$
In the decomposition (3.7) the first matrix on the right hand side acts by
transformation (1.1) as an orthogonal rotation of $\mathbb{T}{}$ or
$\mathbb{D}{}$; and the second one—by transitive family of maps of the unit
disk onto itself.
The standard linearisation procedure [Kirillov76, § 7.1] leads from Möbius
transformations (1.1) to the unitary representation $\rho_{1}$ irreducible on
the _Hardy space_ :
(3.14)
$\rho_{1}(g):f(z)\mapsto\frac{1}{\alpha-{\beta}{z}}\,f\left(\frac{\bar{\alpha}z-\bar{\beta}}{\alpha-{\beta}z}\right)\qquad\textrm{
where }\quad
g^{-1}={\left(\\!\\!\begin{array}[]{cc}\bar{\alpha}&-\bar{\beta}\\\
-\beta&\alpha\end{array}\\!\\!\right)}.$
Möbius transformations provide a natural family of intertwining operators for
$\rho_{1}$ coming from inner automorphisms of $SL_{2}{}(\mathbb{R}{})$ (will
be used later).
We choose [Kisil98a, Kisil01a] $K$-invariant function $v_{0}(z)\equiv 1$ to be
a _vacuum vector_. Thus the associated _coherent states_
$v(g,z)=\rho_{1}(g)v_{0}(z)=(u-z)^{-1}$
are completely determined by the point on the unit disk
$u=\bar{\beta}\bar{\alpha}^{-1}$. The family of coherent states considered as
a function of both $u$ and $z$ is obviously the Cauchy kernel [Kisil97c]. The
_wavelet transform_ [Kisil97c, Kisil98a]
$\mathcal{W}:L_{2}{}(\mathbb{T}{})\rightarrow
H_{2}{}(\mathbb{D}{}):f(z)\mapsto\mathcal{W}f(g)=\left\langle
f,v_{g}\right\rangle$ is the Cauchy integral:
(3.15) $\mathcal{W}f(u)=\frac{1}{2\pi
i}\int_{\mathbb{T}{}}f(z)\frac{1}{u-z}\,dz.$
We start from the following observation reflected in the almost any textbook
on complex analysis:
###### Proposition 3.1.
_Analytic function theory_ in the unit disk $\mathbb{D}{}$ is a manifestation
of the mock discrete series representation $\rho_{1}$ of
$SL_{2}{}(\mathbb{R}{})$:
(3.16)
$\rho_{1}(g):f(z)\mapsto\frac{1}{\alpha-{\beta}{z}}\,f\left(\frac{\bar{\alpha}z-\bar{\beta}}{\alpha-{\beta}z}\right),\quad\textup{
where }{\left(\\!\\!\begin{array}[]{cc}\bar{\alpha}&-\bar{\beta}\\\
-\beta&\alpha\end{array}\\!\\!\right)}\in SL_{2}{}(\mathbb{R}{}).$
Other classical objects of complex analysis (the Cauchy-Riemann equation, the
Taylor series, the Bergman space, etc.) can be also obtained [Kisil97c,
Kisil01a] from representation $\rho_{1}$ as shown below.
### 3.2. The Dirac (Cauchy-Riemann) and Laplace Operators
Consideration of Lie groups is hardly possible without consideration of their
Lie algebras, which are naturally represented by left and right invariant
vectors fields on groups. On a homogeneous space $\Omega=G/H$ we have also
defined a left action of $G$ and can be interested in left invariant vector
fields (first order differential operators). Due to the irreducibility of
$F_{2}{}(\Omega)$ under left action of $G$ every such vector field $D$
restricted to $F_{2}{}(\Omega)$ is a scalar multiplier of identity
$D|_{F_{2}{}(\Omega)}=cI$. We are in particular interested in the case $c=0$.
###### Definition 3.2.
[AtiyahSchmid80, KnappWallach76] A $G$-invariant first order differential
operator
$D_{\tau}:C_{\infty}{}(\Omega,\mathcal{S}\otimes V_{\tau})\rightarrow
C_{\infty}{}(\Omega,\mathcal{S}\otimes V_{\tau})$
such that $\mathcal{W}(F_{2}{}(X))\subset\mathrm{ker}\,D_{\tau}$ is called
_(Cauchy-Riemann-)Dirac operator_ on $\Omega=G/H$ associated with an
irreducible representation $\tau$ of $H$ in a space $V_{\tau}$ and a spinor
bundle $\mathcal{S}$.
The Dirac operator is explicitly defined by the formula [KnappWallach76,
(3.1)]:
(3.17) $D_{\tau}=\sum_{j=1}^{n}\rho(Y_{j})\otimes c(Y_{j})\otimes 1,$
where $Y_{j}$ is an orthonormal basis of
$\mathfrak{p}=\mathfrak{h}^{\perp}$—the orthogonal completion of the Lie
algebra $\mathfrak{h}$ of the subgroup $H$ in the Lie algebra $\mathfrak{g}$
of $G$; $\rho(Y_{j})$ is the infinitesimal generator of the right action of
$G$ on $\Omega$; $c(Y_{j})$ is Clifford multiplication by
$Y_{i}\in\mathfrak{p}$ on the Clifford module $\mathcal{S}$. We also define an
invariant Laplacian by the formula
(3.18) $\Delta_{\tau}=\sum_{j=1}^{n}\rho(Y_{j})^{2}\otimes\epsilon_{j}\otimes
1,$
where $\epsilon_{j}=c(Y_{j})^{2}$ is $+1$ or $-1$.
###### Proposition 3.3.
Let all commutators of vectors of $\mathfrak{h}^{\perp}$ belong to
$\mathfrak{h}$, i.e.
$[\mathfrak{h}^{\perp},\mathfrak{h}^{\perp}]\subset\mathfrak{h}$. Let also
$f_{0}$ be an eigenfunction for all vectors of $\mathfrak{h}$ with eigenvalue
$0$ and let also $\mathcal{W}f_{0}$ be a null solution to the Dirac operator
$D$. Then $\Delta f(x)=0$ for all $f(x)\in F_{2}{}(\Omega)$.
###### Proof.
Because $\Delta$ is a linear operator and $F_{2}{}(\Omega)$ is generated by
$\pi_{0}(s(a))\mathcal{W}f_{0}$ it is enough to check that
$\Delta\pi_{0}(s(a))\mathcal{W}f_{0}=0$. Because $\Delta$ and $\pi_{0}$
commute it is enough to check that $\Delta\mathcal{W}f_{0}=0$. Now we observe
that
$\Delta=D^{2}-\sum_{i,j}\rho([Y_{i},Y_{j}])\otimes c(Y_{i})c(Y_{j})\otimes 1.$
Thus the desired assertion is follows from two identities
$\rho([Y_{i},Y_{j}])\mathcal{W}f_{0}=0$ for $[Y_{i},Y_{j}]\in H$ and
$D\mathcal{W}f_{0}=0$. ∎
###### Example 3.4.
Let $G=SL_{2}{}(\mathbb{R}{})$ and $H$ be its one-dimensional compact subgroup
$K$ generated by an element $Z\in\mathfrak{sl}(2,\mathbb{R}{})$. Then
$\mathfrak{h}^{\perp}$ is spanned by two vectors $Y_{1}=A$ and $Y_{2}=B$. In
such a situation we can use $\mathbb{C}{}$ instead of the Clifford algebra.
Then formula (3.17) takes a simple form $D=r(A+iB)$. Infinitesimal action of
this operator in the upper-half plane follows from calculation in [Lang85,
VI.5(8), IX.5(3)], it is $[D_{\mathbb{H}{}}f](z)=-2iy\frac{\partial
f(z)}{\partial\bar{z}}$, $z=x+iy$. Making the Caley transform we can find its
action in the unit disk $D_{\mathbb{D}{}}$: again the Cauchy-Riemann operator
$\frac{\partial}{\partial\bar{z}}$ is its principal component. We calculate
$D_{\mathbb{H}{}}$ explicitly now to stress the similarity with
$\mathbb{R}^{1,1}{}$ case.
For the upper half plane $\mathbb{H}{}$ we have following formulas:
$\displaystyle s$ $\displaystyle:$ $\displaystyle\mathbb{H}{}\rightarrow
SL_{2}{}(\mathbb{R}{}):z=x+iy\mapsto
g={\left(\\!\\!\begin{array}[]{cc}y^{1/2}&xy^{-1/2}\\\
0&y^{-1/2}\end{array}\\!\\!\right)};$ $\displaystyle s^{-1}$ $\displaystyle:$
$\displaystyle
SL_{2}{}(\mathbb{R}{})\rightarrow\mathbb{H}{}:{\left(\\!\\!\begin{array}[]{cc}a&b\\\
c&d\end{array}\\!\\!\right)}\mapsto z=\frac{ai+b}{ci+d};$
$\displaystyle\rho(g)$ $\displaystyle:$
$\displaystyle\mathbb{H}{}\rightarrow\mathbb{H}{}:z\mapsto s^{-1}(s(z)*g)$
$\displaystyle\qquad\qquad\qquad=s^{-1}{\left(\\!\\!\begin{array}[]{cc}ay^{-1/2}+cxy^{-1/2}&by^{1/2}+dxy^{-1/2}\\\
cy^{-1/2}&dy^{-1/2}\end{array}\\!\\!\right)}$
$\displaystyle\qquad\qquad\qquad=\frac{(yb+xd)+i(ay+cx)}{ci+d}$
Thus the right action of $SL_{2}{}(\mathbb{R}{})$ on $\mathbb{H}{}$ is given
by the formula
$\rho(g)z=\frac{(yb+xd)+i(ay+cx)}{ci+d}=x+y\frac{bd+ac}{c^{2}+d^{2}}+iy\frac{1}{c^{2}+d^{2}}.$
For $A$ and $B$ in $\mathfrak{sl}(2,\mathbb{R}{})$ we have:
$\rho(e^{At})z=x+iye^{2t},\qquad\rho(e^{Bt})z=x+y\frac{e^{2t}-e^{-2t}}{e^{2t}+e^{-2t}}+iy\frac{4}{e^{2t}+e^{-2t}}.$
Thus
$\displaystyle[\rho(A)f](z)$ $\displaystyle=$ $\displaystyle\frac{\partial
f(\rho(e^{At})z)}{\partial t}|_{t=0}=2y\partial_{2}f(z),$
$\displaystyle{}[\rho(B)f](z)$ $\displaystyle=$ $\displaystyle\frac{\partial
f(\rho(e^{Bt})z)}{\partial t}|_{t=0}=2y\partial_{1}f(z),$
where $\partial_{1}$ and $\partial_{2}$ are derivatives of $f(z)$ with respect
to real and imaginary party of $z$ respectively. Thus we get
$D_{\mathbb{H}{}}=i\rho(A)+\rho(B)=2yi\partial_{2}+2y\partial_{1}=2y\frac{\partial}{\partial\bar{z}}$
as was expected.
### 3.3. The Taylor expansion
For any decomposition $f_{a}(x)=\sum_{\alpha}\psi_{\alpha}(x)V_{\alpha}(a)$ of
the coherent states $f_{a}(x)$ by means of functions $V_{\alpha}(a)$ (where
the sum can become eventually an integral) we have the _Taylor expansion_
(3.22) $\displaystyle\widehat{f}(a)$ $\displaystyle=$
$\displaystyle\int_{X}f(x)\bar{f}_{a}(x)\,dx=\int_{X}f(x)\sum_{\alpha}\bar{\psi}_{\alpha}(x)\bar{V}_{\alpha}(a)\,dx$
$\displaystyle=$
$\displaystyle\sum_{\alpha}\int_{X}f(x)\bar{\psi}_{\alpha}(x)\,dx\bar{V}_{\alpha}(a)$
$\displaystyle=$
$\displaystyle\sum_{\alpha}^{\infty}\bar{V}_{\alpha}(a)f_{\alpha},$
where $f_{\alpha}=\int_{X}f(x)\bar{\psi}_{\alpha}(x)\,dx$. However to be
useful within the presented scheme such a decomposition should be connected
with the structures of $G$, $H$, and the representation $\pi_{0}$. We will use
a decomposition of $f_{a}(x)$ by the eigenfunctions of the operators
$\pi_{0}(h)$, $h\in\mathfrak{h}$.
###### Definition 3.5.
Let $F_{2}{}=\int_{A}H_{\alpha}{}\,d\alpha$ be a spectral decomposition with
respect to the operators $\pi_{0}(h)$, $h\in\mathfrak{h}$. Then the
decomposition
(3.23) $f_{a}(x)=\int_{A}V_{\alpha}(a)f_{\alpha}(x)\,d\alpha,$
where $f_{\alpha}(x)\in H_{\alpha}{}$ and
$V_{\alpha}(a):H_{\alpha}{}\rightarrow H_{\alpha}{}$ is called the Taylor
decomposition of the Cauchy kernel $f_{a}(x)$.
Note that the Dirac operator $D$ is defined in the terms of left invariant
shifts and therefor commutes with all $\pi_{0}(h)$. Thus it also has a
spectral decomposition over spectral subspaces of $\pi_{0}(h)$:
(3.24) $D=\int_{A}D_{\delta}\,d\delta.$
We have obvious property
###### Proposition 3.6.
If spectral measures $d\alpha$ and $d\delta$ from (3.23) and (3.24) have
disjoint supports then the image of the Cauchy integral belongs to the kernel
of the Dirac operator.
For discrete series representation functions $f_{\alpha}(x)$ can be found in
$F_{2}{}$ (as in Example 3.7), for the principal series representation this is
not the case. To overcome confusion one can think about the Fourier transform
on the real line. It can be regarded as a continuous decomposition of a
function $f(x)\in L_{2}{}(\mathbb{R}{})$ over a set of harmonics $e^{i\xi x}$
neither of those belongs to $L_{2}{}(\mathbb{R}{})$. This has a lot of common
with the Example 3.10(b) in [Kisil97c].
###### Example 3.7.
Let $G=SL_{2}{}(\mathbb{R}{})$ and $H=K$ be its maximal compact subgroup and
$\pi_{1}$ defined in (3.14). $H$ acts on $\mathbb{T}{}$ by rotations. It is
one dimensional and eigenfunctions of its generator $Z$ are parametrized by
integers (due to compactness of $K$). Moreover, on the irreducible Hardy space
these are positive integers $n=1,2,3\ldots$ and corresponding eigenfunctions
are $f_{n}(\phi)=e^{i(n-1)\phi}$. Negative integers span the space of anti-
holomorphic function and the splitting reflects the existence of analytic
structure given by the Cauchy-Riemann equation. The decomposition of coherent
states $f_{a}(\phi)$ by means of this functions is well known:
$f_{a}(\phi)=\frac{\sqrt[]{1-\left|a\right|^{2}}}{\bar{a}e^{i\phi}-1}=\sum_{n=1}^{\infty}\sqrt[]{1-\left|a\right|^{2}}\bar{a}^{n-1}e^{i(n-1)\phi}=\sum_{n=1}^{\infty}V_{n}(a)f_{n}(\phi),$
where $V_{n}(a)=\sqrt[]{1-\left|a\right|^{2}}\bar{a}^{n-1}$. This is the
classical Taylor expansion up to multipliers coming from the invariant
measure.
## 4\. Functional Calculus
United in the trinity functional calculus, spectrum, and spectral mapping
theorem play the exceptional rôle in functional analysis and could not be
substituted by anything else. All traditional definitions of functional
calculus are covered by the following rigid template based on _algebra
homomorphism_ property:
###### Definition 4.1.
An _functional calculus_ for an element $a\in\mathfrak{A}$ is a continuous
linear mapping $\Phi:\mathcal{A}\rightarrow\mathfrak{A}$ such that
1. (i)
$\Phi$ is a unital _algebra homomorphism_
$\Phi(f\cdot g)=\Phi(f)\cdot\Phi(g).$
2. (ii)
There is an initialisation condition: $\Phi[v_{0}]=a$ for for a fixed function
$v_{0}$, e.g. $v_{0}(z)=z$.
Most typical definition of the spectrum is seemingly independent and uses the
important notion of resolvent:
###### Definition 4.2.
A _resolvent_ of element $a\in\mathfrak{A}$ is the function
$R(\lambda)=(a-\lambda e)^{-1}$, which is the image under $\Phi$ of the Cauchy
kernel $(z-\lambda)^{-1}$.
A _spectrum_ of $a\in\mathfrak{A}$ is the set $\mathbf{sp}\,a$ of singular
points of its resolvent $R(\lambda)$.
Then the following important theorem links spectrum and functional calculus
together.
###### Theorem 4.3 (Spectral Mapping).
For a function $f$ suitable for the functional calculus:
(4.1) $f(\mathbf{sp}\,a)=\mathbf{sp}\,f(a).$
However the power of the classic spectral theory rapidly decreases if we move
beyond the study of one normal operator (e.g. for quasinilpotent ones) and is
virtually nil if we consider several non-commuting ones. Sometimes these
severe limitations are seen to be irresistible and alternative constructions,
i.e. model theory [Nikolskii86], were developed.
Yet the spectral theory can be revived from a fresh start. While three
components—functional calculus, spectrum, and spectral mapping theorem—are
highly interdependent in various ways we will nevertheless arrange them as
follows:
1. (i)
Functional calculus is an _original_ notion defined in some independent terms;
2. (ii)
Spectrum (or spectral decomposition) is derived from previously defined
functional calculus as its _support_ (in some appropriate sense);
3. (iii)
Spectral mapping theorem then should drop out naturally in the form (4.1) or
some its variation.
Thus the entire scheme depends from the notion of the functional calculus and
our ability to escape limitations of Definition 4.1. The first known to the
present author definition of functional calculus not linked to algebra
homomorphism property was the Weyl functional calculus defined by an integral
formula [Anderson69]. Then its intertwining property with affine
transformations of Euclidean space was proved as a theorem. However it seems
to be the only “non-homomorphism” calculus for decades.
The different approach to whole range of calculi was given in [Kisil95i] and
developed in [Kisil98a] in terms of _intertwining operators_ for group
representations. It was initially targeted for several non-commuting operators
because no non-trivial algebra homomorphism with a commutative algebra of
function is possible in this case. However it emerged later that the new
definition is a useful replacement for classical one across all range of
problems.
In the present note we will support the last claim by consideration of the
simple known problem: characterisation a $n\times n$ matrix up to similarity.
Even that “freshman” question could be only sorted out by the classical
spectral theory for a small set of diagonalisable matrices. Our solution in
terms of new spectrum will be full and thus unavoidably coincides with one
given by the Jordan normal form of matrix. Other more difficult questions are
the subject of ongoing research.
### 4.1. Another Approach to Analytic Functional Calculus
Anything called “ _functional_ calculus” uses properties of _functions_ to
model properties of _operators_. Thus changing our viewpoint on functions, as
was done in Section 3, we could get another approach to operators.
The representation (3.16) is unitary irreducible when acts on the Hardy space
$H_{2}{}$. Consequently we have one more reason to abolish the template
definition 4.1: $H_{2}{}$ is _not_ an algebra. Instead we replace the
homomorphism property by a symmetric covariance:
###### Definition 4.4.
An _analytic functional calculus_ for an element $a\in\mathfrak{A}$ and an
$\mathfrak{A}$-module $M$ is a continuous linear mapping
$\Phi:A{}(\mathbb{D}{})\rightarrow A{}(\mathbb{D}{},M)$ such that
1. (i)
$\Phi$ is an _intertwining operator_
$\Phi\rho_{1}=\rho_{a}\Phi$
between two representations of the $SL_{2}{}(\mathbb{R}{})$ group $\rho_{1}$
(3.16) and $\rho_{a}$ defined below in (4.2).
2. (ii)
There is an initialisation condition: $\Phi[v_{0}]=m$ for $v_{0}(z)\equiv 1$
and $m\in M$, where $M$ is a left $\mathfrak{A}$-module.
Note that our functional calculus released form the homomorphism condition can
take value in any left $\mathfrak{A}$-module $M$, which however could be
$\mathfrak{A}$ itself if suitable. This add much flexibility to our
construction.
The earliest functional calculus, which is _not_ an algebraic homomorphism,
was the Weyl functional calculus and was defined just by an integral formula
as an operator valued distribution [Anderson69]. In that paper (joint)
spectrum was defined as support of the Weyl calculus, i.e. as the set of point
where this operator valued distribution does not vanish. We also define the
spectrum as a support of functional calculus, but due to our Definition 4.4 it
will means the set of non-vanishing intertwining operators with primary
subrepresentations.
###### Definition 4.5.
A corresponding _spectrum_ of $a\in\mathfrak{A}$ is the support of the
functional calculus $\Phi$, i.e. the collection of intertwining operators of
$\rho_{a}$ with _prime representations_ [Kirillov76, § 8.3].
More variations of functional calculi are obtained from other groups and their
representations [Kisil95i, Kisil98a].
### 4.2. Representations of $SL_{2}{}(\mathbb{R}{})$ in Banach Algebras
A simple but important observation is that the Möbius transformations (1.1)
can be easily extended to any Banach algebra. Let $\mathfrak{A}$ be a Banach
algebra with the unit $e$, an element $a\in\mathfrak{A}$ with
$\left\|a\right\|<1$ be fixed, then
(4.2) $g:a\mapsto g\cdot a=(\bar{\alpha}a-\bar{\beta}e)(\alpha e-\beta
a)^{-1},\qquad g\in SL_{2}{}(\mathbb{R}{})$
is a well defined $SL_{2}{}(\mathbb{R}{})$ action on a subset
$\mathbb{A}{}=\\{g\cdot a\,\mid\,g\in
SL_{2}{}(\mathbb{R}{})\\}\subset\mathfrak{A}$, i.e. $\mathbb{A}{}$ is a
$SL_{2}{}(\mathbb{R}{})$-homogeneous space. Let us define the _resolvent_
function $R(g,a):\mathbb{A}{}\rightarrow\mathfrak{A}$:
$R(g,a)=(\alpha e-\beta a)^{-1}\quad$
then
(4.3)
$R(g_{1},\mathsf{a})R(g_{2},g_{1}^{-1}\mathsf{a})=R(g_{1}g_{2},\mathsf{a}).$
The last identity is well known in representation theory [Kirillov76, §
13.2(10)] and is a key ingredient of _induced representations_. Thus we can
again linearise (4.2) (cf. (3.14)) in the space of continuous functions
$C{}(\mathbb{A}{},M)$ with values in a left $\mathfrak{A}$-module $M$,
e.g.$M=\mathfrak{A}$:
$\displaystyle\rho_{a}(g_{1}):f(g^{-1}\cdot a)$ $\displaystyle\mapsto$
$\displaystyle R(g_{1}^{-1}g^{-1},a)f(g_{1}^{-1}g^{-1}\cdot a)$
$\displaystyle\quad=(\alpha^{\prime}e-\beta^{\prime}a)^{-1}\,f\left(\frac{\bar{\alpha}^{\prime}\cdot
a-\bar{\beta}^{\prime}e}{\alpha^{\prime}e-\beta^{\prime}a}\right).$
For any $m\in M$ we can again define a $K$-invariant _vacuum vector_ as
$v_{m}(g^{-1}\cdot a)=m\otimes v_{0}(g^{-1}\cdot a)\in C{}(\mathbb{A}{},M)$.
It generates the associated with $v_{m}$ family of _coherent states_
$v_{m}(u,a)=(ue-a)^{-1}m$, where $u\in\mathbb{D}{}$.
The _wavelet transform_ defined by the same common formula based on coherent
states (cf. (3.15)):
$\mathcal{W}_{m}f(g)=\left\langle f,\rho_{a}(g)v_{m}\right\rangle,\qquad$
is a version of Cauchy integral, which maps $L_{2}{}(\mathbb{A}{})$ to
$C{}(SL_{2}{}(\mathbb{R}{}),M)$. It is closely related (but not identical!) to
the Riesz-Dunford functional calculus: the traditional functional calculus is
given by the case:
$\Phi:f\mapsto\mathcal{W}_{m}f(0)\qquad\textrm{ for }M=\mathfrak{A}\textrm{
and }m=e.$
The both conditions—the intertwining property and initial value—required by
Definition 4.4 easily follows from our construction.
### 4.3. Jet Bundles and Prolongations of $\rho_{1}$
Spectrum was defined in 4.5 as the _support_ of our functional calculus. To
elaborate its meaning we need the notion of a _prolongation_ of
representations introduced by S. Lie, see [Olver93, Olver95] for a detailed
exposition.
###### Definition 4.6.
[Olver95, Chap. 4] Two holomorphic functions have $n$th _order contact_ in a
point if their value and their first $n$ derivatives agree at that point, in
other words their Taylor expansions are the same in first $n+1$ terms.
A point $(z,u^{(n)})=(z,u,u_{1},\ldots,u_{n})$ of the _jet space_
$\mathbb{J}^{n}{}\sim\mathbb{D}{}\times\mathbb{C}^{n}{}$ is the equivalence
class of holomorphic functions having $n$th contact at the point $z$ with the
polynomial:
(4.5) $p_{n}(w)=u_{n}\frac{(w-z)^{n}}{n!}+\cdots+u_{1}\frac{(w-z)}{1!}+u.$
For a fixed $n$ each holomorphic function
$f:\mathbb{D}{}\rightarrow\mathbb{C}{}$ has $n$th _prolongation_ (or _$n$
-jet_) $\mathrm{j}_{n}f:\mathbb{D}{}\rightarrow\mathbb{C}^{n+1}{}$:
(4.6) $\mathrm{j}_{n}f(z)=(f(z),f^{\prime}(z),\ldots,f^{(n)}(z)).$
The graph $\Gamma^{(n)}_{f}$ of $\mathrm{j}_{n}f$ is a submanifold of
$\mathbb{J}^{n}{}$ which is section of the _jet bundle_ over $\mathbb{D}{}$
with a fibre $\mathbb{C}^{n+1}{}$. We also introduce a notation $J_{n}$ for
the map $J_{n}:f\mapsto\Gamma^{(n)}_{f}$ of a holomorphic $f$ to the graph
$\Gamma^{(n)}_{f}$ of its $n$-jet $\mathrm{j}_{n}f(z)$ (4.6).
One can prolong any map of functions $\psi:f(z)\mapsto[\psi f](z)$ to a map
$\psi^{(n)}$ of $n$-jets by the formula
(4.7) $\psi^{(n)}(J_{n}f)=J_{n}(\psi f).$
For example such a prolongation $\rho_{1}^{(n)}$ of the representation
$\rho_{1}$ of the group $SL_{2}{}(\mathbb{R}{})$ in $H_{2}{}(\mathbb{D}{})$
(as any other representation of a Lie group [Olver95]) will be again a
representation of $SL_{2}{}(\mathbb{R}{})$. Equivalently we can say that
$J_{n}$ _intertwines_ $\rho_{1}$ and $\rho^{(n)}_{1}$:
$J_{n}\rho_{1}(g)=\rho_{1}^{(n)}(g)J_{n}\quad\textrm{ for all }g\in
SL_{2}{}(\mathbb{R}{}).$
Of course, the representation $\rho^{(n)}_{1}$ is not irreducible: any jet
subspace $\mathbb{J}^{k}{}$, $0\leq k\leq n$ is $\rho^{(n)}_{1}$-invariant
subspace of $\mathbb{J}^{n}{}$. However the representations $\rho^{(n)}_{1}$
are _primary_ [Kirillov76, § 8.3] in the sense that they are not sums of two
subrepresentations.
The following statement explains why jet spaces appeared in our study of
functional calculus.
###### Proposition 4.7.
Let matrix $a$ be a Jordan block of a length $k$ with the eigenvalue
$\lambda=0$, and $m$ be its root vector of order $k$, i.e. $a^{k-1}m\neq
a^{k}m=0$. Then the restriction of $\rho_{a}$ on the subspace generated by
$v_{m}$ is equivalent to the representation $\rho_{1}^{k}$.
### 4.4. Spectrum and the Jordan Normal Form of a Matrix
Now we are prepared to describe a spectrum of a matrix. Since the functional
calculus is an intertwining operator its support is a decomposition into
intertwining operators with prime representations (we could not expect
generally that these prime subrepresentations are irreducible).
Recall the transitive on $\mathbb{D}{}$ group of inner automorphisms of
$SL_{2}{}(\mathbb{R}{})$, which can send any $\lambda\in\mathbb{D}{}$ to $0$
and are actually parametrised by such a $\lambda$. This group extends
Proposition 4.7 to the complete characterisation of $\rho_{a}$ for matrices.
###### Proposition 4.8.
Representation $\rho_{a}$ is equivalent to a direct sum of the prolongations
$\rho_{1}^{(k)}$ of $\rho_{1}$ in the $k$th jet space $\mathbb{J}^{k}{}$
intertwined with inner automorphisms. Consequently the spectrum of $a$
(defined via the functional calculus $\Phi=\mathcal{W}_{m}$) labelled exactly
by $n$ pairs of numbers $(\lambda_{i},k_{i})$, $\lambda_{i}\in\mathbb{D}{}$,
$k_{i}\in\mathbb{Z}_{+}{}$, $1\leq i\leq n$ some of whom could coincide.
Obviously this spectral theory is a fancy restatement of the _Jordan normal
form_ of matrices.
(a) (b) (c)
Figure 8. Classical spectrum of the matrix from the Ex. 4.9 is shown at (a).
Covariant spectrum of the same matrix in the jet space is drawn at (b). The
image of the covariant spectrum under the map from Ex. 4.11 is presented (c).
###### Example 4.9.
Let $J_{k}(\lambda)$ denote the Jordan block of the length $k$ for the
eigenvalue $\lambda$. On the Fig. 8 there are two pictures of the spectrum for
the matrix
$a=J_{3}\left(\lambda_{1}\right)\oplus J_{4}\left(\lambda_{2}\right)\oplus
J_{1}\left(\lambda_{3}\right)\oplus J_{2}\left(\lambda_{4}\right),$
where
$\lambda_{1}=\frac{3}{4}e^{i\pi/4},\quad\lambda_{2}=\frac{2}{3}e^{i5\pi/6},\quad\lambda_{3}=\frac{2}{5}e^{-i3\pi/4},\quad\lambda_{4}=\frac{3}{5}e^{-i\pi/3}.$
Part (a) represents the conventional two-dimensional image of the spectrum,
i.e. eigenvalues of $a$, and (b) describes spectrum $\mathbf{sp}\,{}a$ arising
from the wavelet construction. The first image did not allow to distinguish
$a$ from many other essentially different matrices, e.g. the diagonal matrix
$\mathop{\operator@font
diag}\nolimits\left(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\right),$
which even have a different dimensionality. At the same time the Fig. 8(b)
completely characterise $a$ up to a similarity. Note that each point of
$\mathbf{sp}\,a$ on Fig. 8(b) corresponds to a particular root vector, which
spans a primary subrepresentation.
### 4.5. Spectral Mapping Theorem
As was mentioned in the Introduction a resonable spectrum should be linked to
the corresponding functional calculus by an appropriate spectral mapping
theorem. The new version of spectrum is based on prolongation of $\rho_{1}$
into jet spaces (see Section 4.3). Naturally a correct version of spectral
mapping theorem should also operate in jet spaces.
Let $\phi:\mathbb{D}{}\rightarrow\mathbb{D}{}$ be a holomorphic map, let us
define its action on functions $[\phi_{*}f](z)=f(\phi(z))$. According to the
general formula (4.7) we can define the prolongation $\phi_{*}^{(n)}$ onto the
jet space $\mathbb{J}^{n}{}$. Its associated action
$\rho_{1}^{k}\phi_{*}^{(n)}=\phi_{*}^{(n)}\rho_{1}^{n}$ on the pairs
$(\lambda,k)$ is given by the formula:
(4.8)
$\phi_{*}^{(n)}(\lambda,k)=\left(\phi(\lambda),\left[\frac{k}{\deg_{\lambda}\phi}\right]\right),$
where $\deg_{\lambda}\phi$ denotes the degree of zero of the function
$\phi(z)-\phi(\lambda)$ at the point $z=\lambda$ and $[x]$ denotes the integer
part of $x$.
###### Theorem 4.10 (Spectral mapping).
Let $\phi$ be a holomorphic mapping $\phi:\mathbb{D}{}\rightarrow\mathbb{D}{}$
and its prolonged action $\phi_{*}^{(n)}$ defined by (4.8), then
$\mathbf{sp}\,\phi(a)=\phi_{*}^{(n)}\mathbf{sp}\,a.$
The explicit expression of (4.8) for $\phi_{*}^{(n)}$, which involves
derivatives of $\phi$ upto $n$th order, is known, see for example
[HornJohnson94, Thm. 6.2.25], but was not recognised before as form of
spectral mapping.
###### Example 4.11.
Let us continue with Example 4.9. Let $\phi$ map all four eigenvalues
$\lambda_{1}$, …, $\lambda_{4}$ of the matrix $a$ into themselves. Then Fig.
8(a) will represent the classical spectrum of $\phi(a)$ as well as $a$.
However Fig. 8(c) shows mapping of the new spectrum for the case $\phi$ has
orders of zeros at these points as follows: the order $1$ at $\lambda_{1}$,
exactly the order $3$ at $\lambda_{2}$, an order at least $2$ at
$\lambda_{3}$, and finally any order at $\lambda_{4}$.
## 5\. Open Problems
In this section we indicate several directions for further work which go
through three main areas described in the paper..
### 5.1. Geometry
Geometry is most elaborated area so far, yet many directions are waiting for
further exploration.
1. (i)
Möbius transformations (1.1) with three types of imaginary units appear from
the action of the group $SL_{2}{}(\mathbb{R}{})$ on the homogeneous space
$SL_{2}{}(\mathbb{R}{})/H$ [Kisil09c], where $H$ is any subgroup $A$, $N$, $K$
from the Iwasawa decomposition (1.3). Which other actions and hypercomplex
numbers can be obtained from semisimple Lie groups and their subgroups?
2. (ii)
Lobachevsky geometry of the upper half-plane is extremely beautiful and well-
developed subject [Beardon05a] [CoxeterGreitzer]. However the traditional
study is limited to one subtype out of nine possible: with the complex numbers
for Möbius transformation and the complex imaginary unit used in FSCc (2.2).
The remaining eight cases shall be explored in various directions, notably in
the context of discrete subgroups [Beardon95].
3. (iii)
The Filmore-Springer-Cnops construction, see subsection 2.2, is closely
related to the orbit method [Kirillov99] applied to $SL_{2}{}(\mathbb{R}{})$.
An extension of the orbit method from the Lie algebra dual to matrices
representing cycles may be fruitful for semisimple Lie groups.
### 5.2. Analytic Functions
It is known that in several dimensions there are different notions of
analyticity, e.g. several complex variables and Clifford analysis. However,
analytic functions of a complex variable are usually thought to be the only
options in a plane domain. The following seems to be promising:
1. (i)
Development of the basic components of analytic function theory (the Cauchy
integral, the Taylor expansion, the Cauchy-Riemann and Laplace equations,
etc.) from the same construction and principles in the elliptic, parabolic and
hyperbolic cases and subcases.
2. (ii)
Identification of Hilbert spaces of analytic functions of Hardy and Bergman
types, investigation of their properties. Consideration of the corresponding
Töplitz operators and algebras generated by them.
3. (iii)
Application of analytic methods to elliptic, parabolic and hyperbolic
equations and corresponding boundary and initial values problems.
4. (iv)
Generalisation of the results obtained to higher dimensional spaces. Detailed
investigation of physically significant cases of three and four dimensions.
### 5.3. Functional Calculus
The functional calculus of a finite dimensional operator considered in Section
4 is elementary but provides a coherent and comprehensive treatment. It shall
be extended to further cases where other approaches seems to be rather
limited.
1. (i)
Nilpotent and quasinilpotent operators have the most trivial spectrum possible
(the single point $\\{0\\}$) while their structure can be highly non-trivial.
Thus the standard spectrum is insufficient for this class of operators. In
contract, the covariant calculus and the spectrum give complete description of
nilpotent operators—the basic prototypes of quasinilpotent ones. For
quasinilpotent operators the construction will be more complicated and shall
use analytic functions mentioned in 5.2.i.
2. (ii)
The version of covariant calculus described above is based on the _discrete
series_ representations of $SL_{2}{}(\mathbb{R}{})$ group and is particularly
suitable for the description of the _discrete spectrum_ (note the remarkable
coincidence in the names).
It is interesting to develop similar covariant calculi based on the two other
representation series of $SL_{2}{}(\mathbb{R}{})$: _principal_ and
_complementary_ [Lang85]. The corresponding versions of analytic function
theories for principal [Kisil97c] and complementary series [Kisil05a] were
initiated within a unifying framework. The classification of analytic function
theories into elliptic, parabolic, hyperbolic [Kisil05a, Kisil06a] hints the
following associative chains:
Representations of $SL_{2}{}(\mathbb{R}{})$ — | Function Theory — | Type of Spectrum
---|---|---
discrete series — | elliptic — | discrete spectrum
principal series — | hyperbolic — | continuous spectrum
complementary series — | parabolic — | residual spectrum
3. (iii)
Let $a$ be an operator with $\mathbf{sp}\,a\in\bar{\mathbb{D}{}}$ and
$\left\|a^{k}\right\|<Ck^{p}$. It is typical to consider instead of $a$ the
_power bounded_ operator $ra$, where $0<r<1$, and consequently develop its
$H_{\infty}{}$ calculus. However such a regularisation is very rough and hides
the nature of extreme points of $\mathbf{sp}\,{a}$. To restore full
information a subsequent limit transition $r\rightarrow 1$ of the
regularisation parameter $r$ is required. This make the entire technique
rather cumbersome and many results have an indirect nature.
The regularisation $a^{k}\rightarrow a^{k}/k^{p}$ is more natural and accurate
for polynomially bounded operators. However it cannot be achieved within the
homomorphic calculus Defn. 4.1 because it is not compatible with any algebra
homomorphism. Albeit this may be achieved within the covariant calculus Defn.
4.4 and Bergman type space from 5.2.ii.
4. (iv)
Several non-commuting operators are especially difficult to treat with
functional calculus Defn. 4.1 or a joint spectrum. For example, deep insights
on joint spectrum of commuting tuples [JTaylor72] refused to be generalised to
non-commuting case so far. The covariant calculus was initiated [Kisil95i] as
a new approach to this hard problem and was later found useful elsewhere as
well. Multidimensional covariant calculus [Kisil04d] shall use analytic
functions described in 5.2.iv.
### 5.4. Quantum Mechanics
Due to the space restrictions we did not mentioned connections with quantum
mechanics [Kisil96a] [Kisil02e] [Kisil05c] [Kisil04a] [Kisil09a] [Kisil10a].
In general Erlangen approach is much more popular among physicists rather than
mathematicians. Nevertheless its potential is not exhausted even there.
1. (i)
There is a possibility to build representation of the Heisenberg group using
characters of its centre with values in dual and double numbers rather than in
complex ones. This will naturally unifies classical mechanics, traditional QM
and hyperbolic QM [Khrennikov08a].
2. (ii)
Representations of nilpotent Lie groups with multidimensional centres in
Clifford algebras as a framework for consistent quantum filed theories based
on De Donder–Weyl formalism [Kisil04a].
###### Remark 5.1.
This work is performed within the “Erlangen programme at large” framework
[Kisil06a, Kisil05a], thus it would be suitable to explain the numbering of
various papers. Since the logical order may be different from chronological
one the following numbering scheme is used:
Prefix | Branch description
---|---
“0” or no prefix | Mainly geometrical works, within the classical field of Erlangen programme by F. Klein, see [Kisil05a] [Kisil09c]
“1” | Papers on analytical functions theories and wavelets, e.g. [Kisil97c]
“2” | Papers on operator theory, functional calculi and spectra, e.g. [Kisil02a]
“3” | Papers on mathematical physics, e.g. [Kisil10a]
For example, this is the first paper in the mathematical physics area.
## References
|
arxiv-papers
| 2010-06-10T18:44:17 |
2024-09-04T02:49:10.839473
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vladimir V. Kisil",
"submitter": "Vladimir V Kisil",
"url": "https://arxiv.org/abs/1006.2115"
}
|
1006.2201
|
# Thermal nuclear pairing within the self-consistent quasiparticle RPA
N. Dinh Dang1,2 and N. Quang Hung1,3 1 Theoretical Nuclear Physics Laboratory,
RIKEN Nishina Center for Accelerator-Based Science, 2-1 Hirosawa, Wako City,
351-0198 Saitama, Japan
2 Institute for Nuclear Science and Technique, Hanoi, Vietnam
3 Institute of Physics, Hanoi, Vietnam dang@riken.jp (N.D.D.), nqhung@riken.jp
(N.Q.H.)
###### Abstract
The self-consistent quasiparticle RPA (SCQRPA) is constructed to study the
effects of fluctuations on pairing properties in nuclei at finite temperature
and $z$-projection $M$ of angular momentum. Particle-number projection (PNP)
is taken into account within the Lipkin-Nogami method. Several issues such as
the smoothing of superfluid-normal phase transition, thermally assisted
pairing in hot rotating nuclei, extraction of the nuclear pairing gap using an
improved odd-even mass difference are discussed. A novel approach of embedding
the PNP SCQRPA eigenvalues in the canonical and microcanonical ensembles is
proposed and applied to describe the recent empirical thermodynamic quantities
for iron, molybdenum, dysprosium, and ytterbium isotopes.
## 1 Introduction
Sharp phase transitions such as the superfluid-normal (SN) or shape ones are
prominent features of infinite systems such as metal superconductors, ultra-
cold gases, liquid helium, etc. They are well described by many-body theories
such as the BCS, RPA or quasiparticle RPA (QRPA). The situation changes in
finite small systems such as atomic nuclei, where strong quantal and thermal
fluctuations strongly or completely smooth out these sharp phase transitions.
It is well known that the conventional BCS, RPA or QRPA theories fail in a
number of cases in the description of the ground states as well as excited
states of these systems. The reason is that strong fluctuations invalidate the
assumptions, based on which the main equations of these theories have been
derived. Amongst these assumptions are the Cooper pairs, which violate the
particle-number conservation, and the closely related quasiboson-approximation
(QBA) used in the (Q)RPA, which violates the Pauli principle between the
fermion pairs. These assumptions cause the BCS and QRPA to break down at a
certain critical value $G_{c}$ of the pairing interaction parameter $G$, below
which the BCS theory only has a trivial solution with zero pairing gap
$\Delta=$ 0\. The same is true in the weak coupling region, where the
particle-particle RPA is valid but its solution also breaks down at $G\geq
G_{c}$. Meanwhile, the exact solution of the pairing problem exposes no
singularity at any $G$ [1]. Similarly, at finite temperature $T\neq$ 0, the
omission of quasiparticle-number fluctuations (QNF) within the BCS theory
leads to the collapse of the pairing gap at the critical temperature $T_{c}$,
corresponding to the temperature of the SN phase transition in infinite
systems. Meanwhile, the exact eigenvalues of the pairing problem embedded in
the canonical ensemble (CE) shows a smooth decreasing pairing energy with
increasing $T$ due to thermal fluctuations incorporated in the CE [2]. In
rotating nuclei, strong fluctuations also smear out the Mottelson-Valatin
effect, according to which the pairing gap, existing at zero angular momentum
$M=$ 0, would collapse at a certain critical angular momentum $M_{c}$. This
situation means that, in order to be reliable, the BCS, RPA, and/or QRPA
theories need to be corrected to include these effects of fluctuations when
applied to nuclei, in particular, the light ones. This is done within the
framework of the self-consistent QRPA (SCQRPA) presented in this work.
## 2 Formalism
We consider the pairing Hamiltonian $H=\sum_{k>0}\epsilon_{k}\hat{N}_{\pm
k}-G\sum_{kk^{\prime}}\hat{P}_{k}^{\dagger}\hat{P}_{k^{\prime}}~{}$, where
$\hat{N}_{\pm k}=a_{\pm k}^{\dagger}a_{\pm k}$ is the particle-number
operator, and
$\hat{P}_{k}=a_{k}^{\dagger}a_{-k}^{\dagger},\hat{P}_{j}=(\hat{P}_{j}^{\dagger})^{\dagger}$
are the pairing operators. The operators $a_{k}^{\dagger}$ and $a_{k}$ are
respectively the single-particle creation and destruction operators. This
Hamiltonian has been diagonalized exactly in [1]. The exact partition function
is constructed by embedding the exact eigenvalues into the CE as $Z_{\rm
Exact}(\beta)=\sum_{S}d_{S}\exp({-\beta\varepsilon_{S}^{\rm Exact}})$ , with
the degeneracy $d_{S}=2^{S}$, inverse temperature $\beta=1/T$, and
$S=0,2,...N$ being the total seniority of the system. Knowing the partition
function $Z$, one calculates the free energy $F$, entropy $S$, total energy
${\cal E}$, heat capacity $C$, and pairing gap $\Delta$ as $F=-T{\rm ln}Z(T)$,
$S=-{\partial F}/{\partial T}$, ${\cal E}=F+TS$, $C={\partial{\cal
E}}/{\partial T}$, and $\Delta=[-G({\cal
E}-2\sum_{k}\epsilon_{k}f_{k}+G\sum_{k}f_{k}^{2})]^{1/2}$, where $f_{k}$ is
the single-particle occupation number on the $k$th level obtained by averaging
the state-dependent occupation numbers $f_{k}^{(S)}$ within the CE [2].
The SCQRPA theory [3, 4] includes a set of BCS-based equations, corrected by
the effects of QNF, namely
$\Delta_{k}=\Delta+\delta\Delta_{k}~{},\hskip
5.69054pt\Delta=G\sum_{k^{\prime}}\langle{\cal D}_{k^{\prime}}\rangle
u_{k^{\prime}}v_{k^{\prime}}~{},\hskip
5.69054pt\delta\Delta_{k}=2G\frac{\delta{\cal N}_{k}^{2}}{\langle{\cal
D}_{k}\rangle}u_{k}v_{k}~{}.$ (1) $N=2\sum_{k}\bigg{[}v_{k}^{2}\langle{\cal
D}_{k}\rangle+\frac{1}{2}\big{(}1-\langle{\cal
D}_{k}\rangle\big{)}\bigg{]}~{}.$ (2)
where $u_{k}$ and $v_{k}$ are the Bogoliubov’s coefficients,
$u_{k}^{2}=\frac{1}{2}\bigg{(}1+\frac{\epsilon^{\prime}_{k}-Gv_{k}^{2}-\lambda}{E_{k}}\bigg{)}~{},\hskip
14.22636ptv_{k}^{2}=\frac{1}{2}\bigg{(}1-\frac{\epsilon^{\prime}_{k}-Gv_{k}^{2}-\lambda}{E_{k}}\bigg{)}~{},\hskip
5.69054ptE_{k}=\sqrt{(\epsilon^{\prime}_{k}-Gv_{k}^{2}-\lambda)^{2}+\Delta_{k}^{2}}~{},$
(3)
with the renormalized single-particle energies $\epsilon^{\prime}_{k}$
$\epsilon_{k}^{\prime}=\epsilon_{k}+\frac{G}{\langle{\cal
D}_{k}\rangle}\sum_{k^{\prime}}(u_{k^{\prime}}^{2}-v_{k^{\prime}}^{2})\bigg{(}\langle{\cal
A}_{k}^{\dagger}{\cal A}_{k^{\prime}\neq k}^{\dagger}\rangle+\langle{\cal
A}_{k}^{\dagger}{\cal A}_{k^{\prime}}\rangle\bigg{)}~{},$ (4)
$\langle{\cal D}_{k}\rangle=1-2n_{k}$, the quasiparticle-pair operators ${\cal
A}_{k}^{\dagger}=\alpha_{k}^{\dagger}\alpha_{-k}^{\dagger}$, ${\cal
A}_{k}=({\cal A}_{k}^{\dagger})^{\dagger}$, and $\delta{\cal N}_{k}^{2}\equiv
n_{k}(1-n_{k})$ is the QNF on $k$th level. To avoid level-dependent gaps
$\Delta_{k}$, the level-weighted gap
$\bar{\Delta}_{k}=\sum_{k}\Delta_{k}/\Omega$ ($\Omega$ is the number of
levels) is considered in the numerical results. Because of coupling to
collective vibrations beyond the quasiparticle mean field, the quasiparticle
occupation number $n_{k}$ is not given by a Fermi-Dirac distribution of free
fermions, but is found from the integral equation [4]
$n_{k}=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{\gamma_{k}(\omega)(e^{\beta\omega}+1)^{-1}}{[\omega-
E_{k}-M_{k}(\omega)]^{2}+\gamma_{k}^{2}(\omega)}d\omega~{},$ (5)
where the mass operator $M_{k}(\omega)$ and the quasiparticle damping
$\gamma_{k}(\omega)$ are functions of $n_{k}$, SCQRPA eigenvalues
$\omega_{\mu}$, SCQRPA ${\cal X}_{k}^{\mu}$ and ${\cal Y}_{k}^{\mu}$
amplitudes, SCQRPA phonon occupations numbers $\nu_{\mu}$, as well as $u_{k}$
and $v_{k}$. The SCQRPA submatrices $A$ and $B$ contain the screening factors
$\langle{\cal A}_{k}^{\dagger}{\cal A}^{\dagger}_{k^{\prime}}\rangle$ and
$\langle{\cal A}_{k}^{\dagger}{\cal A}_{k^{\prime}}\rangle$ so that the set of
SCQRPA equations should be solved self-consistently with Eqs. (1), (2) and (5)
to simultaneously determine $\bar{\Delta}$, chemical potential $\lambda$,
$n_{k}$, $\omega_{\mu}$, ${\cal X}_{k}^{\mu}$ and ${\cal Y}_{k}^{\mu}$. To
eliminate particle-number fluctuations inherent in the BCS theory, the Lipkin-
Nogami (LN) particle-number projection (PNP) [5] is applied on top of Eqs. (1)
and (2). The ensuing theory, called the LNSCQRPA theory, has also been
extended to include the finite $z$-projection $M$ of angular momentum
(noncollective rotation) [6]. The set of obtained equations is formally the
same except that now, depending on the single-particle spin projections
$\mp\gamma m_{k}$ with $\gamma$ being the angular velocity, one has two types
of quasiparticle occupation number, $n_{k}^{\pm}$, so that $\langle{\cal
D}_{k}\rangle=1-n_{k}-n_{-k}$. At $T=$ 0 and $M=$ 0 the SCQRPA theory reduces
to its zero temperature and non-rotating limit, where $\langle{\cal
D}_{k}\rangle=1/[1+2\sum_{\mu}({\cal Y}_{k}^{\mu})^{2}]$ [3].
## 3 Numerical results and discussions
Figure 1: Energies of the ground state (a) and first excited states for the
$N=\Omega=$ 10 as functions of $G$ at $T=M=$ 0\. $\omega_{ppRPA}={\cal
E}_{1}(N+2)-{\cal E}_{0}(N+2)$ with the ppRPA eigenvalues ${\cal E}_{i}$.
Shown in Fig. 1 are the energies of the ground state (a) and first excited
state (b) obtained at $T=M=$ 0 within several approximations as well as by
exactly diagonalizing the pairing Hamiltonian for the schematic model, which
consists of $\Omega$ doubly-folded equidistant levels with the single-particle
energies chosen as $\epsilon_{k}=k-(\Omega+1)/2$ MeV. The displayed results
are for the half-filled case with $N=\Omega=$ 10, and plotted as functions of
the pairing interaction parameter $G$. It is seen that the LNSCQRPA describes
rather well the exact energies of both the ground and first excited states
without any discontinuity in the region around $G_{c}$, where all other
approaches such as the RPA, QRPA, and SCQRPA collapse.
Figure 2: Level-weighted pairing gap $\bar{\Delta}$, total energy $E$, and
heat capacity $C$, as functions of $T$ for $N=\Omega=$ 10 (a - c) and 50 (d -
f) obtained within the FTBCS (dotted), FTBCS1 (thin solid), FTLN1 (thin
dashed), SCQRPA (thick solid), LNSCQRPA (thick dashed). The dash-dotted lines
for $N=$ 10 are the exact CE results.
The level-weighted gap, total energy, and heat capacity obtained for the
systems with $N=\Omega=$ 10 and 50 are shown as functions of temperature $T$
in Fig. 2. Beside the predictions by the SCQRPA, LNSCQRPA, as well as by their
corresponding limits, FTBCS1 and FTLN1, where coupling to QRPA is omitted
(i.e. $n_{k}$ is described by the Fermi-Dirac distribution for free fermions),
and the finite-temperature (FT) BCS results, the exact CE results are also
shown. This figure clearly demonstrates how QNF smooth out the sharp SN phase
transition in finite systems. The pairing gap never collapses, but decreases
monotonously with increasing $T$, whereas the spike at $T_{c}$ in the heat
capacity, which serves as a signature of sharp SN phase transition within the
FTBCS, becomes strongly depleted to a broad bump.
Figure 3: Level-weighted pairing gaps $\bar{\Delta}$ for $N=\Omega=$ 10 as a
functions of temperature $T$ at various values of $M/M_{c}$ and angular
momentum $M$ at various values of $T/T_{c}$ within the FTBCS (a, c) and FTBCS1
(b, d) theories.
At finite angular momentum $M\neq$ 0, the FTBCS theory predicts the Mottelson-
Valatin effect, according to which, the zero-temperature pairing gap decreases
with increasing $M$ and collapses at $M=M_{c}$ because the angular momentum
blocks the levels close to the Fermi surface [Fig. 3 (a) and 3 (c)]. Thermal
effects relax the blocking, opening some levels around the Fermi surface for
pairing. This leads to the thermally assisted pairing gap (or pairing
reentrance), according to which at a certain $T=T_{1}$ the pairing gap becomes
finite even at $M>M_{c}$ [7, 8]. With increasing $T$ thermal effects again
break the pairs so that the gap disappears at $T=T_{2}>T_{1}$ [See Fig. 3 (a)
for $M/M_{c}\geq$ 1]. In finite systems, the QNF smooth out both the
Mottelson-Valatin transition and thermal assisted pairing, for instance, for
$N=\Omega=$ 10 with $G=$ 0.5 MeV at $T/T_{c}\geq$ 1, the gap only decreases
monotonously with increasing $M$ but never vanishes [See Fig. 3 (d) for
$M/M_{c}\geq$ 1], whereas at $M/M_{c}\geq$ 3, the pairing gap reappears at
$T>T_{1}$ but remains finite with further increasing $T$ [See Fig. 3 (b) for
$M/M_{c}\geq$ 3].
The odd-even mass difference contains the admixture with the contribution from
uncorrelated single-particle configurations, which increases with $T$.
Therefore, the simple extensions of this formula to obtain the three-point and
four-point gaps, in principle, do not hold at finite temperature. We propose
an improved odd-even mass difference formula at $T\neq$ 0, namely
$\widetilde{\Delta}^{(3)}(\beta,N)=\frac{G}{2}\bigg{[}(-1)^{N}+\sqrt{1-4\frac{S^{\prime}}{G}}\bigg{]}~{},\hskip
5.69054ptS^{\prime}=\frac{1}{2}\big{[}\langle{\cal
E}(N+1)\rangle_{\alpha}+\langle{\cal
E}(N-1)\rangle_{\alpha}\big{]}-\langle{\cal E}(N)\rangle_{\alpha}^{(0)}~{},$
(6)
where $\langle{\cal E}(N)\rangle_{\alpha}$ is the total energy of the system
with $N$ particles within the grand canonical ensemble (GCE) ($\alpha=$GC) or
CE ($\alpha=$C); $\langle{\cal E}\rangle_{\alpha}^{(0)}\equiv
2\sum_{k}\big{[}\epsilon_{k}-Gf_{k}^{(\alpha)}/2\big{]}f_{k}^{(\alpha)}$ with
$-G\sum_{k}[f_{k}^{(\alpha)}]^{2}$ coming from uncorrelated single-particle
configurations.
Figure 4: Pairing gaps extracted from the odd-even mass differences as
functions of $T$ for $N=$ 10 (a,c) and $N=$ 9 (b,d) ($\Omega=$ 10, $G=$ 0.9
MeV). The thin solid and thick solid lines denote the gap $\Delta^{(3)}(N)$
($\Delta^{(4)}(N)=[\Delta^{(3)}(N)+\Delta^{(3)}(N-1)]/2)$, and the improved
gap $\widetilde{\Delta}^{(i)}(\beta,N)$ ($i=$ 3, 4) from Eq. (6),
respectively. The dash-dotted lines are the canonical gaps $\Delta^{(i)}_{\rm
C}$.
Shown in Fig. 4 are the pairing gaps $\Delta^{(i)}(\beta,N)$ ($i=$ 3 and 4),
obtained for $N=$ 9 and 10 ($\Omega=$ 10) by using the simple extension of the
odd-even mass formula to $T\neq$ 0 as well as the modified gaps
$\widetilde{\Delta}^{(i)}(\beta,N)$ from Eq. (6), and the canonical gaps
$\Delta^{(i)}_{\rm C}$. It is seen in Fig. 4 that the naive extension of the
odd-even mass formula to $T\neq$ 0 fails to match the temperature-dependence
of the canonical gap $\Delta^{(i)}_{\rm C}$. The gap $\Delta^{(3)}(\beta,N=9)$
even turns negative at $T>$ 2.4 MeV, suggesting that such simple extension of
the odd-even mass difference to finite $T$ is invalid. The modified gap
$\widetilde{\Delta}^{(i)}(\beta)$ is found in much better agreement with the
canonical one, whereas the modified four-point gaps
$\widetilde{\Delta}^{(4)}(\beta)$ practically coincide with the canonical
gaps. The comparison in Fig. 4 suggests that formula (6) is a much better
candidate for the experimental gap at $T\neq$ 0, rather than the simple odd-
even mass difference.
Figure 5: CE heat capacities $C$ as functions of $T$ and MCE entropies $S$ as
functions of excitation energy $E^{*}$ for 56Fe, 94Mo, 162Dy, and 172Yb.
Experimental data are taken from [11].
In order to construct a feasible description for pairing within the CE, the
eigenvalues of the LNBCS and LNSCQRPA, obtained for each total seniority $S$
at $T=$ 0, are embedded into the CE by using the partition function
$Z_{\gamma}(\beta)=\sum_{S}d_{S}e^{-\beta\varepsilon_{S}^{\gamma}}$ ($\gamma=$
LNBCS, LNSCQRPA). The resulting approaches are called the CE-LNBCS and CE-
LNSCQRPA, respectively [10]. These solutions are also embedded into the
microcanonical ensemble (MCE) by using the Boltzmann’s definition for entropy,
$S({\cal E})={\rm ln}{W}({\cal E})$, where ${W}({\cal E})$ is the number of
accessible states within the energy interval (${\cal E},{\cal E}+\delta{\cal
E}$). The corresponding approaches are called the MCE-LNBCS and MCE-LNSCQRPA,
respectively [10].
The CE heat capacities and MCE entropies for several nuclei are shown in Fig.
5 as functions of $T$ and excitation energy $E^{*}$, respectively. The single-
particle energies are calculated within the deformed Woods-Saxon potentials.
In order to have a consistent comparison with the recent experimental data in
[11], we carried out calculations by using the CE-LNBCS and CE-LNSCQRPA for
56Fe, where pairing is included within the complete $pf+g_{9/2}$ shell above
the 40Ca core. For Mo isotopes, pairing is included in the 22 degenerated
single-particle levels above the 48Ca core. For Dy and Yb the same is done on
top of the 132Sn core. It is clear from this figure that the CE-LNSCQRPA
results agree quite well with the experimental data [11], which are also
deducted from the CE. The MCE entropies, obtained within the MCE-LNBCS and
MCE-LNSCQRPA using ${\delta\cal E}=$1 MeV, are plotted versus the experimental
data. It is seen that the MCE-LNSCQRPA entropy not only offers the best fit to
the experimental data but also predicts the results up to higher $E^{*}>$ 10
MeV.
## 4 Conclusions
The proposed LNSCQRPA theory can describe without discontinuity the pairing
properties of hot noncollectively rotating nuclei at any values of pairing
interaction parameter $G$, temperature $T$, and angular momentum $M$. In the
limit of zero temperature and zero angular momentum, it offers the best fits
to the exact solutions in the weak coupling region with large particle numbers
for the energy of the first excited state, whereas the SCQRPA reproduces well
the exact one in the strong coupling region. In the limit of very large $G$
all the approximations predict nearly the same value as that of the exact one.
Under the effect of QNF, the paring gaps obtained at different values $M$ of
angular momentum decrease monotonously as $T$ increases, and do not collapse
even at hight $T$. The effect of thermally assisted pairing (pairing
reentrance) shows up in such a way that the pairing gap reappears at a given
$T_{1}>$ 0 and remains finite at $T>T_{1}$, in qualitative agreement with the
results of Ref. [9]. We suggest a novel formula to extract the pairing gap at
$T\neq$ 0 from the difference of total energies of even and odd systems, where
the contribution of uncorrelated single-particle motion is subtracted. Its
prediction is found in much better agreement with the canonical gap than the
simple extension of the odd-even mass formula to $T\neq$ 0\. Finally, we
embedded the solutions of the LNBCS and LNSCQRPA into the CE and MCE, and
found that the CE-LNSCQRPA predictions are in quite good agreements with the
exact results as well as the recent experimental data. The present approach
can also describe simultaneously and self-consistently the experimentally
extracted total energy, heat capacity, and entropy within both CE and MCE
treatments. It is simple and feasible for the application to larger finite
systems, where the exact matrix diagonalization and/or solving the Richardson
equation are impracticable to find all eigenvalues, whereas other methods,
such as the quantum Monte-Carlo calculations, are time consuming.
## References
## References
* [1] Volya A, Brown B A and Zelevinsky V 2001 Phys. Lett. B 509 37\.
* [2] Hung N Q and Dang N D 2009 Phys. Rev. C 79 054328\.
* [3] Hung N Q and Dang N D 2007 Phys. Rev. C 76 054302; 2008 Ibid. 77 029905(E).
* [4] Dang N D and Hung N Q 2008 Phys. Rev. C 77 064315\.
* [5] Lipkin H J 1960 Ann. Phys. (NY) 9 272; Nogami Y 1965 Phys. Lett. 15 4\.
* [6] Hung N Q and Dang N D 2008 Phys. Rev. C 78 064315\.
* [7] Moretto L G 1971 Nucl. Phys. A 185 145\.
* [8] Balian R, Flocard H and Vénéroni M 1999 Phys. Rep. 317 251\.
* [9] Frauendorf S, Kuzmenko N K, Mikhajlov V M and Sheikh J A 2003 Phys. Rev. B 68 024518\.
* [10] Hung N Q and Dang N D 2010 Phys. Rev. C 81 057302\.
* [11] Melby E et al. 1999 Phys. Rev. Lett. 83 3150; Guttormsen M et al. 2000 Phys. Rev. C 62 024306; Schiller A et al. 2001 Phys. Rev. C 63 021306(R); Algin E et al. 2008 Phys. Rev. C 78 054321\.
|
arxiv-papers
| 2010-06-11T05:31:13 |
2024-09-04T02:49:10.850800
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "N. Dinh Dang and N. Quang Hung",
"submitter": "Nguyen Quang Hung",
"url": "https://arxiv.org/abs/1006.2201"
}
|
1006.2249
|
# Integrality Gap of the Hypergraphic Relaxation of Steiner Trees: a short
proof of a $1.55$ upper bound
Deeparnab Chakrabarty Jochen Könemann David Pritchard
###### Abstract
Recently, Byrka et al. [1] gave a $1.39$-approximation for the Steiner tree
problem, using a hypergraph-based LP relaxation. They also upper-bounded its
integrality gap by $1.55$. We describe a shorter proof of the same integrality
gap bound, by applying some of their techniques to a randomized loss-
contracting algorithm.
## 1 Introduction
In the Steiner tree problem, we are given an undirected graph $G=(V,E)$ with
costs $c$ on edges and its vertex set partitioned into terminals (denoted
$R\subset V$) and Steiner vertices ($V\setminus R$). A _Steiner tree_ is a
tree spanning all of $R$ plus any subset of $V\backslash R$, and the problem
is to find a minimum-cost such tree. The Steiner tree problem is
$\mathsf{APX}$-hard, thus the best we can hope for is a constant-factor
approximation algorithm.
The best known ratio is a result of Byrka, Grandoni, Rothvoß and Sanità [1]:
their randomized iterated rounding algorithm gives approximation ratio
$\ln(4)+\epsilon\approx 1.39$. The prior best was a $1+\frac{\ln
3}{2}+\epsilon\approx 1.55$ ratio, via the deterministic loss-contracting
algorithm of Robins and Zelikovsky [6]. The algorithm of [1] differs from
previous work in that it uses a linear programming (LP) relaxation; the LP is
based on hypergraphs, and it has several different-looking but equivalent [2,
5] nice formulations. A second result of [1] concerns the LP’s _integrality
gap_ , which is defined as the worst-case ratio (max over all instances) of
the optimal Steiner tree cost to the LP’s optimal value. Byrka et al. show the
integrality gap is at most $1.55$, and their proof builds on the analysis of
[6]. In this note we give a shorter proof of the same bound using a simple LP-
rounding algorithm.
Figure 1: In (i) we show a Steiner tree; circles are terminals and squares are
Steiner nodes. In (ii) we show its decomposition into full components, and
their losses in bold. In (iii) we show the full components after loss
contraction.
We now describe one formulation for the hypergraphic LP. Given a set $K\subset
R$ of terminals, a full component on $K$ is a tree whose leaf set is $K$ and
whose internal nodes are Steiner vertices. Every Steiner tree decomposes in a
unique edge-disjoint way into full components; Figure 1(i) shows an example.
Moreover, one can show that a set of full components on sets
$(K_{1},\dotsc,K_{r})$ forms a Steiner tree if and only if the hypergraph
$(V,(K_{1},\dotsc,K_{r}))$ is a hyper-spanning tree. Let ${\tt F}(K)$ denote a
minimum-cost full component for terminal set $K\subset R$, and let $C_{K}$ be
its cost. The hypergraphic LP is as follows:
$\displaystyle\min$ $\displaystyle\qquad\sum_{K}C_{K}x_{K}:$ ($\mathcal{S}$)
$\displaystyle\forall\varnothing\neq S\subseteq R:$
$\displaystyle\qquad\sum_{K:K\cap S\neq\varnothing}x_{K}(|K\cap
S|-1)\leq|S|-1$ $\displaystyle\qquad\sum_{K}x_{K}(|K|-1)=|R|-1$
$\displaystyle\forall K:$ $\displaystyle\qquad x_{K}\geq 0$
The integral solutions of ($\mathcal{S}$) correspond to the full component
sets of Steiner trees. As an aside, the _$r$ -restricted full component_
method (e.g. [4]) allows us to assume there are a polynomial number of full
components while affecting the optimal Steiner tree cost by a $1+\epsilon$
factor. Then, it is possible to solve ($\mathcal{S}$) in polynomial time [1,
8]. Here is our goal:
###### Theorem 1.
[1] The integrality gap of the hypergraphic LP ($\mathcal{S}$) is at most
$1+\ln{3}/2\approx 1.55$.
## 2 Randomized Loss-Contracting Algorithm
In this section we describe the algorithm. We introduce some terminology
first. The _loss_ of full component ${\tt F}(K)$, denoted by ${\tt Loss}(K)$,
is a minimum-cost subset of ${\tt F}(K)$’s edges that connects the Steiner
vertices to the terminals. For example, Figure 1(ii) shows the loss of the two
full components in bold. We let ${\tt loss}(K)$ denote the total cost of all
edges in ${\tt Loss}(K)$. The _loss-contracted full component of $K$_, denoted
by ${\tt LC}(K)$, is obtained from ${\tt F}(K)$ by contracting its loss edges
(see Figure 1(iii) for an example).
For clarity we make two observations. First, for each $K$ the edges of ${\tt
LC}(K)$ correspond to the edges of ${\tt F}(K)\backslash{\tt Loss}(K)$.
Second, for terminals $u,v$, there may be a $uv$ edge in several ${\tt
LC}(K)$’s but we think of them as distinct parallel edges.
Our randomized rounding algorithm, RLC, is shown below. We choose $M$ to have
value at least $\sum_{K}x_{K}$ such that $t=M\ln 3$ is integral. ${\tt
MST}(\cdot)$ denotes a minimum spanning tree and ${\tt mst}$ its cost.
Algorithm RLC. 1: Let $T_{1}$ be a minimum spanning tree of the induced graph
$G[R]$. 2: $x\leftarrow$ Solve ($\mathcal{S}$) 3: for $1\leq i\leq t$ do 4:
Sample $K_{i}$ from the distribution111$K_{i}\leftarrow\varnothing$ with
probability $1-\sum_{K}x_{K}/M$. with probability $\frac{x_{K}}{M}$ for each
full component $K$. 5: $T_{i+1}\leftarrow{\tt MST}(T_{i}\cup{\tt LC}(K_{i}))$
6: end for 7: Output any Steiner tree in
$ALG:=T_{t+1}\cup\bigcup_{i=1}^{t}{\tt Loss}(K_{i})$.
To prove that $ALG$ actually contains a Steiner tree, we must show all
terminals are connected. To see this, note each edge $uv$ of $T_{t+1}$ is
either a terminal-terminal edge of $G[R]$ in the input instance, or else
$uv\in{\tt LC}(K_{i})$ for some $i$ and therefore a $u$-$v$ path is created
when we add in ${\tt Loss}(K_{i})$.
## 3 Analysis
In this section we prove that the tree’s cost is at most $1+\frac{\ln 3}{2}$
times the optimum value of ($\mathcal{S}$). Each iteration of the main loop of
algorithm RLC first samples a full component $K_{i}$ in step 4, and
subsequently recomputes a minimum-cost spanning tree in the graph obtained
from adding the loss-contracted part of $K_{i}$ to $T_{i}$. The new spanning
tree $T_{i+1}$ is no more expensive than $T_{i}$; some of its edges are
replaced by newly added edges in ${\tt LC}(K_{i})$. Bounding the drop in cost
will be the centerpiece of our analysis, and this step will in turn be
facilitated by the elegant Bridge Lemma of Byrka et al. [1]. We describe this
lemma first.
Figure 2: In (i) we show a terminal spanning tree $T$ in red, and a full
component spanning terminal set $K\subset\\{a,b,c,d\\}$ in black; thick edges
are its loss. In (ii) we show $T/K$, and ${\tt Drop}_{T}(K)$ is shown as
dashed edges. In (iii) we show ${\tt MST}(T\cup{\tt LC}(K))$.
We first define the _drop_ of a full component $K$ with respect to a terminal
spanning tree $T$ (it is just a different name for the bridges of [1]). Let
$T/K$ be the graph obtained from $T$ by identifying the terminals spanned by
$K$. Then let
${\tt Drop}_{T}(K):=E(T)\setminus E({\tt MST}(T/K)),$
be the set of edges of $T$ that are not contained in a minimum spanning tree
of $T/K$, and ${\tt drop}_{T}(K)$ be its cost. We illustrate this in Figure 2.
We state the Bridge Lemma here and present its proof for completeness.
###### Lemma 1 (Bridge Lemma [1]).
Given a terminal spanning tree $T$ and a feasible solution $x$ to
($\mathcal{S}$),
$\sum_{K}x_{K}{\tt drop}_{T}(K)\geq c(T).$ (1)
###### Proof.
The proof needs the following theorem of Edmonds [3]: given a graph $H=(R,F)$,
the extreme points of the polytope
$\\{z\in{\mathbb{R}}^{F}_{\geq 0}:\sum_{(u,v)\in F:u\in S,v\in
S}z_{e}\leq|S|-1\quad\forall S\subset R,\quad\sum_{e\in F}z_{e}=|R|-1\\}$
($\mathcal{G}$)
are the indicator variables of spanning trees of $H$. The proof strategy is as
follows. We construct a multigraph $H=(R,F)$ with costs $c$, and
$z\in{\mathbb{R}}^{F}$ such that: the cost of $z$ equals the left-hand side of
(1); $z\in\eqref{graphic}$; and all spanning trees of $H$ have cost at least
$c(T)$. Edmonds’ theorem then immediately implies the lemma. In the rest of
the proof we define $H$ and supply the three parts of this strategy.
For each full component $K$ with $x_{K}>0$, consider the edges in ${\tt
Drop}_{T}(K)$. Contracting all edges of $E(T)\setminus{\tt Drop}_{T}(K)$, we
see that ${\tt Drop}_{T}(K)$ corresponds to edges of a spanning tree of $K$.
These edges are copied (with the same cost $c$) into the set $F$, and the
copies are given weight $z_{e}=x_{K}$. Using the definition of drop, one can
show each $e\in F$ is a maximum-cost edge in the unique cycle of
$T\cup\\{e\\}$.
Having now defined $F$, we see
$\sum_{e\in F}c_{e}z_{e}=\sum_{K}x_{K}{\tt drop}_{T}(K).$ (2)
Note that we introduce $|K|-1$ edges for each full component $K$, and that,
for any $S\subseteq R$, at most $|S\cap K|-1$ of these have both ends in $S$.
These two observations together with the fact that $x$ is feasible for
($\mathcal{S}$) directly imply that $z$ is feasible for ($\mathcal{G}$).
To show all spanning trees of $H$ have cost at least $c(T)$, it suffices to
show $T$ is an MST of $T\cup H$. In turn, this follows (e.g. [7, Theorem
50.9]) from the fact that each $e\in F$ is a maximum-cost edge in the unique
cycle of $T\cup\\{e\\}$. ∎
We also need two standard facts that we summarize in the following lemma. They
rely on the input costs satisfying the triangle inequality, and that internal
nodes of full components have degree at least 3, both of which hold without
loss of generality.
###### Lemma 2.
(a) The value ${\tt mst}(G[R])$ of the initial terminal spanning tree computed
by algorithm RLC is at most twice the optimal value of ($\mathcal{S}$). (b)
For any full component $K$, ${\tt loss}(K)\leq C_{K}/2$.
###### Proof.
See Lemma 4.1 in [4] for a proof of (b). For (a) we use a shortcutting
argument along with Edmonds’ polytope ($\mathcal{G}$) for the graph $H=G[R]$.
In detail, let $x$ be an optimal solution to ($\mathcal{S}$). For each $K$,
shortcut a tour of ${\tt F}(K)$ to obtain a spanning tree of $K$ with $c$-cost
at most twice $C_{K}$ (by the triangle inequality) and add these edges to $F$
with $z$-value $x_{K}$. Like before, since $x$ is feasible for
($\mathcal{S}$), $z$ is feasible for ($\mathcal{G}$), and so there is a
spanning tree of $G[R]$ whose $c$-cost is at most $\sum_{e\in F}c_{e}z_{e}\leq
2\sum_{K}C_{K}x_{K}$. ∎
We are ready to prove the main theorem.
Proof of Theorem 1. Let $x$ be an optimal solution to ($\mathcal{S}$) computed
in step 2, define ${\tt lp}^{*}$ to be its objective value, and
${\tt loss}^{*}=\sum_{K}x_{K}{\tt loss}(K)$
its fractional loss. Our goal will be to derive upper bounds on the expected
cost of tree $T_{i}$ maintained by the algorithm at the beginning of iteration
$i$. After selecting $K_{i}$, one possible candidate spanning tree of
$T_{i}\cup{\tt LC}(K_{i})$ is given by the edges of $T_{i}\setminus{\tt
Drop}_{T_{i}}(K_{i})\cup{\tt LC}(K_{i})$, and thus
$c(T_{i+1})\leq c(T_{i})-{\tt drop}_{T_{i}}(K_{i})+c({\tt LC}(K_{i})).$ (3)
Let us bound the expected value of $T_{i+1}$, given any fixed $T_{i}$. Due to
the distribution from which $K_{i}$ is drawn, and using (3) with linearity of
expectation, we have
$E[c(T_{i+1})]\leq c(T_{i})-\frac{1}{M}\sum_{K}x_{K}{\tt
drop}_{T_{i}}(K)+\frac{1}{M}\sum_{K}x_{K}(C_{K}-{\tt loss}(K)).$
Applying the bridge lemma on the terminal spanning tree $T_{i}$, and using the
definitions of ${\tt lp}^{*}$ and ${\tt loss}^{*}$, we have
$\displaystyle{\bf E}[c(T_{i+1})]$ $\displaystyle\leq(1-\tfrac{1}{M}){\bf
E}[c(T_{i})]+({\tt lp}^{*}-{\tt loss}^{*})/M$
By induction this gives
$\displaystyle{\bf E}[c(T_{t+1})]$
$\displaystyle=(1-\tfrac{1}{M})^{t}c(T_{1})+({\tt lp}^{*}-{\tt
loss}^{*})(1-(1-\tfrac{1}{M})^{t})$ $\displaystyle\leq{\tt
lp}^{*}(1+(1-\tfrac{1}{M})^{t})-{\tt loss}^{*}(1-(1-\tfrac{1}{M})^{t}).$
where the inequality uses Lemma 2(a). The cost of the final Steiner tree is at
most $c(ALG)\leq c(T_{t+1})+\sum_{i=1}^{t}{\tt loss}(K_{i})$. Moreover,
$\displaystyle{\bf E}[c(ALG)]\leq$ $\displaystyle~{}{\bf
E}[c(T_{t+1})]+t\cdot{\tt loss}^{*}/M$ $\displaystyle\leq$
$\displaystyle~{}{\tt lp}^{*}(1+(1-\tfrac{1}{M})^{t})+{\tt
loss}^{*}((1-\tfrac{1}{M})^{t}+\tfrac{t}{M}-1)$ $\displaystyle\leq$
$\displaystyle~{}{\tt
lp}^{*}\bigg{(}\frac{1}{2}+\frac{3}{2}\Big{(}1-\frac{1}{M}\Big{)}^{t}+\frac{t}{2M}\bigg{)}$
$\displaystyle\mathop{\leq}$ $\displaystyle~{}{\tt
lp}^{*}(1/2+3/2\cdot\exp(-t/M)+t/2M)$
where the third inequality uses (a weighted average of) Lemma 2(b). The last
line explains our choice of $t=M\ln 3$ since $\lambda=\ln 3$ minimizes
$\frac{1}{2}+\frac{3}{2}e^{-\lambda}+\frac{\lambda}{2}$, with value
$1+\frac{\ln 3}{2}$. Thus the algorithm outputs a Steiner tree of expected
cost at most $(1+\frac{\ln 3}{2}){\tt lp}^{*}$, which implies the claimed
upper bound of $1+\frac{\ln 3}{2}$ on the integrality gap. $\Box$
We now discuss a variant of the result just proven. A Steiner tree instance is
_quasi-bipartite_ if there are no Steiner-Steiner edges. For quasibipartite
instances, Robins and Zelikovsky tightened the analysis of their algorithm to
show it has approximation ratio $\alpha$, where $\alpha\approx 1.28$ satisfies
$\alpha=1+\exp(-\alpha)$). Here, we’ll show an integrality gap bound of
$\alpha$ (the longer proof of [1] via the Robins-Zelikovsky algorithm can be
similarly adapted). We can refine Lemma 2(a) (like in [6]) to show that in
quasi-bipartite instances, ${\tt mst}(G[R])\leq 2({\tt lp}^{*}-{\tt
loss}^{*})$. Continuing along the previous lines, we obtain
$\displaystyle{\bf E}[c(ALG)]\leq{\tt lp}^{*}(1+\exp(-t/M))+{\tt
loss}^{*}(t/M-1-\exp(-t/M))$
and setting $t=\alpha M$ gives ${\bf E}[c(ALG)]\leq\alpha\cdot{\tt lp}^{*}$,
as needed. We note that in quasi-bipartite instances the hypergraphic
relaxation is equivalent [2] to the so-called _bidirected cut relaxation_ thus
we get an $\alpha$ integrality gap bound there as well.
At the risk of numerology, we conclude by remarking that $1+\frac{\ln 3}{2}$
arose in two very different ways, by analyzing different algorithms (and
similarly for $\alpha\approx 1.28$). A simple explanation for this phenomenon
would be very interesting.
## References
* [1] J. Byrka, F. Grandoni, T. Rothvoß, and L. Sanità. An improved LP-based approximation for Steiner tree. In Proc. 42nd STOC, pages 583–592, 2010.
* [2] Deeparnab Chakrabarty, Jochen Könemann, and David Pritchard. Hypergraphic LP relaxations for Steiner trees. In Proc. 14th IPCO, pages 383–396, 2010. Full version at arXiv:0910.0281.
* [3] J. Edmonds. Matroids and the greedy algorithm. Math. Programming, 1:127–136, 1971.
* [4] C. Gröpl, S. Hougardy, T. Nierhoff, and H. J. Prömel. Approximation algorithms for the Steiner tree problem in graphs. In X. Cheng and D.Z. Du, editors, Steiner trees in industries, pages 235–279. Kluwer Academic Publishers, Norvell, Massachusetts, 2001.
* [5] Tobias Polzin and Siavash Vahdati Daneshmand. On Steiner trees and minimum spanning trees in hypergraphs. Oper. Res. Lett., 31(1):12–20, 2003.
* [6] G. Robins and A. Zelikovsky. Tighter bounds for graph Steiner tree approximation. SIAM J. Discrete Math., 19(1):122–134, 2005. Preliminary version appeared in _Proc. 11th SODA_ , pages 770–779, 2000.
* [7] A. Schrijver. Combinatorial optimization. Springer, New York, 2003.
* [8] D.M. Warme. A new exact algorithm for rectilinear Steiner trees. In P.M. Pardalos and D.-Z. Du, editors, Network Design: Connectivity and Facilities Location, pages 357–395. American Mathematical Society, 1997. (Result therein attributed to M. Queyranne.).
|
arxiv-papers
| 2010-06-11T10:15:05 |
2024-09-04T02:49:10.860784
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Deeparnab Chakrabarty, Jochen Koenemann, David Pritchard",
"submitter": "David Pritchard",
"url": "https://arxiv.org/abs/1006.2249"
}
|
1006.2423
|
arxiv-papers
| 2010-06-12T00:14:05 |
2024-09-04T02:49:10.868621
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hans V. Klapdor-Kleingrothaus and Irina V. Krivosheina",
"submitter": "Hans-Volker Klapdor-Kleingrothaus",
"url": "https://arxiv.org/abs/1006.2423"
}
|
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1006.2428
|
# Integrality Properties of Variations of Mahler Measures
Jian Zhou Department of Mathematical Sciences
Tsinghua University
Beijing, 100084, China jzhou@math.tsinghua.edu.cn
###### Abstract.
We propose some conjectures on the integrality properties related to the
variation of Mahler measures, inspired by the results in the elliptic curve
case by Rodriguez Villegas, Stienstra and Zagier.
In the study of mirror symmetry, there are some amazing integrality results,
including the integrality of mirror maps (Lian-Yau integrality) [10, 11, 17,
7, 8, 9, 2, 18] and the integrality of instanton numbers (including Gopakumar-
Vafa integrality for closed strings and Ooguri-Vafa integrality for open
strings in arbitrary genera), see e.g. [6, 4, 12, 13, 5]. In this note we will
propose some conjectures on the integrality properties related to the
variation of Mahler measures, inspired by the results in the elliptic curve
case in [14, 15, 16]. More precisely, we will identify a quantity $Q(z)$
associated with the variation of Mahler measures with the local mirror map,
and make some conjectures about the integrality properties of its expression
in term of the mirror parameter $q(z)$ and vice versa. Some examples are
presented.
## 1\. variations of Mahler Measures, Periods, Picard-Fuchs Equations and
Mirror Maps
### 1.1. One-parameter deformations of Fermat type hypersurfaces in weighted
projective spaces
The geometric objects we will study are deformations of Fermat type Calabi-Yau
hypersurfaces of the form:
(1) $x_{1}^{k_{1}}+\cdots+x_{n}^{k_{n}}-k\psi x_{1}\cdots x_{n}=0$
in a weighted projective space ${\mathbb{P}}^{n-1}_{w_{1},\dots,w_{n}}$. Here
$k_{1}\leq\dots\leq k_{n}$ are positive integers such that
(2) $\frac{1}{k_{1}}+\cdots+\frac{1}{k_{n}}=1,$
$k$ is the least common multiplier of $k_{1},\dots,k_{n}$, and
$w_{1}=k/k_{1},\dots,w_{n}=k/k_{n}$.
For each $n$, there are only finitely many solutions to (2). They can be found
by the following search algorithm: First $k_{1}$ is bounded between $2$ and
$n$, we search in a reversed order for $k_{1}$ in this range; for fixed
$k_{1}$, $k_{2}$ has the following bound:
$k_{1}\leq k_{2}\leq\frac{n-1}{1-\frac{1}{k_{1}}},$
we search for $k_{2}$ in reversed order in this range; for fixed
$k_{1},k_{2}$, $k_{3}$ has the following bound:
$k_{2}\leq k_{3}\leq\frac{n-2}{1-\frac{1}{k_{1}}-\frac{1}{k_{2}}},$
we search for $k_{3}$ in reversed order in this range, and so on. This
algorithm can be easily implemented by a computer algebra system 111The author
thanks Dr. Fei Yang for providing us the Maple codes that implements the
search algorithm.. The following are the results for $n=2,3,4,5$. For $n=2$,
there is only one solution: $(2,2)$; For $n=3$, there are $3$ solutions:
$(3,3,3)$, $(2,4,4)$, $(2,3,6)$. For $n=4$, there are $13$ solutions:
$(4,4,4,4)$, $(3,4,4,6)$, $(3,3,4,12)$, $(2,6,6,6)$, $(2,5,5,10)$,
$(2,4,8,8)$, $(2,4,6,12)$, $(2,4,5,20)$, $(2,3,12,12)$, $(2,3,10,15)$,
$(2,3,9,18)$, $(2,3,8,24)$, $(2,3,7,42)$. For $n=5$, there are $147$
solutions:
$(5,5,5,5,5)$, | $(4,4,6,6,6)$, | $(4,4,5,5,10)$, | $(4,4,4,8,8)$,
---|---|---|---
$(4,4,4,6,12)$, | $(4,4,4,5,20)$, | $(3,6,6,6,6)$, | $(3,5,5,6,10)$,
$(3,5,5,5,15)$, | $(3,4,6,8,8)$, | $(3,4,6,6,12)$, | $(3,4,5,6,20)$,
$(3,4,5,5,60)$, | $(3,4,4,12,12)$, | $(3,4,4,10,15)$, | $(3,4,4,9,18)$,
$(3,4,4,8,24)$, | $(3,4,4,7,42)$, | $(3,3,9,9,9)$, | $(3,3,8,8,12)$,
$(3,3,7,7,21)$, | $(3,3,6,12,12)$, | $(3,3,6,10,15)$, | $(3,3,6,9,18)$,
$(3,3,6,8,24)$, | $(3,3,6,7,42)$, | $(3,3,5,15,15)$, | $(3,3,5,12,20)$,
$(3,3,5,10,30)$, | $(3,3,5,9,45)$, | $(3,3,5,8,120)$, | $(3,3,4,24,24)$,
$(3,3,4,21,28)$, | $(3,3,4,20,30)$, | $(3,3,4,18,36)$, | $(3,3,4,16,48)$,
$(3,3,4,15,60)$, | $(3,3,4,14,84)$, | $(3,3,4,13,156)$, | $(2,8,8,8,8)$,
$(2,7,7,7,14)$, | $(2,6,9,9,9)$, | $(2,6,8,8,12)$, | $(2,6,7,7,21)$,
$(2,6,6,12,12)$, | $(2,6,6,10,15)$, | $(2,6,6,9,18)$, | $(2,6,6,8,24)$,
$(2,6,6,7,42)$, | $(2,5,10,10,10)$, | $(2,5,8,8,20)$, | $(2,5,7,7,70)$,
$(2,5,6,15,15)$, | $(2,5,6,12,20)$, | $(2,5,6,10,30)$, | $(2,5,6,9,45)$,
$(2,5,6,8,120)$, | $(2,5,5,20,20)$, | $(2,5,5,15,30)$, | $(2,5,5,14,35)$,
$(2,5,5,12,60)$, | $(2,5,5,11,110)$, | $(2,4,12,12,12)$, | $(2,4,10,12,15)$,
$(2,4,10,10,20)$, | $(2,4,9,12,18)$, | $(2,4,9,9,36)$, | $(2,4,8,16,16)$,
$(2,4,8,12,24)$, | $(2,4,8,10,40)$, | $(2,4,8,9,72)$, | $(2,4,7,14,28)$,
$(2,4,7,12,42)$, | $(2,4,7,10,140)$, | $(2,4,6,24,24)$, | $(2,4,6,21,28)$,
$(2,4,6,20,30)$, | $(2,4,6,18,36)$, | $(2,4,6,16,48)$, | $(2,4,6,15,60)$,
$(2,4,6,14,84)$, | $(2,4,5,13,156)$, | $(2,4,5,40,40)$, | $(2,4,5,36,45)$,
$(2,4,5,30,60)$, | $(2,4,5,28,70)$, | $(2,4,5,25,100)$, | $(2,4,5,24,120)$,
$(2,4,5,22,220)$, | $(2,4,5,21,420)$, | $(2,3,18,18,18)$, | $(2,3,16,16,24)$,
$(2,3,15,20,20)$, | $(2,3,15,15,30)$, | $(2,3,14,21,21)$, | $(2,3,14,15,35)$,
$(2,3,14,14,42)$, | $(2,3,13,13,78)$, | $(2,3,12,24,24)$, | $(2,3,12,21,28)$,
$(2,3,12,20,30)$, | $(2,3,12,18,36)$, | $(2,3,12,16,48)$, | $(2,3,12,15,60)$,
$(2,3,12,14,84)$, | $(2,3,12,13,156)$, | $(2,3,11,22,33)$, | $(2,3,11,15,110)$,
$(2,3,11,14,231)$, | $(2,3,10,30,30)$, | $(2,3,10,24,40)$, | $(2,3,10,20,60)$,
$(2,3,10,18,90)$, | $(2,3,10,16,240)$, | $(2,3,9,36,36)$, | $(2,3,9,30,45)$,
$(2,3,9,27,54)$, | $(2,3,9,24,72)$, | $(2,3,9,22,99)$, | $(2,3,9,21,126)$,
$(2,3,9,20,180)$, | $(2,3,9,19,342)$, | $(2,3,8,48,48)$, | $(2,3,8,42,56)$,
$(2,3,8,40,60)$, | $(2,3,8,36,72)$, | $(2,3,8,33,88)$, | $(2,3,8,32,96)$,
$(2,3,8,30,120)$, | $(2,3,8,28,168)$, | $(2,3,8,27,216)$, | $(2,3,8,26,312)$,
$(2,3,8,25,600)$, | $(2,3,7,84,84)$, | $(2,3,7,78,91)$, | $(2,3,7,70,105)$,
$(2,3,7,63,126)$, | $(2,3,7,60,140)$, | $(2,3,7,56,168)$, | $(2,3,7,54,189)$,
$(2,3,7,51,238)$, | $(2,3,7,49,294)$, | $(2,3,7,48,336)$, | $(2,3,7,46,483)$,
$(2,3,7,45,630)$, | $(2,3,7,44,924)$, | $(2,3,7,43,1806)$. |
When $n=6$, there are $3462$ solutions, e.g. $(2,7,43,1807,3263442)$.
There are two ways to count the number of solutions to (2) for each $n$. The
first is a simple count, i.e., each solution is counted as $1$. The second is
a weighted count, i.e., each solution is counted as $1$ over the order of its
automorphism group. By an automorphism of a solution $(k_{1},\dots,k_{n})$, we
mean a permutation $\sigma\in S_{n}$ such that $k_{\sigma(i)}=k_{i}$ for all
$i=1,\dots,n$. It is interesting to study these counting problems.
### 1.2. Variations of Mahler measures
Given a solution $(k_{1},\dots,k_{n})$ to (2), let $k$ be the least common
multiplier of $k_{1},\dots,k_{n}$. Consider a weighted homogeneous polynomial
of the form $k\psi\prod_{i=1}^{n}x_{i}-P(x_{1},\dots,x_{n})$, where
$P(x_{1},\dots,x_{n}):=\sum_{i=1}^{n}x_{i}^{k_{i}}$
with $\psi$ a complex parameter. This is a weighted homogeneous polynomial of
degree
$k_{1}w_{1}=\cdots=k_{n}w_{n}=w_{1}+\cdots+w_{n}=k,$
it defines a Calabi-Yau hypersurface $X_{\psi}$ in the weighted projective
space ${\mathbb{P}}^{n-1}_{{\mathbf{w}}}$, where
${\mathbf{w}}=(w_{1},\dots,w_{n})$. For
${\mathbf{e}}=(\epsilon_{1},\dots,\epsilon_{n-1})\in{\mathbb{R}}_{+}^{n-1}$,
consider the following $(n-1)$-cycle $C_{\mathbf{e}}$ in
${\mathbb{P}}^{n-1}_{\mathbf{w}}$ defined by:
(3) $|x_{1}|=\epsilon_{1},\dots,|x_{n-1}|=\epsilon_{n-1},x_{n}=1.$
Consider the following integral over this cycle:
(4) $\tilde{M}:=\exp\left(-\frac{1}{(2\pi
i)^{n-1}}\oint_{C_{{\mathbf{e}}}}\log(\psi-\frac{P(x_{1},\dots,x_{n-1},1)}{kx_{1}\cdots
x_{n-1}})\,\frac{dx_{1}}{x_{1}}\cdots\frac{dx_{n-1}}{x_{n-1}}\right).$
Recall the _logarithmic Mahler measure_ $m(F)$ and the _Mahler measure_ $M(F)$
of a Laurent polynomial $F(x_{1},\dots,x_{n-1})$ with complex coefficients
are:
(5) $\displaystyle m(F)$ $\displaystyle:=$ $\displaystyle\frac{1}{(2\pi
i)^{n-1}}\oint\cdots\oint_{|x_{1}|=\epsilon_{1},\dots,|x_{n-1}|=\epsilon_{n-1}}\log|F|\,\prod_{i=1}^{n-1}\frac{dx_{i}}{x_{i}},$
(6) $\displaystyle M(F)$ $\displaystyle:=$ $\displaystyle\exp(m(F))\;.$
One then finds
(7) $M(F_{\psi})=|\tilde{M}|^{-1},$
where
$F_{\psi}(x_{1},\dots,x_{n-1})=\psi-\frac{P(x_{1},\dots,x_{n-1},1)}{kx_{1}\cdots
x_{n-1}}$. In the case of elliptic curves [14], the Mahler measure is related
to the special values of the L-function associated to $X_{\psi}$ by Beilinson
Conjectures. Similar relationship is expected in higher dimensions.
By taking expansion in $\xi=\psi^{-1}$, one gets from (4):
$\displaystyle\tilde{M}$ $\displaystyle=$
$\displaystyle\xi\exp\biggl{(}\sum_{m=1}^{\infty}\frac{\xi^{m}}{mk^{m}}\frac{1}{(2\pi
i)^{2}}\oint_{C_{\mathbf{e}}}\frac{(\sum_{i=1}^{n-1}x_{i}^{k_{i}}+1)^{m}}{(x_{1}\cdots
x_{n-1})^{m}}\,\prod_{j=1}^{n-1}\frac{dx_{j}}{x_{j}}\biggr{)}.$
Thus
(8)
$\tilde{M}=\xi\exp\left(\sum_{m=1}^{\infty}c_{m}\frac{\xi^{m}}{mk^{m}}\right)$
with $c_{m}$ the coefficient of $x_{1}^{m}\cdots x_{n-1}^{m}$ in
$(\sum_{i=1}^{n-1}x_{i}^{k_{i}}+1)^{m}$. In particular, $Q$ is independent of
the choices of $\epsilon_{1},\dots,\epsilon_{n-1}$. By multinomial formula,
one easily gets:
(9) $c_{m}=\begin{cases}\frac{m!}{\prod_{i=1}^{n}(m/k_{i})!},&k|m,\\\
0,&\text{otherwise}.\end{cases}$
Let $Q=\tilde{M}^{k}/k^{k}$, and $z=\xi^{k}/k^{k}$, then we have
(10)
$Q=z\exp\biggl{(}\sum_{m=1}^{\infty}\frac{(km)!}{\prod_{i=1}^{n}(w_{i}m)!}\frac{z^{m}}{m}\biggr{)}.$
### 1.3. Periods and Picard-Fuchs equations
Let $\theta=z\frac{d}{dz}$. Differentiating (4) and (10) one finds
(11) $\displaystyle\theta\log Q$ $\displaystyle=$
$\displaystyle\frac{\psi}{(2\pi
i)^{n-1}}\oint_{C_{\epsilon}}\frac{dx_{1}\cdots dx_{n-1}}{k\psi x_{1}\cdots
x_{n-1}-P(x_{1},\dots,x_{n-1},1)}$ (12) $\displaystyle=$
$\displaystyle\sum_{m=0}^{\infty}\frac{(km)!}{\prod_{i=1}^{n}(w_{i}m)!}z^{m}.$
Thus $\theta\log Q$ is a period of a family $\omega_{\psi}$ of holomorphic
forms on $X_{\psi}$.
Write
(13)
$\alpha_{m}:=\frac{(km)!}{\prod_{i=1}^{n}(w_{i}m)!}=\frac{k^{km}\prod_{j=0}^{km-1}(m-\frac{j}{k})}{\prod_{i=1}^{n}[w_{i}^{w_{i}}\prod_{j=0}^{w_{i}m-1}(m-\frac{j}{w_{i}})]},$
Then we have
(14)
$\frac{\alpha_{m}}{\alpha_{m-1}}=\frac{k^{k}\prod_{j=0}^{k-1}(m-\frac{j}{k})}{\prod_{i=1}^{n}[w_{i}^{w_{i}}\prod_{j=0}^{w_{i}-1}(m-\frac{j}{w_{i}})]}=\frac{k^{k}}{\prod_{i=1}^{n}w_{i}^{w_{i}}}\prod_{j=1}^{l}\frac{m-1+a_{j}}{m-b_{j}},$
where in the second equality we remove the common factors of the numerator and
the denominator. The equality
(15)
$\frac{k^{k}}{\prod_{i=1}^{n}w_{i}^{w_{i}}}\prod_{j=1}^{l}\frac{m-1+a_{j}}{m-b_{j}}=\frac{(km)!}{(k(m-1))!}\cdot\prod_{i=1}^{n}\frac{(w_{i}(m-1))!}{(w_{i}m)!}$
and
(16)
$\sum_{j=1}^{l}(\frac{1}{m-1+a_{j}}-\frac{1}{m-b_{j}})=\sum_{j=0}^{k-1}\frac{1}{m-\frac{j}{k}}-\sum_{i=1}^{n}\sum_{j=0}^{w_{i}-1}\frac{1}{m-\frac{j}{w_{i}}}$
will be of use below.
The recursion relation
(17)
$\prod_{j=1}^{l}(m-b_{j})\alpha_{m}=\frac{k^{k}}{\prod_{i=1}^{n}w_{i}^{w_{i}}}\cdot\prod_{j=1}^{l}(m-1+a_{j})\alpha_{m-1}$
is equivalent to the following Picard-Fuchs equation:
(18) $\prod_{j=1}^{l}(\theta-
b_{j})\Phi=z\frac{k^{k}}{\prod_{i=1}^{n}w_{i}^{w_{i}}}\cdot\prod_{j=1}^{l}(\theta+a_{j})\Phi$
satisfied by $\theta\log Q$.
The recursion relation
(19)
$\frac{\alpha_{m}}{\alpha_{m-1}}=\frac{k^{k}\prod_{j=0}^{k-1}(m-\frac{j}{k})}{\prod_{i=1}^{n}[w_{i}^{w_{i}}\prod_{j=0}^{w_{i}-1}(m-\frac{j}{w_{i}})]}$
can also be rewritten as
(20)
$\prod_{i=1}^{n}\prod_{j=0}^{w_{i}-1}(m-\frac{j}{w_{i}})\cdot\alpha_{m}=\frac{k^{k}}{\prod_{i=1}^{n}w_{i}^{w_{i}}}\cdot\prod_{j=0}^{k-1}(m-1+1-\frac{j}{k})\alpha_{m-1}.$
It is equivalent to the Picard-Fuchs equation:
(21)
$\prod_{i=1}^{n}\prod_{j=0}^{w_{i}-1}(\theta-\frac{j}{w_{i}})\cdot\Phi=z\frac{k^{k}}{\prod_{i=1}^{n}w_{i}^{w_{i}}}\cdot\prod_{j=0}^{k-1}(\theta+1-\frac{j}{k})\Phi.$
In some cases one has
(22)
$\frac{\alpha_{m}}{\alpha_{m-1}}=\frac{k^{k}}{\prod_{i=1}^{n}w_{i}^{w_{i}}}\frac{\prod_{j=1}^{n-1}(m-1+a_{j})}{m^{n-1}},$
where $\\{a_{1},\dots,a_{n-1}\\}$ is obtained from the set
$\\{\frac{1}{k},\frac{2}{k},\dots,\frac{k-1}{k}\\}$ by removing integral
multiples of $\frac{1}{k_{i}}=\frac{w_{i}}{k}$, where $w_{i}>1$. In this case
the Picard-Fuchs equation takes the following form:
(23)
$\theta^{n-1}\Phi=\frac{k^{k}z}{\prod_{i=1}^{n}w_{i}^{w_{i}}}\prod_{j=1}^{n-1}(\theta+a_{j})\Phi.$
This happens if and only if $(w_{i},w_{j})=1$ for $i\neq j$.
For $n=2$, the only case $(k_{1},k_{2})=(2,2)$ has this property. The Picard-
Fuchs equation is
(24) $\theta\Phi-z(\theta+\frac{1}{2})\Phi=0.$
For $n=3$, all cases of solutions to (2) has this property. The Picard-Fuchs
operators are:
(25)
$\displaystyle\theta^{2}-z(\theta+\frac{1}{3})(\theta+\frac{2}{3}),\qquad(k_{1},k_{2},k_{3})=(3,3,3),$
(26)
$\displaystyle\theta^{2}-z(\theta+\frac{1}{4})(\theta+\frac{3}{4}),\qquad(k_{1},k_{2},k_{3})=(2,2,4),$
(27)
$\displaystyle\theta^{2}-z(\theta+\frac{1}{6})(\theta+\frac{5}{6}),\qquad(k_{1},k_{2},k_{3})=(2,3,6).$
For $n=4$, we have the following cases:
(28)
$\displaystyle\theta^{3}-z(\theta+\frac{1}{4})(\theta+\frac{2}{4})(\theta+\frac{3}{4}),\qquad(k_{1},k_{2},k_{3},k_{4})=(4,4,4,4),$
(29)
$\displaystyle\theta^{3}-z(\theta+\frac{1}{6})(\theta+\frac{3}{6})(\theta+\frac{5}{6}),\qquad(k_{1},k_{2},k_{3},k_{4})=(2,6,6,6).$
For $n=5$ we have the following cases:
(30)
$\displaystyle\theta^{4}-z(\theta+\frac{1}{5})(\theta+\frac{2}{5})(\theta+\frac{3}{5})(\theta+\frac{4}{5}),\qquad\vec{k}=(5,5,5,5,5),$
(31)
$\displaystyle\theta^{4}-z(\theta+\frac{1}{6})(\theta+\frac{2}{6})(\theta+\frac{4}{4})(\theta+\frac{5}{6}),\qquad\vec{k}=(3,6,6,6,6),$
(32)
$\displaystyle\theta^{4}-z(\theta+\frac{1}{8})(\theta+\frac{3}{8})(\theta+\frac{5}{8})(\theta+\frac{7}{8}),\qquad\vec{k}=(2,8,8,8,8),$
(33)
$\displaystyle\theta^{4}-z(\theta+\frac{1}{10})(\theta+\frac{3}{10})(\theta+\frac{7}{10})(\theta+\frac{9}{10}),\qquad\vec{k}=(2,5,10,10,10).$
### 1.4. Logarithmic solutions and mirror maps
Equation (18) has a solution of logarithmic behavior:
(34) $g_{1}(z)=g_{0}(z)\cdot\log z+h(z),$
where $g_{0}(z)=\theta\log Q=\sum_{m\geq
0}\frac{(km)!}{\prod_{i=1}^{n}(w_{i}m)!}z^{m}$ and $h(z)=\sum_{m\geq
1}\gamma_{m}z^{m}$. Rewrite (23) as
(35) $\displaystyle\prod_{j=1}^{l}(\theta-
b_{j})h(z)=\frac{k^{k}}{\prod_{i=1}^{n}w_{i}^{w_{i}}}z\prod_{j=1}^{l}(\theta+a_{j})h(z)$
$\displaystyle+\frac{k^{k}}{\prod_{i=1}^{n}w_{i}^{w_{i}}}z\sum_{i=1}^{l}\frac{\prod_{j=1}^{l}(\theta+a_{j})}{\theta+a_{i}}g_{0}(z)-\sum_{i=1}^{l}\frac{\prod_{j=1}^{l}(\theta-
b_{j})}{\theta-b_{i}}g_{0}(z).$
This is equivalent to the following initial value
(36)
$\gamma_{1}=\sum_{i=1}^{n}(\frac{1}{a_{i}}-\frac{1}{1-b_{i}})\frac{k!}{\prod_{i=1}^{n}w_{i}!}$
and recursion relation:
(37)
$\displaystyle\prod_{j=1}^{l}(m-b_{j})\cdot\gamma_{m}=\frac{k^{k}}{\prod_{i=1}^{n}w_{i}^{w_{i}}}\prod_{j=1}^{l}(m-1+a_{j})\cdot\gamma_{m-1}$
$\displaystyle-\sum_{i=1}^{l}\frac{\prod_{j=1}^{l}(m-b_{j})}{m-b_{i}}\frac{(km)!}{\prod_{i=1}^{n}(w_{i}m)!}$
$\displaystyle+\frac{k^{k}}{\prod_{i=1}^{n}w_{i}^{w_{i}}}\sum_{i=1}^{l}\frac{\prod_{j=1}^{l}(m-1+a_{j})}{m-1-a_{i}}\frac{k(m-1))!}{\prod_{i=1}^{n}(w_{i}(m-1))!}.$
Dividing both sides by $\prod_{j=1}^{l}(m-b_{j})$ and making use of (15) and
(16), one gets
(38)
$\displaystyle\gamma_{m}=\frac{(km)!}{(k(m-1))!}\cdot\prod_{i=1}^{n}\frac{(w_{i}(m-1))!}{(w_{i}m)!}\cdot\gamma_{m-1}$
$\displaystyle+\sum_{i=1}^{l}(\frac{1}{m-1+a_{i}}-\frac{1}{m-b_{i}})\frac{(km)!}{\prod_{i=1}^{n}(w_{i}m)!}.$
The solution is given by
(39) $\displaystyle\gamma_{m}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{m}\sum_{i=1}^{l}(\frac{1}{j-1+a_{i}}-\frac{1}{j-b_{i}})\cdot\frac{(km)!}{\prod_{i=1}^{n}(w_{i}m)!}$
(40) $\displaystyle=$
$\displaystyle\sum_{j=1}^{m}(\sum_{a=0}^{k-1}\frac{1}{j-\frac{a}{k}}-\sum_{i=1}^{n}\sum_{a=0}^{w_{i}-1}\frac{1}{j-\frac{a}{w_{i}}})\cdot\frac{(km)!}{\prod_{i=1}^{n}(w_{i}m)!}.$
One can also derive this solution from (21).
The _mirror map_ is defined by
(41) $q\>:=\>\exp\left(\frac{g_{1}(z)}{g_{0}(z)}\right)=z\exp(h(z)/g_{0}(z)).$
### 1.5. A related Picard-Fuchs system and its mirror map
In this section we will relate $Q$ to the mirror map of the following Picard-
Fuchs equation related to (21):
(42)
$\prod_{i=1}^{n}\prod_{j=0}^{w_{i}-1}(\theta-\frac{j}{w_{i}})\cdot\Phi=z\frac{k^{k}}{\prod_{i=1}^{n}w_{i}^{w_{i}}}\cdot\prod_{j=0}^{k-1}(\theta+\frac{j}{k})\Phi.$
Clear $\Phi=1$ is a solution, and we have the following logarithmic solution:
(43) $\Phi_{1}=\log
z+\sum_{m=1}^{\infty}\frac{(km)!}{\prod_{i=1}^{n}(w_{i}m)!}\frac{z^{m}}{m}.$
The corresponding mirror map is defined by
$Q=e^{\Phi_{1}}.$
Note this is exactly the map $Q$ defined in (10).
For example, when $(w_{1},w_{2},w_{3})=(1,1,1)$, this is the Picard-Fuch
system associated with the local ${\mathbb{P}}^{2}$ geometry [1], i.e. the
canonical line bundle $\kappa_{{\mathbb{P}}^{2}}$. In general, the Picard-
Fuchs system (42) is associated with the local Calabi-Yau geometry of
$\kappa_{{\mathbb{P}}^{n-1}_{w_{1},\dots,w_{n}}}$. Hence we will refer to the
mirror map $Q$ as the local mirror map.
## 2\. Integrality Properties of Variation of Mahler Measures
It is expected that $z$, $g_{0}(z)$, $\frac{d}{dq}\log Q$ are modular forms
for the monodromy group of the Picard-Fuchs equation, and often they can be
expressed in terms of usual modular forms. See [14, 15, 16] for examples in
the elliptic curve case. Our conjectures below are inspired by the results in
these papers. We focus on the integrality properties in this paper and leave
the modular properties to future investigations.
We have $q=ze^{h(z)/g_{0}(z)}$ and $Q=ze^{f_{n}(z)}$, where
$f_{n}(z)=\sum_{m=1}^{\infty}\frac{(mk)!}{\prod_{i=1}^{n}(w_{i}m)!}\frac{z^{m}}{m}.$
###### Proposition 2.1.
One has $q,Q\in z+z{\mathbb{Z}}[[z]]$.
###### Proof.
By a result in [18], we have $(z^{-1}Q)^{1/k}\in 1+z{\mathbb{Z}}[[z]]$. By the
main result in [2], to see $q\in z+z{\mathbb{Z}}[[z]]$ one has to show that
(44) $[kx]-\sum_{i=1}^{n}[w_{i}x]\geq 1$
for $x\in[\frac{1}{k},1)$, where $[x]$ means the integral part of $x$, i.e.,
$[x]$ is an integer such that $[x]\leq x<[x]+1$, with equality if and only if
$x\in{\mathbb{Z}}$. Therefore,
(45) $\sum_{i=1}^{n}[w_{i}x]\leq\sum_{i=1}^{n}w_{i}x=kx,$
with equality if and only if $w_{i}x\in{\mathbb{Z}}$ for all $i=1,\dots,n$.
Therefore, one has
(46) $[kx]-\sum_{i=1}^{n}[w_{i}x]\geq 0$
for all $x$. This function is right continuous and jumps at $j/k$,
$j=1,\dots,k-1$. So it suffices to check
(47) $j-\sum_{i=1}^{n}[w_{i}j/k]>0$
for all $j=1,\dots,k-1$. If $\sum_{i=1}^{n}[w_{i}j/k]=j$ for some
$j=1,\dots,k-1$, then we have
$w_{i}j/k=a_{i}$
for some integer $a_{i}$ for all $i=1,\dots,n$. This means
(48) $j=a_{i}\frac{k}{w_{i}}=a_{i}k_{i},$
i.e., $j$ is a common multiplier of $k_{1},\dots,k_{n}$, hence $j\geq k$. A
contradiction. ∎
###### Conjecture 1.
We have $(z^{-1}q)^{1/k}\in{\mathbb{Z}}[[z]]$.
Using the Lagrange-Good inversion formula [3] as in [18] one finds
$z=\sum_{m=1}^{\infty}a_{m}q^{m}$ and $z=\sum_{m=1}^{\infty}A_{m}Q^{m}$, where
(49) $a_{m}=\text{the coefficient of $z^{m-1}$
in}\;(1+\theta(h(z)/g_{0}(z))\cdot e^{-mh(z)/g_{0}(z)},$
and
(50) $A_{m}=\text{the coefficient of $z^{m-1}$ in}\;(1+\theta f_{n}(z))\cdot
e^{-mf_{n}(z)}.$
These coefficients are also integers, i.e., $z\in q+q{\mathbb{Z}}[[q]]$ and
$z\in Q+Q{\mathbb{Z}}[[Q]]$. Now we have $Q=z+O(z^{2})$ and $q=z+O(z^{2})$, so
one can eliminate $z$ and use (49) and (50) to express $Q$ as a function of
$q$ and vice versa. It is easy to see that $Q\in q{\mathbb{Z}}[[q]]$ and $q\in
Q{\mathbb{Z}}[[Q]]$. Write
(51)
$g_{0}(z)=1+\sum_{m=0}^{\infty}c_{m}q^{m}=1+\sum_{m=0}^{\infty}C_{m}Q^{m}.$
Then the coefficients $\\{c_{m}\\}_{m\geq 1}$ and $\\{C_{m}\\}_{m\geq 1}$ are
integers.
Note
(52) $q\frac{d}{dq}\log Q=z\frac{d}{dz}\log
Q\cdot\frac{q}{z}\frac{dz}{dq}=g_{0}(z)\cdot\frac{q}{z}\frac{dz}{dq}.$
Because
(53) $z\frac{d\log q}{dz}=1+\theta(\frac{h(z)}{g_{0}(z)})=1+\frac{h(z)\theta
g_{0}(z)-g_{0}(z)\theta h(z)}{g_{0}(z)^{2}}.$
Therefore,
(54) $q\frac{d}{dq}\log
Q=\frac{g_{0}(z)}{1+\theta(\frac{h(z)}{g_{0}(z)})}=\frac{g^{3}_{0}(z)}{g^{2}_{0}(z)+h(z)\theta
g_{0}(z)-g_{0}(z)\theta h(z)}.$
It follows that $q\frac{d}{dq}\log Q$ lies in ${\mathbb{Q}}[[z]]$ hence in
${\mathbb{Q}}[[q]]$. Write
(55) $q\frac{d}{dq}\log Q=1+\sum_{m=1}^{\infty}u_{m}q^{m}$
and define
(56) $b_{m}=-\frac{1}{m^{2}}\sum_{d|m}\mu(n/d)u_{d}$
and
(57) $\hat{b}_{m}=-\frac{1}{m^{2}}\sum_{d|m}\mu(n/d)(-1)^{d}u_{d}.$
Equivalently,
(58) $q\frac{d}{dq}\log Q=1-\sum_{m\geq
1}b_{m}\frac{m^{2}q^{m}}{1-q^{m}}=1-\sum_{m\geq
1}\hat{b}_{m}\frac{m^{2}(-q)^{m}}{1-(-q)^{m}}.$
###### Conjecture 2.
The numbers $b_{m}$ and $\hat{b}_{m}$ are _integers_ so that
(59) $Q=q\prod_{m\geq 1}(1-q^{m})^{mb_{m}}=q\prod_{m\geq
1}(1-(-q)^{m})^{m\hat{b}_{m}}.$
Similarly from
(60) $Q\frac{d}{dQ}\log q=\frac{Q}{z}\frac{dz}{dQ}\cdot z\frac{d}{dz}\log q,$
and
(61) $\frac{z}{Q}\frac{dQ}{dz}=z\frac{d}{dz}\log Q=g_{0}(z)$
we get:
(62) $Q\frac{d}{dQ}\log
q=\frac{1+\theta(\frac{h(z)}{g_{0}(z)})}{g_{0}(z)}=\frac{g^{2}_{0}(z)+h(z)\theta
g_{0}(z)-g_{0}(z)\theta h(z)}{g^{3}_{0}(z)}.$
It follows that $Q\frac{d}{dQ}\log q$ lies in ${\mathbb{Q}}[[z]]$ hence in
${\mathbb{Q}}[[Q]]$. Write
(63) $Q\frac{d}{dQ}\log q=1+\sum_{m=1}^{\infty}v_{m}Q^{m}$
and define
(64) $c_{m}=-\frac{1}{m^{2}}\sum_{d|m}\mu(n/d)v_{d}$
and
(65) $\hat{c}_{m}=-\frac{1}{m^{2}}\sum_{d|m}\mu(n/d)(-1)^{d}v_{d}.$
Equivalently,
(66) $Q\frac{d}{dQ}\log q=1-\sum_{m\geq
1}c_{m}\frac{m^{2}Q^{m}}{1-Q^{m}}=1-\sum_{m\geq
1}\hat{c}_{m}\frac{m^{2}(-Q)^{m}}{1-(-Q)^{m}}.$
###### Conjecture 3.
The numbers $c_{m}$ and $\hat{c}_{m}$ are _integers_ so that
(67) $q=Q\prod_{m\geq 1}(1-Q^{m})^{mc_{m}}=Q\prod_{m\geq
1}(1-(-Q)^{m})^{m\hat{c}_{m}}.$
We have written a Maple algorithm to automate the calculations of the numbers
$b_{m},\hat{b}_{m}c_{m},\hat{c}_{m}$ and verify their integrality in various
cases. Some results are presented in the following sections.
## 3\. Examples
### 3.1. The $n=2$ case
There is only one possibility:
(68) $x_{1}^{2}+x_{2}^{2}=2\psi x_{1}x_{2}.$
Geometrically, $X_{\psi}$ is just two points in ${\mathbb{P}}^{1}$. The
Picard-Fuchs operator is given by
(69) $L=\theta-2^{2}z(\theta+\frac{1}{2}),$
where $z=(2\psi)^{-2}$, $\theta=z\frac{\partial}{\partial z}$. It follows that
(70) $\displaystyle
g_{0}(z)=\sum_{m=0}^{\infty}\frac{(2m)!}{(m!)^{2}}z^{m}=\frac{1}{\sqrt{1-4z}},$
(71) $\displaystyle g_{1}(z)=\log
z\cdot\sum_{m=0}^{\infty}\frac{(2m)!}{(m!)^{2}}z^{m}+\sum_{m=1}^{\infty}\frac{(2m)!}{(m!)^{2}}\cdot\sum_{k=1}^{m}(\frac{1}{k-1/2}-\frac{1}{k})\cdot
z^{m},$ (72) $\displaystyle
Q(z)=z\exp\sum_{m=1}^{\infty}\frac{(2m)!}{(m!)^{2}}\frac{z^{m}}{m}=\frac{4z}{(1+\sqrt{1-4z})^{2}}.$
From the last equality one easily finds
(73) $z=\frac{Q}{(1+Q)^{2}},$
and so
(74) $g_{0}(z)=\frac{1+Q}{1-Q}.$
Our Maple algorithm indicates that
(75) $Q=q.$
I.e.,
(76)
$\sum_{m=1}^{\infty}\frac{(2m)!}{(m!)^{2}}\frac{z^{m}}{m}\cdot\sum_{m=0}^{\infty}\frac{(2m)!}{(m!)^{2}}z^{m}=\sum_{m=1}^{\infty}\frac{(2m)!}{(m!)^{2}}\cdot\sum_{k=1}^{m}(\frac{1}{k-1/2}-\frac{1}{k})\cdot
z^{m},$
or equivalently, for $m\geq 1$,
(77)
$\sum_{a=1}^{m}\frac{1}{a}\binom{2a}{a}\cdot\binom{2m-2a}{m-a}=\binom{2m}{m}\sum_{k=1}^{m}(\frac{1}{k-1/2}-\frac{1}{k}).$
This does not seem to be easy to establish. Another equivalent formulation is
(78)
$\sum_{m=1}^{\infty}\frac{(2m)!}{(m!)^{2}}\cdot\sum_{k=1}^{m}(\frac{1}{k-1/2}-\frac{1}{k})\cdot
z^{m}=\frac{1}{\sqrt{1-4z}}\log\frac{4}{(1+\sqrt{1-4z})^{2}}.$
This does not seem to be easy to establish either.
### 3.2. The $n=3$ case
There are $3$ possibilities, corresponding to elliptic curves in weighted
projective planes. They have been studied in [14, 15, 16], which are the
source of inspirations of this work. For
(79) $x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=3\psi x_{1}x_{2}x_{3}$
we have
$m$ | $b_{m}$ | $\hat{b}_{m}$ | $c_{m}$ | $\hat{c}_{m}$ | $\hat{c}_{m}/m$
---|---|---|---|---|---
1 | 9 | -9 | -9 | 9 | 9
2 | -9 | -9/2 | -63/2 | -36 | -18
3 | 0 | 0 | -243 | 243 | 81
4 | 9 | 9 | -2304 | -2304 | -576
5 | -9 | 9 | -25425 | 25425 | 5085
6 | 0 | 0 | -614061/2 | -307152 | -51192
7 | 9 | -9 | -3957534 | 3957534 | 565362
8 | -9 | -9 | -53475840 | -5347840 | -6684480
9 | 0 | 0 | -749220273 | 749220273 | 83246697
10 | 9 | 9/2 | -21600703575/2 | -10800364500 | -1080036450
For the elliptic curve
(80) $x_{1}^{2}+x_{2}^{4}+x_{3}^{4}=4\psi x_{1}x_{2}x_{3}$ $m$ | $b_{m}$ | $\hat{b}_{m}$
---|---|---
1 | 28 | -28
2 | -134 | -120
3 | 996 | -996
4 | -10720 | -10720
5 | 139292 | -139292
6 | -2019450 | -2018952
7 | 31545316 | -31545316
8 | -520076672 | -520076672
9 | 8930941980 | -8930941980
10 | -158342776966 | -158342707320
$m$ | $c_{m}$ | $c_{m}/m$ | $\hat{c}_{m}$ | $\hat{c}_{m}/m$
---|---|---|---|---
1 | -28 | -28 | 28 | 28
2 | -258 | -129 | -272 | -136
3 | -4860 | -1620 | 4860 | 1620
4 | -116864 | -29216 | -116864 | -29216
5 | -3259600 | -651920 | 3259600 | 651920
6 | -99763218 | -16627203 | -99765648 | -16627608
7 | -3256509228 | -465215604 | 3256509228 | 465215604
8 | -111422514176 | -13927814272 | -111422514176 | -13927814272
9 | -3951764383896 | -439084931544 | 3951764383896 | 439084931544
10 | -144178140979800 | -14417814097980 | -144178142609600 | -14417814260960
For the elliptic curve
(81) $x_{1}^{2}+x_{2}^{3}+x_{3}^{6}=6\psi x_{1}x_{2}x_{3}$ $m$ | $b_{m}$ | $\hat{b}_{m}$
---|---|---
1 | 252 | -252
2 | -13374 | -13248
3 | 1253124 | -1253124
4 | -151978752 | -151978752
5 | 21255487740 | -21255487740
6 | -3255937602498 | -3255936975936
7 | 531216722607876 | -531216722607876
8 | -90773367805541376 | -90773367805541376
9 | 16069733941012586748 | -16069733941012586748
10 | -2925411405456230806590 | -2925411405445603062720
$m$ | $b_{m}/m$ | $\hat{b}_{m}/m$
---|---|---
1 | 252 | -252
2 | -6687 | -6624
3 | 417708 | -417708
4 | -37994688 | -37994688
5 | 4251097548 | -4251097548
6 | -542656267083 | -542656162656
7 | 531216722607876/7 | -531216722607876/7
8 | -11346670975692672 | -11346670975692672
9 | 1785525993445842972 | -1785525993445842972
10 | -292541140545623080659 | -292541140544560306272
$m$ | $c_{m}$ | $\hat{c}_{m}$
---|---|---
1 | -252 | 252
2 | -18378 | -18504
3 | -2545884 | 2545884
4 | -457060032 | -457060032
5 | -94790322000 | 94790322000
6 | -21537521398170 | -21537522671112
7 | -5211710079116940 | 5211710079116940
8 | -1320613559984014848 | -1320613559984014848
9 | -346614112277503632216 | 346614112277503632216
10 | -93531635843711988483000 | -93531635843759383644000
$m$ | $c_{m}/m$ | $\hat{c}_{m}/m$
---|---|---
1 | -252 | 252
2 | -9189 | -9252
3 | -848628 | 848628
4 | -114265008 | -114265008
5 | -18958064400 | 18958064400
6 | -3589586899695 | -3589587111852
7 | -744530011302420 | 744530011302420
8 | -165076694998001856 | -165076694998001856
9 | -38512679141944848024 | 38512679141944848024
10 | -9353163584371198848300 | -9353163584375938364400
### 3.3. The $n=4$ case
For the K3 surface
(82) $x_{1}^{4}+\cdots+x_{4}^{4}=4\psi x_{1}\cdots x_{4}$
we have
$m$ | $b_{m}$ | $\hat{b}_{m}$ | $b_{m}/m$ | $\hat{b}_{m}/m$
---|---|---|---|---
1 | 80 | -80 | 80 | -80
2 | 80 | 120 | 40 | 60
3 | 240 | -240 | 80 | -80
4 | 160 | 160 | 40 | 40
5 | 400 | -400 | 80 | -80
6 | 240 | 360 | 40 | 60
7 | 560 | -560 | 80 | -80
8 | 320 | 320 | 40 | 40
9 | 720 | \- 720 | 80 | -80
10 | 400 | 600 | 40 | 60
$m$ | $c_{m}$ | $\hat{c}_{m}$
---|---|---
1 | -80 | 80
2 | -3280 | -3320
3 | -272240 | 272240
4 | -29945760 | -29945760
5 | -3860155600 | 3860155600
6 | -550279367920 | -550279504040
7 | -84101456589360 | 84101456589360
8 | -13526805760545600 | -13526805760545600
9 | -2262255520889560560 | 2262255520889560560
10 | -390188833066192395600 | -390188833068122473400
$m$ | $c_{m}/m$ | $\hat{c}_{m}/m$
---|---|---
1 | -80 | 80
2 | -1640 | -1660
3 | -272240/3 | 272240/3
4 | -7486440 | -7486440
5 | -772031120 | 772031120
6 | -275139683960/3 | -275139752020/3
7 | -12014493798480 | 12014493798480
8 | -1690850720068200 | -1690850720068200
9 | -754085173629853520/3 | 754085173629853520/3
10 | -39018883306619239560 | -39018883306812247340
We have also verify the case of
(83) $x_{1}^{4}+x_{2}^{3}+x_{2}^{3}+x_{4}^{2}-12\psi x_{1}\cdots x_{4}=0.$
It turns out that $b_{m}/m$, $\hat{b}_{m}/m$, $c_{m}/m$ and $\hat{c}_{m}/m$
are all integers. The numbers are too large to reproduce here. For example,
$b_{5}=31088578606413096899258654040.$
### 3.4. The $n=5$ case
For the case of
(84) $x_{1}^{5}+\cdots+x_{5}^{5}=5\psi x_{1}\cdots x_{5}$
we have checked that $b_{m}$, $\hat{b}_{m}$, $c_{m}$ and $\hat{c}_{m}$ are all
integers divisible by $5$, e.g.,
$b_{5}=25050301099750,$
but not all $b_{m}/m$, $\hat{b}_{m}/m$, $c_{m}/m$ and $\hat{c}_{m}/m$ are
integers. For example,
$b_{7}/7=31249534645239703150/7.$
We have also checked the case of
(85) $x_{1}^{3}+x_{2}^{3}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}=12\psi x_{1}\cdots
x_{5}$
The numbers $b_{m}/m$, $\hat{b}_{m}/m$, $c_{m}/m$ and $\hat{c}_{m}/m$ are all
integers. For example,
$\frac{b_{6}}{6}=-61961714940992690898780121741257228991904436.$
### 3.5. The $n>5$ cases
We have also checked various $n>5$ cases, e.g. the case of
(86) $x_{1}^{6}+\cdots+x_{6}^{6}=6\psi x_{1}\cdots x_{6}$
and the case of
(87) $x_{1}^{7}+\cdots+x_{7}^{7}=7\psi x_{1}\cdots x_{7}.$
We conjecture that all $b_{m}/m$, $\hat{b}_{m}/m$, $c_{m}/m$ and $\hat{c}_{m}$
are integers are divisible by $n$ for the case of
(88) $x_{1}^{n}+\cdots+x_{n}^{n}=n\psi x_{1}\cdots x_{n}.$
### 3.6. Discussions
In this paper we have considered the variation of Mahler measures of some
polynomials and define a function $Q$. We have identified $Q$ with the local
mirror map of a related Picard-Fuchs system, which corresponds to some local
Calabi-Yau geometry. Some conjectures are made about some integrality
properties of the expression of $Q$ in terms of $q$ and the expression of $q$
in terms of $Q$. Their enumerative meaning is not clear at present.
In [15] Beauville’s semistable families of elliptic curves over
${\mathbb{P}}^{1}$ with four singular fibers were considered. It is
interesting to extend the discussion in this paper to semistable families of
Calabi-Yau $n$-folds over ${\mathbb{P}}^{1}$ for $n>1$. In this paper we have
only considered hypergeometric series in one variable. Another direction for
extension is to consider multivariate hypergeometric series. We hope to
address these problems in subsequent research.
Acknowledgements. This research is supported in part by NSFC grants (10425101
and 10631050) and a 973 project grant NKBRPC (2006cB805905).
## References
* [1] T.-M. Chiang, A. Klemm, S.-T. Yau, E. Zaslow, Local Mirror Symmetry: Calculations and Interpretations, Adv.Theor.Math.Phys. 3 (1999), 495-565.
* [2] E. Delaygue, Critére pour l’intégralité des coefficients de Taylor des applications miroir, arXiv:0912.3776.
* [3] I. J. Good, Generalizations to several variables of Lagrange’s expansion, with applications to stochastic processes, Proc. Cambridge Philos. Soc. 56 (1960), 367-380.
* [4] R. Gopakumar, C. Vafa, M-theory and topological strings-II, hep-th/9812127.
* [5] Y. Konishi, Integrality of Gopakumar-Vafa invariants of toric Calabi-Yau threefolds, Publ. Res. Inst. Math. Sci. 42 (2006), no. 2, 605-648, arXiv:math/0504188.
* [6] M. Kontsevich, A. Schwarz, V. Vologodsky, Integrality of instanton numbers and $p$-adic B-model, Phys. Lett. B 637 (2006), no. 1-2, 97-101, hep-th/0603106.
* [7] C. Krattenthaler,T. Rivoal, Multivariate $p$-adic formal congruences and integrality of Taylor coefficients of mirror maps, arXiv:0804.3049.
* [8] C. Krattenthaler, T. Rivoal, On the integrality of the Taylor coefficients of mirror maps, Duke Math. J. 151 (2010), 175-218, arXiv:0907.2577.
* [9] C. Krattenthaler, T. Rivoal, On the integrality of the Taylor coefficients of mirror maps, II, Commun. Number Theory Phys. 3 (2009), 555-591, arXiv:0907.2578.
* [10] B. H. Lian, S.-T. Yau, Mirror maps, modular relations and hypergeometric series I, appeared as Integrality of certain exponential series , in: Lectures in Algebra and Geometry, Proceedings of the International Conference on Algebra and Geometry, Taipei, 1995, M.-C. Kang (ed.), Int. Press, Cambridge, MA, 1998, pp. 215-227.
* [11] B. H. Lian, S.-T. Yau, The nth root of the mirror map, in: Calabi-Yau varieties and mirror symmetry, Proceedings of the Workshop on Arithmetic, Geometry and Physics around Calabi-Yau Varieties and Mirror Symmetry, Toronto, ON, 2001, N. Yui and J. D. Lewis (eds.), Fields Inst. Commun., 38, Amer. Math. Soc., Providence, RI, 2003, pp. 195-199.
* [12] H. Ooguri, C. Vafa, Knot invariants and topological strings, Nuclear Phys. B 577 (2000), 419-438.
* [13] P. Peng, A simple proof of Gopakumar-Vafa conjecture for local toric Calabi-Yau manifolds, Comm. Math. Phys. 276 (2007), no. 2, 551-569, arXiv:math/0410540.
* [14] F. Rodriguez Villegas, Modular Mahler measures I, Topics in number theory (University Park, PA, 1997), S. Ahlgren, G. Andrews, K. Ono (eds) 17–48, Math. Appl., 467, Kluwer Acad. Publ., Dordrecht, 1999.
* [15] J. Stienstra, Mahler measure variations, Eisenstein series and instanton expansions, in Mirror symmetry. V, 139-150, AMS/IP Stud. Adv. Math., 38, Amer. Math. Soc., Providence, RI, 2006. arXiv:math/0502193.
* [16] D. Zagier, Integral solutions of Apéry-like recurrence equations, in Groups and symmetries, 349-366, CRM Proc. Lecture Notes, 47, Amer. Math. Soc., Providence, RI, 2009.
* [17] W. Zudilin, Integrality of power expansions related to hypergeometric series, Mathematical Notes 71.5 (2002), 604-616.
* [18] J. Zhou, Some integrality properties in local mirror symmetry, arXiv:1005.3243.
|
arxiv-papers
| 2010-06-12T02:31:19 |
2024-09-04T02:49:10.872211
|
{
"license": "Public Domain",
"authors": "Jian Zhou",
"submitter": "Jian Zhou",
"url": "https://arxiv.org/abs/1006.2428"
}
|
1006.2493
|
# Diameter Bounds for Planar Graphs
Radoslav Fulek∗ Filip Morić∗ David Pritchard Ecole Polytechnique Fédérale de
Lausanne. Email: $\\{$radoslav.fulek, filip.moric,
david.pritchard$\\}$@epfl.ch
###### Abstract
The _inverse degree_ of a graph is the sum of the reciprocals of the degrees
of its vertices. We prove that in any connected planar graph, the diameter is
at most $5/2$ times the inverse degree, and that this ratio is tight. To
develop a crucial surgery method, we begin by proving the simpler related
upper bounds $(4(|V|-1)-|E|)/3$ and $4|V|^{2}/3|E|$ on the diameter (for
connected planar graphs), which are also tight.
## 1 Introduction
In this paper we examine the relation between “inverse degree” and diameter in
connected planar simple graphs. The diameter $D(G)$ of a graph $G=(V,E)$ is
the maximum distance between any pair of vertices, $D:=\max_{u,v\in
V}dist(u,v),$ where as usual the distance between two vertices is the minimum
number of edges on any $u$-$v$ path. The _inverse degree_ $r(G)$ is a less
well-studied quantity, and is defined equal to the sum of the inverses of the
degrees, $r:=\sum_{v\in V}d^{-1}(v)$.
The history of inverse degree stems from the conjecture-generating program
Graffiti [2]. Let $n$ denote $|V|$ and $m$ denote $|E|$. Graffiti posited that
the _mean distance_ $\tbinom{n}{2}^{-1}\sum_{\\{u,v\\}\subset V}dist(u,v)$ is
always at most the inverse degree $r(G)$. This was disproved by Erdős, Pach &
Spencer [1], who also proved the tight bound $D=O\bigl{(}\frac{\log
n}{\log\log n}\cdot r\bigr{)}$ in the process. Subsequently, Mukwembi [3]
studied the diameter for various kinds of graphs in terms of inverse degrees.
Among other things he conjectured that for any _planar_ graph $G$,
$D(G)\leq\frac{9}{4}r(G)$.
We disprove Mukwembi’s conjecture and establish just how large $D/r$ can be:
###### Theorem 1.
For any planar graph $G$, $D(G)<\frac{5}{2}r(G)$. There is an infinite family
of graphs with $D(G)=\frac{5}{2}r(G)-O(1)$.
The tight family we construct is very simple, but the bound
$D(G)\leq\frac{5}{2}r(G)$ turns out to be quite challenging. A natural
approach is to use the arithmetic-harmonic mean inequality to bound $r$ with
the simpler quantity $r\geq\frac{n^{2}}{2m}$; to this end we prove the tight
bound $D\leq\frac{4n^{2}}{3m}$ using a simple “surgery argument.”
However, the tight examples of graphs with $D=\frac{4n^{2}}{3m}-O(1)$ are non-
regular (about $2/3$ of vertices have degree 5, and $1/3$ have degree 2) and
so they are not tight for the ratio $D/r$ (since our use of the arithmetic-
harmonic mean is tight only for regular graphs). Indeed, the bounds
$D\leq\frac{4n^{2}}{3m}$ and $r\geq\frac{n^{2}}{2m}$ do not imply Theorem 1,
but rather the weaker bound $D\leq\frac{8}{3}r$. To actually prove Theorem 1
(in Section 3) we carefully engineer a more intricate version of the surgery
argument.
## 2 Initial Bounds from Surgery
In this section we focus on proving the less complex bound
$D\leq\frac{4n^{2}}{3m}$, and on proving that the ratio $\frac{4}{3}$ is best
possible, for connected planar graphs. We use the following sneaky attack on
the problem:
###### Theorem 2.
For every connected planar graph, $D\leq\frac{4(n-1)-m}{3}$.
We give the proof later in this section, introducing our surgery approach
along the way. It gives the desired corollary:
###### Corollary 3.
For every connected planar graph, $D\leq\frac{4n^{2}}{3m}$.
###### Proof.
We know $(2(n-1)-m)^{2}\geq 0$; rearranging yields $4(n-1)-m\leq
4\frac{(n-1)^{2}}{m}$, thus Theorem 2 yields
$D(G)\leq\frac{4(n-1)-m}{3}\leq\frac{4(n-1)^{2}}{3m}$, which implies the
corollary. ∎
We give some examples before proving Theorem 2. One example disproves
Mukwembi’s conjecture, and the others demonstrate the tightness of the above
theorems. For any even integer $n\geq 4$, let $L_{n}$ denote the graph with
vertices $v^{i}_{j}$ for $i\in\\{1,2\\},1\leq j\leq n/2$, such that distinct
nodes $v^{i}_{j},v^{i^{\prime}}_{j^{\prime}}$ are joined by an edge whenever
$|j-j^{\prime}|\leq 1$; the left side of Figure 1 illustrates $L_{8}$. Its
diameter is $D(L_{n})=n/2-1$, and its inverse degree is
$r(L_{n})=\frac{n-4}{5}+\frac{4}{3}$. Hence $D=\frac{5}{2}r-O(1)$ for this
family of graphs and the second half of Theorem 1 is proven.
Figure 1: These planar graphs are depicted as if they were drawn on a
cylindrical tube, with the dashed edges hidden on the back. Left: the graph
$L_{8}$. Right: the graph $T_{12}$.
Here is the tight example for Corollary 3: for any $n$ divisible by 3, take
$L_{2n/3}$ and attach a path with $n/3$ additional nodes to $v^{1}_{1}$. The
resulting graph has diameter $\frac{2n}{3}-1$ and
$m=5\frac{n}{3}-4+\frac{n}{3}$ edges, so $\frac{4n^{2}}{3mD}$ tends to 1 as
$n$ tends to infinity.
Finally, Theorem 2 is best possible, up to an additive constant, for all
possible values of $m$ and $n$. _Euler’s bound_ says that in planar graphs
having $n\geq 3$, we have $m\leq 3n-6$; this maximum is achieved only for
triangulations. For $n\geq 6$ divisible by 3, let $T_{n}$ be obtained from
gluing a sequence of $\frac{n}{3}-1$ octahedra at opposite faces; we
illustrate $T_{12}$ in the right side of Figure 1. To demonstrate tightness of
Theorem 2 we start with the extremal values of $m$. For $m=n-1$ we have exact
tightness: the path graph $P_{n}$ has
$D(P_{n})=n-1=\frac{4(n-1)-m(P_{n})}{3}$. For $m=3n-6$ when 3 divides $n$, the
graph $T_{n}$ has $D=\frac{n}{3}-1$ and $3n-6$ edges, which is tight for
Theorem 2 up to an additive constant; other $n$ are similar. More generally,
for any $n$ and any $n-1\leq m\leq 3n-6$, taking $T_{3\lceil(m+2-n)/6\rceil}$
and adding a path of $n-3\lceil(m+2-n)/6\rceil$ more vertices to one end gives
an $n$-node, $m$-edge graph with $D=\frac{4(n-1)-m}{3}-O(1)$.
Now we give the proof of Theorem 2, which has some ingredients used later on:
a _surgery_ operation and decomposition into levels. In the proof, we will let
$st$ be a diameter of $G$, e.g. $dist_{G}(s,t)=D(G)$. We let $V_{i}$, the _$i$
th level_, denote all vertices at distance $i$ from $s$, hence
$\biguplus_{i=0}^{D}V_{i}$ is a partition of $V$. We use the shorthand
$V_{[i,j]}$ to mean $\uplus_{i\leq x\leq j}V_{x}$ and $V_{\geq i}$ is
analogous. Additionally, $G[X]$ denotes an induced subgraph and we will extend
the subscript notation on $V$ to mean induced subgraphs of $G$, for example
$G_{\geq i}=G[V_{\geq i}]$.
###### Proof of Theorem 2.
Assume for the sake of contradiction that $G$ is a graph with
$D(G)>\frac{4(n-1)-m}{3}$, assume that $n$ is minimal over all such graphs; we
may clearly also assume $E$ is _maximal_ in the sense that for any $e\not\in
E$, either $G\cup\\{e\\}$ is non-planar or $D(G\cup\\{e\\})<D(G)$.
Our first step is to show that $G$ is 2-vertex-connected. Otherwise, pick an
articulation vertex $v$, then we can decompose $G$ into graphs $G_{1},G_{2}$
with $V(G_{1})\cap V(G_{2})=\\{v\\},V(G_{1})\cup V(G_{1})=V(G),E(G_{1})\cup
E(G_{2})=E(G)$, and $n(G_{1}),n(G_{2})<n(G)$ (a _1-sum_). By our choice of
$G$, both $G_{i}$’s satisfy the conclusion of Theorem 2. Moreover it is easy
to see $m(G)=m(G_{1})+m(G_{2})$ and $D(G)\leq D(G_{1})+D(G_{2})$. Hence
$D(G)\leq
D(G_{1})+D(G_{2})\leq\tfrac{4(n(G_{1})-1)-m(G_{1})}{3}+\tfrac{4(n(G_{2})-1)-m(G_{2})}{3}=\tfrac{4(n(G)-1)-m(G)}{3},$
contradicting the fact that $G$ was chosen to be a counterexample. Thus $G$ is
indeed 2-vertex-connected.
We now consider the diameter $st$ and the level decomposition mentioned
previously. Note that there are no edges between any pair of vertices in
$V_{i}$ and $V_{j}$ if $|i-j|>1$. It is easy to see that if $|V_{i}|=1$ for
some $0<i<D$ then $V_{i}$ is an articulation point, so we have (by 2-vertex-
connectivity) that $|V_{i}|\geq 2$ for all $0<i<D$.
To begin, suppose $|V_{i}|\leq 2$ for all $i\neq 0$. Since each vertex can
only connect to neighbours in $V_{i-1},V_{i},V_{i+1}$ the maximum degree is 5
(and 2 for $s$, 3 for $t$, 4 in $V_{1}$). Thus (assuming $n\geq 4$ which is
easy to justify) we have $D=\lfloor\frac{n}{2}\rfloor$ and
$m\leq\lfloor\frac{5n-7}{2}\rfloor$, whence it is easy to verify
$D\leq(4(n-1)-m)/3$ as needed.
Hence, there exists a level of size $\geq 3$. We need one well-known fact and
a technical claim.
###### Fact 4.
Let $G_{1},G_{2}$ be planar graphs with $V(G_{1})\cap V(G_{2})=\\{u,v\\}$ and
$uv\in E(G_{1}),E(G_{2})$. Define their _2-sum_ $G$ by $V(G)=V(G_{1})\cup
V(G_{2})$, $E(G)=E(G_{1})\cup E(G_{2})$. Then $G$ is planar.
###### Claim 5.
If $|V_{i}|=2$, $i<D$, then there is an edge joining the two vertices of
$V_{i}$.
###### Proof.
Suppose otherwise. Let $V_{i}=\\{u,v\\}.$ We will show $uv$ can be added to
$G$ without violating planarity, which will complete the proof, since $G$ was
chosen edge-maximal (and adding $uv$ does not change $D$).
Since $G$ is 2-vertex-connected, $u$ is not an articulation vertex, so
$G[\\{v\\}\cup V_{>i}]$ is connected, and similarly for $G[\\{u\\}\cup
V_{>i}]$. Thus there is a path $P_{R}$ from $u$ to $v$ all of whose internal
vertices lie in $V_{>i}$. Likewise there is a $u$-$v$ path $P_{L}$ all of
whose internal vertices lie in $V_{<i}$ (e.g. concatenate shortest $u$-$s$ and
$s$-$v$ paths).
Consider a drawing of $G$. The sub-drawing of $G_{\leq i}$ must have $u$, $v$
on the same face due to $P_{R}$, so $G_{\leq i}\cup\\{uv\\}$ is planar.
Likewise $G_{\geq i}\cup\\{uv\\}$ is planar and using Fact 4, $G\cup\\{uv\\}$
is planar as needed. ∎
Recall there exists a level of size at least 3, let $L$ be chosen minimal with
$|V_{L+1}|\geq 3$. Let $R$ be chosen maximal such that $R>L$, and all of the
levels $V_{L+1},V_{L+2},\dotsc,V_{R-1}$ have size 3. Thus either $R=D+1$, or
$R\leq D$ and $|V_{R}|<3$. We break into several similar cases now.
Case $L>0,R<D$. Thus $|V_{L}|=|V_{R}|=2$. Consider the graph $G^{\prime}$
obtained by “surgery” from $G$ by deleting all edges in $G_{[L,R]}$, deleting
the isolated vertices $V_{[L+1,R-1]}$, then adding a clique on $V_{L}\cup
V_{R}$. This is a planar graph by Fact 4 and Claim 5: it is obtained by two
2-sums from $G_{\leq L}$, $K_{4}$, and $G_{\geq R}$. We illustrate in Figure
2. Now $G^{\prime}$ is smaller than $G$; write $\Delta
D=D(G)-D(G^{\prime}),\Delta m=m(G)-m(G^{\prime}),\Delta n=n(G)-n(G^{\prime})$.
We have $\Delta n\geq 3\Delta D$ since all deleted levels had size at least 3.
Moreover, since $G_{[L,R]}$ is a planar graph Euler’s bound gives that we
deleted at most $3(\Delta n+4)-6$ edges and added 6 to the new clique, so
$\Delta m\leq 3\Delta n$. Thus $\frac{4(\Delta n)-\Delta m}{3}\geq\frac{\Delta
n}{3}\geq\Delta D$ and from this it is easy to verify that $G^{\prime}$ is a
smaller counterexample to Theorem 2, contradicting our choice of $G$.
Figure 2: Depiction of how surgery changes a graph $G$ (left) into
$G^{\prime}$ (right). Note the $V_{i},G_{i}$ labels are with respect to the
original graph. Gray parts are unaltered.
Case $L>0,R\in\\{D,D+1\\}$. Let $X=V_{>L}\backslash\\{t\\}$. We delete all
edges in $G_{\geq L}$, then the isolated vertices $X$, then we join the three
vertices $V_{L}\cup\\{t\\}$ by a clique. Thus $\Delta m\leq 3(\Delta
n+3)-6-3=3\Delta n$ and we proceed as before.
Case $L=0,R<D$ is the mirror image of the previous case (e.g. the clique is
added to $V_{R}\cup\\{s\\}$).
Case $L=0,R\in\\{D,D+1\\}$. We have $n\geq 3D-1$ since all levels in
$V_{[1,D-1]}$ have size at least 3. Using Euler’s bound, $4(n-1)-m\geq n+2>3D$
and $D<\frac{4(n-1)-m}{3}$ as needed. ∎
## 3 Proof that $r(G)\geq\frac{2}{5}D(G)$ for Planar Graphs
The general idea in the proof of Theorem 1 is similar to what we did in the
previous section, but the devil is in the details, because the terms $1/d(v)$
change in quite complex ways when we perform surgery on the graph. For
example, it is no longer possible to easily argue that the selected
counterexample $G$ is 2-vertex-connected. Here is the sketch of how we prove
$r(G)\geq\frac{2}{5}D(G)$.
* •
Define the _fitness_ of a planar connected graph $G$ to be
$\mathcal{F}(G):=\frac{2}{5}D(G)-r(G)$. So we want to show no graph has
positive fitness.
* •
Let $n$ be minimal such that some $n$-vertex planar connected graph has
positive fitness. Subject to this minimal $n$, take such a graph $G$ having
maximal fitness. If another graph $G^{\prime}$ exists such that
$|V(G^{\prime})|\leq|V(G)|$ and $\mathcal{F}(G^{\prime})\geq\mathcal{F}(G)$
and at least one of the these two inequalities is strict, this contradicts our
choice of $G$. Therefore, the proof strategy uses several parts, and in each
part we either find such a $G^{\prime}$, or impose additional structure on
$G$.
* •
Let $st$ be any diameter of $G$. We show that except for $s$ and $t$, every
vertex has degree at least 3, and that $s$ and $t$ have degree 2 or more.
* •
We lay out the graph $G$ in levels, as in the previous proof: level $V_{i}$
consists of all vertices at distance $i$ from $s$, hence
$\uplus_{i=0}^{D}V_{i}$ is a partition of $V$.
* •
We arrive at a general “cornerstone” theorem (Theorem 20) showing that in many
cases, a surgery like in Section 2 finds the desired $G^{\prime}$.
* •
We clean up some additional cases, and thereby prove that $G$ has at most 3
nodes per level, that no size-3 levels are adjacent, that for every size-2
level the contained nodes share an edge, and that the last level $V_{D}$ has
size 1.
* •
We use a computation (Section 3.7) to prove that this structured graph has
$\mathcal{F}(G)<0$, completing the proof.
### 3.1 Preliminaries
We reiterate the main tool in the proof.
###### Claim 6.
If $G^{\prime}$ is another graph obtained from $G$, with $n(G^{\prime})<n(G)$,
such that $D(G^{\prime})\geq D(G)-\Delta D$, $r(G^{\prime})\leq r(G)-\Delta
r$, and $\Delta r\geq\frac{2}{5}\Delta D$, then $G^{\prime}$ is smaller but at
least as fit as $G$, contradicting our choice of $G$.
Since adding an edge decreases $r$ and increases fitness, we also have the
following.
###### Claim 7 (Maximality).
If $uv\not\in E$ then either $G\cup\\{uv\\}$ is non-planar or
$D(G\cup\\{uv\\})<D(G)$. In particular, when $u$ and $v$ are in the same
levels or adjacent levels, since adding $uv$ would not change the diameter, we
have that $G\cup\\{uv\\}$ is non-planar.
We will repeatedly make use of the arithmetic-harmonic mean in the following
way.
###### Proposition 8.
For any set $S$ of vertices, $\sum_{v\in S}1/d(v)\geq|S|^{2}/(\sum_{v\in
S}d(v))$.
Thus, the contribution to $r$ by any set is at least as big as what it would
give “on average” by counting all endpoints incident on $S$. Later, we will
count $\sum_{v\in S}d(v)$ as twice the number of edges of $G[S]$, plus the
number of edges with exactly one endpoint in $S$.
Suppose that every level of $G$, except possibly the first and last ($V_{0}$
and $V_{D}$) have size 3. Then $n\geq 3(D-1)+2$ and the following proposition
shows such graphs are not problematic.
###### Proposition 9.
If $n\geq 3(D-1)+2$, then $r(G)\geq\frac{2}{5}D$.
###### Proof.
The case that $|n|<3$ is easy to verify, so assume $|E|\leq 3n-6$. Proposition
8 applied to $S=V$ implies that $r\geq n^{2}/(6n-12)$, and by hypothesis
$D\leq(n+1)/3$. Therefore it is enough to prove
$n^{2}/(6n-12)\geq\frac{2}{5}(n+1)/3$, which is easy to verify by cross-
multiplying and solving the resulting quadratic. ∎
### 3.2 Small-Degree Vertices and Articulation Points
###### Proposition 10.
$G$ does not have a degree-1 vertex.
###### Proof.
Let $v$ be a degree-1 vertex with neighbour $z$. We may assume $|V|\geq 3$ so
$d(z)\geq 2$. How do $r$ and $D$ change if we get another graph $G^{\prime}$
by deleting $v$? Clearly $D$ decreases by at most 1; and
$r(G^{\prime})=r(G)-\frac{1}{1}-\frac{1}{d(z)}+\frac{1}{d(z)-1}\leq r(G)-1/2$.
In Claim 6 take $\Delta D=1$ and $\Delta r=1/2$, we are done. ∎
A repeated issue is that $r$ is not monotonic, i.e. sometimes we can decrease
$r$ in a graph by adding extra vertices (e.g. consider the complete bipartite
graphs, where $r(K_{2,10}<r(K_{1,10})$). The following proposition is a first
attack against this issue and shows that adding extra blocks (2-vertex-
connected components) cannot decrease $r$.
###### Proposition 11.
If $v$ is an articulation vertex of $G$, then $G\backslash v$ has exactly two
connected components, one containing $s$ and one containing $t$.
###### Proof.
If the proposition is false, there is an articulation vertex $v$ such that a
connected component $H$ of $G\backslash\\{v\\}$ contains neither $s$ nor $t$.
Thus $G\backslash H$ contains $s$ and $t$, moreover $D(G\backslash H)=D(G)$
since any simple $s$-$t$ path goes through $v$ at most once and hence does not
use any vertex of $H$.
We want to argue that $r(G\backslash H)\leq r(G)$, which will complete the
proof using Claim 6 with $\Delta D=\Delta r=0$. It is enough to use very crude
degree estimates. Let $|V(H)|=k$. Each vertex of $H$ has degree at most $k$ in
$G$ since each $u\in V(H)$ can only have neighbours in
$V(H)\cup\\{v\\}\backslash\\{u\\}$. Moreover, the difference between
$r(G\backslash H)$ and $r(G)$ is due only to vertices in $\\{v\\}\cup V(H)$.
Clearly $v$ has at least one neighbour not in $H$. Then
$r(G)=r(G\backslash H)+\sum_{u\in
H}\frac{1}{d_{G}(u)}+\frac{1}{d_{G}(v)}-\frac{1}{d_{G\backslash H}(v)}\geq
r(G\backslash H)+\frac{k}{k}+0-1=r(G\backslash H),$
as needed. ∎
###### Proposition 12.
Except possibly $s$ and $t$, $G$ does not have a degree-2 vertex.
###### Proof.
Let $v\not\in\\{s,t\\}$ be a degree-2 vertex, with neighbours $a,b$. If $a$
and $b$ are non-adjacent, we can remove $v$ and directly connect them, which
decreases $r$ by 1/2 and decreases $D$ by at most 1, which yields a
contradiction by Claim 6.
Therefore assume $a$ and $b$ are adjacent. If both $a$ and $b$ have degree 2
then $G=K_{3}$ and $\mathcal{F}(G)<0$, so we are done. If both $a$ and $b$
have degree at least 3, since $v\not\in\\{s,t\\}$, $G\backslash\\{v\\}$ is a
connected planar graph with diameter at least as large as $G$ and
$r(G^{\prime})\geq r(G)-1/2+1/6+1/6\geq r(G)$, so we are done by using Claim 6
with $\Delta D=\Delta r=0$.
The final case is that $a$ has degree 2 (w.l.o.g.) and $b$ has degree at least
3. Then $b$ is an articulation vertex, implying by Proposition 11 that
$a\in\\{s,t\\}$, say w.l.o.g. $a=s$, and $t\not\in\\{v,a,b\\}$. But this
contradicts edge-maximality in the following way: let $by$ for
$y\not\in\\{a,v\\}$ be an edge on a common face with $bv$ (see Figure 3), then
adding $vy$ to $G$ does not change the diameter. ∎
Figure 3: Dashed edges are added without violating planarity. (a) The edge
$vy$ contradicting the edge-maximality. (b) The distance 2 neighbourhood of
$s$ after $\omega$-$\mu$ surgery and the added edges.
### 3.3 Basic Surgery: Case Analysis and Bonuses
The central idea for surgery comes from the first case of Theorem 2’s proof.
###### Definition 13.
Given two levels $V_{L}$ and $V_{R}$, to apply _surgery at $V_{L}$ and
$V_{R}$_ means to delete all nodes in $V_{[L+1,R-1]}$ (and their incident
edges) and then to connect each $u\in V_{L}$ to each $v\in V_{R}$ (we “add a
biclique”).
We say a level of size 2 is _connected_ if its vertices share an edge, and
that a level of size 1 is always connected. Assuming the levels are connected
and of size at most 2, Definition 13 is indeed the same surgery as in Section
2. As before we get:
###### Proposition 14.
Suppose $|V_{L}|,|V_{R}|\leq 2$ are connected levels with $L<R$. Surgery at
$V_{L}$ and $V_{R}$ yields a connected planar graph $G^{\prime}$ with
$D(G^{\prime})=D(G)-(R-L-1)$.
We need a collection of _types_ (cases) for our analysis. There are 7 types
and $V_{L}$ may satisfy one or none of them (i.e. the cases are not
exhaustive; nonetheless they form the core of our arguments). Analogous cases
for $V_{R}$ are explained afterwards. Here are the 7 types for $V_{L}$:
* $\omega$:
$L=0$, i.e. the level contains one end of the diameter $st$; for all other
cases, $L>0$.
* $\alpha$:
$|V_{L}|=1$ and the node in $V_{L}$ has 1 neighbour in $V_{L-1}$
* $\beta$:
$|V_{L}|=1$ and the node in $V_{L}$ has 2 neighbours in $V_{L-1}$
* $\beta^{\prime}$:
$|V_{L}|=1$ and the node in $V_{L}$ has $\geq 2$ neighbours in $V_{L-1}$ and
$\geq 2$ neighbours in $V_{L+1}$
* $\mu$:
$|V_{L}|=2$, $V_{L}$ is connected, and each node of $V_{L}$ has 1 neighbour in
$V_{L-1}$, in fact the same one
* $\nu$:
$|V_{L}|=2$, $V_{L}$ is connected, and each node of $V_{L}$ has 2 neighbours
in $V_{L-1}$
* $\nu^{\prime}$:
$|V_{L}|=2$, $V_{L}$ is connected, and each node of $V_{L}$ has $\geq 2$
neighbours in $V_{L-1}$ and $\geq 2$ neighbours in $V_{L+1}$
The analogous cases for the right-hand side are the same with $L=0,L>0$
replaced by $R=D,R<D$, $V_{L}$ replaced by $V_{R}$, $V_{L-1}$ replaced by
$V_{R+1}$, and $V_{L+1}$ replaced by $V_{R-1}$ (note the sign changes).
Fix $V_{L},V_{R}$ each of size $\leq 2$ with $L<R$, such that all levels in
between have size at least 3. Our proof’s cornerstone, which we complete at
the end of Section 3.5, is to show that when $L$ and $R$ are each of one of
the 7 types, provided there are at least 4 nodes between $V_{L}$ and $V_{R}$,
we can get a smaller $G^{\prime}$ which is at least as fit as $G$, by using
surgery and some other “bonus” operations, contradicting our choice of $G$.
After this cornerstone we deal with cases outside the 7 types.
First note that if both $L$ and $R$ are of type $\omega$, Proposition 9
already ensures $r(G)\geq\frac{2}{5}D(G)$. If $V_{L}$ is of type $\lambda$ and
$V_{R}$ is of type $\xi$, we call the surgery type $\lambda$-$\xi$; we call
$\omega$-$\omega$ the _unneeded type_ of surgery since we don’t need to
analyze it. It is essential to increase post-surgery fitness when possible. We
now establish some values $bonus(\\{\lambda,\xi\\})$ (which are symmetric in
$\lambda$ and $\xi$) such that, after a $\lambda$-$\xi$ surgery, we can
increase the fitness by at least $bonus(\\{\lambda,\xi\\})$.
* •
We may take
$bonus(\\{\alpha,\beta\\})=bonus(\\{\alpha,\beta^{\prime}\\})=\frac{1}{10}$
because this surgery results in a degree-2 vertex, which may be shortcutted to
decrease $D$ by 1 and decrease $r$ by 1/2, giving a $\frac{1}{2}-\frac{2}{5}$
increase in fitness.
* •
Similarly we may take $bonus(\\{\alpha,\alpha\\})=\frac{2}{10}$.
* •
We may take
$bonus(\\{\omega,\beta\\})=bonus(\\{\omega,\beta^{\prime}\\})=\frac{13}{30}$
as follows. Consider a $\omega$-$\beta$ (or $\beta^{\prime}$) surgery, so
$V_{R}$ is a singleton $\\{v\\}$. After surgery $s$ has only one neighbour,
$v$, and $v$ has degree at least 3. Then deleting $s$ decreases the diameter
by 1 and decreases $r$ by at least $1-1/6$. Therefore there is a bonus of at
least $1-1/6-2/5=\frac{13}{30}$.
* •
Similarly we can get $bonus(\\{\omega,\alpha\\})=13/30+1/10=8/15$ because
(w.l.o.g. in a $\omega$-$\alpha$ surgery) the $\alpha$ vertex’s right
neighbour has degree at least 3 in the original and post-operation graphs,
using Proposition 12.
* •
Finally we can get $bonus(\\{\omega,\mu\\})=1/12$ as follows. Consider a
(w.l.o.g.) $\mu$-$\omega$ surgery, where $V_{L}=\\{u,v\\}$ and the common
neighbour of $u,v$ in $V_{L-1}$ is $w$. Post-surgery, the distance-2
neighbourhood of $s$ is as shown in Figure 3. Add a new vertex and connect it
to $u,v,w,s$; it is not hard to argue this preserves planarity. Not counting
the increased degree at $w$, we decreased $r$ by
$\frac{1}{2}+\frac{2}{3}-\frac{1}{3}-\frac{3}{4}=\frac{1}{12}$ and preserved
$D$. (Although this adds a vertex, the surgery theorems later on always delete
at least 2 vertices, so overall the total number of vertices always
decreases.)
### 3.4 First Analysis of Surgery
Now we give a lower bound on fitness increase due to surgery. It is convenient
to assume when $V_{L}$ is in cases $\beta^{\prime},\nu^{\prime}$ that each
node in $V_{L}$ has _exactly_ two neighbours in $V_{L-1}$ — call the rest
_ghost neighbours_. Why is this ok? Keep in mind we want to lower bound the
fitness increase from surgery. Due to the “$\geq 2$ neighbours in $V_{L+1}$”
condition in these cases, surgery does not increase the degree of nodes in
$V_{L}$. Further, by the convexity of $d(v)\mapsto\frac{1}{d(v)}$, the actual
$r$ increase including ghost neighbours will be no more than the “virtual $r$
increase” ignoring ghost neighbours made by our analysis.
Here are the details. Let $n_{L}$ denote $|V_{L}|$ and similarly for $n_{R}$.
Let $o_{L}$ denote, for each node in $V_{L}$, the number of “outside”
neighbours such nodes have in $V_{L-1}$; define $o_{R}$ similarly with
$V_{R+1}$ in place of $V_{L-1}$. Thus $n_{L}$ and $o_{L}$ depend only on the
type of $L$, and abusing notation, we write
$n_{\omega}=n_{\alpha}=n_{\beta}=n_{\beta^{\prime}}=1,n_{\mu}=2,n_{\nu}=n_{\nu^{\prime}}=2$
and
$o_{\omega}=0,o_{\alpha}=1,o_{\beta}=o_{\beta^{\prime}}=2,o_{\mu}=1,o_{\nu}=o_{\nu^{\prime}}=2$.
Let $\overline{o}$ denote the number of neighbours each vertex of $V_{L}$ has
in $V_{L}\cup V_{L-1}$, so $\overline{o}=o+(n-1)$. Let $w=R-L-1$ denote the
number of levels in between, and recall that each of these $w$ levels has at
least 3 nodes. Let $x$ denote the number of nodes in the deleted levels, hence
we have $x\geq 3w$.
Before surgery, the sum of the degrees of the nodes in $V_{[L,R]}$ is at most
$n_{L}o_{L}+2(3(n_{L}+x+n_{R})-6)+n_{R}o_{R}$ — the terms count edges from
$V_{L-1}$ to $V_{L}$, in $G_{[L,R]}$, and from $V_{R}$ to $V_{R+1}$
respectively. We thereby use Proposition 8 to lower-bound the initial sum of
the inverse degrees in $V_{[L,R]}$. Post-surgery, we know the degrees of the
nodes in $V_{L}$ are $\overline{o}_{L}+n_{R}$ and similarly for $V_{R}$.
Therefore, if $G^{\prime}$ indicates the result of applying surgery and bonus
operations, we have
$\mathcal{F}(G^{\prime})-\mathcal{F}(G)\geq\eqref{eq:mega}$ defined by
$\frac{(n_{L}+x+n_{R})^{2}}{n_{L}o_{L}+2(3(n_{L}+x+n_{R})-6)+n_{R}o_{R}}-\frac{n_{L}}{\overline{o}_{L}+n_{R}}-\frac{n_{R}}{\overline{o}_{R}+n_{L}}+bonus(L,R)-\frac{2}{5}w.$
($\ast$)
It is easy to verify that
$\eqref{eq:mega}>\frac{x}{6}-4-\frac{2}{5}w\geq\frac{x}{6}-\frac{2x}{15}-4$ so
it is clearly positive for $x\geq 120$. In fact the following precise
statement is true and gives what we want in almost all needed cases; we also
need some $w=0$ cases for later even though they don’t make sense in the
context provided above.
###### Claim 15.
Let $x,w$ be integers with $x\geq 3w,x\geq 2,w\geq 0$. Except for
$(w,x)\in\\{(1,3),(2,6)\\}$, the value ($\ast$ ‣ 3.4) is positive for all
types of $L,R$ (except the unneeded $L=R=\omega$).
###### Proof.
We use a publicly posted Sage worksheet [5] to verify the needed cases. (Note
we’ve chosen things so that a $\lambda$-$\xi$ surgery has the same analysis as
a $\xi$-$\lambda$ surgery, and such that the pairs
$\\{\beta,\beta^{\prime}\\}$ and $\\{\nu,\nu^{\prime}\\}$ are analyzed in the
same way. So our computation involves 14 surgery cases.)∎
More generally, the exact same proof gives the following generalization, which
is needed later.
###### Theorem 16.
Let $V^{\prime}_{R}\subseteq V_{R}$, $L<R$, so that every $s$-$t$ path
intersects $V^{\prime}_{R}$. Let $X$ be the nodes not connected to $s$ or $t$
in $G\backslash V_{L}\backslash V^{\prime}_{R}$ and let $x=|X|$. Let $V_{L}$
be any of the 7 types. Let $V^{\prime}_{R}$ be of one of the 7 types, modified
so that “in $V_{R-1}$” is replaced by “in $X$” and “in $V_{R+1}$” is replaced
by “in $V_{R+1}\backslash X$.” Assume that at least one of $L,R$ is not of
type $\omega$. Let $w=R-L-1$. If we delete $X$ and connect $V_{L}$ to
$V^{\prime}_{R}$ by a biclique, then perform bonus operations, we get a
smaller graph at least as fit as $G$, provided $w\geq 0,x\geq 2$, $x\geq 3w$
and $(w,x)\not\in\\{(1,3),(2,6)\\}$.
### 3.5 Completing the Cornerstone: The Case $w=2,x=6$
If $w=2,x=6$ then $R=L+3$ and $|V_{L+1}|=|V_{L+2}|=3$, since all levels
between $V_{L}$ and $V_{R}$ have size at least 3. We need:
###### Claim 17.
Let $V_{i}$ be a level of size 2, whose vertices are connected by an edge, and
let $j=i+1$ or $j=i-1$, with $|V_{j}|=3$. Then the two vertices of $V_{i}$ do
not have three common neighbours in $V_{j}$.
Figure 4: (a) If we delete $uvb$ the remainder will have at least 3 connected
components. (b) One of these connected components, $H$, does not contain $s$
or $t$; we will delete it.
###### Proof.
The goal of the proof is similar to the result in Proposition 11: assume the
opposite for the sake of contradiction, then show there is some part of the
graph that can be deleted while decreasing $r$ and leaving $D$ unchanged. To
do this, we need to establish some structure.
Let $V_{i}=\\{u,v\\}$. To simplify the notation we handle the case $j=i+1$ but
the proof of the other case is identical. Since $G_{\leq i}$ is planar we can
draw it with the edge $uv$ on the outer face. Likewise, draw $G_{\geq i}$ with
$uv$ on the outer face. Each vertex of $V_{j}$ forms a triangle with $uv$ so
for some labelling $V_{j}=\\{a,b,c\\}$, the drawing of $G_{\geq i}$ has
triangle $uva$ containing vertex $b$ and triangle $uvb$ containing vertex $c$,
as pictured in Figure 4.
We claim by maximality $ab$ is an edge of $G$: indeed, since $u$ has no
neighbours other than $v,a,b,c$ in the drawing of $G_{\geq i}$, if $ab$ is not
present we can add it in a planar way by going next to the path $aub$.
Similarly $bc\in E(G)$.
Now note that $G\backslash\\{u,b,v\\}$ has at least 3 components: one
containing $a$, one containing $c$, and one containing $V_{<i}$. One of the
first two does _not_ contain $t$. Assume the first (the second case is
analogous): denote the component containing $a$ in $G\backslash\\{u,b,v\\}$ by
$H$, so $H\not\ni t$ (see Figure 4). It’s not hard to see any shortest $s$-$t$
path avoids $H$, hence $D(G\backslash H)=D(G)$. Moreover we claim
$r(G\backslash H)<r(G)$, contradicting our choice of $G$. To see this, let $k$
denote $|V(H)|$, note that each vertex in $H$ has degree at most $k+2$, and
that we drop the degrees of $u,b,v$ by at most $k$, thus
$\displaystyle r(G)-r(G\backslash H)$ $\displaystyle\geq
k\frac{1}{k+2}+\sum_{i\in\\{u,b,v\\}}\frac{1}{\deg_{G}(i)}-\frac{1}{\deg_{G\backslash
H}(i)}$
$\displaystyle\geq\frac{k}{k+2}+\sum_{i\in\\{u,b,v\\}}\frac{1}{\deg_{G\backslash
H}(i)+k}-\frac{1}{\deg_{G\backslash H}(i)}$
$\displaystyle\geq\frac{k}{k+2}+3(1/(k+3)-1/3)=\frac{k}{(k+2)(k+3)}>0$
where in the second-to-last inequality we used the fact that
$\deg_{G\backslash H}(i)\geq 3$ and $\frac{1}{\cdot}$ is convex. ∎
This allows us to bound the number of edges between a level $\\{u,v\\}$ with
$uv\in E$ and an adjacent level of size 3: there are at most 5. It’s also
obvious that between a singleton level and an adjacent level of size 3, there
are at most 3 edges. Accordingly, let $z_{L}$ be 3 (resp. 5) when $n_{L}$ is 1
(resp. 2) and similarly define $z_{R}$. In the situation that there are
exactly two levels, each of size-3, between $V_{L}$ and $V_{R}$, we can
replace the quantity $\eqref{eq:mega}$ from the previous section by grouping
the vertices in a different way; specifically we have
$\mathcal{F}(G^{\prime})-\mathcal{F}(G)\geq\eqref{eq:mega2}$ with (✠ ‣ 3.5)
defined by
$\frac{n_{L}^{2}}{n_{L}\overline{o}_{L}+z_{L}}+\frac{x^{2}}{z_{L}+2(3x-6)+z_{R}}+\frac{n_{R}^{2}}{n_{R}\overline{o}_{R}+z_{R}}-\frac{n_{L}}{\overline{o}_{L}+n_{R}}-\frac{n_{R}}{\overline{o}_{R}+n_{L}}+bonus(L,R)-\frac{2}{5}w.$
(✠)
Specifically, the first three terms lower-bound the contribution to $r(G)$ by
vertices in $V_{L}$, in $V_{L+1}\cup V_{L+2}$, and $V_{L+3}=V_{R}$
respectively.
###### Claim 18.
The quantity (✠ ‣ 3.5) is positive when $w=2,x=6$ for all types of $L,R$
(except the unneeded type $L=R=\omega$).
###### Proof.
This calculation is also done via computer at [5].∎
###### Corollary 19.
Let $V_{L}$ and $V_{R}$ be levels of one of the 7 types (except the unneeded
type $L=R=\omega$), with $R=L+3$ and $|V_{L+1}|=|V_{L+2}|=3$. Applying surgery
at $V_{L}$ and $V_{R}$ gives a smaller which is smaller and more fit than $G$.
Together with Theorem 16 this gives the heart of our proof:
###### Theorem 20 (Cornerstone).
Let $V_{L},V_{R}$ be levels of size $\leq 2$, with all levels between them of
size $\geq 3$. If $V_{L}$ and $V_{R}$ are each one of the $7$ types, and there
are at least $4$ nodes between them, this contradicts our choice of $G$.
### 3.6 Sufficiency of the 7 Cases
The structure we want to establish in $G$ is that every level has size at most
3, and that two size-3 levels are never adjacent. We now show how to get from
the cornerstone (Theorem 20) to this structure. We start with a general
observation (which motivated our definition of the 7 cases).
###### Claim 21.
Suppose $V_{i}=\\{u,v\\}$ and $uv\in E$. Suppose $j=i\pm 1$, that $u$ has 1 or
fewer neighbours in $V_{j}$, and that $v$ has at least one neighbour in
$V_{j}$ which is not a neighbour of $u$. Then this violates maximality.
###### Proof.
Take $j=i+1$, the other case is analogous. Embed $G_{\geq i}$ with $uv$ on the
outer face. First if $u$ has no neighbours in $V_{i+1}$ then note $u$ and a
neighbour of $v$ are on the outer face, hence we can add an edge between them
without violating planarity in $G_{\geq i}$ (and hence without violating
planarity in $G$, by Fact 4). Second, suppose $u$ has exactly one neighbour
$x$ in $V_{i+1}$; at least one endpoint emanating from $v$ adjacently to $vu$
is of the form $vy$ with $y\neq u,v,x$; then the path $uvy$ lies on a face and
the edge $uy$ can be added without violating planarity. ∎
In the remainder of the section, we ensure all size-2 levels are connected,
show that $V_{L}$ always is in one of the 7 cases, deal with $V_{R}$’s that
fall outside the 7 cases, and then show the last level $V_{D}$ has size 1.
###### Claim 22.
Any level of size $2$ is connected, except possibly for the last level
$V_{D}$.
###### Proof.
Let $V_{R}$ be minimal, $R<D$, such that $V_{R}=\\{u,v\\}$ is of size 2 and
$uv$ is not an edge. If both $u$ and $v$ are connected to $t$ in $G_{\geq R}$
then using the proof method of Claim 5, $uv$ can be added without violating
planarity, which contradicts maximality. Therefore assume only $u$ has a path
to $t$ in $G_{\geq R}$. It now follows that $v$ is an isolated vertex in
$G_{\geq R}$, or else Proposition 11 is violated because of the articulation
point $v$.
Since $v$ has degree at least 3 (by Proposition 12) and these neighbours are
only in $V_{R-1}$, it follows that $|V_{R-1}|\geq 3$. Let $L$ be maximal with
$L<R$ such that $|V_{L}|\leq 2$. By our choice of $R$, we see $V_{L}$ is
connected if it has size 2. Moreover, each vertex in $V_{L}$ has at least two
neighbours in $V_{L+1}$, using $|V_{L+1}|\geq 3$ and Claim 21. So $V_{L}$ is
of one of the 7 cases.
Now look at $u$. If $u$ has 2 or more neighbours in $V_{R-1}$, we can use
surgery at $V_{L}$ and $u$ which is of type $\beta^{\prime}$ (Theorem 16:
cutting out $R-L-1\geq 1$ levels of size 3, plus $v$). Otherwise, we can use
surgery at $V_{L}$ and the unique neighbour of $u$ in $V_{R-1}$, which is an
articulation vertex of type $\alpha$ (Theorem 16: cutting out $R-L-2\geq 0$
levels of size 3, plus $v$ and at least two nodes from $V_{R-1}$).∎
The following corollary follows from the previous proof and induction:
###### Corollary 23.
Every level $V_{L}$ such that $|V_{L}|\leq 2,|V_{L+1}|\geq 3$ falls in one of
the 7 cases.
###### Proposition 24.
Let $V_{R}$, $R<D$, be such that $|V_{R}|\leq 2$, and either $|V_{R-1}|\geq
4$, or both $|V_{R-2}|,|V_{R-1}|\geq 3$. Then we can perform surgery to
increase the fitness of $G$.
###### Proof.
Let $L<R$ be maximal with $|V_{L}|\leq 2$. Using Corollary 23 (along with
Corollary 19 or Theorem 16) we may assume $V_{R}$ falls outside of the 7
types; using Claim 22 and Claim 21 this means that either $|V_{R}|=1$ and it
has one neighbour in $V_{R-1}$ but $\geq 3$ neighbours in $V_{R+1}$, or
$|V_{R}|=2$ and these vertices each have one neighbour (the same one) in
$V_{R-1}$ and one vertex of $V_{R}$ has $\geq 3$ neighbours in $V_{R+1}$.
In either case, only one vertex in $V_{R-1}$, call it $v$, is adjacent to
$V_{R}$. Since $v$ is an articulation vertex we can do surgery on $V_{L}$ and
$v$ — we apply Theorem 16 to levels $L$ and $R^{\prime}=R-1$, on sets $V_{L}$
and $V^{\prime}_{R^{\prime}}=\\{v\\}$ (here $V^{\prime}_{R^{\prime}}$ is of
type $\alpha$ if $|V_{R}|=1$ or $\beta$ if $|V_{R}|=2$). The set $X$ is
$V_{[L+1,R-1]}\backslash\\{v\\}$, and $w=R^{\prime}-L-1$ so $x=|X|\geq
3w+2,w\geq 0$. This indeed satisfies the conditions of Theorem 16 so we are
done. ∎
###### Proposition 25.
The size of the last level $V_{D}$ is 1.
###### Proof.
Suppose $|V_{D}|>1$ for the sake of contradiction. Let $V_{L}$ be the
rightmost level of size at most 2, which we know is one of the 7 types by
Corollary 23. Let $v\in V_{D}\backslash\\{t\\}$. If $L=D-1$ then it is not
hard to see some face contains $v$ and a vertex from $V_{D-2}$; adding an edge
between this pair does not decrease the diameter, so contradicts edge-
maximality. Otherwise ($L<D-1$) apply surgery to $V_{L}$ and $t$: we cut out 1
or more levels of size at least 3, plus the vertices of
$V_{D}\backslash\\{t\\}$. Thus $x\geq 3w+1,w\geq 1$ and Theorem 16 is
satisfied. ∎
Combining the results just proven, we have the desired structure theorem: $G$
is a graph where the first and last level have size 1, all levels have size at
most 3, every level of size 2 is connected, and no two levels of size 3 are
adjacent.
### 3.7 Computation
We finish by showing that our hypothetical $G$ has $r\geq\frac{2}{5}D$.
###### Theorem 26.
Let $G$ be a graph where the first and last level have size 1, all levels have
size at most 3, every level of size 2 is connected, and no two levels of size
3 are adjacent. Then $r(G)\geq\frac{2}{5}D+\frac{37}{60}$.
###### Proof.
The most important fact about the structure is that, given the sizes of levels
$i-1,i,i+1$, we can determine (or upper bound, depending on how you look at
it) the degrees of the nodes in level $i$, which we use to get a lower bound
on the sum of the inverse degrees for that level.
Given any two adjacent levels, we may upper bound the edges they share by a
biclique. Furthermore, if a level of size 2 and a level of size 3 are
adjacent, by Claim 17 we can upper bound their shared edges as being one edge
short of a biclique. Hence let $\mathcal{S}(i,j)=i\cdot j$ unless
$\\{i,j\\}=\\{2,3\\}$ in which case $\mathcal{S}(i,j)=5$. Thus:
* •
$\sum_{v\in V_{0}}1/d(v)\geq 1/|V_{1}|$
* •
$\sum_{v\in V_{D}}1/d(v)\geq 1/|V_{D-1}|$
* •
For $0<i<D$ there are at most
$\mathcal{E}:=\mathcal{S}(|V_{i-1}|,|V_{i}|)+2\tbinom{|V_{i}|}{2}+\mathcal{S}(|V_{i}|,|V_{i+1}|)$
endpoints incident on $V_{i}$; considering the degrees are integral and using
convexity we see
$\sum_{v\in
V_{i}}1/d(v)\geq\frac{\mathcal{E}\bmod|V_{i}|}{\lceil\mathcal{E}/|V_{i}|\rceil}+\frac{|V_{i}|-(\mathcal{E}\bmod|V_{i}|)}{\lfloor\mathcal{E}/|V_{i}|\rfloor}=:\mathcal{C}.$
Since $\mathcal{C}$ is determined only by $|V_{i-1}|,|V_{i}|,|V_{i+1}|$, we
write it as $\mathcal{C}(|V_{i-1}|,|V_{i}|,|V_{i+1}|)$. We therefore deduce
for any sequence $(n_{0},n_{1},\dotsc,n_{D})$ of level sizes of a graph $G$
that
$r(G)\geq\mathcal{R}(n_{0},n_{1},\dotsc,n_{D}):=1/n_{1}+1/n_{D-1}+\sum_{i=1}^{D-1}\mathcal{C}(|V_{i-1}|,|V_{i}|,|V_{i+1}|).$
Finally, we want to determine which valid sequence minimizes
$\mathcal{R}(n_{0},n_{1},\dotsc,n_{D})-\frac{2}{5}D$. Because $\mathcal{C}$ is
a sum of local contributions, and because each level contributes 1 to the
diameter, we can think of this last step as shortest path problem, as follows.
Define a new digraph with vertex set
$\\{s,(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),t\\},$
where the $(i,j)$-vertices represent a pair of adjacent levels, $s$ represents
the start, and $t$ the end. The intuition: we insert an arc from $(i,j)$ to
$(k,\ell)$ whenever $j=k$, representing three consecutive levels. The cost of
such an edge should account for the $r$-contribution of the level
corresponding to $j$, minus the contribution from lengthening the diameter.
Formally, we add an arc $(i,j)\rightarrow(j,k)$ for all $i,j,k$ (with no
consecutive 3s) having cost $\mathcal{C}(i,j,k)-\frac{2}{5}$; we add an arc
$s\rightarrow(1,i)$ for all $i$ having cost $1/i$; and we add an arc $(i,1)\to
t$ for all $i$ having cost $1/i-\frac{2}{5}.$ Then it’s easy to see that for
any sequence of $n_{i}$’s, $\mathcal{R}-\frac{2}{5}D$ is given by the cost of
the $(D+1)$-edge path
$s\to(n_{0},n_{1})\to(n_{1},n_{2})\to\dotsb(n_{D-1},n_{D})\to t$ in the new
digraph. Executing a shortest-path algorithm such as Bellman-Ford (see the
worksheet at [5]) establishes that the shortest path from $s$ to $t$ has cost
$\frac{37}{60}$, hence $r\geq\mathcal{R}\geq\frac{2}{5}D+\frac{37}{60}$ for
these graphs (and that there are no negative dicycles). ∎
In fact $r\geq\frac{2}{5}D+\frac{37}{60}$ holds for all graphs, is best
possible, and the unique graph with $r=\frac{2}{5}D+\frac{37}{60}$ is
$K_{5}^{-}$. To establish this precise result, small adjustments to our proofs
are necessary, as well as exhaustive searching on all planar graphs with up to
9 vertices.
## 4 Conclusion
The main techniques underlying our diameter bounds for planar graphs were the
surgery operation (which preserves planarity), and the fact that every planar
graph has at most a linear number of edges. One might try the same approach on
the family of graphs excluding any fixed $k$-clique minor, since such graphs
have $O(nk\sqrt{\log k})$ edges (e.g., see [4]). A perpendicular avenue for
future research would be to find a tight relation in connected planar graphs
between the mean distance and the diameter.
## References
* [1] P. Erdős, J. Pach and J. Spencer:On the mean distance between points of a graph, Congr. Numer. 64 (1988), 121 -124.
* [2] S. Fajtlowicz: On conjectures of graffiti II, Congr. Numer. 60 (1987), 189 -197.
* [3] S. Mukwembi: On diameter and inverse degree of a graph, Discrete Mathematics Volume 310, 4, 2010, 940–946.
* [4] A. Thomason: The Extremal Function for Complete Minors, Journal of Combinatorial Theory, Series B Volume 81, 2, 2001, 318–338.
* [5] http://sagenb.org/home/pub/2050
|
arxiv-papers
| 2010-06-12T21:25:05 |
2024-09-04T02:49:10.880139
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Radoslav Fulek, Filip Mori\\'c, David Pritchard",
"submitter": "David Pritchard",
"url": "https://arxiv.org/abs/1006.2493"
}
|
1006.2648
|
# Universal infrared conductivity of graphite
L.A. Falkovsky L.D. Landau Institute for Theoretical Physics, Moscow 117334,
Russia Institute of the High Pressure Physics, Troitsk 142190, Russia
###### Abstract
The conductivity of graphite is analytically evaluated in the range of 0.1-1.5
eV, where the electron relaxation processes can be neglected, and the low
energy excitations at the ”Dirac” points are most essential. The value of
conductivity calculated per one graphite layer is close to the universal
conductivity of graphene. The features of the conductivity are explained in
terms of singularities of the electron dispersion in graphite.
###### pacs:
78.67.-n, 81.05.Bx, 81.05.Uw
Since the pioneering experimental investigations of a single atomic layer of
graphite (graphene) Novo ; ZSA , its properties attract much attention. Among
them, the optical response is of particular interest. Recently the
transmittance of light throw the graphene monolayer has been measured Na ; Li
; Ma . The transmittance
$T=1-\pi\alpha$
was found to be frequency independent in a broad range of photon energy. The
result of the experiments is remarkable because it involves the fine structure
constant $\alpha$. It was discovered that the real part of the optical
conductance of graphene takes the universal value
$G=\frac{e^{2}}{4\hbar}$
which does not depend on any parameters of graphene. This value agrees
perfectly with the calculations GSC ; FV ignoring the Coulomb interactions
between electrons. The agreement shows that the poorly screened Coulomb
interaction does not play any role in graphene for infrared photon frequencies
Mi ; SS .
The intermediate place between 2d graphene and 3d semiconductors belongs to
multilayer graphenes KA and graphite, which have a layered structure with the
interlayer distance $c_{0}=3.35\AA$ much larger than the nearest-neighbor
distance $a_{0}=1.42\AA$ in the layer. In the study of graphite KHC , it was
found that its optical conductivity per one layer is very closed to the
universal conductivity of graphene and has evident peculiarities. The analytic
calculation of the in-plane optical response of graphite done previously Pe
has ignored coupling between layers and no peculiarities have appeared for the
infrared region.
In the present paper, we evaluate analyticaly the conductance of graphite in
the infrared region of the photon frequencies. It is known that the low energy
electron excitations in graphene can be described very well with the
Slonczewski-Weiss-McClure theory SW . The largest parameter of the theory,
$\gamma_{0}=3.1$ eV PP , describes the electron dispersion for in-layer
directions ${\bf k}$. If the photon energy is less than $\gamma_{0}$, we can
use the linear expansion of the in-layer hopping term in the Hamiltonian and
introduce the constant velocity parameter $v=10^{8}$ cm/s. The second
parameter in the rang is the interlayer hopping $\gamma_{1}$ of the order of
0.4 eV which is known from experiments on bilayer graphene KCM ; Ba . The
parameters $\gamma_{3}$ and $\gamma_{4}$ give the corrections of the order of
10% to the in-layer velocity $v$. The electron-hole overlap of the order of
0.02 eV is determined by parameters $\gamma_{2}$ and $\gamma_{5}$ (see Fig.
2). Therefore, for the photon frequencies larger than 0.1 eV, we can neglect
the terms with $\gamma_{2}$ and $\gamma_{5}$. Calculating such the integral
property as conductivity in the region of the infrared frequencies between 0.1
eV and 1.5 eV, we can, first, neglect the small parameters of the theory and,
second, use the $k$-expansion of the in-layer hopping term. Our results have
the evident analytic form.
In this approximation, the effective Hamiltonian writes near the K-G-H lines
of the Brillouin zone in the simple form
$H(\mathbf{k})=\left(\begin{array}[]{cccc}0&k_{+}&\gamma(z)&0\\\
k_{-}&0&0&0\\\ \gamma(z)&0&0&k_{-}\\\ 0&0&k_{+}&0\end{array}\right),$ (1)
determined only by two constants. One is $v=10^{8}$cm/s included in the
definition of the in-plane momentum components, $k_{\pm}=v(\mp ik_{x}-k_{y})$,
and another is the inter-layer interaction $\gamma_{1}$ involved in the
function $\gamma(z)=2\gamma_{1}\cos{z}$. The momentum component $z=k_{z}c_{0}$
is limited by the Brillouin half-zone, $0<z<\pi/2$ in relative units.
The corresponding eigenenergies are
$\varepsilon_{1,2}=\frac{\gamma(z)}{2}\pm\sqrt{\frac{1}{4}\gamma^{2}(z)+k^{2}},$
$\varepsilon_{3,4}=-\frac{\gamma(z)}{2}\pm\sqrt{\frac{1}{4}\gamma^{2}(z)+k^{2}}.$
On the K-G-H lines, $k=0$, these equations determine two bands
$\varepsilon_{1,4}=\pm\gamma(z)$ and two degenerate (electron and hole) bands
with the energy $\varepsilon_{2,3}=0$. We have to emphasize that this
degeneracy results from $C_{3v}$ symmetry on the K-G-H line.
Figure 1: The dispersion of the low energy electron bands in graphite.
In order to calculate the conductivity, we use the general expression
$\displaystyle\sigma^{ij}(\omega)=\frac{2ie^{2}}{(2\pi)^{3}}\int
d^{3}k\sum_{k,n\geq
m}\left\\{-\frac{df}{d\varepsilon_{n}}\frac{v_{n}^{i}v_{n}^{j}}{\omega+i\nu}\right.$
(2)
$\displaystyle\left.+2\omega\frac{v_{nm}^{i}v_{mn}^{j}\\{f[\varepsilon_{n}(\mathbf{k})]-f[\varepsilon_{m}(\mathbf{k})]\\}}{[\varepsilon_{m}(\mathbf{k})-\varepsilon_{n}(\mathbf{k})]\\{(\omega+i\nu)^{2}-[\varepsilon_{n}(\mathbf{k})-\varepsilon_{m}(\mathbf{k})]^{2}\\}}\right\\}\,,$
valid in the collisionless limit $\omega\gg\nu$, where $\nu$ is the collision
rate of the carriers,
$f(\varepsilon)=[\exp(\frac{\varepsilon-\mu}{T})-1]^{-1}$ is the Fermi-Dirac
distribution function, and the integral is over the Brillouin zone.
Here, the first term is the Drude-Boltzmann conductivity negligible for
frequencies larger than the electron-hole overlap. The second term represents
the optical interband transitions of electrons from the valence 2,4 to
conductive 1,3 bands. The real part of the interband contributions into
conductivity arises from the bypass around the pole at
$\varepsilon_{n}(\mathbf{k})-\varepsilon_{m}(\mathbf{k})=\pm\omega$. The
imaginary part is given by the principal value of the integral.
The velocity operator
${\bf v}=\frac{\partial H({\bf k})}{\partial{\bf k}}$
near the K-G-H lines is determined by the Hamiltonian (1). The corresponding
matrix elements should be calculated in the representation, where the
Hamiltonian has a diagonal form. The operator transforming the Hamiltonian to
this form can be written as follows
${U}=\left(\begin{array}[]{cccc}\varepsilon_{1}/N_{1}&\varepsilon_{2}/N_{2}&-\varepsilon_{3}/N_{3}&-\varepsilon_{4}/N_{4}\\\
k_{-}/N_{1}&k_{-}/N_{2}&-k_{-}/N_{3}&-k_{-}/N_{4}\\\
\varepsilon_{1}/N_{1}&\varepsilon_{2}/N_{1}&\varepsilon_{3}/N_{3}&\varepsilon_{4}/N_{4}\\\
k_{+}/N_{1}&k_{+}/N_{2}&k_{+}/N_{3}&k_{+}/N_{4}\end{array}\right)\,,$
where $N_{n}^{2}=2(\varepsilon_{n}^{2}+k^{2})$ . In this representation, the
velocity operator
$U^{-1}{\bf v}U$
has the matrix elements
$\begin{array}[]{c}\mathbf{v}_{nn}=\partial\varepsilon_{n}/\partial{\bf
k}\,,\\\ \mathbf{v}_{23}=2i(\varepsilon_{3}-\varepsilon_{2})(-k_{x}{\bf
e}_{y}+k_{y}{\bf e}_{x})]/N_{2}N_{3}\,,\\\
\mathbf{v}_{12}=2(\varepsilon_{1}+\varepsilon_{2})(k_{x}{\bf e}_{x}+k_{y}{\bf
e}_{y})]/N_{1}N_{2}\,,\\\
\mathbf{v}_{14}=2i(\varepsilon_{4}-\varepsilon_{1})(-k_{x}{\bf
e}_{y}+k_{y}{\bf e}_{x})]/N_{1}N_{4}\,,\\\ \end{array}$
where ${\bf e}_{i}$ are the unit vectors directed along the coordinate axes.
For the real part of conductivity, the integration in Eq. (2) is easily taken
at zero temperatures $T=0$ in cylindrical coordinates $(k_{z},k,\phi)$ over
the angle $\phi$ and over $k$ with the help of the $\delta$-function,
$(\omega-x+i\nu)^{-1}\rightarrow-i\pi\delta(\omega-x)$. One obtains for
contributions of the transitions between the corresponding valence and
conduction bands into the diagonal components of conductivity (off-diagonal
ones equal zero) the following integrals over $z=k_{z}/c_{0}$:
$\text{Re}~{}\sigma_{23}=\frac{e^{2}}{4\pi\hbar
c_{0}}\int_{0}^{\pi/2}dz\frac{2\gamma(z)+\omega}{\gamma(z)+\omega}\,,$
$\displaystyle\text{Re}~{}\sigma_{21}=\frac{e^{2}}{4\pi\hbar
c_{0}}\int_{0}^{\pi/2}dz\frac{\gamma^{2}(z)}{\omega^{2}}\theta[\omega-\gamma(z)]\,,$
(3)
$\text{Re}~{}\sigma_{41}=\frac{e^{2}}{4\pi\hbar
c_{0}}\int_{0}^{\pi/2}dz\frac{2\gamma(z)-\omega}{\gamma(z)-\omega}\theta[\omega-2\gamma(z)]\,,$
$\sigma_{43}=\sigma_{21}\,,$
where $\gamma(z)=2\gamma_{1}\cos{z}$ and $\theta(x)$ is the step function.
It is evident from Eqs. (3) (see also Fig. 2) that the conductivity
$\sigma_{23}$ tends to $e^{2}/4\hbar c_{0}$ at the low frequencies $\omega\ll
2\gamma_{1}$, whereas other contributions go to zero in the limit of low
frequencies. At larger frequencies $\omega\gg 2\gamma_{1}$, the total
conductivity (the sum of $\sigma_{23}$ and $\sigma_{41}$) tends again to
$e^{2}/4\hbar c_{0}$. Therefore, $\sigma_{0}=e^{2}/4\hbar c_{0}$ can be
considered as the universal conductivity of graphite, where $e^{2}/4\hbar$ is
the conductivity of monolayer graphene and the factor $1/c_{0}$ is the number
of the layers per the length unit in the z-direction of graphite.
Figure 2: The real part of the graphite conductivity per layer (in units of
$e^{2}/4\hbar$) versus the frequency (in units of $2\gamma_{1}=0.84$ eV); the
experimental data KHC are shown in the solid line, results of the present
theory in the dashed line. The insert shows the contributions of various
electron transitions.
Integrating in Eqs. (3), we get finally
$\displaystyle\text{Re}~{}\frac{\sigma_{23}}{\sigma_{0}}=1-\frac{2t}{\pi\sqrt{t^{2}-1}}\arctan{\sqrt{\frac{t-1}{t+1}}},\,t>1\,,$
(4)
$\displaystyle\text{Re}~{}\frac{\sigma_{23}}{\sigma_{0}}=1-\frac{t}{\pi\sqrt{1-t^{2}}}\ln{\frac{\sqrt{1+t}+\sqrt{1-t}}{\sqrt{1+t}-\sqrt{1-t}}},\,t<1\,,$
$\displaystyle\text{Re}~{}\frac{\sigma_{21}}{\sigma_{0}}=\frac{1}{4t^{2}}\left\\{\begin{array}[]{ll}1,&t>1\,,\\\
1-\frac{2}{\pi}(\arccos{t}+t\sqrt{1-t^{2}}),&t<1\,.\end{array}\right.$ (7)
$\displaystyle\text{Re}~{}\frac{\sigma_{41}}{\sigma_{0}}=1-\frac{2t}{\pi\sqrt{t^{2}-1}}\arctan{\sqrt{\frac{t+1}{t-1}}},\,t>2,$
$\displaystyle\text{Re}~{}\frac{\sigma_{41}}{\sigma_{0}}=1-\frac{2z_{1}}{\pi}-\frac{2t}{\pi\sqrt{t^{2}-1}}\left[\arctan{\sqrt{\frac{t+1}{t-1}}}\right.$
$\displaystyle\left.-\arctan{\left(\sqrt{\frac{t+1}{t-1}}\tan\frac{z_{1}}{2}\right)}\right],\,1<t<2\,,$
$\displaystyle\text{Re}~{}\frac{\sigma_{41}}{\sigma_{0}}=1-\frac{2z_{1}}{\pi}+\frac{t}{\pi\sqrt{1-t^{2}}}\left[\ln{\frac{\sqrt{1+t}+\sqrt{1-t}}{\sqrt{1+t}-\sqrt{1-t}}}\right.$
$\displaystyle\left.+\ln{\frac{\sqrt{1+t}\tan\frac{z_{1}}{2}-\sqrt{1-t}}{\sqrt{1+t}\tan\frac{z_{1}}{2}+\sqrt{1-t}}}\right],\,t<1\,,$
where $t=\omega/2\gamma_{1}$ and $z_{1}=\arccos(t/2)$.
The peculiarity as a kink can be seen in Fig. 2. The expression (7) shows that
this kink is located at $\omega=2\gamma_{1}$. Taking into account the kink
position $\omega=0.84$ eV determined experimentally, the value of
$\gamma_{1}=0.42$ eV is found in excellent agreement with experiments on
bilayer graphene.
The contributions of the electron interband transitions into the imaginary
part of conductivity can be integrated over $k$ at the zero temperature. The
results are obtained in the form of integrals over $k_{z}$
$\displaystyle\text{Im}~{}\frac{\sigma_{23}}{\sigma_{0}}=\frac{2}{\pi^{2}}\int_{0}^{\pi/2}dz\frac{\omega\gamma(z)}{\gamma^{2}(z)-\omega^{2}}\ln{[\gamma(z)/\omega]}\,,$
$\displaystyle\text{Im}~{}\frac{\sigma_{21}}{\sigma_{0}}=\frac{1}{\pi^{2}}\int_{0}^{\pi/2}dz\frac{\gamma(z)}{\omega}\left(2+\frac{\gamma(z)}{\omega}\ln{\frac{|\gamma(z)-\omega|}{\gamma(z)+\omega}}\right)\,,$
$\displaystyle\text{Im}~{}\frac{\sigma_{41}}{\sigma_{0}}=\frac{1}{\pi^{2}}\int_{0}^{\pi/2}dz\left(\frac{2\gamma(z)-\omega}{\gamma(z)-\omega}\ln{|2-\omega/\gamma(z)|}\right.$
$\displaystyle\left.-\frac{2\gamma(z)+\omega}{\gamma(z)+\omega}\ln{(2+\omega/\gamma(z))}\right)\,$
and shown in Fig. 3. Here, the peculiarity looks like a threshold at
$\omega=2\gamma_{1}$ and it is more clearly marked in comparison with the kink
in the real conductivity. Both peculiarities result due to the electron
transitions between the bands $2\rightarrow 1$ and $4\rightarrow 3$. We should
emphasize that the peculiarities become broader with the temperatures and the
collision processes included.
Figure 3: The imaginary part of the graphite conductivity per layer (in units
of $e^{2}/4\hbar$) versus the frequency (in units of $2\gamma_{1}=0.84$ eV).
So far the in-layer conductivity was considered. The estimate of the inter-
layer conductivity can be also done. Since the conductivity is determined by
the ratio of the corresponding velocities squared, we have to write
$v_{z}=\frac{\partial\varepsilon_{3}}{\partial
k_{z}}\sim\gamma_{1}c_{0}\sin(k_{z}c_{0})\,.$
Then, integrating over $k_{z}$, we get
$\sigma_{z}/\sigma_{0}\sim(\gamma_{1}c_{0}/\hbar v)^{2}/2\sim 0.05\,.$
In conclusion, our calculations reveals that the optical conductance of
graphite can be estimated for frequencies between 0.1 and 1.5 eV multiplying
the graphene conductivity $e^{2}/4\hbar$ by the number of the layers $1/c_{0}$
per the length unit. The Drude-Boltzmann contribution is essential at lower
frequencies, whereas others interband transitions, e.g. at the M point of the
Brillouin zone contribute into the conductivity at higher frequencies. The
similar estimate are applicable for other graphite materials such as
nanoribbons. The kink in the real part of conductivity and the threshold in
the imaginary part appear at the frequency $\omega=2\gamma_{1}$ determined by
the interlayer coupling. The sharpness of the features are smeared with the
relaxation processes and temperatures included.
This work was supported by the Russian Foundation for Basic Research (grant
No. 10-02-00193-a) and by the SCOPES grant IZ73Z0$\\_$128026 of the Swiss NSF.
The author is grateful to the Max Planck Institute for the Physics of Complex
Systems for hospitality in Dresden.
## References
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* (4) Z.Q. Li, E.A. Henriksen, Z. Jiang, Z. Hao, M.C. Martin, P. Kim, H.L. Stormer, D.N. Basov, Nature Physics 4, 532 (2008).
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|
arxiv-papers
| 2010-06-14T09:35:19 |
2024-09-04T02:49:10.891157
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L.A. Falkovsky",
"submitter": "L. A. Falkovsky",
"url": "https://arxiv.org/abs/1006.2648"
}
|
1006.2937
|
8cm
1]Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Serrano 121, 28006
Madrid, Spain 2]School of Mathematics, University of Bristol, University Walk,
Bristol BS8 1TW, UK 3]Institut de Ciències del Mar, CSIC, Passeig Marítim de
la Barceloneta 37-49, 08003 Barcelona, Spain 4]Instituto de Física
Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), Campus Universitat de
les Illes Balears,
07122 Palma de Mallorca, Spain ] ]Nonlin. Processes Geophys. 17, 283 -285
(2010)
www.nonlin-processes-geophys.net/17/283/2010/
doi:10.5194/npg-17-283-2010
Published under a Creative Commons Attribution 3.0 License
A. M. Mancho
(a.m.mancho@icmat.es)
# Preface
“Nonlinear processes in oceanic and atmospheric flows”
A. M. Mancho S. Wiggins A. Turiel E. Hernández-García C. López E. García-
Ladona [ [ [ [ [ [
(31 May 2010)
###### Abstract
Nonlinear phenomena are essential ingredients in many oceanic and atmospheric
processes, and successful understanding of them benefits from
multidisciplinary collaboration between oceanographers, meteorologists,
physicists and mathematicians. The present Special Issue on “Nonlinear
Processes in Oceanic and Atmospheric Flows” contains selected contributions
from attendants to the workshop which, in the above spirit, was held in Castro
Urdiales, Spain, in July 2008. Here we summarize the Special Issue
contributions, which include papers on the characterization of ocean transport
in the Lagrangian and in the Eulerian frameworks, generation and variability
of jets and waves, interactions of fluid flow with plankton dynamics or heavy
drops, scaling in meteorological fields, and statistical properties of El Niño
Southern Oscillation.
##
In recent years atmospheric and oceanic data sets arising from new
observational and computational capabilities have become widely available.
These data sets, and the variety of geophysical nonlinear phenomena that they
reveal, are giving rise to new challenges and opportunities that are
benefiting greatly from a multidisciplinary approach. Methods from diverse
areas of mathematics such as dynamical systems theory and statistics have been
combined with sophisticated computational methods and have been brought to
bear on a variety of data sets taken in very diverse physical settings. These
new approaches have been developed in collaborations between mathematicians,
physicists, oceanographers, and meteorologists.
On 2–4 July 2008 such a group gathered for a workshop entitled “Nonlinear
Processes in Oceanic and Atmospheric Flows” held at Castro Urdiales,
Cantabria, Spain, with the generous support of the United States Office of
Naval Research (ONR Global), Consejo Superior de Investigaciones Científicas
(CSIC), Ministerio de Educación y Ciencia (MEC), Centro Internacional de
Encuentros Matemáticos (CIEM), Consolider i-MATH and SIMUMAT. The fourteen
papers in this Special Issue describe the breadth of ideas and results
discussed at this workshop and illustrate the exciting opportunities for
multidisciplinary collaborations in the oceanic and atmospheric sciences.
The paper of Boucharelet al. (2009) introduces novel ideas from statistics,
taken from the field of financial mathematics, to perform a more detailed
diagnosis of the properties of the El Nino Southern Oscillation (ENSO). The
authors analyze data from the Zebiak-Cane model, models used by the
Intergovernmental Panel for Climate Change (IPCC), as well as in situ data.
Their analysis raises a number of provocative points and conclusions that
should be considered in the context of the fidelity of climate models in
general.
Rossi et al. (2009) analyze the interaction between eddy induced mixing and
phytoplankton distributions on small scale (1–100 km) processes using
satellite data. Their analysis leads to the surprising conclusion that strong
mixing in nutrient-rich waters along Eastern Boundary Upwelling Systems (e.g.
the Benguela and Canary currents in the Atlantic Ocean, and the Humboldt and
California currents in the Pacific) appear to reduce, rather than stimulate,
growth of phytoplankton.
The paper by Sánchez-Garrido and Vlasenko (2009) addresses the behaviour of
internal solitary waves in a rotating and laterally confined domain, with
emphasis in the non-linear regime. According to the classical weakly non-
linear theory, energy is damped through radiation of secondary Poincaré
gravity waves due to rotational dispersion. However, under strong non-
linearity conditions, the energy damping is partially suppressed due to non-
linear wave-wave interactions. This leads to a regime where internal solitary
waves evolve into a slowly decaying packet of Kelvin waves that may propagate
for a long time. An understanding of phenomena of this type is fundamental for
obtaining a deeper insight into energy pathways in the oceans.
Using a variety of meteorological variables (derived either directly from
numerical simulations or from re-analysis which combine observed values with
numerical models assimilating them), Stolle et al. (2009) demonstrate that the
scaling properties of these variables can be explained in terms of underlying
multifractal cascades, beyond the usual, single-exponent characterization.
Their findings can be applied to improve the parametrization of numerical
models, as well as to validate the correctness of the implementation of non-
linear effects.
Using a simple idealized plankton model, McKiver at al. (2009) analyze the
importance of horizontal advection on phytoplankton biomass. They use a single
species model with multiple steady states depending on the values of the
carrying capacity, and show that small changes in the ratio of biological to
hydrodynamic time scales can greatly modify plankton production. As a
consequence, they argue that this effect may be a possible mechanism for
explaining plankton blooms or regime shifts in some oceanic regions.
Dellnitz et al. (2009) consider the fundamental issue of detecting regions in
the ocean that are coherent over an extended period of time. These structures,
such as gyres, are important with respect to the movement of heat around the
planet, distribution of nutrients, etc. The authors use a realistic numerical
model to study a 3-D coherent structure in the Southern Ocean using a
methodology based on transfer operators. They show that transfer operators are
a useful tool for identifying circulating pathways across these structures.
Pierini and Dijkstra (2009) review the proposed ways to understand the bimodal
characteristics of the low-frequency variability of the Kuroshio System: a
state with the presence of a zonally elongated energetic meandering jet
alternating, on decadal time scales, with a state of a weaker jet with reduced
zonal penetration. The origin of such bimodality can be either in the ocean
response to changes of wind stress fields, and then due basically to the
atmospheric forcing of the ocean, or identified as intrinsic ocean
variability. As expected both aspects should be taken into account, but what
is remarkable is that the non-linear behavior of the bimodal system is quite
well reproduced and understood both quantitatively and qualitatively just by
considering the internal variability caused from homoclinic transitions
involving multiple equilibrium states of an ocean reduced gravity model under
steady wind forcing.
Zahnow and Feudel (2009) consider the effects of collision, coagulation and
fragmentation processes on the size distribution of heavy drops moving in a
turbulent fluid. The problem is relevant, for example, to the growth of cloud
droplets. The particle-based approach goes beyond simple transport models of
inertial particles, without the complications of a fully hydrodynamic
simulation. Scaling laws of mean sizes and distributions with respect to the
different flow and particle parameters are obtained by a combination of
numerical and theoretical arguments.
Branicki and Wiggins (2010) give a critical analysis of the use of hyperbolic
trajectories, their stable and unstable manifolds, and finite time Lyapunov
exponents for revealing flow barriers and organized structures in
aperiodically time-dependent flows that exist only for a finite time. This is
a rapidly developing area due to the explosion in the availability of
observational and computational data sets for geophysical flows. This paper
takes a different point of view and describes a series of specific examples
that highlight different phenomena and their interpretation, as well as
problems and pathologies that can arise. Consequently, this paper provides
“benchmarks” for the necessary further development of the theory and for the
application of these methods to complex geophysical flows.
Koszalka et al. (2010) explore how vertical transport within wind-forced
eddies is affected by stratification. They show that the wind energy injected
at the surface is transferred to depth through two stratification-dependent
mechanisms: vortex Rossby waves and near-inertial internal oscillations. In
view of their results on the role of wind-forced mesoscale vortices in the
transmission of wind energy into the ocean and vertical transport, the authors
stress the need to resolve the vertical transport and mixing by mesoscale
eddies in models designed to study oceanic circulation under different
climatological conditions.
Marié (2010) studies mechanisms for the generation of zonal jets by
$\beta$-plane turbulence. The work begins with a simple situation – a study of
linear perturbations of Rossby waves by zonal flow in an infinite
$\beta$-plane. He then considers a more realistic situation consisting of a
reduced-gravity model in a quasi-geostrophic setting and shows that
essentially the same results hold. This work provides insight into a complex
phenomenon resulting from a turbulence-mediated, subtle interaction, between
two very different scales.
The paper by Mendoza et al. (2010) applies a combination of Lagrangian tools,
some of them new and others well established, for studying transport in
velocity data sets obtained from altimetry over the Kuroshio current region.
The study shows how distinguished hyperbolic trajectories and their stable and
unstable manifolds can be computed in realistic data sets. It also addresses
how to achieve an accurate analysis of transport from the stable and unstable
manifolds. The method successfully characterizes the turnstile mechanism
across this area and this mechanism is shown to persist over the spring months
of year 2003.
Branicki and Malek-Madani (2010) consider transport in a realistic time-
dependent-velocity data set obtained from a shallow water model of the
Chesapeake Bay. In this context they assess the limit of validity of 2-D
Lagrangian tools for analyzing estuarine flows. The 2-D Lagrangian analysis of
the surface flow captures the spatio-temporal variability of the freshwater
outflow events. The computation of finite time Lyapunov exponents reveals a
network of ridges, but these are often too short for a meaningful transport
analysis, while computation of stable and unstable manifolds of relevant
hyperbolic trajectories has the comparable challenge of first computing the
hyperbolic trajectories on a sufficiently long time interval. It is
anticipated that a symbiotic combination of these Lagrangian diagnostics might
overcome these difficulties. Their work points out that further development of
3-D Lagrangian techniques is still required for reliable transport analysis of
complex coastal flows.
Hydrodynamic forcing is known to play an important role in plankton dynamics.
Pérez-Muñuzuri and Huhn (2010) consider the influence of the spatial and
temporal scales of the flow on the spatial extension of a plankton bloom using
a reaction-diffusion-advection equation in which the reaction part models a
Nutrient-Phytoplankton-Zooplankton biological dynamics. Their analysis shows
that the bloom size is larger for certain length and time scales of the flow.
This is related to the fact that the balance of two processes, trapping fluid
inside eddies on the one hand, and mixing and diluting on the other hand, is
optimal for bloom growth at these particular length and time scales.
###### Acknowledgements.
The workshop held at Castro Urdiales was possible thanks to the commitment of
its Organizing Committee: C. López, A. M. Mancho, A. Turiel, E. García-Ladona,
E. Hernández-García, J. A. Jiménez-Madrid. Also thanks to Ismael Hernández-
Carrasco and Oriol Pont for their assistance during the event. The warm
hospitality and support of the Cultural Centre “La Residencia” is also
acknowledged. The organization of the workshop was possible thanks to support
from grants: ONR Global (N00014-08-1-1035), CSIC Oceantech (PIF-0059-2006),
Consolider i-MATH C3-0103, CIEM, SIMUMAT S-0505-ESP-0158, MEC FIS2007-30844-E,
CSIC MP-38-AR.
## References
* Boucharelet al. (2009) Boucharel, J., Dewitte, B., Garel, B., and du Penhoat, Y.: ENSO’s non-stationary and non-Gaussian character: the role of climate shifts, Nonlin. Processes Geophys., 16, 453–473, doi:10.5194/npg-16-453-2009, 2009.
* Rossi et al. (2009) Rossi, V., López, C., Hernández-García, E., Sudre, J., Garçon, V., and Morel, Y.: Surface mixing and biological activity in the four Eastern Boundary Upwelling Systems, Nonlin. Processes Geophys., 16, 557–568, doi:10.5194/npg-16-557-2009, 2009.
* Sánchez-Garrido and Vlasenko (2009) Sánchez-Garrido, J. C. and Vlasenko, V.: Long-term evolution of strongly nonlinear internal solitary waves in a rotating channel, Nonlin. Processes Geophys., 16, 587–598, doi:10.5194/npg-16-587-2009, 2009.
* Stolle et al. (2009) Stolle, J., Lovejoy, S., and Schertzer, D.: The stochastic multiplicative cascade structure of deterministic numerical models of the atmosphere, Nonlin. Processes Geophys., 16, 607–621, doi:10.5194/npg-16-607-2009, 2009.
* McKiver at al. (2009) McKiver, W., Neufeld, Z., and Scheuring, I.: Plankton bloom controlled by horizontal stirring, Nonlin. Processes Geophys., 16, 623–630, doi:10.5194/npg-16-623-2009, 2009.
* Dellnitz et al. (2009) Dellnitz, M., Froyland, G., Horenkamp, C., Padberg-Gehle, K., and Sen Gupta, A.: Seasonal variability of the subpolar gyres in the Southern Ocean: a numerical investigation based on transfer operators, Nonlin. Processes Geophys., 16, 655–663, doi:10.5194/npg-16-655-2009, 2009.
* Pierini and Dijkstra (2009) Pierini, S. and Dijkstra, H. A.: Low-frequency variability of the Kuroshio Extension, Nonlin. Processes Geophys., 16, 665–675, doi:10.5194/npg-16-665-2009, 2009.
* Zahnow and Feudel (2009) Zahnow, J. C. and Feudel, U.: What determines size distributions of heavy drops in a synthetic turbulent flow?, Nonlin. Processes Geophys., 16, 677–690, doi:10.5194/npg-16-677-2009, 2009.
* Branicki and Wiggins (2010) Branicki, M. and Wiggins, S.: Finite-time Lagrangian transport analysis: stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents, Nonlin. Processes Geophys., 17, 1–36, doi:10.5194/npg-17-1-2010, 2010.
* Koszalka et al. (2010) Koszalka, I., Ceballos, L., and Bracco, A.: Vertical mixing and coherent anticyclones in the ocean: the role of stratification, Nonlin. Processes Geophys., 17, 37–47, doi:10.5194/npg-17-37-2010, 2010.
* Marié (2010) Marié, L.: A study of the phase instability of quasi-geostrophic Rossby waves on the infinite $\beta$-plane to zonal flow perturbations, Nonlin. Processes Geophys., 17, 49–63, doi:10.5194/npg-17-49-2010, 2010.
* Mendoza et al. (2010) Mendoza, C., Mancho, A. M., and Rio, M.-H.: The turnstile mechanism across the Kuroshio current: analysis of dynamics in altimeter velocity fields, Nonlin. Processes Geophys., 17, 103–111, doi:10.5194/npg-17-103-2010, 2010.
* Branicki and Malek-Madani (2010) Branicki, M. and Malek-Madani, R.: Lagrangian structure of flows in the Chesapeake Bay: challenges and perspectives on the analysis of estuarine flows, Nonlin. Processes Geophys., 17, 149–168, doi:10.5194/npg-17-149-2010, 2010\.
* Pérez-Muñuzuri and Huhn (2010) Pérez-Muñuzuri, V. and Huhn, F.: The role of mesoscale eddies time and length scales on phytoplankton production, Nonlin. Processes Geophys., 17, 177–186, doi:10.5194/npg-17-177-2010, 2010.
|
arxiv-papers
| 2010-06-15T09:44:41 |
2024-09-04T02:49:10.903574
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. M. Mancho, S. Wiggins, A. Turiel, E. Hernandez-Garcia, C. Lopez,\n and E. Garcia-Ladona",
"submitter": "Emilio Hernandez-Garcia",
"url": "https://arxiv.org/abs/1006.2937"
}
|
1006.3048
|
# Large-time Behavior of Solutions to the Inflow Problem of Full Compressible
Navier-Stokes Equations
Xiaohong Qin Yi Wang Department of Mathematics, Nanjing University of Science
and Technology, Nanjing 210094, China. X. Qin is supported in part by NSFC
(grant No. 10901083). E-mail: xqin@amss.ac.cn.Institute of Applied
Mathematics, AMSS, CAS, Beijing 100190, China. Y. Wang is supported by NSFC
(grant No. 10801128). E-mail: wangyi@amss.ac.cn.
###### Abstract
Large-time behavior of solutions to the inflow problem of full compressible
Navier-Stokes equations is investigated on the half line
$\mathbf{R}_{+}=(0,+\infty)$. The wave structure which contains four waves:
the transonic(or degenerate) boundary layer solution, 1-rarefaction wave,
viscous 2-contact wave and 3-rarefaction wave to the inflow problem is
described and the asymptotic stability of the superposition of the above four
wave patterns to the inflow problem of full compressible Navier-Stokes
equations is proven under some smallness conditions. The proof is given by the
elementary energy analysis based on the underlying wave structure. The main
points in the proof are the degeneracies of the transonic boundary layer
solution and the wave interactions in the superposition wave.
Key words: compressible Navier-Stokes equations, inflow problem, boundary
layer solution, rarefaction wave, viscous contact wave
AMS SC2000: 35L60, 35L65
## 1 Introduction
In this paper, we consider an initial-boundary-value problem for full
compressible Navier-Stokes equations in _Eulerian_ coordinates on the half
line $\mathbf{R}_{+}=(0,+\infty)$
$\displaystyle\begin{cases}\rho_{t}+(\rho u)_{x}=0,\cr(\rho u)_{t}+\big{(}\rho
u^{2}+p\big{)}_{x}=(\mu
u_{x})_{x},&x>0,~{}t>0,\cr\left[\rho\left(e+\frac{1}{2}u^{2}\right)\right]_{t}+\left[\rho
u\left(e+\frac{1}{2}u^{2}\right)+pu\right]_{x}=(\kappa\theta_{x}+\mu
uu_{x})_{x}\end{cases}$ (1.1)
where $\rho(t,x)>0$, $u(t,x)$, $\theta(t,x)>0$, $p(t,x)>0$ and $e(t,x)>0$
represent the mass density, the velocity, the absolute temperature, the
pressure, and the specific internal energy of the gas respectively and $\mu>0$
is the coefficient of viscosity, $\kappa>0$ is the coefficient of heat
conduction. Here we assume that both $\mu$ and $\kappa$ are positive
constants. Let $v=\frac{1}{\rho}(>0)$ and $s$ denote the specific volume and
the entropy of the gas, respectively. Then by the second law of
thermodynamics, we have for the ideal polytropic gas
$\displaystyle
p=Rv^{-1}\theta=Av^{-\gamma}\exp\left(\frac{\gamma-1}{R}s\right),~{}~{}~{}e(v,\theta)=\frac{R}{\gamma-1}\theta,~{}~{}$
(1.2)
where $\gamma>1$ denotes the adiabatic exponent of gas, and $A$ and $R$ are
positive constants.
We consider the initial-boundary-value problem (1.1) with the initial values
$(\rho,u,\theta)(0,x)=(\rho_{0},u_{0},\theta_{0})(x)\rightarrow(\rho_{+},u_{+},\theta_{+})~{}~{}\text{as}~{}~{}x\rightarrow+\infty,~{}~{}\inf\limits_{x\in\mathbf{R}_{+}}(\rho_{0},\theta_{0})(x)>0$
(1.3)
where $\rho_{+}>0$, $u_{+}$ and $\theta_{+}>0$ are given constants.
As pointed out by [15], the boundary conditions to the half space problem
(1.1) can be proposed as one of the following three cases:
Case I. outflow problem (negative velocity on the boundary):
$u(t,x)|_{x=0}=u_{-}<0,~{}~{}\theta(t,x)|_{x=0}=\theta_{-}.$ $None$
Case II. impermeable wall problem (zero velocity on the boundary):
$u(t,x)|_{x=0}=0,~{}~{}\theta(t,x)|_{x=0}=\theta_{-}.$ $None$
Case III. inflow problem (positive velocity on the boundary):
$u(t,x)|_{x=0}=u_{-}>0,~{}~{}\rho(t,x)|_{x=0}=\rho_{-},~{}~{}\theta(t,x)|_{x=0}=\theta_{-}.$
$None$
Here all the $\rho_{-}>0$, $u_{-}$ and $\theta_{-}>0$ in (1.4) are prescribed
constants and of course we assume that the initial values (1.3) and the
boundary conditions (1.4) satisfy the compatibility condition at the origin.
Notice that in Cases I and II, the density $\rho_{-}$ on the boundary
$\\{x=0\\}$ could not be given, but in Case III, $\rho_{-}$ must be imposed
due to the well-posedness theory of the hyperbolic equation
$\eqref{(1.1)}_{1}$.
In the present paper, we are concerned with the large-time behavior of the
solutions to the inflow problem (Case III) of the full compressible Navier-
Stokes equations (1.1), (1.3) and $(1.4)_{3}$. The large-time behavior of the
solutions to the compressible Navier-Stokes equations (1.1) is closely related
to the corresponding Euler system
$\begin{cases}\rho_{t}+(\rho u)_{x}=0,\cr(\rho u)_{t}+\big{(}\rho
u^{2}+p\big{)}_{x}=0,\cr\big{[}\rho\big{(}e+\frac{u^{2}}{2}\big{)}\big{]}_{t}+\big{[}\rho
u\big{(}e+\frac{u^{2}}{2}\big{)}+pu\big{]}_{x}=0.\end{cases}$ (1.5)
The Euler system (1.5) is a typical example of the hyperbolic conservation
laws. It is well-known that the main feature of the solutions to the
hyperbolic conservation laws is the formation of the shock wave no matter how
smooth the initial values are. The Euler system (1.5) contains three basic
wave patterns, that is, two nonlinear waves, called shock wave and rarefaction
wave and one linear wave called contact discontinuity in the solutions to the
Riemann problem. The above three dilation invariant wave solutions and their
linear superpositions in the increasing order of characteristic speed, i.e.,
Riemann solutions, govern both local and large-time behavior of solutions to
the Euler system and so govern the large-time behavior of the solutions to the
compressible Navier-Stokes equations (1.1).
There have been a large amount of literature on the large-time behavior of
solutions to the Cauchy problem of the compressible fluid system (1.1) towards
the viscous version of the basic wave patterns. We refer to [1], [2], [5],
[7], [8], [11], [13], [14], [16], [20], [23], [24] and some references
therein. All these works show that the large-time behavior of the solutions to
the Cauchy problem is basically governed by the Riemann solutions to its
corresponding hyperbolic system.
Recently, the initial-boundary value problem of (1.1) attracts increasing
interest because it has more physical meanings and of course produces some new
mathematical difficulties due to the boundary effect. Not only basic wave
patterns but also a new wave, which is called boundary layer solution (BL-
solution for brevity) [15], may appear in the IVBP case. Matsumura [15]
proposes a criterion on the question when the BL-solution forms to the
isentropic Navier-Stokes equations, where the entropy of the gas is assumed to
be constant and the equation $\eqref{(1.1)}_{3}$ for the energy conservation
is neglected. The argument in [15] for the isentropic Navier-Stokes equations
can also be applied to the full Navier-Stokes equations (1.1), see [3] for
details. Consider the Riemann problem to the Euler equations (1.5), where the
initial right state of the Riemann data is given by the far field state
$(\rho_{+},u_{+},\theta_{+})$ in (1.3), and the left end state
$(\rho_{-},u_{-},\theta_{-})$ is given by the all possible states which are
consistent with the boundary condition (1.4) at $\\{x=0\\}$. Note that to the
outflow problem, $\rho_{-}$ can not be prescribed and is free on the boundary.
On one hand, when the left end state is uniquely determined so that the value
at the boundary $\\{x=0\\}$ of the solution to the Riemann problem is
consistent with the boundary condition, we expect that no BL-solution occurs.
On the other hand, if the value of the solution to the Riemann problem on the
boundary is not consistent with the boundary condition for any admissible left
end state, we expect a BL-solution which compensates the gap comes up. Such
BL-solution could be constructed by the stationary solution to Navier-Stokes
equations. The existence and stability of the BL-solution (to the inflow or
outflow problems, to the isentropic or full Navier-Stokes equations) are
studied extensively by many authors, see [3], [4], [6], [10], [15] [18], [21],
[25], etc.
Now we review some recent works on the large-time behavior of the solutions to
the inflow problem of the full Naiver-Stokes equation (1.1), (1.3),
$(1.4)_{3}$ by Huang-Li-Shi [3] and Qin-Wang [21]. In [21], we rigorously
prove the existence (or non-existence) of BL-solution to the inflow problem
(1.1), (1.3), $(1.4)_{3}$ when the right end state
$(\rho_{+},u_{+},\theta_{+})$ belongs to the subsonic, transonic and
supersonic regions respectively. When $(\rho_{\pm},u_{\pm},\theta_{\pm})$ both
belong to the subsonic region, the BL-solution is expected and the stability
of this BL-solution and its superposition with the 3-rarefaction wave is
proved under some smallness assumptions in [3]. The stability of the
superposition of the subsonic BL-solution, the viscous 2-contact wave and
3-rarefaction wave is shown in [21] under the condition that the amplitude of
BL-solution and the contact wave is small enough but the amplitude of the
rarefaction wave is not necessarily small. The stability of the single viscous
contact wave is also obtained in [21] if the contact wave is weak enough. It
should be remarked that the subsonic BL-solution decays exponentially with
respect to $\xi=x-\sigma_{-}t$, which is good enough to get the desired
estimates. When the boundary value $(\rho_{-},u_{-},\theta_{-})$ belongs to
the supersonic region, there is no BL-solution. Thus the large-time behavior
of the solution is expected to be same as that of the Cauchy problem and the
stability of the 3-rarefaction waves is also given in [3].
In the present paper, we are interested in the stability of wave patterns to
the inflow problem (1.1), (1.3) and $(1.4)_{3}$ when
$(\rho_{-},u_{-},\theta_{-})$ belongs to the transonic region. In this case, a
new wave structure which contains four waves: the transonic(or degenerate) BL-
solution, 1-rarefaction wave, viscous 2-contact wave and 3-rarefaction wave,
occurs. Due to the fact that the first characteristic speed on the boundary is
coincident with the speed of the moving boundary in the transonic BL-solution
case, the nonlinear waves in the first characteristic field may appear, which
is quite different from the the regime that $(\rho_{-},u_{-},\theta_{-})$
belongs to the subsonic region in our previous result [21], where the waves in
the first characteristic field must be absent. Here we just assume that the
1-rarefaction wave appear in the first characteristic field. Correspondingly,
some new mathematical difficulties occur due to the degeneracy of the
transonic BL-solution and its interactions with other wave patterns in the
superposition wave. In particular, the transonic boundary layer solution is
attached with 1-rarefaction wave for all time, so the interaction of these two
waves should be carefully treated in the stability analysis.
Because the system $(\ref{(1.1)})$ we consider is in one dimension of the
space variable $x$, it is convenient to use the following Lagrangian
coordinate transformation:
$(t,x)\Rightarrow\left(t,\int^{(t,x)}_{(0,0)}\rho(\tau,y)\,dy-\rho
u(\tau,y)\,d\tau\right).$
Thus the system $(\ref{(1.1)})$ can be transformed into the following moving
boundary problem of Navier-Stokes equations in the Lagrangian coordinates
[18]:
$\begin{cases}v_{t}-u_{x}=0,\cr
u_{t}+p_{x}=\mu\left(\frac{u_{x}}{v}\right)_{x},\qquad\qquad\qquad\qquad\qquad~{}~{}~{}~{}~{}t>0,x>\sigma_{-}t,\cr\left(\frac{R}{\gamma-1}\theta+\frac{1}{2}u^{2}\right)_{t}+(pu)_{x}=\kappa\left(\frac{\theta_{x}}{v}\right)_{x}+\mu\left(\frac{uu_{x}}{v}\right)_{x},\cr(v,u,\theta)(0,x)=(v_{0},u_{0},\theta_{0})(x)\rightarrow(v_{+},u_{+},\theta_{+}),~{}~{}~{}~{}{\rm
as}~{}~{}x\rightarrow+\infty,\cr(v,u,\theta)(t,x=\sigma_{-}t)=(v_{-},u_{-},\theta_{-}),~{}~{}u_{-}>0\end{cases}$
(1.6)
where $\sigma_{-}:=-\frac{u_{-}}{v_{-}}<0$ is the speed of the moving
boundary.
In order to fix the moving boundary $x=\sigma_{-}t$, we introduce a new
variable $\xi=x-\sigma_{-}t$. Then we have the half-space problem
$\begin{cases}v_{t}-\sigma_{-}v_{\xi}-u_{\xi}=0,\cr
u_{t}-\sigma_{-}u_{\xi}+p_{\xi}=\mu\left(\frac{u_{\xi}}{v}\right)_{\xi},\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
t>0,\xi\in\mathbf{R}_{+},\cr\Big{(}\frac{R}{\gamma-1}\theta+\frac{1}{2}u^{2}\Big{)}_{t}-\sigma_{-}\left(\frac{R}{\gamma-1}\theta+\frac{1}{2}u^{2}\right)_{\xi}+(pu)_{\xi}=\kappa\left(\frac{\theta_{\xi}}{v}\right)_{\xi}+\mu\left(\frac{uu_{\xi}}{v}\right)_{\xi},\cr(v,u,\theta)(t=0,\xi)=(v_{0},u_{0},\theta_{0})(\xi)\rightarrow(v_{+},u_{+},\theta_{+})~{}~{}{\rm
as}~{}~{}\xi\rightarrow+\infty,\cr(v,u,\theta)(t,\xi=0)=(v_{-},u_{-},\theta_{-}),~{}~{}u_{-}>0.\end{cases}$
(1.7)
Given the right end state $(v_{+},u_{+},\theta_{+})$, we can define the
following wave curves in the phase space $(v,u,\theta)$ with $v>0$ and
$\theta>0$.
$\bullet$ Transonic(or degenerate) boundary layer curve:
$BL(v_{+},u_{+},\theta_{+}):=\left\\{(v,u,\theta)\bigg{|}\frac{u}{v}=-\sigma_{-}=\frac{u_{-}}{v_{-}},(u,\theta)\in\Sigma(u_{+},\theta_{+})\right\\},$
(1.8)
where
$(v_{+},u_{+},\theta_{+})\in\Gamma_{trans}^{+}=\\{(u,\theta)|u=\sqrt{R\gamma\theta}>0\,\\}$
is the transonic region defined in (2.4) with positive gas velocity and
$\Sigma(u_{+},\theta_{+})$ is the trajectory at the point $(u_{+},\theta_{+})$
defined in Case II of Lemma 2.1 below.
$\bullet$ Contact wave curve:
$CD(v_{+},u_{+},\theta_{+}):=\\{(v,u,\theta)|u=u_{+},p=p_{+},v\not\equiv
v_{+}\\},$ (1.9)
$\bullet$ $i-$Rarefaction wave curve $(i=1,3)$:
$R_{i}(v_{+},u_{+},\theta_{+}):=\left\\{(v,u,\theta)\bigg{|}\lambda_{i}<\lambda_{i+},~{}u=u_{+}-\int^{v}_{v_{+}}\lambda_{i}(\eta,s_{+})\,d\eta,~{}s(v,\theta)=s_{+}\right\\},$
(1.10)
where $s_{+}=s(v_{+},\theta_{+})$ and $\lambda_{i}=\lambda_{i}(v,s)$ is the
$i-$th characteristic speed given in (2.2).
Our main stability result is, roughly speaking, as follows:
$\bullet$ Assume that $(v_{-},u_{-},\theta_{-})\in{\rm
BL\texttt{-}R_{1}\texttt{-}CD\texttt{-}R_{3}}(v_{+},u_{+},\theta_{+})$, that
is, there exist the unique medium states
$(v_{*},u_{*},\theta_{*})\in\Gamma_{trans}^{+}$, $(v_{m},u_{m},\theta_{m})$
and $(v^{*},u^{*},\theta^{*})$, such that $(v_{-},u_{-},\theta_{-})\in{\rm
BL}(v_{*},u_{*},\theta_{*})$, $(v_{*},u_{*},\theta_{*})\in
R_{1}(v_{m},u_{m},\theta_{m})$, $(v_{m},u_{m},\theta_{m})\in{\rm
CD}(v^{*},u^{*},\theta^{*})$ and $(v^{*},u^{*},\theta^{*})\in{\rm
R_{3}}(v_{+},u_{+},\theta_{+})$, then the superposition of the four wave
patterns: the transonic (or degenerate) BL-solution, 1-rarefaction wave,
2-viscous contact wave and 3-rarefaction wave is time-asymptotically stable
provided that the wave strength
$\delta=|(v_{+}-v_{-},u_{+}-u_{-},\theta_{+}-\theta_{-})|$ is suitably small
and the conditions in Theorem 2.1 hold.
This paper is organized as follows. In Section 2, after giving some
preliminaries on boundary layer solution, viscous 2-contact wave, rarefaction
waves and their superposition, we state our main result. In Section 3, first
the wave interaction estimations are shown, then the desired energy estimates
are performed and finally our main result is proven.
_Notations._ Throughout this paper, several positive generic constants are
denoted by $c,C$ without confusion, and $C(\cdot)$ stands for some generic
constant(s) depending only on the quantity listed in the parenthesis. For
function spaces, $L^{p}(\mathbf{R}_{+}),1\leq p\leq\infty$, denotes the usual
Lebesgue space on $\mathbf{R}_{+}$. $W^{k,p}(\mathbf{R}_{+})$ denotes the
$k^{th}$ order Sobolev space, and if $p=2$, we note
$H^{k}(\mathbf{R}_{+}):=W^{k,2}(\mathbf{R}_{+})$,
$\|\cdot\|:=\|\cdot\|_{L^{2}(\mathbf{R}_{+})}$, and
$\|\cdot\|_{k}:=\|\cdot\|_{H^{k}(\mathbf{R}_{+})}$ for simplicity. The domain
$\mathbf{R}_{+}$ will be often abbreviated without confusion.
## 2 Preliminaries and Main Result
It is well known that the hyperbolic system (1.5) has three characteristic
speeds
$\displaystyle\lambda_{1}(v,\theta)=-\frac{\sqrt{R\gamma\theta}}{v},~{}~{}~{}\lambda_{2}=0,~{}~{}~{}\lambda_{3}(v,\theta)=\frac{\sqrt{R\gamma\theta}}{v}.$
(2.1)
The first and the third characteristic field is genuinely nonlinear, which may
have nonlinear waves, shock wave and rarefaction wave, while the second
characteristic field is linearly degenerate, where contact discontinuity may
occur.
Let
$\displaystyle
c(v,s):=\sqrt{-v^{2}p_{v}(v,s)}=\sqrt{R\gamma\theta}=:c(v,\theta),\quad
M(v,u,\theta):=\frac{|u|}{c(v,\theta)}$ (2.2)
be the sound speed and the Mach number at the state $(v,u,\theta)$.
Correspondingly, set
$\displaystyle c_{+}:=c(v_{+},\theta_{+})=\sqrt{R\gamma\theta_{+}},\quad
M_{+}:=M(v_{+},u_{+},\theta_{+})=\frac{|u_{+}|}{c_{+}}$ (2.3)
be the sound speed and the Mach number at the far field $\\{x=+\infty\\}$. We
divide the phase space $\\{(v,u,\theta)|\,v>0,\theta>0\\}$ into three parts:
$\displaystyle\begin{cases}~{}~{}\Omega_{sub}:=\left\\{(v,u,\theta)~{}|~{}M<1\,\right\\},\cr\Gamma_{trans}:=\left\\{(v,u,\theta)~{}|~{}M=1\,\right\\},\cr\Omega_{super}:=\left\\{(v,u,\theta)~{}|~{}M>1\,\right\\}.\end{cases}$
(2.4)
Call them subsonic, transonic and supersonic region, respectively. Obviously,
if we add the alternative condition $u>0$ or $u\leq 0$, then we have six
regions $\Omega_{sub}^{\pm}$, $\Gamma_{trans}^{\pm}$, and
$\Omega_{super}^{\pm}$.
### 2.1 Boundary layer solution
When $(v_{-},u_{-},\theta_{-})\in\Omega_{sub}^{+}\cup\Gamma^{+}_{trans}$, we
have
$\displaystyle\lambda_{1}(v_{-},\theta_{-})=-\frac{\sqrt{R\gamma\theta_{-}}}{v_{-}}\leq-\frac{u_{-}}{v_{-}}=\sigma_{-}<0,$
(2.5)
hence a stationary solution $\big{(}V^{b},U^{b},\Theta^{b}\big{)}(\xi)$ to the
inflow problem (1.7) is expected
$\displaystyle\begin{cases}-\sigma_{-}V^{b}_{\xi}-U^{b}_{\xi}=0,\cr-\sigma_{-}U^{b}_{\xi}+P^{b}_{\xi}=\mu\Big{(}\frac{U^{b}_{\xi}}{V^{b}}\Big{)}_{\xi},\cr-\sigma_{-}\left(\frac{R}{\gamma-1}\Theta^{b}+\frac{1}{2}\left(U^{b}\right)^{2}\right)_{\xi}+\left(P^{b}U^{b}\right)_{\xi}=\kappa\Big{(}\frac{\Theta^{b}_{\xi}}{V^{b}}\Big{)}_{\xi}+\mu\Big{(}\frac{U^{b}U^{b}_{\xi}}{V^{b}}\Big{)}_{\xi},\cr\big{(}V^{b},U^{b},\Theta^{b}\big{)}(0)=(v_{-},u_{-},\theta_{-}),~{}~{}~{}\big{(}V^{b},U^{b},\Theta^{b}\big{)}(+\infty)=(v_{+},u_{+},\theta_{+}),\end{cases}$
(2.6)
where $P^{b}:=p\big{(}V^{b},\Theta^{b}\big{)}=\frac{R\Theta^{b}}{V^{b}}$. We
call this stationary solution $\big{(}V^{b},U^{b},\Theta^{b}\big{)}(\xi)$ the
boundary layer solution (simply, BL-solution) to the inflow problem (1.7).
From the fact that $V^{b}(\xi)>0$ and $u_{-}>0$, then
$u_{+}>0,\qquad\frac{U^{b}}{V^{b}}=\frac{u_{+}}{v_{+}}=\frac{u_{-}}{v_{-}}=-\sigma_{-}.$
(2.7)
Thus (2.6) is equivalent to (2.7) and the following ODE system
$\displaystyle\begin{cases}\left(U^{b}\right)^{\prime}=-\frac{\sigma_{-}}{\mu}V^{b}\big{(}U^{b}-u_{+}\big{)}+\frac{R}{\mu}\left(\Theta^{b}-\frac{\theta_{+}}{v_{+}}V^{b}\right)\qquad\quad^{\prime}=\frac{d}{d\xi},\cr\left(\Theta^{b}\right)^{\prime}=-\frac{R\sigma_{-}}{\kappa(\gamma-1)}V^{b}\big{(}\Theta^{b}-\theta_{+}\big{)}+\frac{p_{+}}{\kappa}V^{b}\big{(}U^{b}-u_{+}\big{)}+\frac{\sigma_{-}}{2\kappa}V^{b}\big{(}U^{b}-u_{+}\big{)}^{2},\cr\left(U^{b},\Theta^{b}\right)(0)=(u_{-},\theta_{-}),~{}~{}~{}\left(U^{b},\Theta^{b}\right)(+\infty)=(u_{+},\theta_{+}),\end{cases}$
(2.8)
where $p_{+}:=p(v_{+},\theta_{+})$.
We can compute that the
Now we state the existence results of the BL-solution to (2.8) while its proof
has been shown in [21].
Lemma 2.1 (Existence of BL-solution) [21] _Suppose that $v_{\pm}>0$,
$u_{-}>0$, $\theta_{\pm}>0$ and let
$\delta_{b}:=|(u_{+}-u_{-},\theta_{+}-\theta_{-})|$. If $u_{+}\leq 0$, then
there is no solution to $(\ref{(2.8)})$. If $u_{+}>0$, then there exists a
suitably small constant $\delta_{0}>0$ such that if
$0<\delta^{b}\leq\delta_{0}$, then the existence and non-existence of
solutions to (2.8) is divided into three cases according to the location of
$(u_{+},\theta_{+})$: _
_Case I : $(u_{+},\theta_{+})\in\Omega_{sup}^{+}$. Then there is no solution
to (2.8)._
_Case II : $(u_{+},\theta_{+})\in\Gamma_{trans}^{+}$. Then
$(u_{+},\theta_{+})$ is a saddle-knot point to (2.8). Precisely, there exists
a unique trajectory $\Sigma$ tangent to the straight line_
$\displaystyle\mu u_{+}(u-u_{+})-\kappa(\gamma-1)(\theta-\theta_{+})=0$ (2.9)
_at the point $(u_{+},\theta_{+}).$ For each
$(u_{-},\theta_{-})\in\Sigma(u_{+},\theta_{+})$, there exists a unique
solution $\big{(}U^{b},\Theta^{b}\big{)}$ satisfying_
$U^{b}_{\xi}>0,\qquad\Theta^{b}_{\xi}>0,$
_and_
$\displaystyle\left|\frac{d^{n}}{d\xi^{n}}\big{(}U^{b}-u_{+},\Theta^{b}-\theta_{+}\big{)}\right|=O(1)\frac{\delta_{b}^{n+1}}{(1+\delta_{b}\xi)^{n+1}},~{}~{}~{}n=0,1,2,\dots.$
(2.10)
_Case III : $(u_{+},\theta_{+})\in\Omega_{sub}^{+}$. Then $(u_{+},\theta_{+})$
is a saddle point to (2.8). PPrecisely, there exists a center-stable manifold
$\mathcal{M}$ tangent to the line_
$(1+a_{2}c_{2}u_{+})(U^{B}-u_{+})-a_{2}(\Theta^{B}-\theta_{+})=0$
_on the opposite directions at the point $(u_{+},\theta_{+})$. Here $c_{2}$ is
one of the solutions to the equation_
$y^{2}+\Bigg{(}\frac{M_{+}^{2}\gamma-1}{M_{+}^{2}R\gamma}-\frac{\mu}{\kappa(\gamma-1)}\Bigg{)}y-\frac{\mu}{M_{+}^{2}R\gamma\kappa}=0$
_and $a_{2}=-\frac{R}{\mu(\lambda_{J}^{1}-\lambda_{J}^{2})}$ with
$\lambda_{J}^{1}>0,~{}~{}\lambda_{J}^{2}<0$ are the two eigenvalues of the
linearized matrix of ODE (2.8). Only when
$(u_{-},\theta_{-})\in\mathcal{M}(u_{+},\theta_{+})$, does there exist a
unique solution
$\big{(}U^{b},\Theta^{b}\big{)}\subset\mathcal{M}(u_{+},\theta_{+})$
satisfying_
$\displaystyle\left|\frac{d^{n}}{d\xi^{n}}\big{(}U^{b}-u_{+},\Theta^{b}-\theta_{+}\big{)}\right|=O(1)\delta_{b}e^{-c\xi},~{}~{}~{}n=0,1,2,\dots.$
(2.11)
### 2.2 Viscous Contact Wave
If $(v_{-},u_{-},\theta_{-})\in CD(v_{+},u_{+},\theta_{+})$, then the
following Riemann problem
$\displaystyle\begin{cases}v_{t}-u_{x}=0,\cr u_{t}+p_{x}=0,\qquad\qquad\qquad
t>0,x\in\mathbf{R},\cr\left(\frac{R}{\gamma-1}\theta+\frac{1}{2}u^{2}\right)_{t}+(pu)_{x}=0,\cr(v,u,\theta)(0,x)=\begin{cases}(v_{-},u_{-},\theta_{-}),\quad
x<0,\cr(v_{+},u_{+},\theta_{+}),\quad x>0\end{cases}\end{cases}$ (2.12)
admits a contact discontinuity solution
$(v,u,\theta)(t,x)=\left\\{\begin{array}[]{ll}(v_{-},u_{-},\theta_{-}),&x<0,~{}t>0,\\\
(v_{+},u_{+},\theta_{+}),&x>0,~{}t>0.\end{array}\right.$
From [7], the viscous version of the above contact discontinuity, called
viscous contact wave $\big{(}V^{d},U^{d},\Theta^{d}\big{)}(t,x)$ can be
defined by
$\displaystyle\begin{cases}V^{d}(t,x)=\frac{R\Theta^{\rm sim}(t,x)}{p_{+}},\cr
U^{d}(t,x)=u_{+}+\frac{(\gamma-1)\kappa\Theta^{\rm
sim}_{x}(t,x)}{\gamma\Theta^{\rm sim}(t,x)},\cr\Theta^{d}(t,x)=\Theta^{\rm
sim}\Big{(}\frac{x}{\sqrt{1+t}}\Big{)}+R\Big{(}\mu-\frac{(\gamma-1)\kappa}{R\gamma}\Big{)}\Theta^{\rm
sim}_{t}\end{cases}$ (2.13)
where $\Theta^{\rm sim}\left(\frac{x}{\sqrt{1+t}}\right)$ is the unique self-
similar solution to the following nonlinear diffusion equation
$\displaystyle\begin{cases}\Theta_{t}=\frac{(\gamma-1)\kappa
p_{+}}{R^{2}\gamma}\left(\frac{\Theta_{x}}{\Theta}\right)_{x},\cr\Theta(t,\pm\infty)=\theta_{\pm}.\end{cases}$
(2.14)
Note that $\xi=x-\sigma_{-}t$, we have the following Lemma:
Lemma 2.2. [7] _The viscous contact wave
$\big{(}V^{d},U^{d},\Theta^{d}\big{)}(t,x),~{}(x=\xi+\sigma_{-}t)$ defined in
(2.13) satisfies_
* i)
$\partial_{\xi}^{n}\big{(}\Theta^{d}-\theta_{\pm}\big{)}=O(1)\delta_{d}(1+t)^{-\frac{n}{2}}\exp\left(-\frac{C_{d}(\xi+\sigma_{-}t)^{2}}{1+t}\right),\quad
n=0,1,2,\cdots$_;_
* ii)
$U^{d}_{\xi}(t,\xi)=O(1)\delta_{d}(1+t)^{-1}\exp\left(-\frac{C_{d}(\xi+\sigma_{-}t)^{2}}{1+t}\right);$__
* iii)
$\big{(}V^{d},U^{d},\Theta^{d}\big{)}(t,\xi=0)-(v_{-},u_{-},\theta_{-})=O(1)\delta_{d}e^{-ct}$_._
_where $\delta_{d}=|\theta_{+}-\theta_{-}|$ is the amplitude of the viscous
contact wave and $C_{d},c>0$ are constants. _
Then the viscous contact wave $\big{(}V^{d},U^{d},\Theta^{d}\big{)}$ defined
in (2.13) satisfies the system
$\displaystyle\begin{cases}V^{d}_{t}-\sigma_{-}V^{d}_{\xi}-U^{d}_{\xi}=0,\cr
U^{d}_{t}-\sigma_{-}U^{d}_{\xi}+P^{d}_{\xi}=\mu\Big{(}\frac{U^{d}_{\xi}}{V^{d}}\Big{)}_{\xi},\qquad\qquad~{}~{}~{}~{}t>0,\xi\in\mathbf{R}_{+},\cr\frac{R}{\gamma-1}\big{(}\Theta^{d}_{t}-\sigma_{-}\Theta^{d}_{\xi}\big{)}+P^{d}U^{d}_{\xi}=\kappa\bigg{(}\frac{\Theta^{d}_{\xi}}{V^{d}}\bigg{)}_{\xi}+\mu\frac{(U^{d}_{\xi})^{2}}{V^{d}}+H^{d}\end{cases}$
(2.15)
where $P^{d}:=p\big{(}V^{d},\Theta^{d}\big{)}$ and
$H^{d}=O(1)\delta_{d}(1+t)^{-2}\exp\left(-\frac{C_{d}(\xi+\sigma_{-}t)^{2}}{1+t}\right)$
due to Lemma 2.2.
### 2.3 Rarefaction waves
It is well known that if $(v_{-},u_{-},\theta_{-})\in
R_{i}(v_{+},u_{+},\theta_{+}),~{}(i=1,3)$, then there exist a $i-$rarefaction
wave $(v^{r_{i}},u^{r_{i}},\theta^{r_{i}})(x/t)$ which is the global weak
solution to the following Riemann problem
$\displaystyle\begin{cases}v_{t}-u_{x}=0,\cr
u_{t}+p_{x}=0,\quad\quad\quad\quad\quad\quad\,t>0,x\in\mathbf{R},\cr\left(\frac{R}{\gamma-1}\theta+\frac{1}{2}u^{2}\right)_{t}+(pu)_{x}=0,\cr(v,u,\theta)(0,x)=\begin{cases}(v_{-},u_{-},\theta_{-}),\quad
x<0,\cr(v_{+},u_{+},\theta_{+}),\quad x>0.\end{cases}\end{cases}$ (2.16)
Consider the following Burgers equation
$\displaystyle\begin{cases}w_{t}+ww_{x}=0,\quad\,t>0,x\in\mathbf{R},\cr
w_{0}(x):=w(0,x)=\begin{cases}w_{-},~{}\quad\quad\quad\quad\quad\quad\quad\quad
x<0,\cr\displaystyle
w_{-}+C_{q}(w_{+}-w_{-})\int^{x}_{0}y^{q}e^{-y}\,dy,~{}x\geq
0.\end{cases}\end{cases}$ (2.17)
Here $q\geq 14$ is a constant to be determined, and $C_{q}$ is a constant such
that $\displaystyle C_{q}\int^{+\infty}_{0}y^{q}e^{-y}dy=1$. If $w_{-}<w_{+},$
then the solution to the above Burgers equation can be expressed by
$\displaystyle w(t,x)=w_{0}(x_{0}(t,x)),\quad\quad
x=x_{0}(t,x)+w_{0}(x_{0}(t,x))t.$ (2.18)
Moreover, we have
$\bullet$ $w(t,x)=w_{-}$, if $x\leq w_{-}t$.
$\bullet$ For any positive constant $\sigma_{0}>0$ and for $x\geq 0$
$\displaystyle|w(t,x)-w_{+}|$ $\displaystyle=$
$\displaystyle|w_{0}(x_{0}(t,x))-w_{+}|$ (2.19) $\displaystyle=$
$\displaystyle C_{q}(w_{+}-w_{-})\int_{x_{0}(t,x)}^{+\infty}y^{q}e^{-y}\,dy$
(2.21) $\displaystyle=$ $\displaystyle
C_{q}(w_{+}-w_{-})\int_{x-w_{0}(x_{0}(t,x))t}^{+\infty}y^{q}e^{-y}\,dy$ (2.23)
$\displaystyle\leq$ $\displaystyle
C_{q}(w_{+}-w_{-})\int_{x-w_{+}t}^{+\infty}y^{q}e^{-y}\,dy$ (2.25)
$\displaystyle\leq$ $\displaystyle
C_{q}(w_{+}-w_{-})e^{-\sigma_{0}t},\qquad\text{if}~{}~{}x\geq(2\sigma_{0}+w_{+})t.$
(2.27)
Note that the estimation in (2.19) play an important role in the wave
interaction estimates, which is motivated by [12] and [16] .
Now the $i-$rarefaction wave
$(V^{r_{i}},U^{r_{i}},\Theta^{r_{i}})(t,x)~{}(i=1,3)$ to the inflow problem
(1.7) can be defined by
$\displaystyle\begin{cases}\lambda_{i}(V^{r_{i}},\Theta^{r_{i}})(t,x)=w(1+t,x+\sigma_{-}),\cr
s(V^{r_{i}},\Theta^{r_{i}})(t,x)=s_{+}=s(v_{+},\theta_{+}),\cr\displaystyle
U^{r_{i}}(t,x)=u_{+}-\int^{V^{r_{i}}(t,x)}_{v_{+}}\lambda_{i}(\eta,s_{+})d\eta.\end{cases}$
(2.28)
Then the $i-$rarefaction wave
$(V^{r_{i}},U^{r_{i}},\Theta^{r_{i}})(t,x),~{}(i=1,3)$ defined in (2.28)
satisfies the system
$\displaystyle\begin{cases}V^{r_{i}}_{t}-\sigma_{-}V^{r_{i}}_{\xi}-U^{r_{i}}_{\xi}=0,\cr
U^{r_{i}}_{t}-\sigma_{-}U^{r_{i}}_{\xi}+P^{r_{i}}_{\xi}=0,\cr\left[\frac{R}{\gamma-1}\Theta^{r_{i}}+\frac{1}{2}(U^{{r_{i}}})^{2}\right]_{t}-\sigma_{-}\left[\frac{R}{\gamma-1}\Theta^{r_{i}}+\frac{1}{2}(U^{{r_{i}}})^{2}\right]_{\xi}+(P^{r_{i}}U^{r_{i}})_{\xi}=0,\cr(V^{r_{i}},U^{r_{i}},\Theta^{r_{i}})(t,\xi=0)=(v_{-},u_{-},\theta_{-}),\cr(V^{r_{i}},U^{r_{i}},\Theta^{r_{i}})(t,\xi)\rightarrow(v_{+},u_{+},\theta_{+})~{}~{}\text{
as}~{}~{}\xi\rightarrow+\infty\end{cases}$ (2.29)
where $P^{r_{i}}:=p(V^{r_{i}},\Theta^{r_{i}})$.
Lemma 2.3 _ $i-$rarefaction wave
$(V^{r_{i}},U^{r_{i}},\Theta^{r_{i}})(t,\xi),~{}(i=1,3)$ defined in (2.28)
satisfies_
* i)
_$U^{r_{i}}_{\xi}(t,\xi)
>0,~{}~{}(|V^{r_{i}}_{\xi}|,|\Theta_{\xi}^{r_{i}}|)\leq CU^{r_{i}}_{\xi}$_;
* ii)
_For any $p$_ ($1\leq p\leq\infty$), _there exists a constant $C_{pq}$ such
that_
$\displaystyle\|(V^{r_{i}}_{\xi},U^{r_{i}}_{\xi},\Theta^{r_{i}}_{\xi})(t)\|_{L^{p}}\leq
C_{p}\min\big{\\{}\delta_{r_{i}},\delta_{r_{i}}^{1/p}(1+t)^{-1+1/p}\big{\\}},$
$\displaystyle\|(V^{r_{i}}_{\xi\xi},U^{r_{i}}_{\xi\xi},\Theta^{r_{i}}_{\xi\xi})(t)\|_{L^{p}}\leq
C_{p}\min\big{\\{}\delta_{r_{i}},\delta_{r_{i}}^{1/p+1/q}(1+t)^{-1+1/q}\big{\\}};$
(2.30)
* iii)
_For_ $\forall\,\sigma_{0}>0$, if
$\xi\geq\left[-\sigma_{-}+\lambda_{1}(v_{+},\theta_{+})+2\sigma_{0}\right](1+t)$,
then
$\Big{|}\partial_{\xi}^{n}\big{\\{}(V^{r_{1}},U^{r_{1}},\Theta^{r_{1}})(t,\xi)-(v_{+},u_{+},\theta_{+})\big{\\}}\Big{|}\leq
C\delta_{r_{1}}e^{-\sigma_{0}t},~{}n=0,1,2,\cdots;$
* iv)
_For_ $\xi\leq\left[-\sigma_{-}+\lambda_{3}(v_{-},\theta_{-})\right](1+t)$,
$(V^{r_{3}},U^{r_{3}},\Theta^{r_{3}})-(v_{-},u_{-},\theta_{-})\equiv 0;$
* v)
$\lim\limits_{t\rightarrow\infty}\sup\limits_{\xi\in\mathbf{R}_{+}}\big{|}(V^{r_{i}},U^{r_{i}},\Theta^{r_{i}})(t,\xi)-(v^{r_{i}},u^{r_{i}},\theta^{r_{i}})\big{(}\frac{\xi}{1+t}\big{)}\big{|}=0$_._
Remark: The statement ${\rm iii)}$ is a direct consequence of the (2.19).
### 2.4 Superposition of transonic BL-solution, 1-rarefaction wave, 2-viscous
contact wave and 3-rarefaction wave
In this subsection, we consider the case that $(v_{-},u_{-},\theta_{-})\in
BL$-$R_{1}$-$CD$-$R_{3}(v_{+},u_{+},\theta_{+})$, that is, there exist
uniquely three medium states $(v_{*},u_{*},\theta_{*})\in\Gamma_{trans}^{+}$,
$(v_{m},u_{m},\theta_{m})$ and $(v^{*},u^{*},\theta^{*})$ such that
$(v_{*},u_{*},\theta_{*})\in BL(v_{-},u_{-},\theta_{-})$,
$(v_{*},u_{*},\theta_{*})\in R_{1}(v_{m},u_{m},\theta_{m})$,
$(v_{m},u_{m},\theta_{m})\in CD(v^{*},u^{*},\theta^{*})$ and
$(v^{*},u^{*},\theta^{*})\in R_{3}(v_{+},u_{+},\theta_{+})$. In fact, three
medium states $(v_{*},u_{*},\theta_{*})\in\Gamma_{trans}^{+}$,
$(v_{m},u_{m},\theta_{m})$ and $(v^{*},u^{*},\theta^{*})$ can be expressed
explicitly and uniquely by the following nine equations
$\displaystyle\begin{cases}\displaystyle\frac{u_{-}}{v_{-}}=\frac{u_{*}}{v_{*}},\quad
u_{*}=\sqrt{R\gamma\theta_{*}},\quad(u_{-},\theta_{-})\in\Sigma(u_{*},\theta_{*}),\cr\displaystyle
u_{*}=u_{m}-\int_{v_{*}}^{v_{m}}\sqrt{R\gamma
v_{+}^{\gamma-1}\theta_{+}}~{}\eta^{-\frac{\gamma+1}{2}}\,d\eta,\quad~{}v_{*}^{\gamma-1}\theta_{*}=v_{m}^{\gamma-1}\theta_{m},\cr\displaystyle
u_{m}=u^{*},\quad\frac{\theta_{m}}{v_{m}}=\frac{\theta^{*}}{v^{*}},\cr\displaystyle
u^{*}=u_{+}+\int_{v^{*}}^{v_{+}}\sqrt{R\gamma
v_{+}^{\gamma-1}\theta_{+}}~{}\eta^{-\frac{\gamma+1}{2}}\,d\eta,\quad~{}v^{*\gamma-1}\theta^{*}=v_{+}^{\gamma-1}\theta_{+}.\end{cases}$
(2.31)
Define the superposition wave $(V,U,\Theta)(t,\xi)$ by
$\displaystyle\left(\begin{array}[]{cc}V\\\ U\\\
\Theta\end{array}\right)(t,\xi)=\left(\begin{array}[]{cc}V^{b}+V^{r_{1}}+V^{d}+V^{r_{3}}\\\
U^{b}+U^{r_{1}}+U^{d}+U^{r_{3}}\\\
\Theta^{b}+\Theta^{r_{1}}+\Theta^{d}+\Theta^{r_{3}}\end{array}\right)(t,\xi)-\left(\begin{array}[]{cc}v_{*}+v_{m}+v^{*}\\\
u_{*}+u_{m}+u^{*}\\\ \theta_{*}+\theta_{m}+\theta^{*}\end{array}\right)$
(2.41)
where $(V^{b},U^{b},\Theta^{b})(\xi)$ is the transonic BL-solution defined in
Case II of Lemma 2.1 with the right state $(v_{+},u_{+},\theta_{+})$ replaced
by $(v_{*},u_{*},\theta_{*})$, $(V^{r_{1}},U^{r_{1}},\Theta^{r_{1}})(t,\xi)$
is the 1-rarefaction wave defined in (2.28) with the states
$(v_{-},u_{-},\theta_{-})$ and $(v_{+},u_{+},\theta_{+})$ replaced by
$(v_{*},u_{*},\theta_{*})$ and $(v_{m},u_{m},\theta_{m})$ respectively,
$(V^{d},U^{d},\Theta^{d})(t,\xi)$ is the viscous contact wave defined in
(2.13) with the states $(v_{-},u_{-},\theta_{-})$ and
$(v_{+},u_{+},\theta_{+})$ replaced by $(v_{m},u_{m},\theta_{m})$ and
$(v^{*},u^{*},\theta^{*})$, respectively, and
$(V^{r_{3}},U^{r_{3}},\Theta^{r_{3}})(t,\xi)$ is the 3-rarefaction wave
defined in (2.28) with the left state $(v_{-},u_{-},\theta_{-})$ replaced by
$(v^{*},u^{*},\theta^{*})$.
Now we state the main result of the paper as follows.
Theorem 2.1 (Stability of superposition of four waves) _Assume that
$(v_{-},u_{-},\theta_{-})\in BL$-$R_{1}$-$CD$-$R_{3}(v_{+},u_{+},\theta_{+})$.
Let $(V,U,\Theta)(t,\xi)$ be the superposition of the transonic BL-solution,
1-rarefaction wave, viscous 2-contact wave and 3-rarefaction wave defined in
(2.41). Then there exists a small positive constant $\delta_{0}$ such that if
the initial values and the wave strength
$\delta=|(v_{+}-v_{-},u_{+}-u_{-},\theta_{+}-\theta_{-})|$ satisfy_
$\displaystyle\delta+\|(v_{0}-V_{0},u_{0}-U_{0},\theta_{0}-\Theta_{0})\|_{1}\leq\delta_{0}.$
(2.42)
_the inflow problem $(\ref{(1.7)})$ has a unique global-in-time solution
$(v,u,\theta)(t,\xi)$ satisfying_
$\displaystyle\begin{cases}(v-V,u-U,\theta-\Theta)(t,\xi)\in
C\big{(}[0,\infty);H^{1}(\mathbf{R}^{+})\big{)},\cr(v-V)_{\xi}(t,\xi)\in
L^{2}\big{(}0,\infty;L^{2}(\mathbf{R}^{+})\big{)},\cr(u-U,\theta-\Theta)_{\xi}(t,\xi)\in
L^{2}\big{(}0,\infty;H^{1}(\mathbf{R}^{+})\big{)}.\end{cases}$ (2.43)
_Furthermore,_
$\displaystyle\lim_{t\rightarrow\infty}\sup_{\xi\in\mathbf{R}_{+}}|(v-V,u-U,\theta-\Theta)(t,\xi)|=0.$
(2.44)
Remark. In Theorem 2.1, we assume that
$\delta=|(v_{+}-v_{-},u_{+}-u_{-},\theta_{+}-\theta_{-})|$ is suitably small.
This assumption is equivalent to the one that the amplitudes of the four waves
are all suitably small. In fact, from the relations in (2.31) and the facts
$U^{b}_{\xi}>0$, $U^{r_{1}}_{\xi}>0$, $U^{r_{3}}_{\xi}>0$, we have
$\displaystyle\begin{cases}|v_{*}-v_{-}|+|\theta_{*}-\theta_{-}|=O(1)(u_{*}-u_{-}),\cr|v_{m}-v_{*}|+|\theta_{m}-\theta_{*}|=O(1)(u_{m}-u_{*}),\cr|v_{+}-v^{*}|+|\theta_{+}-\theta^{*}|=O(1)(u_{+}-u^{*}).\end{cases}$
(2.45)
Thus $\delta_{b}=O(1)(u_{*}-u_{-}),\delta_{r_{1}}=O(1)(u_{m}-u_{*})$,
$\delta_{r_{3}}=O(1)(u_{+}-u^{*})$. Due to $u_{m}=u^{*}$ by the contact
discontinuity curve, we have if $\delta$ is small, then
$\delta_{b},\delta_{r_{1}}$ and $\delta_{r_{3}}$ are all small. Furthermore,
we have
$\delta_{d}=|\theta^{*}-\theta_{m}|\leq\delta_{b}+\delta_{r_{1}}+\delta_{r_{3}}+\delta$
is small.
## 3 Stability Analysis
### 3.1 Wave interaction estimates
Recalling the definition of the superposition wave $(V,U,\Theta)(t,\xi)$
defined in (2.41), we have
$\displaystyle\begin{cases}V_{t}-\sigma_{-}V_{\xi}-U_{\xi}=0,\cr
U_{t}-\sigma_{-}U_{\xi}+P_{\xi}=\mu\Big{(}\frac{U_{\xi}}{V}\Big{)}_{\xi}+G,\qquad\qquad\qquad~{}~{}~{}~{}~{}t>0,\xi\in\mathbf{R}_{+},\cr\frac{R}{\gamma-1}\left(\Theta_{t}-\sigma_{-}\Theta_{\xi}\right)+PU_{\xi}=\kappa\Big{(}\frac{\Theta_{\xi}}{V}\Big{)}_{\xi}+\mu\frac{(U_{\xi})^{2}}{V}+H,\cr(V,U,\Theta)(t,\xi=0)=(v_{-},u_{-},\theta_{-})+\left(V^{d},U^{d},\Theta^{d}\right)(t,\xi=0)-(v_{m},u_{m},\theta_{m}).\end{cases}$
(3.1)
where $P:=p(V,\Theta)$ and
$\displaystyle\begin{cases}G=\big{(}P-P^{b}-P^{r_{1}}-P^{d}-P^{r_{3}}\big{)}_{\xi}-\mu\bigg{(}\frac{U_{\xi}}{V}-\frac{U^{b}_{\xi}}{V^{b}}-\frac{U^{d}_{\xi}}{V^{d}}\bigg{)}_{\xi}=:G_{1}+G_{2},\cr
H=(PU_{\xi}-P^{b}U^{b}_{\xi}-P^{r_{1}}U^{r_{1}}_{\xi}-P^{d}U^{d}_{\xi}-P^{r_{3}}U^{r_{3}}_{\xi})\cr~{}~{}~{}~{}~{}~{}-\bigg{[}\kappa\bigg{(}\frac{\Theta_{\xi}}{V}-\frac{\Theta^{b}_{\xi}}{V^{b}}-\frac{\Theta^{d}_{\xi}}{V^{d}}\bigg{)}_{\xi}+\mu\Bigg{(}\frac{(U_{\xi})^{2}}{V}-\frac{\big{(}U^{b}_{\xi}\big{)}^{2}}{V^{b}}-\frac{\big{(}U^{d}_{\xi}\big{)}^{2}}{V^{d}}\Bigg{)}-H^{d}\bigg{]}=:H_{1}+H_{2}.\end{cases}$
(3.2)
To control the interaction terms coming from different wave patterns, we give
the following lemma which will be critical in the energy estimate in
Subsection 3.3.
Lemma 3.1 (Wave interaction estimates) __
$\displaystyle\begin{cases}\displaystyle\int_{\mathbf{R}_{+}}\big{|}V^{b}_{\xi}\big{(}V^{r_{1}}-v_{*}\big{)}\big{|}+\big{|}V^{r_{1}}_{\xi}\big{(}V^{b}-v_{*}\big{)}\big{|}\,d\xi=O(1)\delta^{1/8}(1+t)^{-13/16},\cr\displaystyle\int_{\mathbf{R}_{+}}\big{|}V^{b}_{\xi}\big{(}V^{d}-v_{m}\big{)}\big{|}+\big{|}V^{d}_{\xi}\big{(}V^{b}-v_{*}\big{)}\big{|}\,d\xi=O(1)\delta(1+t)^{-1},\cr\displaystyle\int_{\mathbf{R}_{+}}\big{|}V^{b}_{\xi}\big{(}V^{r_{3}}-v^{*}\big{)}\big{|}+\big{|}V^{r_{3}}_{\xi}\big{(}V^{b}-v_{*}\big{)}\big{|}\,d\xi=O(1)\delta^{1/8}(1+t)^{-7/8},\cr\displaystyle\int_{\mathbf{R}_{+}}\big{|}V^{d}_{\xi}\big{(}V^{r_{1}}-v_{m}\big{)}\big{|}+\big{|}V^{r_{1}}_{\xi}\big{(}V^{d}-v_{m}\big{)}\big{|}\,d\xi=O(1)\delta
e^{-ct},\cr\displaystyle\int_{\mathbf{R}_{+}}\big{|}V^{d}_{\xi}\big{(}V^{r_{3}}-v^{*}\big{)}\big{|}+\big{|}V^{r_{3}}_{\xi}\big{(}V^{d}-v^{*}\big{)}\big{|}\,d\xi=O(1)\delta
e^{-ct},\cr\displaystyle\int_{\mathbf{R}_{+}}\big{|}V^{r_{1}}_{\xi}\big{(}V^{{r_{3}}}-v^{*}\big{)}\big{|}+\big{|}V^{r_{3}}_{\xi}\big{(}V^{{r_{1}}}-v_{m}\big{)}\big{|}\,d\xi=O(1)\delta
e^{-ct},\end{cases}$ (3.3)
$\displaystyle\begin{cases}\displaystyle\int_{\mathbf{R}_{+}}\big{|}V^{b}_{\xi}V^{d}_{\xi}\big{|}\,d\xi=O(1)\delta(1+t)^{-2},\qquad\int_{\mathbf{R}_{+}}\big{|}V^{b}_{\xi}V^{r_{1}}_{\xi}\big{|}\,d\xi=O(1)\delta(1+t)^{-1},\cr\displaystyle\int_{\mathbf{R}_{+}}\big{|}V^{b}_{\xi}V^{r_{3}}_{\xi}\big{|}\,d\xi=O(1)\delta(1+t)^{-1},\,~{}\quad\int_{\mathbf{R}_{+}}\big{|}V^{d}_{\xi}V^{r_{1}}_{\xi}\big{|}\,d\xi=O(1)\delta
e^{-ct},\cr\displaystyle\int_{\mathbf{R}_{+}}\big{|}V^{d}_{\xi}V^{r_{3}}_{\xi}\big{|}\,d\xi=O(1)\delta
e^{-ct},\quad~{}~{}\qquad\int_{\mathbf{R}_{+}}\big{|}V^{r_{1}}_{\xi}V^{r_{3}}_{\xi}\big{|}\,d\xi=O(1)\delta
e^{-ct},\end{cases}$ (3.4)
Proof. First we prove $(\ref{3.3})_{1}$, that is
$\bullet$ Interaction of transonic boundary layer solution and 1-rarefaction
wave:
Since $V^{r_{1}}_{\xi}\geq 0$ and $V^{b}_{\xi}\geq 0$, we have
$V^{r_{1}}-v_{*}\geq 0$ and $v_{*}-V^{b}\geq 0$. Thus we have
$\displaystyle\int_{\mathbf{R}_{+}}\big{|}V^{b}_{\xi}\big{(}V^{r_{1}}-v_{*}\big{)}\big{|}+\big{|}V^{r_{1}}_{\xi}\big{(}V^{b}-v_{*}\big{)}\big{|}\,d\xi$
(3.5) $\displaystyle=$ $\displaystyle
2\left\\{\int_{0}^{\left[-\sigma_{-}+\frac{\lambda_{1}(v_{m},\theta_{m})}{2}\right](1+t)}+\int_{\left[-\sigma_{-}+\frac{\lambda_{1}(v_{m},\theta_{m})}{2}\right](1+t)}^{+\infty}\right\\}V^{r_{1}}_{\xi}\big{(}v_{*}-V^{b}\big{)}\,d\xi$
(3.6) $\displaystyle:=$ $\displaystyle J_{1}+J_{2}.$ (3.7)
Note that
$\begin{array}[]{ll}\displaystyle-\sigma_{-}+\frac{\lambda_{1}(v_{m},\theta_{m})}{2}\\\
\displaystyle=\frac{u_{-}}{v_{-}}+\frac{\lambda_{1}(v_{m},\theta_{m})}{2}=\frac{u_{*}}{v_{*}}+\frac{\lambda_{1}(v_{m},\theta_{m})}{2}\\\
=\frac{\sqrt{R\gamma\theta_{*}}}{v_{*}}+\frac{\lambda_{1}(v_{m},\theta_{m})}{2}=-\lambda_{1}(v_{*},\theta_{*})+\frac{\lambda_{1}(v_{m},\theta_{m})}{2}\\\
=\left[\lambda_{1}(v_{m},\theta_{m})-\lambda_{1}(v_{*},\theta_{*})\right]-\frac{\lambda_{1}(v_{m},\theta_{m})}{2}\\\
\geq-\frac{\lambda_{1}(v_{m},\theta_{m})}{2}>0.\end{array}$
Now we can compute that
$\displaystyle J_{1}$ $\displaystyle=$
$\displaystyle\int_{0}^{\left[-\sigma_{-}+\frac{\lambda_{1}(v_{m},\theta_{m})}{2}\right](1+t)}V^{r_{1}}_{\xi}\big{(}v_{*}-V^{b}\big{)}\,d\xi$
(3.8) $\displaystyle=$ $\displaystyle
O(1)\|V_{\xi}^{r_{1}}(t)\|_{L^{\infty}}\int_{0}^{\left[-\sigma_{-}+\frac{\lambda_{1}(v_{m},\theta_{m})}{2}\right](1+t)}\frac{\delta_{b}}{1+\delta_{b}\xi}\,d\xi$
(3.9) $\displaystyle=$ $\displaystyle
O(1)\delta_{r_{1}}^{\frac{1}{8}}(1+t)^{-\frac{7}{8}}\ln(1+\delta_{b}t)$ (3.10)
$\displaystyle=$ $\displaystyle
O(1)\delta_{r_{1}}^{\frac{1}{8}}(1+t)^{-\frac{13}{16}},$ (3.11)
and
$\displaystyle J_{2}$ $\displaystyle=$
$\displaystyle\int_{\left[-\sigma_{-}+\frac{\lambda_{1}(v_{m},\theta_{m})}{2}\right](1+t)}^{\infty}V^{r_{1}}_{\xi}\big{(}v_{*}-V^{b}\big{)}\,d\xi$
(3.12) $\displaystyle=$ $\displaystyle
O(1)\delta_{b}(v_{m}-V^{r_{1}}(t,\xi))|_{\xi=\left[-\sigma_{-}+\frac{\lambda_{1}(v_{m},\theta_{m})}{2}\right](1+t)}$
(3.13) $\displaystyle=$ $\displaystyle O(1)\delta_{b}e^{-\sigma_{0}t}.$ (3.14)
due to the statement ${\rm iii)}$ in Lemma 2.3 by taking
$\sigma_{0}=-\frac{\lambda_{1}(v_{m},\theta_{m})}{2}>0$. So the combination of
(3.8) and (3.12) gives $\eqref{3.3}_{1}$.
Then we prove $(\ref{3.3})_{2}$:
$\bullet$ Interaction of transonic boundary layer solution and viscous
2-contact wave:
$\begin{array}[]{ll}\displaystyle\int_{\mathbf{R}_{+}}\big{|}V^{b}_{\xi}\big{(}V^{d}-v_{m}\big{)}\big{|}+\big{|}V^{d}_{\xi}\big{(}V^{b}-v_{*}\big{)}\big{|}\,d\xi\\\
\displaystyle=\left\\{\int_{0}^{-\frac{\sigma_{-}t}{2}}+\int_{-\frac{\sigma_{-}t}{2}}^{+\infty}\right\\}\big{|}V^{b}_{\xi}\big{(}V^{d}-v_{m}\big{)}\big{|}+\big{|}V^{d}_{\xi}\big{(}V^{b}-v_{*}\big{)}\big{|}\,d\xi\\\
\displaystyle:=J_{3}+J_{4}.\end{array}$
We calculate
$\displaystyle J_{3}$ $\displaystyle=$
$\displaystyle\int_{0}^{-\frac{\sigma_{-}t}{2}}\big{|}V^{b}_{\xi}\big{(}V^{d}-v_{m}\big{)}\big{|}+\big{|}V^{d}_{\xi}\big{(}V^{b}-v_{*}\big{)}\big{|}\,d\xi$
(3.15) $\displaystyle=$ $\displaystyle
O(1)\delta_{d}\int_{0}^{-\frac{\sigma_{-}t}{2}}\exp\left(-\frac{C_{d}(\xi+\sigma_{-}t)^{2}}{1+t}\right)d\xi$
(3.16) $\displaystyle=$ $\displaystyle O(1)\delta_{d}e^{-ct}.$ (3.17)
Also, we have
$\begin{array}[]{l}\displaystyle
J_{4}=\int_{-\frac{\sigma_{-}t}{2}}^{+\infty}\big{|}V^{b}_{\xi}\big{(}V^{d}-v_{m}\big{)}\big{|}+\big{|}V^{d}_{\xi}\big{(}V^{b}-v_{*}\big{)}\big{|}\,d\xi\\\
\displaystyle\quad:=J_{4}^{1}+J_{4}^{2}.\end{array}$
We can estimate
$\displaystyle J_{4}^{1}$ $\displaystyle=$
$\displaystyle\int_{-\frac{\sigma_{-}t}{2}}^{\infty}\big{|}V^{b}_{\xi}\big{(}V^{d}-v_{m}\big{)}\big{|}\,d\xi$
(3.18) $\displaystyle=$ $\displaystyle
O(1)\delta_{d}\delta_{b}^{2}(1+\delta_{b}t)^{-2}\int_{-\frac{\sigma_{-}t}{2}}^{\infty}\exp\left(-\frac{C_{d}(\xi+\sigma_{-}t)^{2}}{1+t}\right)d\xi$
(3.19) $\displaystyle=$ $\displaystyle
O(1)\delta_{d}(1+t)^{-3/2}\int_{-\infty}^{\infty}\exp\left(-C_{d}\eta^{2}\right)d\eta$
(3.20) $\displaystyle=$ $\displaystyle O(1)\delta_{d}(1+t)^{-3/2},$ (3.21)
and
$\displaystyle J_{4}^{2}$ $\displaystyle=$
$\displaystyle\int_{-\frac{\sigma_{-}t}{2}}^{\infty}\big{|}V^{d}_{\xi}\big{(}V^{b}-v_{*}\big{)}\big{|}\,d\xi$
(3.22) $\displaystyle=$ $\displaystyle
O(1)\delta_{d}\delta_{b}(1+\delta_{b}t)^{-1}(1+t)^{-1/2}\int_{-\frac{\sigma_{-}t}{2}}^{\infty}\exp\left(-\frac{C_{d}(\xi+\sigma_{-}t)^{2}}{1+t}\right)d\xi$
(3.23) $\displaystyle=$ $\displaystyle O(1)\delta_{d}(1+t)^{-1}.$ (3.24)
Thus we proved $\eqref{3.3}_{2}.$
Now we compute $(\ref{3.3})_{3}$:
$\bullet$ Interaction of transonic boundary layer solution and 3-rarefaction
wave:
$\displaystyle\int_{\mathbf{R}_{+}}\big{|}V^{b}_{\xi}\big{(}V^{r_{3}}-v^{*}\big{)}\big{|}+\big{|}V^{r_{3}}_{\xi}\big{(}V^{b}-v_{*}\big{)}\big{|}\,d\xi$
(3.25) $\displaystyle=$
$\displaystyle\int_{\left[-\sigma_{-}+\lambda_{3}(v^{*},\theta^{*})\right](1+t)}^{\infty}V^{b}_{\xi}\big{(}v^{*}-V^{r_{3}}\big{)}+V^{r_{3}}_{\xi}\big{(}V^{b}-v_{*}\big{)}\,d\xi$
(3.26) $\displaystyle=$ $\displaystyle O(1)\delta_{b}(1+\delta_{b}t)^{-1}$
(3.27) $\displaystyle=$ $\displaystyle
O(1)\min\big{\\{}\delta,(1+t)^{-1}\big{\\}}$ (3.28) $\displaystyle=$
$\displaystyle O(1)\delta^{\frac{1}{8}}(1+t)^{-\frac{7}{8}}.$ (3.29)
where in the first equality we have used the fact ${\rm iv)}$ in Lemma 2.3.
Then we verify $(\ref{3.3})_{4}$:
$\bullet$ Interaction of 1-rarefaction wave and viscous 2-contact wave:
First we have
$\begin{array}[]{ll}\displaystyle\int_{\mathbf{R}_{+}}\big{|}V^{d}_{\xi}(v_{m}-V^{r_{1}})\big{|}\,d\xi\\\
\displaystyle=\left\\{\int_{0}^{\left[-\sigma_{-}+\frac{\lambda_{1}(v_{m},\theta_{m})}{2}\right](1+t)}+\int_{\left[-\sigma_{-}+\frac{\lambda_{1}(v_{m},\theta_{m})}{2}\right](1+t)}^{+\infty}\right\\}\big{|}V^{d}_{\xi}\big{(}v_{m}-V^{r_{1}}\big{)}\big{|}\,d\xi\\\
\displaystyle:=J_{5}+J_{6}.\end{array}$
Then we can compute
$\displaystyle J_{5}$ $\displaystyle=$
$\displaystyle\int_{0}^{\left[-\sigma_{-}+\frac{\lambda_{1}(v_{m},\theta_{m})}{2}\right](1+t)}\big{|}V^{d}_{\xi}\big{(}v_{m}-V^{r_{1}}\big{)}\big{|}\,d\xi$
(3.30) $\displaystyle=$ $\displaystyle
O(1)\delta_{d}(1+t)^{-\frac{1}{2}}\int_{0}^{\left[-\sigma_{-}+\frac{\lambda_{1}(v_{m},\theta_{m})}{2}\right](1+t)}\exp\left(-\frac{C_{d}(\xi+\sigma_{-}t)^{2}}{1+t}\right)d\xi$
(3.31) $\displaystyle=$ $\displaystyle O(1)\delta_{d}e^{-ct},$ (3.32)
and
$\displaystyle J_{6}$ $\displaystyle=$
$\displaystyle\int_{\left[-\sigma_{-}+\frac{\lambda_{1}(v_{m},\theta_{m})}{2}\right](1+t)}^{\infty}\big{|}V^{d}_{\xi}\big{(}v_{m}-V^{r_{1}}\big{)}\big{|}\,d\xi$
(3.33) $\displaystyle=$ $\displaystyle
O(1)\delta_{d}\sup_{\xi\geq\left[-\sigma_{-}+\frac{\lambda_{1}(v_{m},\theta_{m})}{2}\right](1+t)}\big{(}v_{m}-V^{r_{1}}(t,\xi)\big{)}=O(1)\delta_{d}\,e^{-ct}.$
(3.34)
Similarly, we can estimate the interaction term
$\displaystyle\int_{\mathbf{R}_{+}}\big{|}V^{r_{1}}_{\xi}\big{(}V^{d}-v_{m}\big{)}\big{|}\,d\xi=O(1)\delta_{d}\,e^{-ct}.$
(3.35)
So $(\ref{3.3})_{4}$ is verified.
For $(\ref{3.3})_{5}$, that is
$\bullet$ Interaction of 3-rarefaction wave and viscous 2-contact wave, which
can be done similarly as $(\ref{3.3})_{4}$, we omit the details for
simplicity.
Finally, we prove $(\ref{3.3})_{6}$:
$\bullet$ Interaction of 1-rarefaction wave and 3-rarefaction wave:
Since $V^{r_{1}}_{\xi}\geq 0$, $V^{r_{3}}_{\xi}\leq 0$ and the facts ${\rm
iii)}$ and ${\rm iv)}$ in Lemma 2.3, one has
$\displaystyle\int_{\mathbf{R}_{+}}\big{|}V^{r_{1}}_{\xi}\big{(}V^{{r_{3}}}-v^{*}\big{)}\big{|}+\big{|}V^{r_{3}}_{\xi}\big{(}V^{{r_{1}}}-v_{m}\big{)}\big{|}\,d\xi$
(3.36) $\displaystyle=$ $\displaystyle
2\int_{\left[-\sigma_{-}+\lambda_{3}(v^{*},\theta^{*})\right](1+t)}^{+\infty}V^{r_{1}}_{\xi}\big{(}v^{*}-V^{{r_{3}}}\big{)}\,d\xi$
(3.38) $\displaystyle=$ $\displaystyle O(1)\delta_{r_{1}}e^{-ct}=O(1)\delta
e^{-ct}.$ (3.39)
Thus we justified (3.3). The proof of (3.4) can be done similarly, but the
decay rates with respect to the time $t$ may be higher. Therefore, we complete
the proof of the wave interaction estimates in Lemma 3.1. $\blacksquare$
With the wave interaction estimation Lemma 3.1 in hand, we have the following
Lemma:
Lemma 3.2.
$\displaystyle\displaystyle\|G(t)\|_{L^{1}}+\|H(t)\|_{L^{1}}=O(1)\delta^{\frac{1}{8}}(1+t)^{-\frac{13}{16}},$
(3.40) $\displaystyle\displaystyle\|G(t)\|+\|H(t)\|=O(1)\delta(1+t)^{-1}.$
(3.41)
Proof. We can compute
$\displaystyle G_{1}$ $\displaystyle=$
$\displaystyle\big{|}\big{(}P-P^{b}-P^{r_{1}}-P^{d}-P^{r_{3}}\big{)}_{\xi}\big{|}$
(3.42) $\displaystyle=$ $\displaystyle
O(1)\big{|}V^{b}_{\xi}\big{|}\big{(}|V^{r_{1}}-v_{*}|+\big{|}V^{d}-v_{m}\big{|}+|V^{r_{3}}-v^{*}|\big{)}$
(3.46)
$\displaystyle+O(1)\big{|}V^{d}_{\xi}\big{|}\big{(}\big{|}V^{b}-v_{*}\big{|}+|V^{r_{1}}-v_{m}|+|V^{r_{3}}-v^{*}|\big{)}$
$\displaystyle+O(1)\big{|}V^{r_{1}}_{\xi}\big{|}\big{(}\big{|}V^{b}-v_{*}\big{|}+\big{|}V^{d}-v_{m}\big{|}+|V^{r_{3}}-v^{*}|\big{)}$
$\displaystyle+O(1)\big{|}V^{r_{3}}_{\xi}\big{|}\big{(}\big{|}V^{b}-v_{*}\big{|}+|V^{r_{1}}-v_{m}|+\big{|}V^{d}-v^{*}\big{|}\big{)}.$
Thus by the wave interaction estimation Lemma 3.1, we have
$\|G_{1}\|_{L^{1}}=O(1)\delta^{\frac{1}{8}}(1+t)^{-\frac{13}{16}}.$
Similarly, $\|H_{1}\|_{L^{1}}=O(1)\delta^{\frac{1}{8}}(1+t)^{-\frac{13}{16}}$
can be obtained.
Now we estimate $\|G_{2}\|_{L^{1}}$ and $\|H_{2}\|_{L^{1}}$. Note that in
$G_{2}$, besides the wave interaction terms, there are the error terms due to
the $i-$rarefaction waves $(i=1,3).$ So we can write $G_{2}$ as
$\begin{array}[]{ll}G_{2}&\displaystyle=-\mu\bigg{(}\frac{U_{\xi}}{V}-\frac{U^{b}_{\xi}}{V^{b}}-\frac{U^{d}_{\xi}}{V^{d}}-\sum_{i=1,3}\frac{U^{r_{i}}_{\xi}}{V^{r_{i}}}\bigg{)}_{\xi}-\mu\bigg{(}\sum_{i=1,3}\frac{U^{r_{i}}_{\xi}}{V^{r_{i}}}\bigg{)}_{\xi}\\\
&\displaystyle:=G_{21}+G_{22}.\end{array}$
Since the wave interaction terms $G_{21}$ can be verified similarly as
$G_{1}$, we only compute the error terms $G_{22}$ due to rarefaction waves.
$\begin{array}[]{ll}\displaystyle\|G_{22}\|_{L^{1}}&\displaystyle=O(1)\sum_{i=1,3}(\|U^{r_{i}}_{\xi\xi}\|_{L^{1}}+\|(U^{r_{i}}_{\xi},V^{r_{i}}_{\xi})\|^{2})\\\
&\displaystyle=O(1)\delta^{\frac{1}{8}}(1+t)^{-\frac{13}{16}}\end{array}$
if we choose $q\geq 14$ in Lemma 2.3.
In $H_{2}$, besides the wave interaction terms and the error terms due to the
$i-$rarefaction waves $(i=1,3)$, there exists the error terms $H^{d}$ due to
the viscous $2-$contact wave. We can compute that
$\begin{array}[]{ll}\displaystyle\|H^{d}\|_{L^{1}}&\displaystyle=O(1)\delta_{d}(1+t)^{-2}\int_{\mathbf{R}_{+}}\exp{\left(-\frac{C_{d}(\xi+\sigma_{-}t)^{2}}{1+t}\right)}d\xi\\\
&\displaystyle=O(1)\delta(1+t)^{-\frac{3}{2}}.\end{array}$
The estimation of $\|G\|$ and $\|H\|$ can be done similarly, thus the details
are omitted. $\blacksquare$
### 3.2 Reformulation of the Problem
Put the perturbation $(\phi,\psi,\vartheta)(t,\xi)$ around the superposition
wave $(V,U,\Theta)(t,\xi)$ by
$\displaystyle(\phi,\psi,\vartheta)(t,\xi)=(v,u,\theta)(t,\xi)-(V,U,\Theta)(t,\xi),$
(3.47)
then by (1.7) and (3.1), the system for the perturbation
$(\phi,\psi,\vartheta)(t,\xi)$ becomes
$\displaystyle\begin{cases}\phi_{t}-\sigma_{-}\phi_{\xi}-\psi_{\xi}=0,\cr\psi_{t}-\sigma_{-}\psi_{\xi}+(p-P)_{\xi}=\mu\Big{(}\frac{u_{\xi}}{v}-\frac{U_{\xi}}{V}\Big{)}_{\xi}-G,\quad\quad\quad\quad~{}~{}~{}~{}t>0,~{}\xi>0,\cr\frac{R}{\gamma-1}\big{(}\vartheta_{t}-\sigma_{-}\vartheta_{\xi}\big{)}+\big{(}pu_{\xi}-PU_{\xi}\big{)}=\kappa\left(\frac{\theta_{\xi}}{v}-\frac{\Theta_{\xi}}{V}\right)_{\xi}+\mu\Big{(}\frac{(u_{\xi})^{2}}{v}-\frac{(U_{\xi})^{2}}{V}\Big{)}-H,\cr(\psi_{0},\psi_{0},\vartheta_{0})(\xi):=(\phi,\psi,\vartheta)(0,\xi)\rightarrow(0,0,0),~{}~{}\text{as}~{}~{}\xi\rightarrow+\infty,\cr(\phi,\psi,\vartheta)(t,\xi=0)=(V^{d},U^{d},\Theta^{d})(t,\xi=0)-(v_{m},u_{m},\theta_{m}).\end{cases}$
(3.48)
Define the solution space $\mathbf{X}(0,T)$ to the above system by
$\displaystyle\mathbf{X}(0,T)$ $\displaystyle:=$
$\displaystyle\Big{\\{}~{}(\phi,\psi,\vartheta)(t,\xi)\,\Big{|}\,(\phi,\psi,\vartheta)\in
C\left([0,T];H^{1}\right),~{}\phi_{\xi}\in L^{2}\left(0,T;L^{2}\right),$
(3.50) $\displaystyle~{}~{}~{}\big{(}\psi_{\xi},\vartheta_{\xi}\big{)}\in
L^{2}\left(0,T;H^{1}\right),~{}N(T)=:\sup_{0\leq t\leq
T}\|(\phi,\psi,\vartheta)(t)\|_{1}\leq\varepsilon_{0}\Big{\\}},$
Here
$\varepsilon_{0}\leq\frac{1}{4}\min\bigg{\\{}\inf\limits_{\mathbf{R}_{+}\times\mathbf{R}_{+}}V(t,\xi),\inf\limits_{\mathbf{R}_{+}\times\mathbf{R}_{+}}\Theta(t,\xi)\bigg{\\}}$
is a suitably small and positive constant to be determined.
Since the proof for the local existence of the solution to the system
$(\ref{31})$ is standard, the details are omitted. To prove Theorem 2.1, it is
sufficient to prove the following _a priori_ estimate by combining the local
existence of the solution and the continuation process.
Proposition 3.1 (_A priori_ estimate) _Let
$(\phi,\psi,\vartheta)\in\mathbf{X}(0,T)$ be a solution to the system
$(\ref{31})$ in the time interval $[0,T)$ with suitably small
$\varepsilon_{0}$, and the conditions in Theorem 2.1 hold. Then there exist a
positive constant $C$ independent of $T$ such that _
$\displaystyle\|(\phi,\psi,\vartheta)(t)\|^{2}_{1}+\int^{t}_{0}\left[\|\phi_{\xi}(\tau)\|^{2}+\|(\psi_{\xi},\vartheta_{\xi})(\tau)\|^{2}_{1}\right]\,d\tau$
(3.51)
$\displaystyle{}+\int^{t}_{0}\|\sqrt{(U^{b}_{\xi},U^{r_{1}}_{\xi},U^{r_{3}}_{\xi})}(\phi,\vartheta)(\tau)\|^{2}d\tau\leq
C\left(\|(\phi_{0},\psi_{0},\vartheta_{0})\|^{2}_{1}+\delta^{\frac{1}{6}}\right).$
(3.52)
### 3.3 Energy estimates
To prove Proposition 3.1, we need the following several lemmas. First we give
the following boundary estimates whose proof can be found in [21].
Lemma 3.3 (Boundary Estimates)[21] _There exists the positive constant $C$
such that for any_ $t>0$,
$\displaystyle\int^{t}_{0}|(\phi,\psi,\vartheta)(\tau,0)|^{2}\,d\tau\leq
C\delta,$
$\displaystyle\int^{t}_{0}\big{(}\big{|}\psi\psi_{\xi}\big{|}+\big{|}\vartheta\vartheta_{\xi}\big{|}\big{)}(\tau,0)\,d\tau\leq
C\delta+C\delta\int^{t}_{0}\|(\psi_{\xi},\vartheta_{\xi})(\tau)\|_{1}^{2}d\tau.$
$\displaystyle\int^{t}_{0}\big{(}|\phi_{\tau}\psi|+\phi_{\xi}^{2}\big{)}(\tau,0)\,d\tau\leq
C\delta+\epsilon\int^{t}_{0}\|\psi_{\xi\xi}(\tau)\|^{2}d\tau+C_{\epsilon}\int_{0}^{t}\|\psi_{\xi}(\tau)\|^{2}d\tau,$
$\displaystyle\int^{t}_{0}\big{(}\big{|}\psi_{\tau}\psi_{\xi}\big{|}+\psi_{\xi}^{2}\big{)}(\tau,0)\,d\tau\leq
C\delta+\epsilon\int^{t}_{0}\|\psi_{\xi\xi}(\tau)\|^{2}d\tau+C_{\epsilon}\int_{0}^{t}\|\psi_{\xi}(\tau)\|^{2}d\tau,$
$\displaystyle\int^{t}_{0}\big{(}\big{|}\vartheta_{\tau}\vartheta_{\xi}\big{|}+\vartheta_{\xi}^{2}\big{)}(\tau,0)\,d\tau\leq
C\delta+\epsilon\int^{t}_{0}\|\vartheta_{\xi\xi}(\tau)\|^{2}d\tau+C_{\epsilon}\int_{0}^{t}\|\vartheta_{\xi}(\tau)\|^{2}d\tau,$
_where $\epsilon>0$ is a constant to be determined and $C_{\epsilon}$ is the
constant depending on $\epsilon$_.
Lemma 3.4 _Let $(\phi,\psi,\vartheta)\in\mathbf{X}(0,T)$ be a solution to the
system $(\ref{31})$ for some positive T and suitably small
$\varepsilon_{0}>0$, and the conditions in Theorem 2.1 hold. Then there exist
a positive constant $C$ such that_
$\displaystyle\|(\phi,\psi,\vartheta)(t)\|^{2}_{1}+\int^{t}_{0}\|\phi_{\xi}(\tau)\|^{2}+\|(\psi_{\xi},\vartheta_{\xi})(\tau)\|^{2}_{1}d\tau$
(3.54)
$\displaystyle{}+\int^{t}_{0}\|\sqrt{(U^{b}_{\xi},U^{r_{1}}_{\xi},U^{r_{3}}_{\xi})}(\phi,\vartheta)(\tau)\|^{2}d\tau$
$\displaystyle\leq$ $\displaystyle
C\left(\|(\phi_{0},\psi_{0},\vartheta_{0})\|^{2}_{1}+\delta^{\frac{1}{6}}\right)+C\delta^{\frac{1}{8}}\int^{t}_{0}(1+\tau)^{-\frac{13}{12}}\|(\phi,\psi,\vartheta)(\tau)\|^{2}d\tau$
(3.56)
$\displaystyle{}+C\delta\int^{t}_{0}\int_{\mathbf{R}_{+}}(1+\tau)^{-1}\exp\left(-\frac{C_{d}(\xi+\sigma_{-}\tau)^{2}}{1+\tau}\right)|(\phi,\vartheta)|^{2}d\xi
d\tau.$
_Proof_. Step 1. Define
$\displaystyle\Phi(\eta):=\eta-\ln\eta-1.$ (3.57)
Under the a priori assumption, there exist a positive constant $C$ such that
$\displaystyle C^{-1}\eta^{2}\leq\Phi(\eta)\leq C\eta^{2}.$ (3.58)
Let
$\displaystyle
E:=R\Theta\Phi\left(\frac{v}{V}\right)+\frac{1}{2}\psi^{2}+\frac{R}{\gamma-1}\Theta\Phi\left(\frac{\theta}{\Theta}\right),$
(3.59) $\displaystyle
F:=\sigma_{-}E+(P-p)\psi+\mu\bigg{(}\frac{u_{\xi}}{v}-\frac{U_{\xi}}{V}\bigg{)}\psi+\kappa\bigg{(}\frac{\theta_{\xi}}{v}-\frac{\Theta_{\xi}}{V}\bigg{)}\frac{\vartheta}{\theta}.$
(3.60)
Then a complicated but direct computation gives
$\displaystyle
E_{t}-F_{\xi}+\frac{\mu\Theta}{v\theta}\psi_{\xi}^{2}+\frac{\kappa\Theta}{v\theta^{2}}\vartheta_{\xi}^{2}+P(U^{b}_{\xi}+U^{r_{1}}_{\xi}+U^{r_{3}}_{\xi})\left[\gamma\Phi\left(\frac{v}{V}\right)+\Phi\left(\frac{\theta
V}{v\Theta}\right)\right]=Q,$ (3.61)
where
$\displaystyle Q$ $\displaystyle=$ $\displaystyle-
PU^{d}_{\xi}\left[\gamma\Phi\left(\frac{v}{V}\right)+\Phi\left(\frac{\theta
V}{v\Theta}\right)\right]-\bigg{(}G\psi+H\frac{\vartheta}{\theta}\bigg{)}$
(3.64) $\displaystyle+\bigg{[}\frac{\mu U_{\xi}\phi\psi_{\xi}}{vV}+\frac{2\mu
U_{\xi}\vartheta\psi_{\xi}}{v\theta}+\frac{\kappa\Theta\Theta_{\xi}\phi\vartheta_{\xi}}{vV\theta^{2}}+\kappa\frac{\Theta_{\xi}\vartheta\vartheta_{\xi}}{v\theta^{2}}-\frac{\mu(U_{\xi})^{2}\phi\vartheta}{vV\theta}-\frac{\kappa(\Theta_{\xi})^{2}\phi\vartheta}{vV\theta^{2}}\bigg{]}$
$\displaystyle+\bigg{[}\kappa\left(\frac{\Theta_{\xi}}{V}\right)_{\xi}+\mu\frac{(U_{\xi})^{2}}{V}+H\bigg{]}\left[(\gamma-1)\Phi\left(\frac{v}{V}\right)+\Phi\left(\frac{\theta}{\Theta}\right)-\frac{\vartheta^{2}}{\Theta\theta}\right]$
$\displaystyle=:$ $\displaystyle\sum_{i=1}^{i=4}Q_{i}.$ (3.65)
Integrating $(\ref{32})$ over $[0,t]\times\mathbf{R}_{+}$ yields
$\displaystyle\|(\phi,\psi,\vartheta)\|^{2}+\int^{t}_{0}\|(\psi_{\xi},\vartheta_{\xi})(\tau)\|^{2}d\tau+\int_{0}^{t}\|\sqrt{(U^{b}_{\xi},U^{r_{1}}_{\xi},U^{r_{3}}_{\xi})}(\phi,\vartheta)(\tau)\|^{2}d\tau$
(3.66) $\displaystyle\leq$ $\displaystyle
C\|(\phi_{0},\psi_{0},\vartheta_{0})\|^{2}+C\int_{0}^{t}|F(\tau,\xi=0)|d\tau+\sum_{i=1}^{i=4}I_{i},$
(3.67)
where $\displaystyle I_{i}=O(1)\int^{t}_{0}\int_{\mathbf{R}_{+}}Q_{i}\,d\xi
d\tau$.
From the boundary estimates in Lemma 3.3, we have
$\int_{0}^{t}|F(\tau,\xi=0)|d\tau\leq
C\delta+C\delta\int^{t}_{0}\|(\psi_{\xi},\vartheta_{\xi})(\tau)\|_{1}^{2}d\tau.$
(3.68)
We can compute that
$\displaystyle I_{1}\leq
C\delta\int^{t}_{0}\int_{\mathbf{R}_{+}}(1+\tau)^{-1}\exp\left(-\frac{C_{d}(\xi+\sigma_{-}\tau)^{2}}{1+\tau}\right)|(\phi,\vartheta)|^{2}d\xi
d\tau$ (3.69)
and
$\displaystyle I_{2}$ $\displaystyle\leq$ $\displaystyle
C\int^{t}_{0}\|(\psi,\vartheta)(\tau)\|_{L^{\infty}}(\|G(\tau)\|_{L^{1}}+\|H(\tau)\|_{L^{1}})\,d\tau$
(3.70) $\displaystyle\leq$ $\displaystyle
C\delta^{\frac{1}{8}}\int^{t}_{0}(1+\tau)^{-\frac{13}{16}}\|(\psi_{\xi},\vartheta_{\xi})(\tau)\|^{\frac{1}{2}}\|(\psi,\vartheta)(\tau)\|^{\frac{1}{2}}d\tau$
(3.71) $\displaystyle\leq$
$\displaystyle\epsilon\int^{t}_{0}\|(\psi_{\xi},\vartheta_{\xi})(\tau)\|^{2}d\tau+C_{\epsilon}\delta^{\frac{1}{6}}\bigg{(}1+\int^{t}_{0}(1+\tau)^{-\frac{13}{12}}\|(\psi,\vartheta)(\tau)\|^{2}d\tau\bigg{)}$
(3.72)
where and in the sequel $\epsilon>0$ is a small constant to be determined and
$C_{\epsilon}$ is the positive constant depending on $\epsilon$.
Now we calculate $I_{3}$. By Cauchy inequality, we have
$I_{3}\leq\epsilon\int_{0}^{t}\|(\psi_{\xi},\vartheta_{\xi})\|^{2}d\tau+C_{\epsilon}\int_{0}^{t}\int_{\mathbf{R}_{+}}|(U_{\xi},\Theta_{\xi})|^{2}\cdot|(\phi,\vartheta)|^{2}d\xi
d\tau.$ (3.73)
By Lemma 2.1-Lemma2.3, one has
$\displaystyle|(U_{\xi},\Theta_{\xi})|^{2}\leq
C\bigg{[}\delta^{\frac{1}{2}}(1+t)^{-\frac{3}{2}}+\frac{\delta^{4}}{(1+\delta\xi)^{4}}+\delta(1+t)^{-1}\exp\left(-\frac{C_{d}(\xi+\sigma_{-}t)^{2}}{1+t}\right)\bigg{]}.$
(3.74)
By the techniques in [19]
$\begin{array}[]{ll}|f(t,\xi)|&\displaystyle=|f(t,\xi=0)+\int_{0}^{\xi}f_{\xi}(t,\xi)d\xi|\\\
&\displaystyle\leq|f(t,\xi=0)|+\sqrt{\xi}\|f_{\xi}\|,\end{array}$
we have
$\displaystyle\displaystyle\int^{t}_{0}\int_{\mathbf{R}_{+}}\frac{\delta^{4}}{(1+\delta\xi)^{4}}|(\phi,\vartheta)|^{2}d\xi
d\tau$ (3.75) $\displaystyle\leq$ $\displaystyle
C\delta^{3}\int^{t}_{0}|(\phi,\vartheta)(\tau,\xi=0)|^{2}d\tau+C\int_{0}^{t}\left[\|(\phi_{\xi},\vartheta_{\xi})\|^{2}\int_{\mathbf{R}_{+}}\frac{\delta^{4}\xi}{(1+\delta\xi)^{4}}d\xi\right]d\tau$
(3.76) $\displaystyle\leq$ $\displaystyle
C\delta\left(1+\int_{0}^{t}\|(\phi_{\xi},\vartheta_{\xi})(\tau)\|^{2}d\tau\right).$
(3.77)
Substituting (3.74) and (3.75) into (3.73) yields
$\displaystyle I_{3}$ $\displaystyle\leq$ $\displaystyle
C(\epsilon+\delta)\int^{t}_{0}\|(\psi_{\xi},\vartheta_{\xi})(\tau)\|^{2}d\tau+C\delta\int_{0}^{t}\|\phi_{\xi}(\tau)\|^{2}d\tau$
(3.80)
$\displaystyle{}+C\delta+C\delta^{\frac{1}{2}}\int^{t}_{0}(1+\tau)^{-\frac{3}{2}}\|(\phi,\vartheta)(\tau)\|^{2}d\tau$
$\displaystyle{}+C\delta\int^{t}_{0}\int_{\mathbf{R}_{+}}(1+\tau)^{-1}\exp\left(-\frac{C_{d}(\xi+\sigma_{-}\tau)^{2}}{1+\tau}\right)|(\phi,\vartheta)|^{2}d\xi
d\tau.$
Then we have
$\displaystyle I_{4}$ $\displaystyle=$ $\displaystyle
O(1)\int_{0}^{t}\int_{\mathbf{R}_{+}}|(\Theta_{\xi\xi},V_{\xi}^{2},U_{\xi}^{2},\Theta_{\xi}^{2},H)||(\phi,\vartheta)|^{2}d\xi
d\tau.$ (3.81)
So $I_{4}$ can be estimated similarly as $I_{2}$ and $I_{3}$.
Combining (3.68), (3.69), (3.70), (3.73), (3.80) and (3.81), and then choosing
$\delta$ and $\epsilon$ suitably small yield that
$\displaystyle\|(\phi,\psi,\vartheta)(t)\|^{2}+\int^{t}_{0}\|(\psi_{\xi},\vartheta_{\xi})(\tau)\|^{2}d\tau+\int^{t}_{0}\|\sqrt{(U^{b}_{\xi},U^{r_{1}}_{\xi},U^{r_{3}}_{\xi})}(\phi,\vartheta)(\tau)\|^{2}d\tau$
(3.82) $\displaystyle\leq$ $\displaystyle
C\left(\|(\phi_{0},\psi_{0},\vartheta_{0})\|^{2}+\delta^{\frac{1}{6}}\right)+C\delta^{\frac{1}{8}}\int^{t}_{0}\|(\phi_{\xi},\psi_{\xi\xi},\vartheta_{\xi\xi})(\tau)\|^{2}d\tau$
(3.85)
$\displaystyle{}+C\delta^{\frac{1}{8}}\int^{t}_{0}(1+\tau)^{-\frac{13}{12}}\|(\phi,\psi,\vartheta)(\tau)\|^{2}d\tau$
$\displaystyle{}+C\delta\int^{t}_{0}\int_{\mathbf{R}_{+}}(1+\tau)^{-1}\exp\left(-\frac{C_{d}(\xi+\sigma_{-}\tau)^{2}}{1+\tau}\right)|(\phi,\vartheta)|^{2}d\xi
d\tau.$
Step 2. Differentiating $(\ref{31})_{1}$ w.r.t. $\xi$ and multiplying it by
$\frac{\phi_{\xi}}{v^{2}}$ yield
$\displaystyle\left(\frac{\phi_{\xi}^{2}}{2v^{2}}\right)_{t}-\sigma_{-}\left(\frac{\phi_{\xi}^{2}}{2v^{2}}\right)_{\xi}+\frac{u_{x}\phi_{\xi}^{2}}{v^{3}}-\frac{\phi_{\xi}\psi_{\xi\xi}}{v^{2}}=0.$
(3.86)
Multiplying $(\ref{31})_{2}$ by $\frac{\phi_{\xi}}{v}$ gives
$\displaystyle\left(\frac{\phi_{\xi}\psi}{v}\right)_{t}-\left(\frac{\phi_{t}\psi}{v}\right)_{\xi}+\frac{(p-P)_{\xi}\phi_{\xi}}{v}$
(3.87) $\displaystyle=$
$\displaystyle-\frac{U_{\xi}\phi_{\xi}\psi}{v^{2}}+\frac{V_{\xi}\psi\psi_{\xi}}{v^{2}}+\sigma_{-}\frac{\phi_{\xi}\psi_{\xi}}{v}+\mu\left(\frac{u_{\xi}}{v}-\frac{U_{\xi}}{V}\right)_{\xi}\frac{\phi_{\xi}}{v}-G\frac{\phi_{\xi}}{v}.$
(3.88)
$\mu\times(\ref{34})-(\ref{35})$ gives
$\displaystyle\left(\frac{\mu\phi_{\xi}^{2}}{2v^{2}}-\frac{\phi_{\xi}\psi}{v}\right)_{t}-\left(\frac{\sigma_{-}\mu\phi_{\xi}^{2}}{2v^{2}}-\frac{\phi_{t}\psi}{v}\right)_{\xi}-\frac{p_{v}}{v}\phi_{\xi}^{2}$
(3.89) $\displaystyle=$
$\displaystyle\frac{U_{\xi}\phi_{\xi}\psi}{v^{2}}-\frac{V_{\xi}\psi\psi_{\xi}}{v^{2}}-\sigma_{-}\frac{\phi_{\xi}\psi_{\xi}}{v}+\mu\frac{V_{\xi}\phi_{\xi}\psi_{\xi}}{v^{3}}-\mu\frac{U_{\xi}\phi_{\xi}^{2}}{v^{3}}+\mu\bigg{(}\frac{U_{\xi}\phi}{vV}\bigg{)}_{\xi}\frac{\phi_{\xi}}{v}$
(3.91)
$\displaystyle{}+\frac{p_{\theta}\phi_{\xi}\vartheta_{\xi}}{v}+\frac{V_{\xi}(p_{v}-P_{V})\phi_{\xi}}{v}+\frac{\Theta_{\xi}(p_{\theta}-P_{\Theta})\phi_{\xi}}{v}+G\frac{\phi_{\xi}}{v}.$
Integrating $(\ref{70})$ over $[0,t]\times\mathbf{R}_{+}$, using the boundary
estimations in Lemma3.3 and choosing $\delta$ suitably small yield
$\displaystyle\|\phi_{\xi}(t)\|^{2}+\int^{t}_{0}\|\phi_{\xi}(\tau)\|^{2}d\tau$
(3.92) $\displaystyle\leq$ $\displaystyle
C\left(\|(\psi_{0},\phi_{0\xi})\|^{2}+\delta^{\frac{1}{6}}\right)+C\delta^{\frac{1}{8}}\int^{t}_{0}(1+\tau)^{-\frac{13}{12}}\|(\phi,\psi,\vartheta)(\tau)\|^{2}d\tau$
(3.95)
$\displaystyle{}+\int^{t}_{0}\Big{\\{}C\Big{(}\delta^{\frac{1}{8}}+\epsilon\Big{)}\|(\psi_{\xi\xi},\vartheta_{\xi\xi})(\tau)\|^{2}+C_{\epsilon}\|\psi_{\xi}(\tau)\|^{2}\Big{\\}}\,d\tau$
$\displaystyle{}+C\delta\int^{t}_{0}\int_{\mathbf{R}_{+}}(1+\tau)^{-1}\exp\left(-\frac{C_{d}(\xi+\sigma_{-}\tau)^{2}}{1+\tau}\right)|(\phi,\vartheta)|^{2}d\xi
d\tau.$
Step 3. Multiplying $(\ref{31})_{2}$ by $-\psi_{\xi\xi}$, then
$\displaystyle\left(\frac{\psi_{\xi}^{2}}{2}\right)_{t}-\left(\psi_{t}\psi_{\xi}-\frac{\sigma_{-}}{2}\psi_{\xi}^{2}\right)_{\xi}+\mu\frac{\psi_{\xi\xi}^{2}}{v}=\bigg{[}(p-P)_{\xi}+\frac{\mu
v_{\xi}\psi_{\xi}}{v^{2}}+\mu\left(\frac{U_{\xi}\phi}{vV}\right)_{\xi}+G\bigg{]}\psi_{\xi\xi}.$
(3.96)
Integrating $(\ref{36})$ over $[0,t]\times\mathbf{R}_{+}$ yields
$\displaystyle\|\psi_{\xi}(t)\|^{2}+\int^{t}_{0}\|\psi_{\xi\xi}(\tau)\|^{2}d\tau$
(3.97) $\displaystyle\leq$ $\displaystyle
C\left(\|(\phi_{0},\psi_{0},\vartheta_{0})\|^{2}_{1}+\delta^{\frac{1}{6}}\right)++C\delta^{\frac{1}{8}}\int^{t}_{0}(1+\tau)^{-\frac{13}{12}}\|(\phi,\psi,\vartheta)(\tau)\|^{2}d\tau$
(3.100)
$\displaystyle{}+\int^{t}_{0}\Big{\\{}C\Big{(}\delta^{\frac{1}{8}}+\epsilon\Big{)}\|(\psi_{\xi\xi},\vartheta_{\xi\xi})(\tau)\|^{2}+C_{\epsilon}\|\psi_{\xi}(\tau)\|^{2}\Big{\\}}\,d\tau$
$\displaystyle{}+C\delta\int^{t}_{0}\int_{\mathbf{R}_{+}}(1+\tau)^{-1}\exp\left(-\frac{C_{d}(\xi+\sigma_{-}\tau)^{2}}{1+\tau}\right)\left(\phi^{2}+\vartheta^{2}\right)d\xi
d\tau$
where we use the following estimate
$\displaystyle\int^{t}_{0}\int_{\mathbf{R}_{+}}\big{|}\phi_{\xi}\psi_{\xi}\psi_{\xi\xi}\big{|}\,d\xi
d\tau$ $\displaystyle\leq$
$\displaystyle\int^{t}_{0}\|\phi_{\xi}(\tau)\|\|\psi_{\xi\xi}(\tau)\|\|\psi_{\xi}(\tau)\|_{L^{\infty}}d\tau$
(3.101) $\displaystyle\leq$
$\displaystyle\int^{t}_{0}\|\phi_{\xi}(\tau)\|\|\psi_{\xi\xi}(\tau)\|^{\frac{3}{2}}\|\psi_{\xi}(\tau)\|^{\frac{1}{2}}d\tau$
(3.102) $\displaystyle\leq$
$\displaystyle\epsilon\int^{t}_{0}\|\psi_{\xi\xi}(\tau)\|^{2}d\tau+C_{\epsilon}\varepsilon_{0}^{4}\int^{t}_{0}\|\psi_{\xi}(\tau)\|^{2}d\tau.$
(3.103)
Multiplying $(\ref{31})_{3}$ by $-\vartheta_{\xi\xi}$, then
$\displaystyle\frac{R}{\gamma-1}\Bigg{[}\left(\frac{\vartheta_{\xi}^{2}}{2}\right)_{t}-\left(\vartheta_{t}\vartheta_{\xi}-\frac{\sigma_{-}}{2}\vartheta_{\xi}^{2}\right)_{\xi}\Bigg{]}+\frac{\kappa}{v}\vartheta_{\xi\xi}^{2}$
(3.104) $\displaystyle=$
$\displaystyle\Bigg{[}\big{(}pu_{\xi}-PU_{\xi}\big{)}+\frac{\kappa
v_{\xi}\vartheta_{\xi}}{v^{2}}+\kappa\left(\frac{\Theta_{\xi}\phi}{vV}\right)_{\xi}-\mu\bigg{(}\frac{(u_{\xi})^{2}}{v}-\frac{(U_{\xi})^{2}}{V}\bigg{)}+H\Bigg{]}\vartheta_{\xi\xi}.$
(3.105)
Integrating $(\ref{37})$ over $[0,t]\times\mathbf{R}_{+}$ yields
$\displaystyle\|\vartheta_{\xi}(t)\|^{2}+\int^{t}_{0}\|\vartheta_{\xi\xi}(\tau)\|^{2}d\tau$
(3.106) $\displaystyle\leq$ $\displaystyle
C\left(\|(\phi_{0},\psi_{0},\vartheta_{0})\|^{2}_{1}+\delta^{\frac{1}{6}}\right)++C\delta^{\frac{1}{8}}\int^{t}_{0}(1+\tau)^{-\frac{13}{12}}\|(\phi,\psi,\vartheta)(\tau)\|^{2}d\tau$
(3.109)
$\displaystyle{}+\int^{t}_{0}\Big{\\{}C\Big{(}\delta^{\frac{1}{8}}+\epsilon\Big{)}\|(\psi_{\xi\xi},\vartheta_{\xi\xi})(\tau)\|^{2}+C_{\epsilon}\|\vartheta_{\xi}(\tau)\|^{2}\Big{\\}}\,d\tau$
$\displaystyle{}+C\delta\int^{t}_{0}\int_{\mathbf{R}_{+}}(1+\tau)^{-1}\exp\left(-\frac{C_{d}(\xi+\sigma_{-}\tau)^{2}}{1+\tau}\right)|(\phi,\vartheta)|^{2}d\xi
d\tau,$
where we use the following estimate
$\displaystyle\int^{t}_{0}\int_{\mathbf{R}_{+}}\big{|}\phi_{\xi}\vartheta_{\xi}\vartheta_{\xi\xi}\big{|}+\big{|}\psi_{\xi}^{2}\vartheta_{\xi\xi}\big{|}\,d\xi
d\tau$ (3.110) $\displaystyle\leq$
$\displaystyle\epsilon\int^{t}_{0}\|(\psi_{\xi\xi},\vartheta_{\xi\xi})(\tau)\|^{2}d\tau+C_{\epsilon}\varepsilon_{0}^{4}\int^{t}_{0}\|(\psi_{\xi},\vartheta_{\xi})(\tau)\|^{2}d\tau.$
(3.111)
Combining $(\ref{81}),(\ref{82}),(\ref{83})$ and $(\ref{84})$ and choosing
$\delta$, $\epsilon$ and $\varepsilon_{0}$ suitably small, we can complete the
proof of Lemma 3.4. $\blacksquare$
Now to close the a priori estimates, the remaining thing is to compute the
last term in the right-hand side of $(\ref{(3.22)})$ which comes from the
viscous contact wave. Here we use the method of the heat kernel estimation
invented in [2].
Lemma 3.5.[2] _Suppose that $h(t,\xi)$ satisfies_
$\displaystyle h\in
L^{\infty}\left(0,T;L^{2}(\mathbf{R}_{+})\right),~{}~{}h_{\xi}\in
L^{2}\left(0,T;L^{2}(\mathbf{R}_{+})\right),~{}~{}h_{t}-\sigma_{-}h_{\xi}\in
L^{2}\left(0,T;H^{-1}(\mathbf{R}_{+})\right),$ (3.112)
_then_
$\displaystyle\int_{0}^{t}\int_{\mathbf{R}_{+}}(1+\tau)^{-1}\exp\left(-\frac{2a(\xi+\sigma_{-}\tau)^{2}}{1+\tau}\right)h^{2}\,d\xi
d\tau$ (3.113) $\displaystyle\leq$ $\displaystyle
C_{a}\left\\{\|h(0,\cdot)\|^{2}+\int_{0}^{t}\Big{[}h^{2}(\tau,0)+\|h_{\xi}(\tau,\cdot)\|^{2}+\big{\langle}h_{\tau}-\sigma_{-}h_{\xi},(w^{a})^{2}h\big{\rangle}_{H^{-1}\times
H^{1}}\Big{]}d\tau\right\\}$ (3.114)
_where_
$\displaystyle
w^{a}(t,\xi)=-(1+t)^{-\frac{1}{2}}\int_{\xi+\sigma_{-}t}^{\infty}\exp\left(-\frac{ay^{2}}{1+t}\right)dy,$
(3.115)
_and $a>0$ is a constant to be determined_.
Based on Lemma 3.5, we have the desired estimates in the following Lemma.
Lemma 3.6 _There exist a uniform constant $C>0$ such that if $\delta$ and
$\varepsilon_{0}$ are small enough, then we have_
$\displaystyle\int_{0}^{t}\int_{\mathbf{R}_{+}}(1+\tau)^{-1}\exp\left(-\frac{C_{d}(\xi+\sigma_{-}\tau)^{2}}{1+\tau}\right)|(\phi,\psi,\vartheta)|^{2}\,d\xi
d\tau$ (3.116) $\displaystyle\leq$ $\displaystyle
C\left(\|(\phi_{0},\psi_{0},\vartheta_{0})\|^{2}_{1}+\delta^{\frac{1}{6}}\right)+C\delta^{\frac{1}{8}}\int^{t}_{0}(1+\tau)^{-\frac{13}{12}}\|(\phi,\psi,\vartheta)(\tau)\|^{2}d\tau.$
(3.117)
_Proof_. Step 1. First, let
$\displaystyle h=P\phi+\frac{R}{\gamma-1}\vartheta$ (3.118)
in Lemma 3.4. Then we only need to control the last term of (3.113) on the
right hand side.
We have from the energy equation $\eqref{31}_{3}$,
$\displaystyle h_{t}-\sigma_{-}h_{\xi}$ $\displaystyle=$
$\displaystyle(P-p)\psi_{\xi}+U_{\xi}(P-p)+\big{(}P_{t}-\sigma_{-}P_{\xi}\big{)}\phi$
(3.120)
$\displaystyle{}+\kappa\big{(}\frac{\theta_{\xi}}{v}-\frac{\Theta_{\xi}}{V}\big{)}_{\xi}+\mu\big{(}\frac{(u_{\xi})^{2}}{v}-\frac{(U_{\xi})^{2}}{V}\big{)}-H.$
Thus
$\displaystyle\int_{0}^{t}\big{\langle}h_{\tau}-\sigma_{-}h_{\xi},(w^{a})^{2}h\big{\rangle}_{H^{-1}\times
H^{1}}\,d\tau$ (3.121) $\displaystyle=$
$\displaystyle-\kappa\int^{t}_{0}\big{[}(\frac{\theta_{\xi}}{v}-\frac{\Theta_{\xi}}{V})(w^{a})^{2}h\big{]}(\tau,0)\,d\tau-\kappa\int^{t}_{0}\int_{\mathbf{R}_{+}}(\frac{\theta_{\xi}}{v}-\frac{\Theta_{\xi}}{V})[(w^{a})^{2}h]_{\xi}\,d\xi
d\tau$ (3.124)
$\displaystyle{}+\int^{t}_{0}\int_{\mathbf{R}_{+}}\big{[}(P-p)\psi_{\xi}+U_{\xi}(P-p)+\big{(}P_{t}-\sigma_{-}P_{\xi}\big{)}\phi$
$\displaystyle\qquad\qquad\qquad+\mu\big{(}\frac{(u_{\xi})^{2}}{v}-\frac{(U_{\xi})^{2}}{V}\big{)}-H\big{]}\left(w^{a}\right)^{2}h\,d\xi
d\tau.$
Notice that
$\displaystyle\|w^{a}(t)\|_{L^{\infty}}\leq C_{a},\quad
w^{a}_{\xi}=(1+t)^{-\frac{1}{2}}\exp\left(-\frac{a(\xi+\sigma_{-}t)^{2}}{1+t}\right),\quad\big{|}w^{a}_{t}-\sigma_{-}w^{a}_{\xi}\big{|}\leq
C_{a}(1+t)^{-1},$ (3.125)
$\displaystyle\big{|}P_{t}-\sigma_{-}P_{\xi}\big{|}\leq
C\bigg{\\{}U^{b}_{\xi}+U^{r_{1}}_{\xi}+U^{r_{3}}_{\xi}+\delta(1+t)^{-1}\exp\left(-\frac{C_{d}(\xi+\sigma_{-}t)^{2}}{1+t}\right)\bigg{\\}},$
(3.126)
thus to control terms on the right hand side of $(\ref{47})$, we only consider
the term $(w^{a})^{2}(P-p)h\psi_{\xi}$. By using the mass equation
$\eqref{31}_{1}$ and the momentum equation $\eqref{31}_{2}$ again, we have
$\displaystyle\left(w^{a}\right)^{2}(P-p)h\psi_{\xi}$ $\displaystyle=$
$\displaystyle\frac{(w^{a})^{2}[\gamma
P\phi-(\gamma-1)h]h(\phi_{t}-\sigma_{-}\phi_{\xi})}{v}$ (3.127)
$\displaystyle=$ $\displaystyle\frac{\gamma
P(w^{a})^{2}h}{2v}\Big{[}\big{(}\phi^{2}\big{)}_{t}-\sigma_{-}\big{(}\phi^{2}\big{)}_{\xi}\Big{]}-\frac{(\gamma-1)(w^{a})^{2}h^{2}}{v}\big{(}\phi_{t}-\sigma_{-}\phi_{\xi}\big{)}$
(3.128) $\displaystyle=$ $\displaystyle\left(\frac{\gamma
P(w^{a})^{2}h\phi^{2}-2(\gamma-1)(w^{a})^{2}\phi h^{2}}{2v}\right)_{t}$
(3.133) $\displaystyle{}-\sigma_{-}\left(\frac{\gamma
P(w^{a})^{2}h\phi^{2}-2(\gamma-1)(w^{a})^{2}\phi h^{2}}{2v}\right)_{\xi}$
$\displaystyle{}-\frac{\gamma Ph\phi^{2}-2(\gamma-1)\phi
h^{2}}{v}w^{a}\big{(}w^{a}_{t}-\sigma_{-}w^{a}_{\xi}\big{)}-\frac{\gamma(w^{a})^{2}\phi^{2}h}{2v}\big{(}P_{t}-\sigma_{-}P_{\xi}\big{)}$
$\displaystyle{}+\frac{\gamma P(w^{a})^{2}h\phi^{2}-2(\gamma-1)(w^{a})^{2}\phi
h^{2}}{2v^{2}}\big{(}\psi_{\xi}+U_{\xi}\big{)}$
$\displaystyle{}+\frac{(w^{a})^{2}[4(\gamma-1)h-\gamma
P\phi]\phi}{2v}\big{(}h_{t}-\sigma_{-}h_{\xi}\big{)}.$
Now the terms in the right hand side of (3.127) can be estimated directly and
in particular, we have
$\displaystyle\int^{t}_{0}\int_{\mathbf{R}_{+}}\frac{\gamma
P(w^{a})^{2}h\phi^{2}-2(\gamma-1)(w^{a})^{2}\phi h^{2}}{2v^{2}}\psi_{\xi}d\xi
d\tau$ (3.134) $\displaystyle\leq$ $\displaystyle
C\int^{t}_{0}\int_{\mathbf{R}_{+}}|\psi_{\xi}|\big{(}|\phi|^{3}+|\vartheta|^{3}\big{)}\,d\xi
d\tau$ (3.135) $\displaystyle\leq$ $\displaystyle
C\int_{0}^{t}\|(\phi,\vartheta)\|^{2}_{L_{\infty}}\|\psi_{\xi}\|\|(\phi,\vartheta)\|d\tau$
(3.136) $\displaystyle\leq$ $\displaystyle
C\varepsilon_{0}^{2}\int^{t}_{0}\|(\phi_{\xi},\psi_{\xi},\vartheta_{\xi})(\tau)\|^{2}d\tau.$
(3.137)
The other terms can be controlled by the similar procedure as Step 1 of Lemma
3.4. Thus the combination of the above estimates and Lemma 3.5 yield
$\displaystyle\int_{0}^{t}\int_{\mathbf{R}_{+}}(1+\tau)^{-1}\exp\left(-\frac{2a(\xi+\sigma_{-}\tau)^{2}}{1+\tau}\right)\bigg{(}P\phi+\frac{R}{\gamma-1}\vartheta\bigg{)}^{2}\,d\xi
d\tau$ (3.138) $\displaystyle\leq$ $\displaystyle
C_{a}\left(\|(\phi_{0},\psi_{0},\vartheta_{0})\|^{2}_{1}+\delta^{\frac{1}{6}}\right)+C_{a}\delta^{\frac{1}{8}}\int^{t}_{0}(1+\tau)^{-\frac{13}{12}}\|(\phi,\psi,\vartheta)(\tau)\|^{2}d\tau$
(3.141)
$\displaystyle{}+C_{a}(\delta+\varepsilon_{0})\int^{t}_{0}\int_{\mathbf{R}_{+}}(1+\tau)^{-1}\exp\left(-\frac{C_{d}(\xi+\sigma_{-}\tau)^{2}}{1+\tau}\right)|(\phi,\vartheta)|^{2}d\xi
d\tau.$
Step 2. Let
$\displaystyle W^{A}(t,\xi)$ $\displaystyle:=$
$\displaystyle-(1+t)^{-1}\int^{\infty}_{\xi+\sigma_{-}t}\exp\left(-\frac{Ay^{2}}{1+t}\right)\,dy,$
(3.142)
where $A>0$ is a constant to be determined.
Then
$\displaystyle
W^{A}_{\xi}=(1+t)^{-1}\exp\left(-\frac{A(\xi+\sigma_{-}t)^{2}}{1+t}\right),\quad\big{|}W^{A}_{t}-\sigma_{-}W^{A}_{\xi}\big{|}\leq
C_{A}(1+t)^{-\frac{3}{2}}.$ (3.143)
From the fact $p-P=\frac{R\vartheta-P\phi}{v}$, we have
$\displaystyle\frac{(R\vartheta-P\phi)_{\xi}}{v}-\frac{v_{\xi}(R\vartheta-P\phi)}{v^{2}}=-\big{(}\psi_{t}-\sigma_{-}\psi_{\xi}\big{)}+\mu\left(\frac{u_{\xi}}{v}-\frac{U_{\xi}}{V}\right)_{\xi}-G.$
(3.144)
Multiplying (3.144) by $W^{A}(R\vartheta-P\phi)$ implies
$\displaystyle\left(\frac{W^{A}(R\vartheta-P\phi)^{2}}{2v}\right)_{\xi}-\frac{W^{A}_{\xi}(R\vartheta-P\phi)^{2}}{2v}-\frac{W^{A}v_{\xi}(R\vartheta-P\phi)^{2}}{2v^{2}}$
(3.145) $\displaystyle=$
$\displaystyle-W^{A}\bigg{[}\big{(}\psi_{t}-\sigma_{-}\psi_{\xi}\big{)}-\mu\left(\frac{u_{\xi}}{v}-\frac{U_{\xi}}{V}\right)_{\xi}+G\bigg{]}(R\vartheta-P\phi).$
(3.146)
Note that
$\begin{array}[]{ll}\displaystyle
W^{A}(\psi_{t}-\sigma_{-}\psi_{\xi})(R\vartheta-P\phi)&\displaystyle=\big{\\{}W^{A}\psi(R\vartheta-P\phi)\big{\\}}_{t}-\sigma_{-}\big{\\{}W^{A}\psi(R\vartheta-P\phi)\big{\\}}_{\xi}\\\
&\displaystyle-\psi(R\vartheta-P\phi)\big{(}W^{A}_{t}-\sigma_{-}W^{A}_{\xi}\big{)}\\\
&\displaystyle-W^{A}\psi\big{\\{}(R\vartheta-P\phi)_{t}-\sigma_{-}(R\vartheta-P\phi)_{\xi}\big{\\}},\end{array}$
$\displaystyle(R\vartheta-P\phi)_{t}-\sigma_{-}(R\vartheta-P\phi)_{\xi}$
(3.147) $\displaystyle=$
$\displaystyle{}(\gamma-1)\bigg{\\{}(P-p)u_{\xi}+\kappa\left(\frac{\theta_{\xi}}{v}-\frac{\Theta_{\xi}}{V}\right)_{\xi}+\mu\bigg{(}\frac{(u_{\xi})^{2}}{v}-\frac{(U_{\xi})^{2}}{V}\bigg{)}-H\bigg{\\}}$
(3.149) $\displaystyle{}-\gamma
P\psi_{\xi}-\big{(}P_{t}-\sigma_{-}P_{\xi}\big{)}\phi$
and
$\displaystyle\gamma PW^{A}\psi\psi_{\xi}$ $\displaystyle=$
$\displaystyle\frac{\gamma}{2}\big{(}PW^{A}\psi^{2}\big{)}_{\xi}-\frac{\gamma}{2}PW^{A}_{\xi}\psi^{2}-\frac{\gamma}{2}P_{\xi}W^{A}\psi^{2},$
(3.150)
we have
$\displaystyle-\frac{W^{A}_{\xi}}{2v}\big{\\{}(R\vartheta-P\phi)^{2}+\gamma
vP\psi^{2}\\}=-\big{\\{}W^{A}\psi(R\vartheta-P\phi)\big{\\}}_{t}-E^{A}_{\xi}+Q^{A},$
(3.151)
where
$\displaystyle E^{A}:$ $\displaystyle=$
$\displaystyle\frac{W^{A}(R\vartheta-P\phi)^{2}}{2v}+\frac{\gamma}{2}PW^{A}\psi^{2}-\mu
W^{A}(R\vartheta-P\phi)\left(\frac{u_{\xi}}{v}-\frac{U_{\xi}}{V}\right)$
(3.153)
$\displaystyle{}-\sigma_{-}W^{A}\psi(R\vartheta-P\phi)-(\gamma-1)\kappa
W^{A}\psi\left(\frac{\theta_{\xi}}{v}-\frac{\Theta_{\xi}}{V}\right),$
and
$\displaystyle Q^{A}:$ $\displaystyle=$
$\displaystyle\frac{W^{A}v_{\xi}(P-p)^{2}}{2}+\big{(}W^{A}_{t}-\sigma_{-}W^{A}_{\xi}\big{)}(R\vartheta-P\phi)\psi-W^{A}G(R\vartheta-P\phi)$
(3.156)
$\displaystyle{}+W^{A}\psi\Bigg{\\{}(\gamma-1)\Bigg{[}(P-p)u_{\xi}+\mu\Bigg{(}\frac{u_{\xi}^{2}}{v}-\frac{(U_{\xi})^{2}}{V}\Bigg{)}-H\Bigg{]}-\big{(}P_{t}-\sigma_{-}P_{\xi}\big{)}\phi+\frac{\gamma
P_{\xi}\psi}{2}\Bigg{\\}}$
$\displaystyle{}-\mu\big{\\{}W^{A}(R\vartheta-P\phi)\big{\\}}_{\xi}\left(\frac{u_{\xi}}{v}-\frac{U_{\xi}}{V}\right)-(\gamma-1)\kappa\big{(}W^{A}\psi\big{)}_{\xi}\left(\frac{\theta_{\xi}}{v}-\frac{\Theta_{\xi}}{V}\right).$
First, we have
$\displaystyle\bigg{|}\int^{t}_{0}E^{A}(\tau,0)\,d\tau\bigg{|}\leq
C_{A}\delta+C_{A}\delta\int^{t}_{0}\|(\psi_{\xi},\vartheta_{\xi})(\tau)\|_{1}^{2}d\tau.$
(3.157)
The estimations of the terms concerned with $W^{A}$ are similar to those in
Step 1 while the other terms are similar to those of Step 1 in the proof of
Lemma 3.4. Thus integrating $(\ref{49})$ over $[0,t]\times\mathbf{R}_{+}$
yields
$\displaystyle\int^{t}_{0}\int_{\mathbf{R}_{+}}(1+\tau)^{-1}\exp\left(-\frac{A(\xi+\sigma_{-}\tau)^{2}}{1+\tau}\right)\big{\\{}(R\vartheta-P\phi)^{2}+\psi^{2}\big{\\}}\,d\xi
d\tau$ (3.158) $\displaystyle\leq$ $\displaystyle
C_{A}\left(\|(\phi_{0},\psi_{0},\vartheta_{0})\|^{2}_{1}+\delta^{\frac{1}{6}}\right)+C_{A}\delta^{\frac{1}{8}}\int^{t}_{0}(1+\tau)^{-\frac{13}{12}}\|(\phi,\psi,\vartheta)(\tau)\|^{2}d\tau$
(3.161)
$\displaystyle{}+C_{A}(\delta+\varepsilon_{0})\int^{t}_{0}\int_{\mathbf{R}_{+}}(1+\tau)^{-1}\exp\left(-\frac{C_{d}(\xi+\sigma_{-}\tau)^{2}}{1+\tau}\right)|(\phi,\vartheta)|^{2}d\xi
d\tau.$
Step 3. Combining $(\ref{48})$ and $(\ref{50})$, then choosing $A=2a=C_{d}$
and setting $\delta,\,\varepsilon_{0}$ suitably small, we can complete the
proof of Lemma 3.6. $\blacksquare$
_Proof of Proposition 3.1._ Choosing $\delta,\varepsilon_{0}$ suitably small
in Lemmas 3.4 and Lemma 3.6, then using Gronwall inequality yield Proposition
3.1. $\blacksquare$
## References
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* [3] F. Huang, J. Li, X. Shi, Asymptotic behavior of solutions to the full compressible Navier-Stokes equations in the half space. to appear in Commu. Math. Sci.
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|
arxiv-papers
| 2010-06-15T18:33:29 |
2024-09-04T02:49:10.911340
|
{
"license": "Public Domain",
"authors": "Xiaohong Qin, Yi Wang",
"submitter": "Yi Wang",
"url": "https://arxiv.org/abs/1006.3048"
}
|
1006.3093
|
# Dark Matter and Electroweak Symmetry Breaking from $SO(10)$
K. KANNIKE
We consider a minimal model of GUT scalar dark matter (DM) stabilized by the
discrete gauge matter parity $P_{X}$ that arises from breaking of $SO(10)$.
The dark sector comprises the complex singlet $S$ and the inert doublet
$H_{2}$. GUT scale parameters are evaluated to the electroweak scale via
Renormalization Group Equations (RGEs). Experimental and theoretical
constraints limit the DM mass to the 80 GeV to 2 TeV range. The EW symmetry
breaking is radiative and can occur via RGE running and 1-loop matching
corrections from integrating out DM. Because the next-to-lightest scalar is
almost degenerate with DM, it gives a background free displaced decay vertex
at the LHC.
The Standard Model (SM) is a good theory of ordinary matter. Yet the WMAP
measurements of the cosmic microwave background$\\!{}^{{\bf?}}$ show that
$4/5$ of the matter in the Universe is an unknown form of matter – dark matter
(DM) – usually thought to be a thermal relic whose density is determined by
freezeout.
Heavy cold dark matter must be made stable by some symmetry. The simplest such
symmetry is a new mirror symmetry or parity $Z(2)$. The usual way to stabilize
DM is to impose a global parity by hand. In MSSM, the $R$-parity – added by
hand to prevent fast proton decay $\\!{}^{{\bf?},{\bf?},{\bf?}}$ – in addition
stabilizes neutralino DM. Such $Z(2)$-symmetry is also imposed in low energy
phenomenological models of DM with a new scalar singlet
$\\!{}^{{\bf?},{\bf?},{\bf?},{\bf?}}$, doublet
$\\!{}^{{\bf?},{\bf?},{\bf?},{\bf?}}$ or higher multiplets $\\!{}^{{\bf?}}$.
However, global discrete symmetries are violated by Planck scale
operators$\\!{}^{{\bf?}}$. The solution is to get the $Z_{2}$ from breaking a
gauged $U(1)$ embedded in some Grand Unified Theory. One of the most plausible
candidates is the $SO(10)$ group that contains the SM symmetry group and an
extra $U(1)_{X}$ subgroup. Therefore, $SO(10)$ can broken down to the symmetry
group of the Standard Model and the gauged $Z_{2}$ parity
$P_{X}\equiv P_{M}=(-1)^{3(B-L)}$ (1)
that is equivalent to the $R$-parity in supersymmetric theories.
Each generation of SM fermions and the heavy singlet neutrinos needed for the
seesaw mechanism of neutrino mass$\\!{}^{{\bf?},{\bf?},{\bf?},{\bf?},{\bf?}}$
reside in the representation $\bf 16$ of $SO(10)$. They are odd under the
$P_{M}$ parity Eq. (1). The Standard Model Higgs in $\bf 10$ is even. To be
stable, scalar dark matter has to be odd$\\!{}^{{\bf?}}$ under $P_{M}$.
Because the only small representation that is odd under $P_{M}$ is the $\bf
16$, the minimal model of scalar $SO(10)$ DM adds one scalar $\bf 16$ to the
theory.
The $SO(10)$ symmetric scalar potential of one $\mathbf{16}$ and one $\bf 10$
is
$\begin{array}[]{rcl}V&=&\mu_{1}^{2}\;{\bf 10}\;{\bf 10}+\lambda_{1}({\bf
10}\;{\bf 10})^{2}+\mu_{2}^{2}\;\overline{{\bf 16}}\;{\bf
16}+\lambda_{2}(\overline{\bf 16}\,{\bf 16})^{2}\\\ &+&\lambda_{3}({\bf
10}\;{\bf 10})(\overline{\bf 16}\,{\bf 16})+\lambda_{4}({\bf 16\;}{\bf
10})(\overline{\bf 16}\,{\bf 10})\\\
&+&\frac{1}{2}\left(\lambda^{\prime}_{S}{\bf
16}^{4}+\mathrm{h.c.}\right)+\frac{1}{2}\left(\mu^{\prime}_{SH}{\bf 16\;}{\bf
10\;}{\bf 16}+\mathrm{h.c.}\right).\end{array}$ (2)
All the parameters are real with the exception of $\lambda^{\prime}_{S}$ and
$\mu^{\prime}_{SH}$. We assume that $SO(10)$ breaks down to $SU(3)_{c}\times
SU(2)_{L}\times U(1)_{Y}\times P_{M}$ in such a way that only one SM Higgs
boson $H_{1}\in\mathbf{10}$ and the DM candidates complex singlet
$S\in\mathbf{16}$ and the Inert Doublet $H_{2}\in\mathbf{16}$ are light, but
all other particles have masses of order $M_{\mathrm{G}}$.
Below $M_{\mathrm{G}}$, the most general CP-invariant scalar potential
invariant under the $P_{M}$ parity $H_{1}\to H_{1}$, $H_{2}\to-H_{2}$,
$S\to-S$ is
$\begin{array}[]{rcl}V&=&\mu_{1}^{2}H_{1}^{\dagger}H_{1}+\lambda_{1}(H_{1}^{\dagger}H_{1})^{2}+\mu_{2}^{2}H_{2}^{\dagger}H_{2}+\lambda_{2}(H_{2}^{\dagger}H_{2})^{2}+\mu_{S}^{2}S^{\dagger}S\\\
&+&\frac{\mu_{S}^{\prime
2}}{2}\left[S^{2}+(S^{\dagger})^{2}\right]+\lambda_{S}(S^{\dagger}S)^{2}+\frac{\lambda^{\prime}_{S}}{2}\left[S^{4}+(S^{\dagger})^{4}\right]+\frac{\lambda^{\prime\prime}_{S}}{2}(S^{\dagger}S)\left[S^{2}+(S^{\dagger})^{2}\right]\\\
&+&\lambda_{S1}(S^{\dagger}S)(H_{1}^{\dagger}H_{1})+\lambda_{S2}(S^{\dagger}S)(H_{2}^{\dagger}H_{2})\\\
&+&\frac{\lambda^{\prime}_{S1}}{2}(H_{1}^{\dagger}H_{1})\left[S^{2}+(S^{\dagger})^{2}\right]+\frac{\lambda^{\prime}_{S2}}{2}(H_{2}^{\dagger}H_{2})\left[S^{2}+(S^{\dagger})^{2}\right]\\\
&+&\lambda_{3}(H_{1}^{\dagger}H_{1})(H_{2}^{\dagger}H_{2})+\lambda_{4}(H_{1}^{\dagger}H_{2})(H_{2}^{\dagger}H_{1})+\frac{\lambda_{5}}{2}\left[(H_{1}^{\dagger}H_{2})^{2}+(H_{2}^{\dagger}H_{1})^{2}\right]\\\
&+&\frac{\mu_{SH}}{2}\left[S^{\dagger}H_{1}^{\dagger}H_{2}+H_{2}^{\dagger}H_{1}S\right]+\frac{\mu^{\prime}_{SH}}{2}\left[SH_{1}^{\dagger}H_{2}+H_{2}^{\dagger}H_{1}S^{\dagger}\right],\end{array}$
(3)
together with the GUT scale boundary conditions
$\displaystyle\mu_{1}^{2}(M_{\mathrm{G}})$ $\displaystyle>$ $\displaystyle
0,\;\mu_{2}^{2}(M_{\mathrm{G}})=\mu_{S}^{2}(M_{\mathrm{G}})>0,$
$\displaystyle\lambda_{2}(M_{\mathrm{G}})$ $\displaystyle=$
$\displaystyle\lambda_{S}(M_{\mathrm{G}})=\lambda_{S2}(M_{\mathrm{G}}),\;\lambda_{3}(M_{\mathrm{G}})=\lambda_{S1}(M_{\mathrm{G}}),$
(4)
and
$\displaystyle\mu_{S}^{\prime 2},\mu_{SH}^{2}$ $\displaystyle\leq$
$\displaystyle{{\mathcal{O}}\frac{M_{\mathrm{G}}}{M_{\mathrm{P}}}}^{n}\mu^{2}_{1,2},$
$\displaystyle\lambda_{5},\lambda^{\prime}_{S1},\lambda^{\prime}_{S2},\lambda^{\prime\prime}_{S}$
$\displaystyle\leq$
$\displaystyle{{\mathcal{O}}\frac{M_{\mathrm{G}}}{M_{\mathrm{P}}}}^{n}\lambda_{1,2,3,4}.$
(5)
The parameters in Eq. (5) can only be generated by operators suppressed by $n$
powers of the Planck scale $M_{\mathrm{P}}$.
We see that the dimensionful coupling $\mu^{\prime}_{SH}$, not suppressed by
$SO(10)$, can be large and form a “soft portal” to the dark sector
$\\!{}^{{\bf?}}$. It can induce electroweak symmetry breaking$\\!{}^{{\bf?}}$
via the diagrams in Fig. 1. (EWSB via effective potential for the Inert
Doublet Model was considered in $\\!{}^{{\bf?}}$.)
diagrams (3,1)(4,0) (20,15) i1 o1 $H_{1}$i1 $H_{1}$o1 scalar,tension=2i1,v1
scalar,tension=2v2,o1 fermion,label=$t$,label.side=left,leftv1,v2
fermion,label=$t$,label.side=left,leftv2,v1 (4,1)(0,0) (20,15) i1 o1 $H_{1}$i1
$H_{1}$o1 scalar,tension=2i1,v1 scalar,tension=2v2,o1
scalar,label=$S$,label.side=left,leftv1,v2
scalar,label=$H_{2}$,label.side=right,rightv1,v2
Figure 1: Dominant diagrams contributing to the Higgs boson mass.
In the mass spectrum of the dark sector we have the charged Higgs from $H_{2}$
and four neutral mass eigenstates from the mixing of the singlet and doublet
neutral components. The mass matrices of real and imaginary neutral scalars,
respectively, are
$M_{H,A}^{2}=\left(\begin{array}[]{cc}\mu_{S}^{2}\pm\mu_{S}^{\prime
2}+\lambda_{S1}v^{2}/2&\pm\mu^{\prime}_{SH}v/(2\sqrt{2})\\\
\pm\mu^{\prime}_{SH}v/(2\sqrt{2})&\mu_{2}^{2}+(\lambda_{3}+\lambda_{4})v^{2}/2\end{array}\right),$
(6)
where we have neglected all $SO(10)$-suppressed parameters save
$\mu_{S}^{\prime 2}$. The mass spectrum is $M_{\mathrm{DM}}\leq
M_{\mathrm{NL}}\ll M_{\mathrm{NL2}}\leq M_{\mathrm{NL3}}$, where the next-to-
lightest (NL) particle is almost degenerate with DM. There is another,
heavier, pair of states $S_{\mathrm{NL2}}$ and $S_{\mathrm{NL3}}$. The mass
gaps between $M_{\mathrm{DM}},M_{\mathrm{NL}}$ and likewise between
$M_{\mathrm{NL2}},M_{\mathrm{NL3}}$ are proportional to $\mu_{S}^{\prime 2}$.
We give $\mu_{S}^{\prime 2}$ a small positive value to avoid total degeneracy
of real and imaginary mass eigenstates.
Because we have a lot of unknown parameters, we do a Monte Carlo scan over
them at the GUT scale and run them down to the electroweak scale by
renormalization group equations. We integrate out the dark sector particles at
their average mass and the top quark at its mass scale.
At the GUT scale we impose $SO(10)$ boundary conditions (4) and (5). In
addition we demand that the electroweak symmetry breaking must arise from dark
matter. We require perturbativity of dimensionless interactions
($\lambda_{i}<4\pi$) and vacuum stability in the whole range from $M_{Z}$ to
$M_{\mathrm{GUT}}$. LEP2 data says that $H^{+}$ must be heavier than about 80
GeV, and we have a lower bound of $M_{Z}/2$ on dark matter mass from $Z$
invisible width$\\!{}^{{\bf?}}$. Last not least, dark matter must have correct
cosmic density within $3\sigma$, that is
$0.94<\Omega_{\mathrm{DM}}<0.129$$\\!{}^{{\bf?}}$.
Figure 2: An example of running interaction couplings.
Fig. 2 shows an example of running of the dimensionless interaction couplings
from GUT scale to the electroweak scale.
Fig. 3 displays running of mass parameters. Note that we have two distinct
possibilities to induce electroweak symmetry breaking: (i) via the Higgs mass
parameter $\mu_{1}^{2}$ evolving to negative values via RGE running, and (ii)
by integrating out dark matter in the effective
potential$\\!{}^{{\bf?},{\bf?}}$, equivalent to calculating the 1-loop
diagrams shown in Fig. 1. The first possibility is demonstrated on the left
panel and the second one on the right panel of Fig. 3. The loop mechanism is
embedded in the $SO(10)$ GUT context here, but it is a general mechanism that
can as well originate in some low energy effective theory.
The Monte Carlo points that satisfy all constraints are plotted on Fig. 4 as
DM mass vs. its spin-independent direct detection cross section per nucleon.
The solid lines show sensitivities of current experiments like CDMS
$\\!{}^{{\bf?}}$ and Xenon$\\!{}^{{\bf?},{\bf?}}$, the dashed lines are the
expected sensitivities of future experiments. In the low mass region, the
cross section can vary a lot, because there are several different annihilation
reactions, and accidental cancellations can occur in the effective Higgs-DM-DM
coupling
$\lambda_{\mathrm{eff}}\,v=\frac{1}{2}(-\sqrt{2}s\,c\,\mu^{\prime}_{SH}+2s^{2}(\lambda_{3}+\lambda_{4})v+2c^{2}\lambda_{S1}v).$
(7)
If dark matter has a relatively high mass, both the annihilation and direct
detection cross sections are dominated by the dimensionful Higgs-DM-DM
coupling $\mu^{\prime}_{SH}$. The high mass region in which electroweak
symmetry breaking can occur by integrating out dark matter is circled in red.
Figure 3: Examples of running mass parameters. On the left panel, EWSB is
achieved via RGE running, on the right panel, via 1-loop corrections from DM,
as shown on the inset. Figure 4: DM direct detection cross section per nucleon
vs. DM mass. The colour signifies Higgs mass from 130 GeV (yellow) to 185 GeV
(violet). The region circled by red line allows EWSB by integrating out DM.
Figure 5: Dark sector particle production cross sections for LHC with centre
of mass energy $\sqrt{s}=\mathrm{14~{}TeV}$. Red dots mean the process $pp\to
S_{\mathrm{NL}}S_{\mathrm{NL}}$, green lozenges $pp\to
S_{\mathrm{NL}}S_{\mathrm{NL3}}$, blue squares $pp\to
S_{\mathrm{DM}}S_{\mathrm{NL}}$, and black triangles $pp\to
S_{\mathrm{NL}}H^{\pm}$.
If the dark sector particles are relatively light (with masses up to about 700
GeV), they can be produced in the LHC$\\!{}^{{\bf?}}$. The Fig. 5 shows LHC
production cross sections for the processes $pp\to
S_{\mathrm{NL}}S_{\mathrm{NL}}$, $pp\to S_{\mathrm{NL}}S_{\mathrm{NL3}}$,
$pp\to S_{\mathrm{DM}}S_{\mathrm{NL}}$, and $pp\to S_{\mathrm{NL}}H^{\pm}$.
The first three reactions generate dark sector particles from quark-quark
interactions (via $Z^{*}$) or gluon fusion (via $h^{*}$), the last one from
quarks via $W^{\pm*}$. The cross sections are correlated with direct detection
cross sections, so if dark matter is discovered in CDMS II or Xenon100, we can
hope it can be detected at the LHC as well.
Because the mass splitting between dark matter and the next-to-lightest
particle is suppressed by $SO(10)$, the next-to-lightest particle can have a
long lifetime and give a vertex displaced from the collision point by
millimetres to metres, decaying into dark matter and a pair of leptons. This
is a highly distinct signature that is easy to discover.
In conclusion, we consider breaking non-SUSY $SO(10)$ GUT into the SM symmetry
group and the matter parity $P_{M}$. The new parity is not a global symmetry
imposed by hand but a discrete gauge symmetry. The dark matter resides in a
scalar representation $\bf 16$ of $SO(10)$. Because it is odd under $P_{M}$,
it is the scalar analogue of Standard Model fermions. We require DM to induce
electroweak symmetry breaking. This and other constraints predict a DM mass
range between 80 GeV and 2 TeV. The collider signature of the dark sector is a
displaced vertex of two leptons with almost no background.
## Acknowledgments
This work was supported by the ESF Grant 8090, JD164, SF0690030s09 and EU
FP7-INFRA-2007-1.2.3 contract No 223807.
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* [27] Z. Ahmed et al. [The CDMS-II Collaboration], arXiv:0912.3592 [astro-ph.CO].
* [28] J. Angle et al. [XENON Collaboration], “First Results from the XENON10 Dark Matter Experiment at the Gran Sasso Phys. Rev. Lett. 100, 021303 (2008) [arXiv:0706.0039 [astro-ph]].
* [29] J. Angle et al. [XENON10 Collaboration], Phys. Rev. D 80, 115005 (2009) [arXiv:0910.3698 [astro-ph.CO]].
* [30] M. Kadastik, K. Kannike, A. Racioppi and M. Raidal, arXiv:0912.3797 [hep-ph].
|
arxiv-papers
| 2010-06-15T20:56:19 |
2024-09-04T02:49:10.922853
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kristjan Kannike",
"submitter": "Kristjan Kannike",
"url": "https://arxiv.org/abs/1006.3093"
}
|
1006.3108
|
# Entanglement Creation and Storage in Two Qubits Coupling to an Anisotropic
Heisenberg Spin Chain
Chunlei Zhang Shiqun Zhu111Corresponding author, E-mail: szhu@suda.edu.cn
Jie Ren School of Physical Science and Technology, Suzhou University, Suzhou,
Jiangsu 215006, People’s Republic of China
###### Abstract
The time evolution of the entanglement of a pair of two spin qubits is
investigated when the two qubits simultaneously couple to an environment of an
anisotropic Heisenberg $XXZ$ spin chain. The entanglement of the two spin
qubits can be created and is a periodic function of the time if the magnetic
field is greater than a critical value. If the two spin qubits are in the Bell
state, the entanglement can be stored with relatively large value even when
the magnetic field is large.
###### pacs:
03.67.Mn, 42.50.Dv, 03.65.Ud
## I Introduction
Entangled quantum states are used mainly for quantum information processing,
such as quantum teleportation, quantum secret-code and quantum computation
Nielsen ; Bennett ; Murao . Many investigations showed that entanglement
exists naturally in the spin system when the temperature of the system is at
zero Vidal ; Osborne ; Osterloh ; Amico . In recent years, the study of the
dynamics of entanglement Ciccarello ; Konrad ; Yu ; Paz1 ; Paz2 ; Tiersch ;
Hamdouni ; Wang has attracted much attention in the manipulation of quantum
systems. The dynamical properties and the time evolution of the entanglement
in different quantum systems were investigated. These systems included mobile
particle elastically-scattered by static spins Ciccarello , quantum mixed
states Konrad ; Yu , two oscillators coupled to the same environment Paz1 ;
Paz2 , two d-level systems Tiersch , decoherence of a spin-$1/2$ particle
coupled to a spin bath in thermal equilibrium Hamdouni , a spin chain in
driving the the decoherence of a coupled quantum system Wang , etc. Meanwhile,
the effects of the environment were taken into account. The excitation and
quantum information transfer was investigated between two external spins when
they coupled to a one-dimensional spin chain at different sites Hartmann . The
entanglement induced by two external spins could be used to signal the
critical points when they were simultaneously coupled to an environmental $XY$
spin chain Yi ; Yuan . The decay of the Loschmidt echo was enhanced by the
quantum criticality of the surrounding Ising chain when an external spin was
coupled to the environment Quan . When two external spins coupled to a
transverse field Ising chain, the induced entanglement could be enhanced near
quantum criticality and could be used to detect the quantum phase transition
Ai , which occurred in the many-body quantum systems Sachdev . The dynamical
properties of the entanglement in a spin system need to be further
investigated when it is coupled to an external environment.
In this paper, the dynamics and the time evolution of the entanglement of a
pair of two qubits are investigated when the two qubits simultaneously couple
to an environment of an anisotropic antiferromagnetic Heisenberg spin chain
with magnetic field. In Section II, the Hamiltonian of the system and the
effective Hamiltonian of the two qubits coupled to the environment are
presented. In Section III, the time evolution of the system is analyzed for
the simplest case of the environment. The entanglement creation in the coupled
pair of two external qubits is discussed in Section IV. In Section V, the
storage of the entanglement in the coupled pair of two external qubits is
investigated. A discussion concludes the paper.
## II Hamiltonian of the System
When two external spin qubits are coupled with the environment of a one-
dimensional spin chain, the Hamiltonian of this system can be written as
$H=H_{0}+H_{I}.$ (1)
where $H_{0}$ is the Hamiltonian of the environment. If the environment is an
anisotropic Heisenberg $XXZ$ spin chain, one has
$H_{0}=J\sum_{i=1}^{N}(\sigma^{x}_{i}\sigma^{x}_{i+1}+\sigma^{y}_{i}\sigma^{y}_{i+1}+\Delta_{i}\sigma^{z}_{i}\sigma^{z}_{i+1})+B\sum_{i=1}^{N}\sigma^{z}_{i},$
(2)
were $\sigma^{\alpha}_{i}(\alpha=x,y,z)$ are Pauli operators, $N$ is the
number of the spin chain, $J$ is the coupling coefficient between the spins,
$B$ is the magnetic field along the z-axis with the anisotropy
$\Delta_{i}=\Delta$ $\in$ (0,1). In Eq. (1), $H_{I}$ is the interaction
Hamiltonian between the two external spin qubits and the environment and can
be written as,
$H_{I}=J_{p}\sum_{i=1}^{N}(\sigma_{a}\sigma_{i}+\sigma_{b}\sigma_{i}).$ (3)
where $\sigma_{a}$ and $\sigma_{b}$ are the Pauli operators of the qubits $a$
and $b$, $J_{p}$ is the coupling coefficient between the external spin qubits
($a$ and $b$) and the Heisenberg spin chain. In order to facilitate the
calculation, the coupling coefficients are chosen as $J=1$ and $J_{p}=0.2$ in
this paper. That is, the environment is represented by the antiferromagnetic
Heisenberg $XXZ$ spin model. The schematic diagram of the system is shown in
Fig. 1. The two qubits are symmetrically located at the two sides of the spin
chain.
Fröhlich transformation Ai ; Frohlich can be used to solve the problem of
induced effective interaction between two qubits through the medium of the
Heisenberg spin chain. The environment of the antiferromagnetic Heisenberg
spin chain has non-degenerate ground state $|\psi_{0}\rangle$ with ground
state energy $E_{0}$. According to the standard canonical transformation Ai ;
Frohlich ; Ferreira , the effective Hamiltonian of the external spin qubits
can be written as
$H^{ab}_{eff}=\sum_{j=1}^{k}\frac{\langle\psi_{0}|H_{i}P_{j}H_{i}|\psi_{0}\rangle}{E_{j}-E_{0}},$
(4)
where the projector is $P_{j}=|\psi_{j}\rangle\langle\psi_{j}|$ and
$|\psi_{j}\rangle(j=1,2,...k)$ is the time dependent excited state with energy
$E_{j}$. After some straightforward calculations, the effective Hamiltonian
can be reduced to
$H^{ab}_{eff}=-\sum_{j}2J_{p}J_{p}\sum_{\alpha,\beta}\Re(m_{\alpha}n^{\ast}_{\beta})\sigma^{\alpha}_{a}\sigma^{\beta}_{b}+\sum_{\alpha,\beta}\frac{J^{2}_{p}}{4}(|m_{\alpha}|^{2}+|n_{\beta}|^{2}),$
(5)
where the parameters are
$m_{\alpha}=\frac{\langle\psi_{0}|s^{\alpha}_{m}|\psi_{j}\rangle}{\sqrt{E_{k}-E_{0}}},n_{\beta}=\frac{\langle\psi_{0}|s^{\beta}_{n}|\psi_{j}\rangle}{\sqrt{E_{k}-E_{0}}},s^{\alpha,\beta}=\frac{1}{2}\sigma^{\alpha,\beta}$,
and $\Re(m_{\alpha}n^{\ast}_{\alpha})$ means the real part of the product
$(m_{\alpha}n^{\ast}_{\alpha})$ with $\alpha,\beta=x,y,z$. When the eigenstate
$|\psi_{j}\rangle$ and the corresponding eigenvalue $E_{j}$ of $H_{0}$ are
obtained, the effective Hamiltonian $H^{ab}_{eff}$ can be easily calculated.
## III Analysis of Time evolution
In order to describe the time evolution of the entanglement of two-qubit
system, the concurrence is used as a measure of the entanglement. The
concurrence is defined as Hill ; Wootters
$C=\max\\{{\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4},0}\\},$ (6)
where the $\lambda_{i}(i=1,2,3,4)$ are the square roots of the eigenvalues of
the density matrix $\varrho_{ab}$. The density matrix $\varrho_{ab}$ is given
by
$\varrho_{ab}=\rho_{12}(\sigma_{1}^{y}\otimes\sigma_{2}^{y})\rho_{12}^{\ast}(\sigma_{1}^{y}\otimes\sigma_{2}^{y}).$
(7)
The ground state of the environment of the Heisenberg spin chain can be chosen
as $|\phi_{0}\rangle$ while that of the two external spin qubits $a$ and $b$
can be chosen as $|01\rangle$. Under the influence of the environment, the two
external spin qubits have an initial state as follows
$|\psi_{0}\rangle=|\phi_{0}\rangle\otimes|01\rangle_{ab}.$ (8)
The time evolution of the state is
$|\psi(t)\rangle=\exp(-iH^{ab}_{eff}t)|\psi_{0}\rangle_{ab},$ (9)
with the density matrix $\varrho_{ab}=|\psi(t)\rangle\langle\psi(t)|$.
The reduced density matrix $\varrho_{ab}(t)$ can be written as
$\varrho_{ab}(t)=\left(\begin{array}[]{cccc}u(t)&0&0&0\\\ 0&w_{1}(t)&y(t)&0\\\
0&y^{\ast}(t)&w_{2}(t)&0\\\ 0&0&0&v(t)\\\ \end{array}\right)$ (10)
in the standard basis $\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$. The
corresponding concurrence $C(t)$ of the two external spin qubits can be
calculated from the reduced density matrix $\varrho_{ab}(t)$ and given by
$C(t)=2\max\\{|y(t)|-\sqrt{u(t)v(t)},0\\}.$ (11)
## IV Entanglement Creation
For the simplest case of $N=2$ in the anisotropic Heisenberg $XXZ$ spin chain,
the eigenenergies and eigenstates of the system are
$E_{1}=\Delta-2B,E_{2}=\Delta+2B,E_{3}=-\Delta+2,E_{4}=-\Delta-2$ and
$|\varphi_{1}\rangle=|11\rangle,|\varphi_{2}\rangle=|00\rangle,|\varphi_{3}\rangle=\frac{\sqrt{2}}{2}(|01\rangle+|10\rangle),|\varphi_{4}\rangle=\frac{\sqrt{2}}{2}(-|01\rangle+|10\rangle)$
respectively. When $B-\Delta>1$, the ground state is
$|\phi_{0}\rangle=|\varphi_{1}\rangle$. Then the effective Hamiltonian
$H^{ab}_{eff}$ can be written as
$H^{ab}_{eff}=g\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&2&0\\\ 0&2&1&0\\\
0&0&0&1\\\ \end{array}\right),$ (12)
where the parameter $g$ is given by
$g=J_{p}^{2}\frac{\Delta-B}{(B-\Delta+1)(B-\Delta-1)}$. The matrix elements
$u(t),w_{1}(t),y(t),w_{2}(t)$ and $v(t)$ in Eq. (10) are given by
$u(t)=0,w_{1}(t)=\frac{1}{2}+\frac{1}{4}e^{-i4gt}+\frac{1}{4}e^{i4gt},y(t)=-\frac{1}{4}e^{-i4gt}+\frac{1}{4}e^{i4gt},w_{2}(t)=\frac{1}{2}-\frac{1}{4}e^{-i4gt}-\frac{1}{4}e^{i4gt},v(t)=0$.
When $B-\Delta<1$, the effective Hamiltonian $H^{ab}_{eff}=0$. There is no
entanglement between spin qubits $a$ and $b$. When $B-\Delta=1$, the ground
state energy equals the excited state energy, i. e., $E_{1}=E_{4}$. The
energies of the two states are crossed at this point. Since the two states are
degenerate, Eq. (4) is not valid to calculate the effective Hamiltonian
$H^{ab}_{eff}$ when $B-\Delta=1$. That is, there is a critical value of the
magnetic field $B_{C}$. The value of $B_{C}$ is given by $B_{C}=1+\Delta$. If
$B<B_{C}$, the concurrence $C(t)$ is zero. That is, there is no entanglement
when $B<B_{C}$. If $B>B_{C}$, the entanglement appears. That is, the
entanglement can be created when $B>B_{C}$. Then the concurrence $C(t)$ can be
given by
$C(t)=\left\\{\begin{array}[]{ll}0,&(B-\Delta<1);\\\
|\sin(4gt)|,&(B-\Delta>1).\\\ \end{array}\right.$ (13)
The concurrence $C(t)$ as a function of the time $t$ is plotted in Fig. 2 when
the magnetic field $B$ and the anisotropy $\Delta$ are varied. The values of
the anisotropy are $\Delta=0.2,0.4$ and $0.6$ with $B>B_{C}$ in Figs. 2(a),
2(b) and 2(c) respectively. From Fig. 2, it is seen that the concurrence
$C(t)$ is a periodic function of time $t$. It almost oscillates between the
maximum value of one and the minimum value of zero. The period decreases as
the magnetic field $B$ increases.
The anisotropic antiferromagnetic Heisenberg $XXZ$ model was used to
investigate the order-to-disorder transition of the material $Cs_{2}CoCl_{4}$
Kenzelmann . For the material $Cs_{2}CoCl_{4}$, the anisotropy is
$\Delta=0.25$. When the number of spins in the environment of the Heisenberg
$XXZ$ chain is greater than two, there is no approximate analytic solution of
$H^{ab}_{eff}$ and $C(t)$. To calculate $C(t)$, the numerical computation
needs to be performed. In Fig. 3, the concurrence $C(t)$ is plotted as a
function of time $t$ when the spin numbers in the environment are $N=4,6$ and
$8$. From Fig. 3, it is seen that the concurrence $C(t)$ is a periodic
function of $t$ with two different kinds of periods. Both periods decrease as
the spin number $N$ in the environment increase. There is a critical value
$B_{C}$ of the magnetic field. When $B<B_{C}$, the concurrence $C(t)$
oscillates following the large period. The period decreases slightly as $B$
increases. While $B>B_{C}$, $C(t)$ oscillates following the small period. The
period increases as $B$ increases. The concurrence $C(t)$ can be approximately
given by
$C(t)\sim\left\\{\begin{array}[]{ll}|sin[g(N)\sqrt{N/(N+1)}t]|,&(B<B_{C});\\\
|sin[g(N)Nt]|,&(B>B_{C}).\\\ \end{array}\right.$ (14)
Where $g(N)$ is a function of the spin number $N$ in the environment. Though
there is no analytic expression of the critical field $B_{C}$, it can be
numerical calculated. The critical field $B_{C}$ is plotted in Fig. 3(d) as a
function of $1/N$. From Fig. 3(d), it is seen that $B_{C}$ decreases linearly
as $1/N$ decreases. In the thermodynamic limit of $N\rightarrow\infty$,
$B_{C}$ tends to zero. The regime for larger period of oscillation disappears.
## V Entanglement storage
The concurrence $C(t)$ is plotted as a function of magnetic field $B$ and time
$t$ in Fig. 4 when the initial state of the two external spin qubits $a$ and
$b$ is in the Bell state $\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$. The
anisotropy is chosen as $\Delta=0.25$ Kenzelmann . The spin numbers in the
environment of the anisotropic antiferromagnetic Heisenberg chain are
$N=2,4,6$, and $8$. From Fig. 4, it is seen that the concurrence $C(t)$ is a
oscillation function of time $t$. The oscillation period decreases as $B$
increases. Obviously, the concurrence $C(t)$ is divided into several regions
by different critical values of the magnetic field $B_{C}$. The red circles in
Fig. 4 show the critical values of $B_{C}$. At the critical point $B_{C}$, the
energies of the ground and excited states are crossing and the states are
degenerate. For $N=2$, there are two parts in $C(t)$ divided by one $B_{C}$.
For $B<B_{C}$, $C(t)$ is almost a constant of $C(t)=1.0$. For $B>B_{C}$,
$C(t)$ oscillates with a small period [cf. Fig. 4(a)]. For $N=4$, there are
three parts divided by two different values of $B_{C}$ [(marked by two red red
circles in Fig. 4(b)]. In the first part, $C(t)$ is very close to $1.0$. It
oscillates with quite small amplitude. In the second part, $C(t)$ oscillates
with small period. In the third part, $C(t)$ oscillates with even smaller
period [cf. Fig. 4(b)]. When $N=6$ and $8$ in Figs. 4(c) and 4(d), similar
phenomena occurs. Obviously, the concurrence $C(t)$ is divided into $(N/2+1)$
parts by $N/2$ critical values of $B_{C}$. The energy is crossing at the
critical values of $B_{C}$ in the ground state as well as in excited
states.The concurrence $C(t)$ is jumping as the state is switched from an
entangled state to another. In the thermodynamic limit, the continuous energy
level crossings occur Son . The first part of concurrence $C(t)$ disappears.
Other parts of $C(t)$ tends to smooth and continuous. In Fig. 3(d), only the
first critical value of $B_{C}$ as a function of spin number $1/N$ is plotted.
From Fig. 4, it is also clear that the entanglement $C(t)$ can keep large
value even for relatively large magnetic field $B$. If the initial state of
the two external spin qubits is the Bell state
$\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$, similar results as that shown in
Fig. 4 are obtained.
## VI Discussion
The time evolution of the entanglement of two external spin qubits is
investigated when they are coupled to the environment of an anisotropic
antiferromagnetic Heisenberg $XXZ$ spin chain with magnetic field. The
approximate form of the effective Hamiltonian is derived. The concurrence is
used as a measure of the entanglement. When there are two spins in the
environment, there is no entanglement between two external spin qubits when
the magnetic field is smaller than a critical value. When the magnetic field
is greater than the critical value, the entanglement can be created and is a
periodic function of the time. The entanglement almost oscillates between one
and zero. The oscillation period decrease as the anisotropy and the magnetic
field increase. There are $N/2+1$ parts in the entanglement divided by $N/2$
values of critical magnetic fields. The first critical magnetic field tends to
zero when the spin number in the environment tends to infinity. When the
initial state of the two external spin qubits is in one of the Bell state, the
entanglement can be stored. Though there are different regimes in the
entanglement, the entanglement always keeps quite large value when it
oscillates with increasing number of spins in the environment.
Acknowledgments
It is a pleasure to thank Xiang Hao and Tao Song for their many helpful
discussions. The financial support from the National Natural Science
Foundation of China (Grant No. 10774108) is gratefully acknowledged.
## References
* (1) M. A. Nielsen and I. L. Chuang, _Quantum Computation and Quantum Information_ (Cambridge University Press, Cambridge, England, 2000).
* (2) C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895(1993).
* (3) M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral, Phys. Rev. A 59, 156(1999).
* (4) G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003).
* (5) T. J. Osborne, and M. A. Nielsen, Phys. Rev. A 66, 032110 (2002).
* (6) A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature 416, 608 (2002).
* (7) L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. 80, 517 (2008).
* (8) F. Ciccarello, M. Paternostro G. M. Palma, and M. Zarcone, arXiv:quant-ph/0812.0755(2008).
* (9) T. Konrad, F. Melo, M. Tiersch, C. Kasztelan, A. Aragao, and A. Buchleitner, Nat. Phys. 4, 99(2008).
* (10) C.-S. Yu, X. X. Li, and H.-S. Song, Phys. Rev. A 78, 062330(2008).
* (11) J. P. Paz, and A. J. Roncaglia, Phys. Rev. Lett. 100, 220401(2008).
* (12) J. P. Paz, and A. J. Roncaglia, Phys. Rev. A 79, 032102(2009).
* (13) M. Tiersch, F. de Melo, and A. Buchleitner, Phys. Rev. Lett. 101, 170502(2008).
* (14) Y. Hamdouni, F. Petruccione, Phys. Rev. B 76, 174306(2007).
* (15) Z.-H. Wang, B.-S. Wang, and Z.-B. Su, arXiv:quant-ph/0903.0944(2009).
* (16) M. J. Hartmann, M. E. Reuter, and M. B. Plenio, New J. Phys. 8, 94(2006).
* (17) X. X. Yi, H. T. Cui, and L. C. Wang, Phys. Rev. A 74, 054102(2006).
* (18) Z. G. Yuan, P. Zhang, and S. S. Li, Phys. Rev. A 76, 042118(2007).
* (19) H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun, Phys. Rev. Lett. 96, 140604(2006).
* (20) Q. Ai, T. Shi, G. Long, and C. P. Sun, Phys. Rev. A 78, 022327(2008).
* (21) S. Sachdev, _Quantum Phase Transitions_ (Cambridge University Press, Cambridge, England, 1999).
* (22) H. Fröhlich, Phys. Rev. 79, 845(1950).
* (23) A. Ferreira, J. M. B. Lopes dos Santos, Phys. Rev. A 77, 034301(2008).
* (24) S. Hill and W. K. Wootters, Phys. Rev. Lett. 78, 5022(1997).
* (25) W. K. Wootters, Phys. Rev. Lett. 80, 2245(1998).
* (26) M. Kenzelmann, R. Coldea, and D. A. Tennant, Phys. Rev. B. 65, 144432(2002).
* (27) W. Son and V. Vedral, arXiv:quant-ph/0905.3065 (2009).
Figure Captions
Fig. 1
The schematic diagram of two external spin qubits symmetrically coupled to the
environment of an anisotropic Heisenberg spin chain.
Fig. 2
The concurrence $C(t)$ is plotted as a functions of the time $t$ for $N=2$
when the magnetic field $B$ and the anisotropy $\Delta=$ are varied with
$B>B_{C}$. (a). $\Delta=0.2$. (b). $\Delta=0.4$. (c). $\Delta=0.8$.
Fig. 3
The concurrence $C(t)$ is plotted as a function of the magnetic field $B$ and
the time $t$ for different spin numbers $N$ of the environment in (a), (b),
and (c). The anisotropy is $\Delta=0.25$. (a). $N=4$. (b). $N=6$. (c). $N=8$.
(d). The critical field $B_{C}$ is plotted as a function of $1/N$.
Fig. 4
The concurrence $C$ is plotted as a function of the magnetic field $B$ and the
time $t$ for different spin numbers in the environment. The anisotropy is
$\Delta=0.25$ and the Bell state $\frac{1}{\sqrt{2}}(|01>+|10\rangle)$ is
chosen. (a). $N=2$. (b). $N=4$. (c). $N=6$. (d). $N=8$.
Fig. 1
Fig. 2
Fig. 3
Fig. 4
|
arxiv-papers
| 2010-06-15T23:26:56 |
2024-09-04T02:49:10.927724
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chunlei Zhang, Shiqun Zhu, and Jie Ren",
"submitter": "Jie Ren",
"url": "https://arxiv.org/abs/1006.3108"
}
|
1006.3134
|
11institutetext: INRIA Saclay, France
11email: kaustuv.chaudhuri@inria.fr
# Classical and Intuitionistic Subexponential Logics
are Equally Expressive
Kaustuv Chaudhuri
###### Abstract
It is standard to regard the intuitionistic restriction of a classical logic
as increasing the expressivity of the logic because the classical logic can be
adequately represented in the intuitionistic logic by double-negation, while
the other direction has no truth-preserving propositional encodings. We show
here that subexponential logic, which is a family of substructural refinements
of classical logic, each parametric over a preorder over the subexponential
connectives, does not suffer from this asymmetry if the preorder is
systematically modified as part of the encoding. Precisely, we show a
bijection between synthetic (i.e., focused) partial sequent derivations modulo
a given encoding. Particular instances of our encoding for particular
subexponential preorders give rise to both known and novel adequacy theorems
for substructural logics.
## 1 Introduction
In [13], Miller writes:
> “While there is some recognition that logic is a unifying and universal
> discipline underlying computer science, it is far more accurate to say that
> its universal character has been badly fractured …one wonders if there is
> any sense to insisting that there is a core notion of ‘logic’.”
Possibly the oldest such split is along the classical/intuitionistic seam, and
each side can be seen as more universal than the other. Classical logics, the
domain of traditional mathematics, generally have an elegant symmetry in the
connectives that can often be exploited to create sophisticated proof search
and model checking algorithms. On the other hand, intuitionistic logics, which
introduce an asymmetry between multiple hypotheses and single conclusions, can
express the computational notion of _function_ directly, making it the
preferred choice for programming languages and logical frameworks. Can the
rift between these two sides be bridged?
Miller proposes one approach: to use structural proof theory, particularly the
proof theory of focused sequent calculi, as a unifying language for logical
formalisms. There is an important proof theoretic difference between a given
classical logic and its _intuitionistic restriction_ (see defn. 8): the
classical formulas can be encoded using the intuitionistic connectives in such
a way that classical provability is preserved, i.e., a formula is classically
provable if and only if its encoding is intuitionistically provable. In the
other direction, however, there are no such general encodings. The classical
logic will either have to be extended (for example, with terms and
quantifiers) or refined with substructural or modal operators. For this
reason, intuitionistic logics are sometimes considered to be _more expressive_
than their classical counterparts.
In this paper, we compare logical calculi for “universality” using the
specific technical apparatus of _adequate propositional encodings_. That is,
given a formula in a source logic $O$, we must be able to encode it in a
target logic $M$ that must preserve the atomic predicates and must reuse the
reasoning principles of $M$, particularly its notion of provability. An
example of such an encoding would be ordinary classical logic encoded in
ordinary intuitionistic logic where each classical formula $A$ is encoded as
the intuitionistic formula $\lnot\lnot A$. We can go further and also reuse
the proofs of the target calculus; in fact, there are at least the following
_levels_ of adequacy:
###### Definition 1 (levels of adequacy)
An encoding of formulas (equiv. of sequents) from a source to a target
calculus is
* •
_globally adequate_ if a formula is true (equiv. a sequent is derivable) in
the source calculus iff its encoding is true (equiv. the encoding of the
sequent is derivable) in the target calculus;
* •
_adequate_ if the proofs of a formula (equiv. a sequent) in the source
calculus are in bijection with the proofs of the encoding of the formula
(equiv. the sequent) in the target calculus; and
* •
_locally adequate_ if open derivations (i.e., partial proofs with possibly
unproved premises) of a formula (equiv. a sequent) in the source calculus are
in bijection with the open derivations of the formula (equiv. the sequent) in
the target calculus.
Local adequacy is an ideal for encodings because it is a strong justification
for seeing the target calculus as more universal: (partial) proofs in the
source calculus can be recovered at any level of detail. However, it is
unachievable except in trivial situations. Indeed, even adequacy is often
difficult; for instance, the linear formula
${!}a\mathbin{{\multimap}}{!}b\mathbin{{\multimap}}{!}a$ has three sequent
proofs, differing in the order in which the second $\mathbin{{\multimap}}$ and
the two ${!}$s are introduced, but there is only a single sequent proof of
$a\mathbin{{\supset}}b\mathbin{{\supset}}a$.
It is nevertheless possible to define a kind of local adequacy that is more
flexible: adequacy up to permutations of inference rules entirely inside one
of the phases of _focusing_. A focused proof [1] is a proof that makes large
_synthetic_ rules that are maximal chains of positive or negative inference
rules. An inference rule is positive, sometimes called synchronous, if it
involves an essential choice, while it is negative or asynchronous if the
choices it presents (if any) are inessential. The term “focus” describes the
way positive inferences are chained to form synthetic steps: each inference is
applied (read from conclusion to premises) to a single formula _under focus_ ,
and the operands of this connective remain under focus in the premises.
###### Definition 2 (focal adequacy)
An encoding of sequents from a source to a target focused calculus is _focally
adequate_ if they have the same synthetic inference rules.
Since focusing abstracts away the inessential permutations of inference rules,
a focally adequate encoding can be used to compare logics for “essential
universality”. Surprisingly, there are very few known focal adequacy results
(see [4, 11] for practically all such known results). This paper fills in many
of the gaps for existing (substructural) logics by proving a pair of general
encodings (see theorems 12 and 17) about _subexponential_ logics [8, 15]. It
is well known that the exponentials of linear logic are non-canonical. If a
pre-order is imposed upon them with suitable conditions, then the resulting
logic is well-behaved, satisfying identity, admitting cuts, and allowing
focusing. Moreover, classical, intuitionistic, and linear logics can be seen
as _instances_ of subexponential logic for particular collections of
subexponentials. Our encodings are _generic_ , parametric on the
_subexponential signature_ of the source and target logics. As particular
instances, we obtain focal adequacy results for: classical logic (CL) in
intuitionistic logic (IL), IL in classical linear logic (CLL), CLL in
intuitionistic linear logic (ILL), and an indefinite bidirectional chain
between classical and intuitionistic subexponential logics, all of which are
novel. Moreover, our encodings show that any analysis (such as cut-
elimination) or algorithm (such as proof search) that is generic on the
subexponential signature cannot (and _need not_) distinguish between classical
and intuitionistic logics.
The rest of this paper is organized as follows: in sec. 2 classical
subexponential logic is introduced, together with its focused sequent calculus
and well known instances; in sec. 3 its intuitionistic restriction is
presented; then in sec. 4 the bidirectional encoding between classical and
intuitionistic subexponential logic is constructed. Details omitted here for
space reasons can be found in the accompanying technical report [6].
## 2 Classical subexponential logic
Subexponential logic borrows most of its syntax from linear logic [9]. As we
are comparing focused systems, we adopt a polarised syntax from the beginning.
Polarised formulas will have exactly one of two polarities: _positive_
($P,Q,\ldots$) constructed out of the positive atoms and connectives, and
_negative_ ($N,M,\ldots$) constructed out of the negative atoms and
connectives. These two classes of formulas are mutually recursive, mediated by
the indexed subexponential operators ${!}_{z}$ and ${?}_{z}$.
###### Notation 3 (syntax)
_Positive formulas_ ($P,Q$) and _negative formulas_ ($N,M$) have the following
grammar:
$\displaystyle P,Q$ $\displaystyle\Coloneqq p\mathbin{\left.\hbox
to0.0pt{\vbox to7.74997pt{}}\right|}P\mathbin{{\otimes}}Q\mathbin{\left.\hbox
to0.0pt{\vbox to7.74997pt{}}\right|}{\mathbf{1}}\mathbin{\left.\hbox
to0.0pt{\vbox to7.74997pt{}}\right|}P\mathbin{{\oplus}}Q\mathbin{\left.\hbox
to0.0pt{\vbox to7.74997pt{}}\right|}{\mathbf{0}}\mathbin{\left.\hbox
to0.0pt{\vbox to7.74997pt{}}\right|}{!}_{z}{N^{+}}$ (positive) $\displaystyle
N,M$ $\displaystyle\Coloneqq n\mathbin{\left.\hbox to0.0pt{\vbox
to7.74997pt{}}\right|}N\mathbin{{\&}}M\mathbin{\left.\hbox to0.0pt{\vbox
to7.74997pt{}}\right|}{\top}\mathbin{\left.\hbox to0.0pt{\vbox
to7.74997pt{}}\right|}N\mathbin{{\invamp}}M\mathbin{\left.\hbox to0.0pt{\vbox
to7.74997pt{}}\right|}{\bot}\mathbin{\left.\hbox to0.0pt{\vbox
to7.74997pt{}}\right|}P\mathbin{{\multimap}}N\mathbin{\left.\hbox
to0.0pt{\vbox to7.74997pt{}}\right|}{?}_{z}{P^{-}}$ (negative)
Atomic formulas are written in lower case ($a,b,\ldots$), with $p$ and $q$
reserved for positive and $n$ and $m$ reserved for negative atomic formulas.
${P^{-}}$ denotes either a positive formula or a negative atom, and likewise
${N^{+}}$ denotes a negative formula or a positive atom. We write $A,B,\ldots$
for any arbitrary formula (positive or negative).
Because we will eventually consider its intuitionistic restriction, we retain
implication $\mathbin{{\multimap}}$ as a primitive even though it is
classically definable. However, we exclude the non-linear implication
($\mathbin{{\supset}}$) because the unrestricted zones are non-canonical;
i.e., there are many such implications, each defined using a suitable
subexponential (or compositions thereof). The subscript $z$ in exponential
connectives denotes zones drawn from a _subexponential signature_ (using the
terminology of [15]).
###### Definition 4
A _subexponential signature_ $\Sigma$ is a structure $\langle
Z,\leq,{\mathfrak{l}},U\rangle$ where:
* •
$\langle Z,\leq\rangle$ is a non-empty pre-ordered set (the “zones”);
* •
${\mathfrak{l}}\in Z$ is a “ _working_ ” zone;
* •
$U\subseteq Z$ is a set of _unrestricted_ zones that is $\leq$-closed, i.e.,
for every $z_{1},z_{2}\in Z$, if $z_{1}\leq z_{2}$, then $z_{1}\in U$ implies
$z_{2}\in U$. $Z\setminus U$ will be called the _restricted_ zones.
We use $u,v,w$ to denote unrestricted zones and $r,s,t$ to denote restricted
zones.
Unrestricted zones admit both weakening and contraction, while restricted
zones are linear. The logic is parametric on the signature. (Particular
mentions of the signature will be omitted unless necessary to disambiguate, in
which case they will be written in a subscript.) We use use a two-sided
sequent calculus formulation of the logic in order to avoid appeals to De
Morgan duality. This will not only simplify the definition of the
intuitionistic restriction (sec. 3), but will also be crucial to the main
adequacy result. Formulas in contexts are annotated with their subexponential
zones as follows: ${z{\,:\,}A}$ will stand for $A$ occurring in zone denoted
by $z$, and ${z{\,:\,}(A_{1},\ldots,A_{k})}$ for
${z{\,:\,}A_{1}},\ldots,{z{\,:\,}A_{k}}$. Sequents are of the following kinds:
$\UpGamma$ $\vdash$ | $\left[P\right]\ ;\ \UpDelta$ | right focus on $P$
---|---|---
$\UpGamma\ ;\ \left[N\right]$ $\vdash$ | $\UpDelta$ | left focus on $N$
$\UpGamma\ ;\ \UpOmega$ $\vdash$ | $\UpXi\ ;\ \UpDelta$ | active on $\UpOmega$ and $\UpXi$
The contexts in these sequents have the following restrictions:
* •
All elements of the _left passive_ context $\UpGamma$ are of the form
${z{\,:\,}{N^{+}}}$.
* •
All elements of the _right passive_ context $\UpDelta$ are of the form
${z{\,:\,}{P^{-}}}$.
* •
All elements of the _left active_ context $\UpOmega$ are of the form
${P^{-}}$.
* •
All elements of the _right active_ context $\UpXi$ are of the form ${N^{+}}$.
###### Notation 5
We write ${\UpGamma}^{\mathfrak{u}}$ or ${\UpDelta}^{\mathfrak{u}}$ for those
contexts containing only unrestricted elements, i.e., each element is of the
form ${u{\,:\,}A}$ with $u\in U$. Likewise, we write
${\UpGamma}^{\mathfrak{r}}$ or ${\UpDelta}^{\mathfrak{r}}$ for contexts
containing only restricted elements.
(right focus)
r
$\displaystyle\linfer[{\mathbin{{\oplus}}}\text{{r}}_{i}]{\UpGamma\vdash\left[P_{1}\mathbin{{\oplus}}P_{2}\right]\
;\ \UpDelta}{\UpGamma\vdash\left[P_{i}\right]\ ;\
\UpDelta}\quad\UpGamma\vdash\left[{!}_{z}{N^{+}}\right]\ ;\
\UpDelta\lx@proof@logical@and\UpGamma\ ;\ \cdot\vdash{N^{+}}\ ;\
\UpDelta\bigl{(}\forall{{x{\,:\,}A}\in\UpGamma,\UpDelta}.\,z\leq x\bigr{)}$
(left focus)
$\displaystyle\linfer[\text{{nl}}]{{\UpGamma}^{\mathfrak{u}}\ ;\
\left[n\right]\vdash{\UpDelta}^{\mathfrak{u}},{z{\,:\,}n}}{}\quad\linfer[{\mathbin{{\&}}}\text{{l}}_{i}]{\UpGamma\
;\ \left[P_{1}\mathbin{{\&}}P_{2}\right]\vdash\UpDelta}{\UpGamma\ ;\
\left[P_{i}\right]\vdash\UpDelta}\quad{\UpGamma}^{\mathfrak{u}},{\UpGamma}^{\mathfrak{r}}_{1},{\UpGamma}^{\mathfrak{r}}_{2}\
;\
\left[N\mathbin{{\invamp}}M\right]\vdash{\UpDelta}^{\mathfrak{u}},{\UpDelta}^{\mathfrak{r}}_{1},{\UpDelta}^{\mathfrak{r}}_{2}\lx@proof@logical@and{\UpGamma}^{\mathfrak{u}},{\UpGamma}^{\mathfrak{r}}_{1}\
;\
\left[N\right]\vdash{\UpDelta}^{\mathfrak{u}},{\UpDelta}^{\mathfrak{r}}_{1}{\UpGamma}^{\mathfrak{u}},{\UpGamma}^{\mathfrak{r}}_{2}\
;\
\left[M\right]\vdash{\UpDelta}^{\mathfrak{u}},{\UpDelta}^{\mathfrak{r}}_{2}$
$\displaystyle{\UpGamma}^{\mathfrak{u}}\ ;\
\left[{\bot}\right]\vdash{\UpDelta}^{\mathfrak{u}}\quad{\UpGamma}^{\mathfrak{u}},{\UpGamma}^{\mathfrak{r}}_{1},{\UpGamma}^{\mathfrak{r}}_{2}\
;\
\left[P\mathbin{{\multimap}}M\right]\vdash{\UpDelta}^{\mathfrak{u}},{\UpDelta}^{\mathfrak{r}}_{1},{\UpDelta}^{\mathfrak{r}}_{2}\lx@proof@logical@and{\UpGamma}^{\mathfrak{u}},{\UpGamma}^{\mathfrak{r}}_{1}\vdash\left[P\right]\
;\
{\UpDelta}^{\mathfrak{u}},{\UpDelta}^{\mathfrak{r}}_{1}{\UpGamma}^{\mathfrak{u}},{\UpGamma}^{\mathfrak{r}}_{2}\
;\
\left[M\right]\vdash{\UpDelta}^{\mathfrak{u}},{\UpDelta}^{\mathfrak{r}}_{2}$
$\displaystyle\UpGamma\ ;\
\left[{?}_{z}{P^{-}}\right]\vdash\UpDelta\lx@proof@logical@and\UpGamma\ ;\
{P^{-}}\vdash\cdot\ ;\
\UpDelta\bigl{(}\forall{{x{\,:\,}A}\in\UpGamma,\UpDelta}.\,z\leq x\bigr{)}$
(right active)
r $\displaystyle\linfer[{\mathbin{{\invamp}}}\text{{r}}]{\UpGamma\ ;\
\UpOmega\vdash\UpXi,N\mathbin{{\invamp}}M\ ;\ \UpDelta}{\UpGamma\ ;\
\UpOmega\vdash\UpXi,N,M\ ;\
\UpDelta}\quad\linfer[{{\bot}}\text{{r}}]{\UpGamma\ ;\
\UpOmega\vdash\UpXi,{\bot}\ ;\ \UpDelta}{\UpGamma\ ;\ \UpOmega\vdash\UpXi\ ;\
\UpDelta}\quad\linfer[{\mathbin{{\multimap}}}\text{{r}}]{\UpGamma\ ;\
\UpOmega\vdash\UpXi,P\mathbin{{\multimap}}N\ ;\ \UpDelta}{\UpGamma\ ;\
\UpOmega,P\vdash\UpXi,N\ ;\
\UpDelta}\quad\linfer[{{?}_{z}}\text{{r}}]{\UpGamma\ ;\
\UpOmega\vdash\UpXi,{?}_{z}{P^{-}}\ ;\ \UpDelta}{\UpGamma\ ;\
\UpOmega\vdash\UpXi\ ;\ \UpDelta,{z{\,:\,}{P^{-}}}}$
(left active)
$\displaystyle\linfer[\text{{al}}]{\UpGamma\ ;\ \UpOmega,a\vdash\UpXi\ ;\
\UpDelta}{\UpGamma,{{\mathfrak{l}}{\,:\,}a}\ ;\ \UpOmega\vdash\UpXi\ ;\
\UpDelta}\quad\linfer[{\mathbin{{\otimes}}}\text{{l}}]{\UpGamma\ ;\
\UpOmega,P\mathbin{{\otimes}}Q\vdash\UpXi\ ;\ \UpDelta}{\UpGamma\ ;\
\UpOmega,P,Q\vdash\UpXi\ ;\
\UpDelta}\quad\linfer[{{\mathbf{1}}}\text{{l}}]{\UpGamma\ ;\
\UpOmega,{\mathbf{1}}\vdash\UpXi\ ;\ \UpDelta}{\UpGamma\ ;\
\UpOmega\vdash\UpXi\ ;\ \UpDelta}$ ll
(decision)
$\displaystyle\UpGamma\ ;\ \cdot\vdash\cdot\ ;\
\UpDelta,{r{\,:\,}P}\UpGamma\vdash\left[P\right]\ ;\ \UpDelta\quad\UpGamma\ ;\
\cdot\vdash\cdot\ ;\ \UpDelta,{u{\,:\,}P}\UpGamma\vdash\left[P\right]\ ;\
\UpDelta,{u{\,:\,}P}\quad\UpGamma,{r{\,:\,}N}\ ;\ \cdot\vdash\cdot\ ;\
\UpDelta\UpGamma\ ;\ \left[N\right]\vdash\UpDelta\quad\UpGamma,{u{\,:\,}N}\ ;\
\cdot\vdash\cdot\ ;\ \UpDelta\UpGamma,{u{\,:\,}N}\ ;\
\left[N\right]\vdash\UpDelta$
Figure 1: Focused sequent calculus for classical subexponential logic
The rules of the calculus are presented in fig. 2. Focused sequent calculi
presented in this style, which is a stylistic variant of Andreoli’s original
formulation [1], have an intensional reading in terms of _phases_. At the
boundaries of phases are sequents of the form $\UpGamma\ ;\ \cdot\vdash\cdot\
;\ \UpDelta$, which are known as _neutral sequents_. Proofs of neutral
sequents proceed (reading from conclusion to premises) as follows:
1. 1.
_Decision_ : a _focus_ is selected from a neutral sequent, either from the
left or the right context. This focused formula is moved to its corresponding
focused zone using one of the rules rdr, udr, rdl and udl (u/r =
“unrestricted”/“restricted”, d = “decision”, and r/l = “right”/“left”). These
_decision_ rules copy the focused formula iff it occurs in an unrestricted
zone.
2. 2.
_Focused phase_ : for a left or a right focused sequent, left or right focus
rules are applied to the formula under focus. These focused rules are all non-
invertible in the (unfocused) sequent calculus and therefore depend on
essential choices made in the proof. In all cases except ${{!}_{z}}\text{{r}}$
and ${{?}_{z}}\text{{l}}$ the focus persists to the subformulas (if any) of
the focused formula. For binary rules, the restricted portions of the contexts
are separated and distributed to the two premises. This much should be
familiar from focusing for linear logic [1, 7].
The two unusual rules for subexponential logic are ${{!}_{z}}\text{{r}}$ and
${{?}_{z}}\text{{l}}$, which are generalizations of rules for the single
exponential in ordinary linear logic. These rules have a side condition that
no formulas in a strictly $\leq$-smaller zone may be present in the
conclusion. If the working zone ${\mathfrak{l}}$ is $\leq$-minimal (which is
not necessarily the case), then this side condition is trivial and the rules
amount to a pure change of polarities, similar to the $\uparrow$ and
$\downarrow$ connectives of polarised linear logic [10]. For the other zones,
this rule tests for the emptiness of some of the zones. It is this selective
emptiness test that gives subexponential logic its expressive power [15, 14].
3. 3.
_Active phase_ : once the exponential rules ${{!}_{z}}\text{{r}}$ and
${{?}_{z}}\text{{l}}$ are applied, the sequents become active and left and
right active rules are applied. The order of the active rules is immaterial as
all orderings will produce the same list of neutral sequent premises. In
Andreoli’s system the irrelevant non-determinism in the order of these rules
was removed by treating the active contexts $\UpXi$ and $\UpOmega$ as ordered
contexts; however, we do not fix any particular ordering.
In the traditional model of focusing, the above three steps repeat, in that
order, in the entire proof. The focused system can therefore be seen as a
system of _synthetic_ inference rules (sometimes known as _bipoles_) for
neutral sequents. It is possible to give a very general presentation of such
synthetic inference systems, for which we can prove completeness and cut-
elimination in a very general fashion [5]. It is also possible, with some non-
trivial effort, to show completeness of the focused calculus without appealing
to synthetic rules [7, 11]. We do not delve into such proofs in this paper
because this ground is well trodden. Indeed, a focused completeness theorem
for a very similar (but more general) formulation of subexponential logic can
be found in [14, chapter 6]. The synthetic soundness and completeness theorems
are as follows, proof omitted:
###### Fact 6 (synthetic soundness and completeness)
Write $\Vdash$ for the sequent arrow for an unfocused variant of the calculus
of fig. 2, obtained by placing the focused and active formulas in the
${\mathfrak{l}}$ zone and relaxing the focusing discipline.111This is
basically Gentzen’s LK in two-sided form for subexponential logic.
1. 1.
If $\UpGamma\ ;\ \cdot\vdash\cdot\ ;\ \UpDelta$, then $\UpGamma\Vdash\UpDelta$
(synthetic soundness).
2. 2.
If
$\UpGamma,{{\mathfrak{l}}{\,:\,}\UpOmega}\Vdash{{\mathfrak{l}}{\,:\,}\UpXi},\UpDelta$
then $\UpGamma\ ;\ \UpOmega\vdash\UpXi\ ;\ \UpDelta$ (synthetic completeness).
∎
Despite its somewhat esoteric formulation, it is easy to see how
subexponential logic generalizes classical substructural logics.
###### Fact 7 (familiar instances)
* •
_Polarised classical multiplicative additive linear logic_ (MALL) is
determined by
$\mathtt{mall}=\left\langle\\{{\mathfrak{l}}\\},\cdot,{\mathfrak{l}},\emptyset\right\rangle$.
The injections between the two polarised classes, sometimes known as _shifts_
, are as follows: $\downarrow={!}_{{\mathfrak{l}}}$ and
$\uparrow={?}_{\mathfrak{l}}$.
* •
_Polarised classical linear logic_ (CLL) is determined by
$\mathtt{ll}=\left\langle\\{{\mathfrak{l}},{\mathfrak{u}}\\},{\mathfrak{l}}\leq{\mathfrak{u}},{\mathfrak{l}},\\{{\mathfrak{u}}\\}\right\rangle$.
In addition to the injections of mall, we also have the exponentials
${!}={!}_{\mathfrak{u}}$ and ${?}={?}_{\mathfrak{u}}$.
* •
_Polarised classical logic_ (CL) is given by the signature
$\mathtt{l}=\left\langle\\{{\mathfrak{l}}\\},\cdot,{\mathfrak{l}},\\{{\mathfrak{l}}\\}\right\rangle$.
∎
In addition to such instances produced by instantiating the subexponential
signature, it is also possible to get the unpolarised versions of these logics
by applying ${!}_{\mathfrak{l}}$ and ${?}_{\mathfrak{l}}$ to immediate
negative (resp. positive) subformulas of positive (resp. negative) formulas.
## 3 Intuitionistic subexponential logic
One direct way of defining intuitionistic fragments of classical logics is as
follows:
###### Definition 8 (intuitionistic restriction)
Given a two-sided sequent calculus, its _intuitionistic restriction_ is that
fragment where all inference rules are constrained to have exactly a single
formula on the right hand sides of sequents.
The practical import of this restriction is that the connectives
$\mathbin{{\invamp}}$ and ${\bot}$ disappear, because their right rules
require two and zero conclusions, respectively. As a result,
$\mathbin{{\multimap}}$ becomes a primitive because its classical definition
requires $\mathbin{{\invamp}}$ (and De Morgan duals, which are also missing
with the intuitioistic restriction). In a slight break from tradition [9, 16,
2], we retain ${?}_{z}$ in the intuitionistic syntax. The intuitionistic
restriction produces the following kinds of sequents:
$\UpGamma$ $\vdash$ | $\left[P\right]$ | right focus on $P$
---|---|---
$\UpGamma\ ;\ \left[N\right]$ $\vdash$ | ${z{\,:\,}{Q^{-}}}$ | left focus on $N$
$\UpGamma\ ;\ \UpOmega$ $\vdash$ | ${N^{+}}\ ;\ \cdot$ | active on $\UpOmega$ and ${N^{+}}$
$\UpGamma\ ;\ \UpOmega$ $\vdash$ | $\cdot\ ;\ {z{\,:\,}{Q^{-}}}$ | active on $\UpOmega$
We shall use $\gamma$ to stand for the right hand forms—either ${N^{+}}\ ;\
\cdot$ or $\cdot\ ;\ {z{\,:\,}{Q^{-}}}$—for active sequents above. The full
collection of rules is given in fig. 3. As before, we use ${Q^{-}}$ (resp.
${N^{+}}$) to refer to a positive formula or negative atom (resp. negative
formula or positive atom).
The nature of subexponential signatures does not change in moving from
classical to intuitionistic logic. The decision rule udr obviously cannot copy
the right formula in the intuitionistic case. Thus, both the right decision
rules collapse; ${?}_{z}$ takes on an additional modal aspect and is no longer
the perfect dual of ${!}_{z}$. The standard explanation of this loss of
symmetry in the exponentials is the creation of a new _possibility_ judgement
that is weaker than linear truth; see [3] for such a reconstruction of the
intuitionistic ${?}$.
(right focus)
r
$\displaystyle\linfer[{\mathbin{{\oplus}}}\text{{r}}_{i}]{\UpGamma\vdash\left[P_{1}\mathbin{{\oplus}}P_{2}\right]}{\UpGamma\vdash\left[P_{i}\right]}\quad\UpGamma\vdash\left[{!}_{z}{N^{+}}\right]\lx@proof@logical@and\UpGamma\
;\ \cdot\vdash{N^{+}}\ ;\ \cdot\bigl{(}\forall{{x{\,:\,}A}\in\UpGamma}.\,z\leq
x\bigr{)}$
(left focus)
$\displaystyle\linfer[\text{{nl}}]{{\UpGamma}^{\mathfrak{u}}\ ;\
\left[n\right]\vdash{z{\,:\,}n}}{}\quad\linfer[{\mathbin{{\&}}}\text{{l}}_{i}]{\UpGamma\
;\ \left[P_{1}\mathbin{{\&}}P_{2}\right]\vdash{z{\,:\,}{Q^{-}}}}{\UpGamma\ ;\
\left[P_{i}\right]\vdash{z{\,:\,}{Q^{-}}}}\quad{\UpGamma}^{\mathfrak{u}},{\UpGamma}^{\mathfrak{r}}_{1},{\UpGamma}^{\mathfrak{r}}_{2}\
;\
\left[P\mathbin{{\multimap}}M\right]\vdash{z{\,:\,}{Q^{-}}}\lx@proof@logical@and{\UpGamma}^{\mathfrak{u}},{\UpGamma}^{\mathfrak{r}}_{1}\vdash\left[P\right]{\UpGamma}^{\mathfrak{u}},{\UpGamma}^{\mathfrak{r}}_{2}\
;\ \left[M\right]\vdash{z{\,:\,}{Q^{-}}}$ $\displaystyle\UpGamma\ ;\
\left[{?}_{z}{P^{-}}\right]\vdash{y{\,:\,}{Q^{-}}}\lx@proof@logical@and\UpGamma\
;\ {P^{-}}\vdash\cdot\ ;\
{y{\,:\,}{Q^{-}}}\bigl{(}\forall{{x{\,:\,}A}\in\UpGamma,{y{\,:\,}{Q^{-}}}}.\,z\leq
x\bigr{)}$
right active
r $\displaystyle\linfer[{\mathbin{{\multimap}}}\text{{r}}]{\UpGamma\ ;\
\UpOmega\vdash P\mathbin{{\multimap}}N\ ;\ \cdot}{\UpGamma\ ;\
\UpOmega,P\vdash N\ ;\ \cdot}\quad\linfer[{{?}_{z}}\text{{r}}]{\UpGamma\ ;\
\UpOmega\vdash{?}_{z}P\ ;\ \cdot}{\UpGamma\ ;\ \UpOmega\vdash\cdot\ ;\
{z{\,:\,}P}}$
(left active)
$\displaystyle\linfer[\text{{al}}]{\UpGamma\ ;\
\UpOmega,a\vdash\gamma}{\UpGamma,{{\mathfrak{l}}{\,:\,}a}\ ;\
\UpOmega\vdash\gamma}\quad\linfer[{\mathbin{{\otimes}}}\text{{l}}]{\UpGamma\
;\ \UpOmega,P\mathbin{{\otimes}}Q\vdash\gamma}{\UpGamma\ ;\
\UpOmega,P,Q\vdash\gamma}\quad\linfer[{{\mathbf{1}}}\text{{l}}]{\UpGamma\ ;\
\UpOmega,{\mathbf{1}}\vdash\gamma}{\UpGamma\ ;\ \UpOmega\vdash\gamma}$ ll
(decision)
$\displaystyle\UpGamma\ ;\ \cdot\vdash\cdot\ ;\
{z{\,:\,}P}\UpGamma\vdash\left[P\right]\quad\UpGamma,{r{\,:\,}N}\ ;\
\cdot\vdash\cdot\ ;\ {z{\,:\,}{Q^{-}}}\UpGamma\ ;\
\left[N\right]\vdash{z{\,:\,}{Q^{-}}}\quad\UpGamma,{u{\,:\,}N}\ ;\
\cdot\vdash\cdot\ ;\ {z{\,:\,}{Q^{-}}}\UpGamma,{u{\,:\,}N}\ ;\
\left[N\right]\vdash{z{\,:\,}{Q^{-}}}$
Figure 2: Focused sequent calculus for intuitionstic subexponential logic
The proof of completeness for focused intuitionistic subexponential logic has
never been published. However, any similar proof for intuitionistic linear
logic, such as [7, 11], can be adapted. Again, we simply state the synthetic
version of the theorems here without proof.
###### Fact 9 (synthetic soundness and completeness)
Write $\Vdash$ for the sequent arrow for an unfocused variant of the calculus
of fig. 3, obtained by placing the focused and active formulas in the
${\mathfrak{l}}$ zone and relaxing the focusing discipline.
1. 1.
If $\UpGamma\ ;\ \cdot\vdash\cdot\ ;\ {z{\,:\,}{Q^{-}}}$, then
$\UpGamma\Vdash{z{\,:\,}{Q^{-}}}$.
2. 2.
If $\UpGamma,{{\mathfrak{l}}{\,:\,}\UpOmega}\Vdash{z{\,:\,}{Q^{-}}}$ then
$\UpGamma\ ;\ \UpOmega\vdash\cdot\ ;\ {z{\,:\,}{Q^{-}}}$.
3. 3.
If $\UpGamma,{{\mathfrak{l}}{\,:\,}\UpOmega}\Vdash{{\mathfrak{l}}{\,:\,}N}$
then $\UpGamma\ ;\ \UpOmega\vdash N\ ;\ \cdot$. ∎
The intuitionstic restrictions of the familiar instances from defn. 7 simply
use the same subexponential signatures.
## 4 Focally adequate encodings
This section contains the main technical contribution of this paper: focally
adequate encodings (defn. 2) that are generic on subexponential signatures. At
the level of focal adequacy, therefore, the asymmetry in the expressive power
of classical and intuitionistic logics disappears.
### 4.1 Classical in intuitionistic
To introduce the mechanisms of encoding, we first look at the unsurprising
direction: a classical logic in its own intuitionistic restriction. The well
known double negation translation, if performed clumsily, can break even full
adequacy. For example, if $N\mathbin{{\invamp}}M$ is translated as
$\lnot({!}_{\mathfrak{l}}\lnot{!}_{\mathfrak{l}}N\mathbin{{\otimes}}{!}_{\mathfrak{l}}\lnot{!}_{\mathfrak{l}}M)$
where $\lnot P\triangleq P\mathbin{{\multimap}}k$ where $k$ is some fixed
negative atom that is not used in classical logic. In the rule
${\mathbin{{\invamp}}}\text{{r}}$ under this encoding, there are instances of
${!}_{\mathfrak{l}}$ that have no counterpart in the classical side. Indeed,
there is no derived rule in the classical focused calculus that allows one to
conclude $\UpGamma\ ;\ \cdot\vdash N\mathbin{{\invamp}}M\ ;\ \cdot$ from
$\UpGamma\ ;\ \cdot\vdash\cdot\ ;\ {!}_{\mathfrak{l}}N,{!}_{\mathfrak{l}}M$,
which is what would result if the active phase could be suspended arbitrarily
and the subformula property were discarded. Such a rule is certainly
admissible, but admissibile rules do not preserve bijections between proofs,
and are only definable for full proofs in any case.
How does one encode classical logic in its intuitionistic restriction such
that polarities are respected? The above example suggests an obvious answer:
when translating $N\mathbin{{\invamp}}M$ as if it were right-active, do not
also translate the subformulas $M$ and $N$ as if they were right-active, for
they will be sent to the left. Instead, translate them as if they were _left_
-active.222The astute reader might recall that this is the essence of Kuroda’s
encodings.
###### Definition 10 (encoding classical formulas)
* •
The encoding $\left(-\right)^{=}$ from classical positive (resp. negative)
formulas to intuitionistic positive (resp. negative) formulas is as follows:
$\displaystyle\left(p\right)^{=}$ $\displaystyle=p$
$\displaystyle\left({!}_{z}N\right)^{=}$
$\displaystyle={!}_{z}\left(N\right)^{=}$
$\displaystyle\left(P\mathbin{{\otimes}}Q\right)^{=}$
$\displaystyle=\left(P\right)^{=}\mathbin{{\otimes}}\left(Q\right)^{=}$
$\displaystyle\left({\mathbf{1}}\right)^{=}$ $\displaystyle={\mathbf{1}}$
$\displaystyle\left(P\mathbin{{\oplus}}Q\right)^{=}$
$\displaystyle=\left(P\right)^{=}\mathbin{{\oplus}}\left(Q\right)^{=}$
$\displaystyle\left({\mathbf{0}}\right)^{=}$ $\displaystyle={\mathbf{0}}$
$\displaystyle\left(N\right)^{=}$ $\displaystyle=\lnot\left(N\right)^{\neq}$
* •
The encoding $\left(-\right)^{\neq}$ from classical negative (resp. positive)
formulas to intuitionstic positive (resp. negative) formulas is as follows:
$\displaystyle\left(n\right)^{\neq}$ $\displaystyle={n}^{\perp}$
$\displaystyle\left({?}_{z}P\right)^{\neq}$
$\displaystyle={!}_{z}\left(P\right)^{\neq}$
$\displaystyle\left(N\mathbin{{\invamp}}N\right)^{\neq}$
$\displaystyle=\left(N\right)^{\neq}\mathbin{{\otimes}}\left(M\right)^{\neq}$
$\displaystyle\left({\bot}\right)^{\neq}$ $\displaystyle={\mathbf{1}}$
$\displaystyle\left(N\mathbin{{\&}}M\right)^{\neq}$
$\displaystyle=\left(N\right)^{\neq}\mathbin{{\oplus}}\left(M\right)^{\neq}$
$\displaystyle\left({\top}\right)^{\neq}$ $\displaystyle={\mathbf{0}}$
$\displaystyle\left(P\mathbin{{\multimap}}N\right)^{\neq}$
$\displaystyle=\left(P\right)^{=}\mathbin{{\otimes}}\left(N\right)^{\neq}$
$\displaystyle\left(P\right)^{\neq}$ $\displaystyle=\lnot\left(P\right)^{=}$
where for every negative atom $n$, there is a positive atom ${n}^{\perp}$ in
the encoding.
Contexts are translated element-wise.
###### Definition 11 (encoding classical sequents)
The encoding ${\left(\hbox to0.0pt{\vbox to5.42494pt{}}-\right)}^{\perp\perp}$
of classical sequents as intuitionistic sequents is as follows:
$\displaystyle{\left(\hbox to0.0pt{\vbox
to5.42494pt{}}\UpGamma\vdash\left[P\right]\ ;\ \UpDelta\right)}^{\perp\perp}$
$\displaystyle=\left(\UpGamma\right)^{=},\left(\UpDelta\right)^{\neq}\vdash\left[\left(P\right)^{=}\right]\qquad{\left(\hbox
to0.0pt{\vbox to5.42494pt{}}\UpGamma\ ;\
\left[N\right]\vdash\UpDelta\right)}^{\perp\perp}=\left(\UpGamma\right)^{=},\left(\UpDelta\right)^{\neq}\vdash\left[\smash{\left(N\right)^{\neq}}\right]$
$\displaystyle{\left(\hbox to0.0pt{\vbox to5.42494pt{}}\UpGamma\ ;\
\UpOmega\vdash\UpXi\ ;\ \UpDelta\right)}^{\perp\perp}$
$\displaystyle=\left(\UpGamma\right)^{=},\left(\UpDelta\right)^{\neq}\ ;\
\left(\UpOmega\right)^{=},\left(\UpXi\right)^{\neq}\vdash\cdot\ ;\
{{\mathfrak{l}}{\,:\,}k}$
In other words, focused sequents are translated to right-focused sequents, and
active sequents to left-active sequents. The right contexts are dualised and
sent to the left where the intuitionistic restriction does not apply, while
the left focus on negative formulas is turned into a right focus because of
the lack of a multiplicative left-focused rule (except
${\mathbin{{\multimap}}}\text{{l}}$ which would cause an inadvertent polarity
switch).
###### Theorem 12
The encoding of defn. 11 is focally adequate (defn. 2).
###### Proof
We will inventory the classical rules in fig. 2, and in each case compute the
intuitionistic synthetic derivations of the encoding of the conclusion of the
classical rules. Here are the interesting333See [6] for the remaining cases.
cases, with the double inference lines denoting (un)folding of defns. 10 and
11, and the rule names written with the prefix c/ or i/ to distinguish between
classical and intuitionistic respectively.
* •
_cases of c/pr and $\text{{c/}}{{!}}\text{{r}}$_:
$\displaystyle{\left(\hbox to0.0pt{\vbox
to5.42494pt{}}{\UpGamma}^{\mathfrak{u}},{z{\,:\,}p}\vdash\left[p\right]\ ;\
{\UpDelta}^{\mathfrak{u}}\right)}^{\perp\perp}\left({\UpGamma}^{\mathfrak{u}}\right)^{=},\left({z{\,:\,}p}\right)^{=},\left({\UpDelta}^{\mathfrak{u}}\right)^{\neq}\vdash\left[p\right]\left({\UpGamma}^{\mathfrak{u}}\right)^{=},{z{\,:\,}p},\left({\UpDelta}^{\mathfrak{u}}\right)^{\neq}\vdash\left[p\right]\qquad{\left(\hbox
to0.0pt{\vbox to5.42494pt{}}\UpGamma\vdash\left[{!}_{z}N\right]\ ;\
\UpDelta\right)}^{\perp\perp}\left(\UpGamma\right)^{=},\left(\UpDelta\right)^{\neq}\vdash\left[\left({!}_{z}N\right)^{=}\right]\left(\UpGamma\right)^{=},\left(\UpDelta\right)^{\neq}\vdash\left[{!}_{z}\lnot\left(N\right)^{\neq}\right]\left(\UpGamma\right)^{=},\left(\UpDelta\right)^{\neq}\
;\ \cdot\vdash\lnot\left(N\right)^{\neq}\ ;\
\cdot\left(\UpGamma\right)^{=},\left(\UpDelta\right)^{\neq}\ ;\
\left(N\right)^{\neq}\vdash k\ ;\
\cdot\left(\UpGamma\right)^{=},\left(\UpDelta\right)^{\neq}\ ;\
\left(N\right)^{\neq}\vdash\cdot\ ;\ {{\mathfrak{l}}{\,:\,}k}{\left(\hbox
to0.0pt{\vbox to5.42494pt{}}\UpGamma\ ;\ \cdot\vdash N\ ;\
\UpDelta\right)}^{\perp\perp}$
All the logical rules used are invertible. The boxed instance of
$\text{{i/}}{{?}_{\mathfrak{l}}}\text{{r}}$ requires some explanation:
obviously a left active rule on $\left(N\right)^{\neq}$ can be applied before
this rule. However, since they are both active rules, the choice of which to
perform first is immaterial as they will produce the same neutral premises. If
we want local—not focal—adequacy, we will have to impose a right-to-left
ordering on the active rules. The case of c/nl and
$\text{{c/}}{{?}}\text{{l}}$ is similar.
* •
_case of $\text{{c/}}{\mathbin{{\invamp}}}\text{{r}}$_:
$\displaystyle\linfer={{\left(\hbox to0.0pt{\vbox to5.42494pt{}}\UpGamma\ ;\
\UpOmega\vdash\UpXi,N\mathbin{{\invamp}}M\ ;\
\UpDelta\right)}^{\perp\perp}}{\left(\UpGamma\right)^{=},\left(\UpDelta\right)^{\neq}\
;\
\left(\UpOmega\right)^{=},\left(\UpXi\right)^{\neq},\left(N\mathbin{{\invamp}}M\right)^{\neq}\vdash\cdot\
;\
{{\mathfrak{l}}{\,:\,}k}\left(\UpGamma\right)^{=},\left(\UpDelta\right)^{\neq}\
;\
\left(\UpOmega\right)^{=},\left(\UpXi\right)^{\neq},\left(N\right)^{\neq}\mathbin{{\otimes}}\left(M\right)^{\neq}\vdash\cdot\
;\
{{\mathfrak{l}}{\,:\,}k}\left(\UpGamma\right)^{=},\left(\UpDelta\right)^{\neq}\
;\
\left(\UpOmega\right)^{=},\left(\UpXi\right)^{\neq},\left(N\right)^{\neq},\left(M\right)^{\neq}\vdash\cdot\
;\ {{\mathfrak{l}}{\,:\,}k}{\left(\hbox to0.0pt{\vbox to5.42494pt{}}\UpGamma\
;\ \UpOmega\vdash\UpXi,N,M\ ;\ \UpDelta\right)}^{\perp\perp}}$
The cases of $\text{{c/}}{{\bot}}\text{{r}}$, $\text{{c/}}{{!}_{z}}\text{{l}}$
and $\text{{c/}}{{?}_{z}}\text{{r}}$ are similar.
* •
_case of c/rdr_:
$\displaystyle{\left(\hbox to0.0pt{\vbox
to5.42494pt{}}{{\UpGamma}^{\mathfrak{u}}_{1},{\UpGamma}^{\mathfrak{r}}_{2}}\
;\ \cdot\vdash\cdot\ ;\
{{\UpDelta}^{\mathfrak{u}}_{1},{\UpDelta}^{\mathfrak{r}}_{2}},{r{\,:\,}P}\right)}^{\perp\perp}\left({\UpGamma}^{\mathfrak{u}}_{1},{\UpGamma}^{\mathfrak{r}}_{2}\right)^{=},\left({{\UpDelta}^{\mathfrak{u}}_{1},{\UpDelta}^{\mathfrak{r}}_{2}}\right)^{\neq},\left({r{\,:\,}P}\right)^{\neq}\
;\ \cdot\vdash\cdot\ ;\
{{\mathfrak{l}}{\,:\,}k}\left({\UpGamma}^{\mathfrak{u}}_{1},{\UpGamma}^{\mathfrak{r}}_{2}\right)^{=},\left({{\UpDelta}^{\mathfrak{u}}_{1},{\UpDelta}^{\mathfrak{r}}_{2}}\right)^{\neq},{r{\,:\,}\lnot\left(P\right)^{=}}\
;\ \cdot\vdash\cdot\ ;\
{{\mathfrak{l}}{\,:\,}k}\left({\UpGamma}^{\mathfrak{u}}_{1},{\UpGamma}^{\mathfrak{r}}_{2}\right)^{=},\left({{\UpDelta}^{\mathfrak{u}}_{1},{\UpDelta}^{\mathfrak{r}}_{2}}\right)^{\neq}\
;\
\left[\lnot\left(P\right)^{=}\right]\vdash{{\mathfrak{l}}{\,:\,}k}\lx@proof@logical@and\left({\UpGamma}^{\mathfrak{u}}_{1},{\UpGamma}^{\mathfrak{r}}_{2}\right)^{=},\left({{\UpDelta}^{\mathfrak{u}}_{1},{\UpDelta}^{\mathfrak{r}}_{2}}\right)^{\neq}\vdash\left[\left(P\right)^{=}\right]{\left(\hbox
to0.0pt{\vbox
to5.42494pt{}}{{\UpGamma}^{\mathfrak{u}}_{1},{\UpGamma}^{\mathfrak{r}}_{2}}\vdash\left[P\right]\
;\
{{\UpDelta}^{\mathfrak{u}}_{1},{\UpDelta}^{\mathfrak{r}}_{2}}\right)}^{\perp\perp}\left({\UpGamma}^{\mathfrak{u}}_{1}\right)^{=},\left({\UpDelta}^{\mathfrak{u}}_{1}\right)^{\neq}\
;\ \left[k\right]\vdash{{\mathfrak{l}}{\,:\,}k}$
Note that the right premise is forced to terminate in the same phase. This
would not be possible if, instead of $k$, we were to use some other negative
formula such as ${?}_{\mathfrak{l}}{\mathbf{0}}$. In the presence of some
unrestricted subexponential $u$, we might have used ${?}_{u}{\mathbf{0}}$
instead (note that, classically, ${?}_{u}{\mathbf{0}}\equiv{\bot}$). ∎
###### Corollary 13
* •
There is a focally adequate encoding of classical MALL in intuitionistic MALL.
* •
There is a focally adequate encoding of CLL in ILL.
* •
There is a focally adequate encoding of CL in IL.
###### Proof
Instantiate thm. 12 on the subexponential signatures from defn. 7. ∎
These instances are all apparently novel, partly because focal adequacy of
classical logics in their own intuitionistic restrictions has not been deeply
investigated. In the work on LJF [11] there is a focally adequate encoding of
classical logic in intuitionistic linear logic, which can be seen as a
combination of the second and third of the above instances.
### 4.2 Intuitionistic in classical
The previous subsection showed that the intuitionistic restriction of a
classical logic can adequately encode the classical logic itself. This is not
the case in the other direction without further modifications to the
subexponential signature. It is easy to see this: consider just the MALL
fragment and the problem of encoding the
$\text{{i/}}{\mathbin{{\multimap}}}\text{{l}}$ rule. If
$\mathbin{{\multimap}}$ is encoded as itself, then in the classical side we
have the following derived rule (all the zones are ${\mathfrak{l}}$, and
elided):
$\displaystyle\UpGamma\ ;\
\left[P\mathbin{{\multimap}}N\right]\vdash{Q^{-}}\lx@proof@logical@and\UpGamma\vdash\left[P\right]\
;\ {Q^{-}}\UpGamma\ ;\ \left[N\right]\vdash\cdot$
This rule has no intuitionistic counterpart. Therefore, the encoding of
$\mathbin{{\multimap}}$ must prevent the right formula ${Q^{-}}$ from being
sent to the left branch, i.e., to test that the rest of the right context in a
right focus is empty. MALL itself cannot perform this test because it lacks
any truly modal operators. Exactly the same problem exists for the encoding of
IL in CL, which also lacks any true modal operators.
Quite obviously, the encoding of $\mathbin{{\multimap}}$ requires some means
of testing the emptiness of contexts. CLL (defn. 7) has an additional zone
${\mathfrak{u}}$ that is greater than ${\mathfrak{l}}$, and therefore
${!}_{\mathfrak{u}}$ can test for the absence of any
${\mathfrak{l}}$-formulas. It turns out that this is enough to get a focally
adequate encoding of IL as follows: the sole zone ${\mathfrak{l}}$ of IL is
split into two, ${\mathfrak{l}}_{r}$ (restricted) and ${\mathfrak{l}}_{u}$
(unrestricted), and the right hand side of IL sequents is encoded with
${\mathfrak{l}}_{r}$. Then, whenever $P$ is of the form ${!}_{\mathfrak{l}}N$,
the translation of it on the right is of the form ${!}_{{\mathfrak{l}}_{u}}M$.
In the rest of this subsection, we will systematically extend this observation
to an arbitrary subexponential signature.
###### Definition 14 (signature splitting)
Let a subexponential signature $\Sigma=\left\langle
Z,\leq,{\mathfrak{l}},U\right\rangle$ be given. Write:
* •
$\hat{Z}$ for the zone set
$(Z\times\\{\mathtt{l}\\})\cup(Z\times\\{\mathtt{r}\\})$, where $\mathtt{l}$
and $\mathtt{r}$ are distinct labels for the left and the right of the
sequents, respectively, and $\times$ is the Cartesian product.
$Z\times\\{\mathtt{l}\\}$ will be called the _left form_ of $\hat{Z}$, and
$Z\times\\{\mathtt{r}\\}$ will be called its _right form_.
* •
$\hat{U}$ for the unrestricted zone set $U\times\\{\mathtt{l}\\}$.
* •
$\hat{}{\mathfrak{l}}$ for the working zone $({\mathfrak{l}},\mathtt{l})$.
* •
$\mathbin{\hat{\leq}}$ for the smallest relation on $\hat{Z}\times\hat{Z}$ for
which:
* –
$(x,\mathtt{l})\mathbin{\hat{\leq}}(y,\mathtt{l})$ if $x\leq y$;
* –
$(x,\mathtt{r})\mathbin{\hat{\leq}}(y,\mathtt{r})$ if $x\leq y$; and
* –
$(x,\mathtt{r})\mathbin{\hat{\leq}}(x,\mathtt{l})$ and
$(x,\mathtt{l})\mathbin{\hat{\nleq}}(x,\mathtt{r})$.
The subexponential signature
$\hat{\Sigma}=\left\langle\hat{Z},\mathbin{\hat{\leq}},\hat{}{\mathfrak{l}},\hat{U}\right\rangle$
will be called the _split form_ of $\Sigma$.
We intend to treat the right form specially. The zones in the right form are
restricted, which encodes the linearity of the right hand side inherent in the
intuitionistic restriction (defn. 8). Our encoding will guarantee that the
right hand sides of sequents in the encoding contain no zones in the left
form. Thus, when ${!}_{(z,\mathtt{l})}N$ is under right focus, the side
condition on the ${{!}}\text{{r}}$ rule will ensure that there are no other
formulas on the right hand side, because the right forms are made pointwise
smaller than their left forms. Dually, on the left we shall use
${?}_{(z,\mathtt{r})}$ to encode ${?}_{z}$; since the right form zones are
pointwise smaller than the left form zones, but retain the pre-split ordering
inside their own zone, the side conditions enforce the same occurrences as in
the source calculus.
###### Definition 15 (encoding intuitionistic contexts)
* •
The left-passive context $\UpGamma$ is encoded pointwise using the translation
$\left(-\right)^{\mathtt{lp}}$:
$\displaystyle\left({z{\,:\,}{N^{+}}}\right)^{\mathtt{lp}}$
$\displaystyle={(z,\mathtt{l}){\,:\,}\left({N^{+}}\right)^{\mathtt{lp}}}$
$\displaystyle\left(p\right)^{\mathtt{lp}}$ $\displaystyle=p$
$\displaystyle\left(N\right)^{\mathtt{lp}}$
$\displaystyle=\left(N\right)^{\mathtt{lf}}$
* •
A left-focused formula $N$ is encoded using the translation
$\left(-\right)^{\mathtt{lf}}$:
$\displaystyle\left(n\right)^{\mathtt{lf}}$
$\displaystyle=n\qquad\left({?}_{z}{P^{-}}\right)^{\mathtt{lf}}={?}_{(z,\mathtt{r})}\left({P^{-}}\right)^{\mathtt{la}}\quad\left(N\mathbin{{\&}}M\right)^{\mathtt{lf}}=\left(N\right)^{\mathtt{lf}}\mathbin{{\&}}\left(M\right)^{\mathtt{lf}}\quad\left({\top}\right)^{\mathtt{lf}}={\top}$
$\displaystyle\left(P\mathbin{{\multimap}}N\right)^{\mathtt{lf}}$
$\displaystyle=\left(P\right)^{\mathtt{rf}}\mathbin{{\multimap}}\left(N\right)^{\mathtt{lf}}$
* •
A right-focused formula $P$ is encoded using the translation
$\left(-\right)^{\mathtt{rf}}$:
$\displaystyle\left(p\right)^{\mathtt{rf}}$
$\displaystyle=p\qquad\left({!}_{z}{N^{+}}\right)^{\mathtt{rf}}={!}_{(z,\mathtt{l})}\left({N^{+}}\right)^{\mathtt{ra}}\quad\left(P\mathbin{{\otimes}}Q\right)^{\mathtt{rf}}=\left(P\right)^{\mathtt{rf}}\mathbin{{\otimes}}\left(Q\right)^{\mathtt{rf}}\quad\left({\mathbf{1}}\right)^{\mathtt{rf}}={\mathbf{1}}$
$\displaystyle\left(P\mathbin{{\oplus}}Q\right)^{\mathtt{rf}}$
$\displaystyle=\left(P\right)^{\mathtt{rf}}\mathbin{{\oplus}}\left(Q\right)^{\mathtt{rf}}\quad\left({\mathbf{0}}\right)^{\mathtt{rf}}={\mathbf{0}}$
* •
A left-active context $\UpOmega$ is encoded pointwise using the translation
$\left(-\right)^{\mathtt{la}}$:
$\displaystyle\left(a\right)^{\mathtt{la}}$
$\displaystyle={!}_{({\mathfrak{l}},\mathtt{l})}a\qquad\left({!}_{z}{N^{+}}\right)^{\mathtt{la}}={!}_{(z,\mathtt{l})}\left({N^{+}}\right)^{\mathtt{lp}}\quad\left(P\mathbin{{\otimes}}Q\right)^{\mathtt{la}}=\left(P\right)^{\mathtt{la}}\mathbin{{\otimes}}\left(Q\right)^{\mathtt{la}}\quad\left({\mathbf{1}}\right)^{\mathtt{la}}={\mathbf{1}}$
$\displaystyle\left(P\mathbin{{\oplus}}Q\right)^{\mathtt{la}}$
$\displaystyle=\left(P\right)^{\mathtt{la}}\mathbin{{\oplus}}\left(Q\right)^{\mathtt{la}}\quad\left({\mathbf{0}}\right)^{\mathtt{la}}={\mathbf{0}}$
* •
A right-active formula ${N^{+}}$ is encoded using the translation
$\left(-\right)^{\mathtt{ra}}$:
$\displaystyle\left(a\right)^{\mathtt{ra}}$
$\displaystyle={!}_{({\mathfrak{l}},\mathtt{r})}a\qquad\left({?}_{z}{P^{-}}\right)^{\mathtt{ra}}={?}_{(z,\mathtt{r})}\left({P^{-}}\right)^{\mathtt{rp}}\quad\left(N\mathbin{{\&}}M\right)^{\mathtt{ra}}=\left(N\right)^{\mathtt{ra}}\mathbin{{\&}}\left(M\right)^{\mathtt{ra}}\quad\left({\top}\right)^{\mathtt{ra}}={\top}$
$\displaystyle\left(P\mathbin{{\multimap}}N\right)^{\mathtt{ra}}$
$\displaystyle=\left(P\right)^{\mathtt{la}}\mathbin{{\multimap}}\left(N\right)^{\mathtt{ra}}$
* •
A right-passive zoned formula ${z{\,:\,}{P^{-}}}$ is encoded using the
translation $\left(-\right)^{\mathtt{rp}}$:
$\displaystyle\left({z{\,:\,}{P^{-}}}\right)^{\mathtt{rp}}$
$\displaystyle={(z,\mathtt{r}){\,:\,}\left({P^{-}}\right)^{\mathtt{rp}}}$
$\displaystyle\left(n\right)^{\mathtt{rp}}$ $\displaystyle=n$
$\displaystyle\left(P\right)^{\mathtt{rp}}$
$\displaystyle=\left(P\right)^{\mathtt{rf}}$
The cases for $\left({!}_{z}{N^{+}}\right)^{\mathtt{rf}}$ and
$\left({?}_{z}{P^{-}}\right)^{\mathtt{lf}}$ will be crucial for the proof of
thm. 17. Most of the remaining cases can be seen as an abstract interpretation
of the focused rules (fig. 3) on the various contexts. The definition of the
encoding of intuitionistic sequents is now completely systematic.
###### Definition 16 (encoding intuitionistic sequents)
The encoding $\left(\hbox to0.0pt{\vbox to6.58745pt{}}-\right)^{{?}{!}}$ of
intuitionistic sequents as classical sequents is as follows:
$\displaystyle\left(\hbox to0.0pt{\vbox
to6.58745pt{}}\UpGamma\vdash\left[P\right]\right)^{{?}{!}}$
$\displaystyle=\left(\UpGamma\right)^{\mathtt{lp}}\vdash\left[\left(P\right)^{\mathtt{rf}}\right]\
;\ \cdot$ $\displaystyle\left(\hbox to0.0pt{\vbox to6.58745pt{}}\UpGamma\ ;\
\left[N\right]\vdash{z{\,:\,}{Q^{-}}}\right)^{{?}{!}}$
$\displaystyle=\left(\UpGamma\right)^{\mathtt{lp}}\ ;\
\left[\left(N\right)^{\mathtt{lf}}\right]\vdash\left({z{\,:\,}{Q^{-}}}\right)^{\mathtt{rp}}$
$\displaystyle\left(\hbox to0.0pt{\vbox to6.58745pt{}}\UpGamma\ ;\
\UpOmega\vdash{N^{+}}\ ;\ \cdot\right)^{{?}{!}}$
$\displaystyle=\left(\UpGamma\right)^{\mathtt{lp}}\ ;\
\left(\UpXi\right)^{\mathtt{la}}\vdash\left({N^{+}}\right)^{\mathtt{ra}}\ ;\
\cdot$ $\displaystyle\left(\hbox to0.0pt{\vbox to6.58745pt{}}\UpGamma\ ;\
\UpOmega\vdash\cdot\ ;\ {z{\,:\,}{Q^{-}}}\right)^{{?}{!}}$
$\displaystyle=\left(\UpGamma\right)^{\mathtt{lp}}\ ;\
\left(\UpXi\right)^{\mathtt{la}}\vdash\cdot\ ;\
\left({z{\,:\,}{Q^{-}}}\right)^{\mathtt{rp}}$
Observe that the right hand sides of the encoding have the intuitionistic
restriction (defn. 8). This restriction will be enforced at every transtion
from a focused to an active phase, which is enough because the active rules
cannot increase the size of the right contexts.
###### Theorem 17
The encoding of defn. 16 is focally adequate (defn. 2).
###### Proof
As before for thm. 12, we shall prove this by inventorying the intuitionistic
rules of fig. 3, encode the conclusions of each of these rules, and observe
whether the neutral premises of the derived inference rules are in bijection
with those of the fig. 3. All but the following important cases are omitted
here for space reasons.444See [6].
* •
_cases of i/pr and $\text{{i/}}{{!}_{z}}\text{{r}}$_:
$\displaystyle\left(\hbox to0.0pt{\vbox
to6.58745pt{}}{\UpGamma}^{\mathfrak{u}},{z{\,:\,}p}\vdash\left[p\right]\right)^{{?}{!}}\left({\UpGamma}^{\mathfrak{u}}\right)^{\mathtt{lp}},\left({z{\,:\,}p}\right)^{\mathtt{lp}}\vdash\left[\left(p\right)^{\mathtt{rf}}\right]\left({\UpGamma}^{\mathfrak{u}}\right)^{\mathtt{lp}},{z{\,:\,}p}\vdash\left[p\right]\qquad\left(\hbox
to0.0pt{\vbox
to6.58745pt{}}\UpGamma\vdash\left[{!}_{z}{N^{+}}\right]\right)^{{?}{!}}\left(\UpGamma\right)^{\mathtt{lp}}\vdash\left[\left({!}_{z}{N^{+}}\right)^{\mathtt{rf}}\right]\
;\
\cdot\left(\UpGamma\right)^{\mathtt{lp}}\vdash\left[{!}_{(z,\mathtt{l})}\left({N^{+}}\right)^{\mathtt{la}}\right]\
;\ \cdot\left(\UpGamma\right)^{\mathtt{lp}}\ ;\
\cdot\vdash\left({N^{+}}\right)^{\mathtt{la}}\ ;\ \cdot\left(\hbox
to0.0pt{\vbox to6.58745pt{}}\UpGamma\ ;\ \cdot\vdash{N^{+}}\ ;\
\cdot\right)^{{?}{!}}$
The boxed instance of $\text{{c/}}{{!}}\text{{r}}$ is valid because all the
zoned formulas in $\left(\UpGamma\right)^{\mathtt{lp}}$ are in the left form
zones, as is the zone of the ${!}$ itself, so the comparison
$\mathbin{\hat{\leq}}$ is the same as $\leq$ on the intuitionistic zones
(defn. 14).
* •
_case of $\text{{i/}}{\mathbin{{\multimap}}}\text{{l}}$_:
$\displaystyle\left(\hbox to0.0pt{\vbox
to6.58745pt{}}{\UpGamma}^{\mathfrak{u}},{\UpGamma}^{\mathfrak{r}}_{1},{\UpGamma}^{\mathfrak{r}}_{2}\
;\
\left[P\mathbin{{\multimap}}N\right]\vdash{z{\,:\,}{Q^{-}}}\right)^{{?}{!}}\left({\UpGamma}^{\mathfrak{u}},{\UpGamma}^{\mathfrak{r}}_{1},{\UpGamma}^{\mathfrak{r}}_{2}\right)^{\mathtt{lp}}\
;\
\left[\left(P\mathbin{{\multimap}}N\right)^{\mathtt{lf}}\right]\vdash\left({z{\,:\,}{Q^{-}}}\right)^{\mathtt{rp}}\left({\UpGamma}^{\mathfrak{u}},{\UpGamma}^{\mathfrak{r}}_{1},{\UpGamma}^{\mathfrak{r}}_{2}\right)^{\mathtt{lp}}\
;\
\left[\left(P\right)^{\mathtt{rf}}\mathbin{{\multimap}}\left(N\right)^{\mathtt{lf}}\right]\vdash\left({z{\,:\,}{Q^{-}}}\right)^{\mathtt{rp}}\lx@proof@logical@and\left({\UpGamma}^{\mathfrak{u}},{\UpGamma}^{\mathfrak{r}}_{1}\right)^{\mathtt{lp}}\vdash\left[\left(P\right)^{\mathtt{rf}}\right]\
;\ \cdot\left(\hbox to0.0pt{\vbox
to6.58745pt{}}{\UpGamma}^{\mathfrak{u}},{\UpGamma}^{\mathfrak{r}}_{1}\vdash\left[P\right]\right)^{{?}{!}}\left({\UpGamma}^{\mathfrak{u}},{\UpGamma}^{\mathfrak{r}}_{2}\right)^{\mathtt{lp}}\
;\
\left[\left(N\right)^{\mathtt{lf}}\right]\vdash\left({z{\,:\,}{Q^{-}}}\right)^{\mathtt{rp}}\left(\hbox
to0.0pt{\vbox
to6.58745pt{}}{\UpGamma}^{\mathfrak{u}},{\UpGamma}^{\mathfrak{r}}_{2}\ ;\
\left[N\right]\vdash{z{\,:\,}{Q^{-}}}\right)^{{?}{!}}$
The boxed instance of $\text{{c/}}{\mathbin{{\multimap}}}\text{{l}}$ contains
the only split of the right context that can succeed in the same focused
phase, i.e., reach an initial sequent or a phase transition, becaue that
$\left(P\right)^{\mathtt{rf}}$ eventually produces either a positive atom
(which must finish the proof with c/pr and since right form zones are
restricted $\left({z{\,:\,}{Q^{-}}}\right)^{\mathtt{rp}}$ cannot be present)
or a ${!}_{(z,\mathtt{l})}$ which guarantees that the rest of the right
context is empty.
* •
_cases of $\text{{i/}}{{?}_{z}}\text{{l}}$ and dr_:
$\displaystyle\left(\hbox to0.0pt{\vbox to6.58745pt{}}\UpGamma\ ;\
\left[{?}_{z}{P^{-}}\right]\vdash{y{\,:\,}{Q^{-}}}\right)^{{?}{!}}\left(\UpGamma\right)^{\mathtt{lp}}\
;\
\left[\left({?}_{z}{P^{-}}\right)^{\mathtt{lf}}\right]\vdash\left({y{\,:\,}{Q^{-}}}\right)^{\mathtt{rp}}\left(\UpGamma\right)^{\mathtt{lp}}\
;\
\left[{?}_{(z,\mathtt{r})}\left({P^{-}}\right)^{\mathtt{la}}\right]\vdash\left({y{\,:\,}{Q^{-}}}\right)^{\mathtt{rp}}\left(\UpGamma\right)^{\mathtt{lp}}\
;\ \left({P^{-}}\right)^{\mathtt{la}}\vdash\cdot\ ;\
\left({y{\,:\,}{Q^{-}}}\right)^{\mathtt{rp}}\left(\hbox to0.0pt{\vbox
to6.58745pt{}}\UpGamma\ ;\ {P^{-}}\vdash\cdot\ ;\
{y{\,:\,}{Q^{-}}}\right)^{{?}{!}}\qquad\left(\hbox to0.0pt{\vbox
to6.58745pt{}}\UpGamma\ ;\ \cdot\vdash\cdot\ ;\
{z{\,:\,}P}\right)^{{?}{!}}\left(\UpGamma\right)^{\mathtt{lp}}\ ;\
\cdot\vdash\cdot\ ;\
\left({z{\,:\,}P}\right)^{\mathtt{rp}}\left(\UpGamma\right)^{\mathtt{lp}}\ ;\
\cdot\vdash\cdot\ ;\
{(z,\mathtt{r}){\,:\,}\left(P\right)^{\mathtt{rf}}}\left(\UpGamma\right)^{\mathtt{lp}}\vdash\left[\left(P\right)^{\mathtt{rf}}\right]\
;\ \cdot\UpGamma\vdash\left[P\right]$
The boxed instance of $\text{{c/}}{{?}}\text{{l}}$ is justified because the
subscript zone $(z,\mathtt{r})$ is of the right form (in order to compare with
$(y,\mathtt{r})$) which is $\mathbin{\hat{\leq}}$-smaller than its
corresponding left-form zone (defn. 14). Note that it is crucial for soundness
to have $(z,\mathtt{r})$ not be smaller than all left form zones. Since right
form zones are restricted, there is no copying in the boxed instance of c/rdr.
The other decision cases are similar. ∎
We note one important direct corollary of thm. 17.
###### Corollary 18 (intuitionistic logic in classical linear logic)
There is a focally adequate encoding of intuitiontistic logic in classical
linear logic.
It is well known [9] that (classical) linear logic can encode the
intuitionistic implication $\mathbin{{\supset}}$ as follows:
$A\mathbin{{\supset}}B\triangleq{!}A\mathbin{{\multimap}}B$. However, this
encoding is only globally adequate [16]. It is possible to refine this
encoding to obtain a fully adequate encoding [12] in an enriched classical
linear logic which is not apparently an instance of classical subexponential
logic. Corollary 18 further improves our undertanding of encodings of
intuitionistic implicication by permuting ${!}$ into the antecedent of the
implication until there is a phase change, which removes the bureaucratic
polarity switch inherent in this implication.555Note that the polarised
intuitionistic implication $P\mathbin{{\multimap}}N$, if encoded using
Girard’s encoding, would be ${!}{{\uparrow}P}\mathbin{{\multimap}}N$, which
breaks the polarisation of the antecedent.
###### Proof (of cor. 18)
The split of the signature l (defn. 7) is isomorphic to the signature ll, so
apply thm. 17. ∎
## 5 Conclusions
Section 4 shows that any given classical (resp. intuitionistic) subexponential
logic can be encoded in a related intuitionistic (resp. classical)
subexponential logic such that partial synthetic derivations are preserved.
This is a technical result, with at least one of the directions of encoding
being novel. It strongly suggests that one of the fractures in logic
identified by Miller in [13]—the classical/intuitionistic divide—might be
healed by analyses and algorithms that are generic on subexponential
signatures. One might still favour “classical” or “intuitionistic” dialects
for proofs, but neither format is more fundamental.
The results of this paper have two caveats. First, we only consider the
“restricted” or the “unrestricted” flavours of subexponentials; in [8] there
were also subexponentials of the “strict” and “affine” flavours for which our
results here do not extend directly. Second, we do not consider encodings
involving non-propositional kinds, such as terms or frames. Subexponentials
are still useful for such stronger encodings, but _representational adequacy_
may not be as straightforward.
## References
* [1] J.-M. Andreoli. Logic programming with focusing proofs in linear logic. J. of Logic and Computation, 2(3):297–347, 1992.
* [2] A. Barber and G. Plotkin. Dual intuitionistic linear logic. Technical Report ECS-LFCS-96-347, University of Edinburgh, 1996.
* [3] B.-Y. E. Chang, K. Chaudhuri, and F. Pfenning. A judgmental analysis of linear logic. Technical Report CMU-CS-03-131R, Carnegie Mellon University, Dec. 2003\.
* [4] K. Chaudhuri. The Focused Inverse Method for Linear Logic. PhD thesis, Carnegie Mellon University, Dec. 2006. Technical report CMU-CS-06-162.
* [5] K. Chaudhuri. Focusing strategies in the sequent calculus of synthetic connectives. In LPAR-15, volume 5330, pages 467–481, Nov. 2008.
* [6] K. Chaudhuri. Classical and intuitionistic subexponential logics are equally expressive. Technical report, INRIA, 2010.
* [7] K. Chaudhuri, F. Pfenning, and G. Price. A logical characterization of forward and backward chaining in the inverse method. J. of Automated Reasoning, 40(2-3):133–177, Mar. 2008.
* [8] V. Danos, J.-B. Joinet, and H. Schellinx. The structure of exponentials: Uncovering the dynamics of linear logic proofs. In KGC, volume 713, pages 159–171. Springer, 1993.
* [9] J.-Y. Girard. Linear logic. Theoretical Computer Science, 50:1–102, 1987.
* [10] O. Laurent. Etude de la polarisation en logique. Thèse de doctorat, Université Aix-Marseille II, Mar. 2002\.
* [11] C. Liang and D. Miller. Focusing and polarization in linear, intuitionistic, and classical logics. Theoretical Computer Science, 410(46):4747–4768, 2009.
* [12] C. Liang and D. Miller. A unified sequent calculus for focused proofs. In LICS-24, pages 355–364, 2009.
* [13] D. Miller. Finding unity in computational logic. In ACM-BCS-Visions, Apr. 2010.
* [14] V. Nigam. Exploiting non-canonicity in the sequent calculus. PhD thesis, Ecole Polytechnique, Sept. 2009.
* [15] V. Nigam and D. Miller. Algorithmic specifications in linear logic with subexponentials. In PPDP, pages 129–140, 2009.
* [16] H. Schellinx. Some syntactical observations on linear logic. Journal of Logic and Computation, 1(4):537–559, Sept. 1991.
|
arxiv-papers
| 2010-06-16T06:00:23 |
2024-09-04T02:49:10.933646
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kaustuv Chaudhuri",
"submitter": "Kaustuv Chaudhuri",
"url": "https://arxiv.org/abs/1006.3134"
}
|
1006.3158
|
]
# THE SYSTEMS DYNAMICS OF THE STRUCTURED PARTICLES
V.M. Somsikov [ vmsoms@rambler.ru Laboratory of Physics of the
geoheliocosmic relation, Institute of Ionosphere, Almaty, Kazahstan.
###### Abstract
Dynamics of the structured particles consisting of potentially interacting
material points is considered in the framework of classical mechanics.
Equations of interaction and motion of structured particles have been derived.
The expression for friction force has been obtained. It has been shown that
irreversibility of dynamics of structured particles is caused by increase of
their internal energy due to the energy of motion. It has been shown also that
the dynamics of the structured particles is determined by two types of
symmetry: the symmetry of the space and the internal symmetry of the
structured particles. Possibility of theoretical substantiation of the laws of
thermodynamics has been considered.
nonequilibrium, classical mechanics, thermodynamics
###### pacs:
05.45; 02.30.H, J
††preprint: APS/123-QED
## I Introduction
All real bodies in nature are the structured particles ($SP$). But the
existing classical mechanics has been developed for material points($MP$) or
hard bodies [1] which does not exist in the nature and which have no internal
structure. Therefore it is desirable to create the mechanics of $SP$. This
mechanics will be more general than the existing mechanics of unstructured
bodies. Indeed, at the $MP$ motion in non-homogeneity space and their
interaction the energy of $MP$ motion changes only, while for the $SP$
internal energy varies also.
As usually the change of $SP$ internal energy is described empirically by the
classical mechanics for $MP$. So a question arises, whether it is possible to
find rigorous mathematical description of $SP$ dynamics within the frames of
the Newtonian mechanics and if possible then how? We found the answer on this
question by studying the motion equation of $SP$ when $SP$ is an equilibrium
system of potentially interacting $MP$.
It turns out that under certain conditions dynamics of such systems is
irreversible [2-4]. These conditions are formulated as follows:
1). The energy of an $SP$ must be presented as a sum of internal energy and
the energy of $SP$ motion as a whole.
2). Each material point in the system must be connected with a certain $SP$
independent of its motion in space.
3). During all the process the subsystems are considered to be equilibrium.
The first condition is necessary to introduce internal energy in the
description of system dynamics as a new key parameter charactering energy
variations of $SP$. The second condition enables not to redefine $SP$ after
mixing of $MP$. The last condition is taken from thermodynamics. It is
equivalent to the condition of weak interactions in the $SP$, which do not
violate $SP$ equilibrium. Moreover, it implies that each $SP$ contains so many
elements that it can be described using the concept of equilibrium system.
In this paper we consider derivation of the motion equation of interacting
$SP$. With the help of this equation it is shown how the mechanism of friction
can be explained in the frame of laws of the classical mechanics. It is shown
also how based on the hypothesis of local equilibrium, which enables to
represent non-equilibrium systems as an ensemble of equilibrium subsystems,
one can generalize the obtained results for two interacting $SP$. It is also
shown how Lagrange, Hamilton and Liouville equations for non-equilibrium
systems are derived from the equation of motion of a set of equilibrium $SP$.
We consider how such equations are different from their canonic prototypes for
the system of $MP$. We consider why the $SP$ dynamics is determined by the two
types of symmetries: the symmetry of space in which the $SP$ motion and
internal symmetry of distributions of elements of $SP$. It is shown how the
main equation of thermodynamics can be derived from the equation of $SP$
interaction and how the concept of entropy arises in classical mechanics.
## II The motion equation of equilibrium structural particles
It was shown in [4] that for obtaining of the $SP$ motion equation it is
necessary to define the energy of each $SP$ as a sum of internal energy and
energy of its motion. Differentiating energy of system with respect to time
and using a condition of its conservation, the equation for an energy exchange
between $SP$ can be obtained and then with its help the equation of motion of
$SP$ can be found. After that the equation of motion for $SP$ can be obtained
in two stages. At the first stage, based on the condition of energy
conservation, we obtain the equation of motion for the system in the field of
external forces. Then we take a system consisting of $SP$ and obtain their
equations of motion when the external field for one $SP$ is the field of
forces of the other $SP$. Forces acting between the $SP$ can be obtained from
$MP$ potential interaction.
Let us show how the equation of motion for a system of $N$ material points
with weights $m=1$ can be obtained [2-4]. Forces acting between pairs of $MP$
are assumed to be central and potential. The energy of the system $E$ is equal
to the sum of kinetic energies of $MP$. Thus
$T_{N}=\sum\limits_{i=1}^{N}m{v_{i}}^{2}/2$, their potential energy in the
field of external forces, ${U_{N}}^{env}$, and potential energy of their
interaction
${U_{N}}(r_{ij})={\sum\limits_{i=1}^{N-1}}{\sum\limits_{j=i+1}^{N}}U_{ij}(r_{ij})$,
where $r_{ij}=r_{i}-r_{j}$, $r_{i},v_{i}$ are coordinates and velocities of
the $i$-th $MP$. Thus, $E=E_{N}+U^{env}=T_{N}+U_{N}+U^{env}=const$.
By substituting variables we represent the energy of the system as a sum of
the motion energy of the center of mass ($CM$) and the internal energy.
Differentiating this energy with respect to time, we will obtain [3]:
$\displaystyle
V_{N}M_{N}\dot{V}_{N}+{\dot{E}}_{N}^{ins}=-V_{N}F^{env}-\Phi^{env}$ (1)
Here $F^{env}=\sum\limits_{i=1}^{N}F_{i}^{env}(R_{N},\tilde{r}_{i})$,
${\dot{E}}_{N}^{ins}={\dot{T}}_{N}^{ins}(\tilde{v}_{i})+{\dot{U}}_{N}^{ins}(\tilde{r}_{i})$=
$\sum\limits_{i=1}^{N}\tilde{v}_{i}(m\dot{\tilde{v}}_{i}+F(\tilde{r})_{i})$,
$\Phi^{env}=\sum\limits_{i=1}^{N}\tilde{v}_{i}F_{i}^{env}(R_{N},\tilde{r}_{i})$,
$r_{i}=R_{N}+\tilde{r}_{i}$, $M_{N}=mN$, $v_{i}=V_{N}+\tilde{v}_{i}$,
$F_{i}^{env}=\partial{U^{env}}/\partial{\tilde{r}_{i})}$, $\tilde{r}_{i}$,
$\tilde{v}_{i}$ are the coordinates and velocity of $i$-th $MP$ in the $CM$
system, $R_{N},V_{N}$ are the coordinates and velocity of the $CM$ system.
The equation (1) represents the balance of the energy of the system of $MP$ in
the field of external forces.
The first term in the left-hand side of the equation determines the change of
kinetic energy of the system - ${\dot{T}}_{N}^{tr}=V_{N}M_{N}\dot{V}_{N}$. The
second term determines the change of internal energy of the system,
${\dot{E}}_{N}^{ins}$. This energy dependent on coordinates and velocities of
$MP$ relative to the $CM$ of the system.
The right-hand side corresponds to the work of internal forces changing the
energy of the system. The first term changes
${\dot{T}}_{N}^{tr}=V_{N}M_{N}\dot{V}_{N}$. The second term determines the
work of forces changing ${\dot{E}}_{N}^{ins}$.
Let us determine the condition when the work of non-potential forces is not
equal to zero. We must take into account that
$F^{env}=F^{env}(R+\tilde{r}_{i})$ where $R$ is the distance from the source
of force to the $CM$ of the system. Let us assume that $R>>\tilde{r}_{i}$. In
this case the force $F^{env}$ can be expanded with respect to a small
parameter. Leaving in the expansion terms of zero and first order we can
write: $F_{i}^{env}=F_{i}^{env}|_{R}+(\nabla{F_{i}^{env}})|_{R}\tilde{r}_{i}$.
Taking into account that
$\sum\limits_{i=1}^{N}\tilde{v}_{i}=\sum\limits_{i=1}^{N}\tilde{r}_{i}=0$ and
$\sum\limits_{i=1}^{N}F_{i}^{env}|_{R}=NF_{i}^{env}|_{R}=F_{0}^{env}$, we get
from (1):
$\displaystyle
V_{N}(M_{N}\dot{V}_{N})+\sum\limits_{i=1}^{N}m\tilde{v}_{i}(\dot{\tilde{v}}_{i}+F(\tilde{r})_{i})\approx$
$\displaystyle\approx-
V_{N}F_{0}^{env}-({\nabla}F^{env}_{i}|_{R})\sum\limits_{i=1}^{N}\tilde{v}_{i}\tilde{r}_{i}$
(2)
In the right-hand side of equation (2) the force $F_{0}^{env}$ in the first
term depends on $R$. It is a potential force. The second term depending on
coordinates of $MP$ and their velocities relative to the $CM$ of the system
determines changes in the internal energy of the system. It is proportional to
the divergence of the external force. Therefore, in spite of the condition
$R>>\tilde{r}_{i}$ the values of $\tilde{v}_{i}$ may be not small, and the
second term cannot be omitted. Forces corresponding to this term are not
potential forces. So, the change in the internal energy will be not equal to
zero only if the characteristic scale of inhomogeneities of the external field
is commensurable with the system scale.
Thus, inhomogeneity of space leads to the inhomogeneity of time for the
system. It is connected with possibility of increase in internal energy of
system at the expense of energy of its motion and impossibility of returning
of the system’s internal energy into the energy of its motion due to the law
of momentum conservation. But the law of preservation of full energy is
carried out.
Equation (2) confirms assumption of A. Poincare [5] that it is necessary to
take into account structures of interacting bodies at rather small distances
between them.
Dynamics of an individual $MP$ as well as dynamics of a system of $MP$ can be
derived from equation (1). A $MP$ does not have an internal energy, and forces
acting on it are caused by potential forces of interaction with other $MP$ and
the external force. Therefore the motion of a $MP$ is determined by the work
of potential forces transforming the energy of the external field into its
kinetic energy only.
Unlike $MP$, a system has its internal energy. Therefore the work of external
forces over the system causes changes in its $T_{N}^{tr}$ and $E_{N}^{ins}$,
i.e. the external force breaks up into two components. The first component is
a potential force. It changes momentum of the system’s $CM$. The second
component is non-potential. Its work changes $E_{N}^{ins}$. Hence, the motion
of the system is determined by the work of potential and non-potential forces
transforming the external field energy into the energy of $CM$ motion and
internal energy.
Multiplying eq.(1) by $V_{N}$ and dividing by $V_{N}^{2}$ we find the equation
of a system motion [4]:
$\displaystyle M_{N}\dot{V}_{N}=-F^{env}-{\alpha_{N}}V_{N}$ (3)
where $\alpha_{N}=[{\dot{E}}_{N}^{ins}+\Phi^{env}]/V_{N}^{2}$ is a coefficient
determined by the change of internal energy.
The equation (3) is a motion equation for $SP$. The first term in the right-
hand side of the equation determines the system acceleration, and the second
term determines the change of its internal energy. The eq. (3) is reduced to
the Newton equation if it is possible to neglect variation in the internal
energy.
Thus, the system state in the external field is determined by two parameters:
the energy of motion and the internal energy. Each type of energy has its own
force. The change in the motion energy is caused by the potential component of
the force, whereas the change in the internal energy is caused by the non-
potential component.
Let us show how to obtain the equation for interaction two equilibrium $SP$.
For this purpose we take the system consisting of two $ES$-$L$ and $K$. The
$L$ is the number of elements in the $L$-$SP$ and $K$ is the number of
elements in $K$-$SP$, i.e. $L+K=N$. Let $LV_{L}+KV_{K}=0$, where $V_{L}$ and
$V_{K}$ are velocities of $L$ and $K$ equilibrium subsystems relative to the
$CM$ of the system. Differentiating the energy of the system with respect to
time, we obtain:
${\sum\limits_{i=1}^{N}v_{i}{\dot{v}}_{i}}+{\sum\limits_{i=1}^{N-1}}\sum\limits_{j=i+1}^{N}v_{ij}F_{ij}=0$,
where $F_{ij}=U_{ij}=\partial{U}/\partial{r_{ij}}$.
In order to derive the equation for $L$-$SP$, in the left-hand side of the
equation we leave only terms determining change of kinetic and potential
energy of interaction of $L$-$SP$ elements among themselves. All other terms
we displace into the right-hand side of the equation and combine the groups of
terms in such a way that each group contains the terms with identical
velocities. In accordance with Newton equation, the groups which contain terms
with velocities of the elements from $K$-$SP$ are equal to zero. As a result
the right-hand side of the equation will contain only the terms which
determine the interaction of the elements $L$-$SP$ with the elements $K$-$SP$.
Thus we will have:
${\sum\limits_{i_{L}=1}^{L}}v_{i_{L}}{\dot{v}}_{i_{L}}+{\sum\limits_{i_{L}=1}^{L-1}}\sum\limits_{j_{L}=i_{L}+1}^{L}F_{{i_{L}}{j_{L}}}v_{{i_{L}}{j_{L}}}={\sum\limits_{i_{L}=1}^{L}}\sum\limits_{j_{K}=1}^{K}F_{{i_{L}}{j_{K}}}v_{j_{K}}$
where double indexes are introduced to denote that a particle belongs to the
corresponding system. If we make substitution
$v_{i_{L}}=\tilde{v}_{i_{L}}+V_{L}$, where $\tilde{v}_{i_{L}}$ is the velocity
of $i_{L}$ particle relative to the $CM$ of $L$ -$SP$, we obtain the equation
for $L$-$SP$. The equation for $K$-$SP$ can be obtained in the same way. The
equations for two interacting systems can be written as [4]:
$\displaystyle
V_{L}M_{L}\dot{V}_{L}+{\dot{E}_{L}}^{ins}=-{\Phi}_{L}-V_{L}{\Psi}$ (4)
$\displaystyle
V_{K}M_{K}\dot{V}_{K}+{\dot{E}_{K}}^{ins}={\Phi}_{K}+V_{K}{\Psi}$ (5)
Here $M_{L}=mL,M_{K}=mK,\Psi=\sum\limits_{{i_{L}}=1}^{L}F^{K}_{i_{L}}$;
${\Phi}_{L}=\sum\limits_{{i_{L}}=1}^{L}\tilde{v}_{i_{L}}F^{K}_{i_{L}}$;
${\Phi}_{K}=\sum\limits_{{i_{K}}=1}^{K}\tilde{v}_{i_{K}}F^{L}_{i_{K}}$;
$F^{K}_{i_{L}}=\sum\limits_{{j_{K}}=1}^{K}F_{i_{L}j_{K}}$;
$F^{L}_{j_{K}}=\sum\limits_{{i_{L}}=1}^{L}F_{i_{L}j_{K}}$;
${\dot{E}_{L}}^{ins}={\sum\limits_{i_{L}=1}^{L-1}}\sum\limits_{j_{L}=i_{L}+1}^{L}v_{i_{L}j_{L}}[\frac{{m\dot{v}}_{i_{L}j_{L}}}{L}+\\\
+F_{i_{L}j_{L}}]$;
${\dot{E}_{K}}^{ins}={\sum\limits_{i_{K}=1}^{K-1}}\sum\limits_{j_{K}=i_{K}+1}^{K}v_{i_{K}j_{K}}[\frac{{m\dot{v}}_{i_{K}j_{K}}}{K}+\\\
+F_{i_{K}j_{K}}]$.
The equations (4, 5) are equations for interactions two $SP$. They describe
energy exchange between $SP$. Independent variables are macro-parameters and
micro-parameters. Macro-parameters are coordinates and velocities of the
motion of $CM$ of $SP$. Micro-parameters are relative coordinates and
velocities of $MP$.
Therefore the equation of $SP$ interaction binds together two types of
description: on the macrolevel and on the microlevel. The description on the
macrolevel determines dynamics of an $SP$ as a whole and description on the
microlevel determines dynamics of the elements of an $SP$.
The potential force, $\Psi$, determines the motion of an $SP$ as a whole. This
force is the sum of potential forces acting on the elements of one $SP$ from
the other $SP$.
The forces determined by terms ${\Phi}_{L}$ and ${\Phi}_{K}$ transform the
motion energy of $SP$ into their internal energy as a result of chaotic motion
of elements of one $SP$ in the field of forces of the other $SP$. As in the
case of the system in the external field, these terms are not zero only if the
characteristic scale of inhomogeneity of forces of one system is commeasurable
with the scale of the other system. The work of such forces causes violation
of time symmetry for $SP$ dynamics.
The equations for $SP$ motion corresponding to the equations (4,5) can be
written as [4]:
$M_{L}\dot{V}_{L}=-\Psi-{\alpha}_{L}V_{L}$ (6)
$M_{K}\dot{V}_{K}=\Psi+{\alpha}_{K}V_{K}$ (7)
where ${\alpha}_{L}=(\dot{E}^{ins}_{L}+{\Phi}_{L})/V^{2}_{L}$,
${\alpha}_{K}=({\Phi}_{K}-\dot{E}^{ins}_{K})/V^{2}_{K}$,
The equations (6, 7) are motion equations for interacting $SP$. The second
terms in the right-hand side of the equations determine the forces changing
the internal energy of the $SP$. These forces are equivalent to the friction
forces. Their work is a sum of works of forces acting on the $MP$ of one $SP$
from the other $SP$.
The coefficients ”$\alpha_{L}$”, ”$\alpha_{K}$” determine efficiency of
transformation of the energy of $SP$ motion into their internal energy. These
coefficients are friction coefficients. Therefore equations (6, 7) enable to
determine analytical form of non-potential forces in the non-equilibrium
system causing changes in the internal energy of the $SP$.
## III The generals of Lagrange, Hamilton and Liouville equations for
equilibrium systems
Let us show qualitative difference of Lagrange, Hamilton and Liouville
equations for the systems of $MP$ from similar equations for $SP$.
Using Newton equation one can derive Hamilton principle for $MP$ from
differential D’Alambert principle [6]. For this purpose the time integral of
virtual work done by effective forces is equated to zero. Integration over
time is carried out provided that external forces possess a power function. It
means that the canonical principle of Hamilton is valid only for cases when
$\sum F_{i}\delta R_{i}=-\delta U$, where $i$ is a particle number, and
$F_{i}$ is a force acting on this particle. But for interacting $SP$ the
condition of conservation of forces is not fulfilled because of the presence
of a non-potential component. Therefore Hamiltonian principle for $SP$ as well
as Lagrange, Hamilton and Liouville equations must be derived using eq. (3).
Liouville equation for non-equilibrium system consisting from a set of
equilibrium $SP$ is written as [2, 4]:
$df/dt=-\sum\limits_{L=1}^{R}{\partial}{F_{L}}/{\partial}V_{L}$ (8)
Here $f$ is a distribution function for a set of $SP$, $F_{L}$ is a non-
potential part of collective forces acting on the $SP$, $V_{L}$ is the
velocity of $L$-$SP$.
The right-hand side of the equation is determined by the efficiency of
transformation of the $SP$ motion energy into their internal energy. For non-
equilibrium systems the right-hand side is not equal to zero because of non-
potentiality of forces changing the internal energy.
The state of the system as a set of $SP$ can be defined in the phase space
which consists of $6R-1$ coordinates and momentums of $SP$, where $R$ is the
number of $SP$. Location of each $SP$ is given by three coordinates and their
moments. Let us call this space an $S$-space for $SP$ in order to distinguish
it from the usual phase space for $MP$. Unlike the usual phase space [7,8] the
$S$-space is not conserved. It is caused by transformation of the energy of
$SP$ relative motion into their internal energy. The $SP$ internal energy
cannot be transformed into the $SP$ energy of motion as $SP$ momentum cannot
change due to the motion of its $MP$ [7]. Therefore $S$-space is compressible.
## IV The dynamics geometry of $SP$
The task of mechanics is definition of trajectories of material bodies in
space with the help of dynamics laws. Therefore the geometry is included
naturally into a formalism of classical mechanics. The interrelation of
geometry and mechanics is carried out through concept of an interval. This
concept lies in bases of the formalism, both classical, and the relativistic
mechanics [6]. We will consider in what difference of an interval for $MP$
from an interval for $SP$.
Let’s consider a point in the configuration space, corresponding to the system
of $MP$. Through an interval time of $dt\longrightarrow 0$ the $MP$ will move
on distance $ds$. The volume of $ds$ is an interval. The interval for a set of
$MP$ is possible to express through the kinetic energies as follows [6]:
$\displaystyle
d\overline{s}^{2}=2T_{N}dt^{2}=\sum^{N}_{i=1}\breve{v}^{2}_{i}dt^{2}=\sum^{N}_{i=1}(d\breve{x}^{2}_{i}+d\breve{y}^{2}_{i}+d\breve{z}^{2}_{i})$
(9)
where ${d\overline{s}}$ is interval displaying infinitesimal distance between
two points of configuration space; ${\breve{x}=\surd{m_{i}x_{i}}}$,
${\breve{y}=\surd{m_{i}y_{i}}}$, ${\breve{z}=\surd{m_{i}z_{i}}}$ are
coordinates of the $i$ element; $m_{i}$ is a mass of the $i$ -element. The
configuration space is $3N$ dimensional Euclidian spaces for $N$ $MP$. In
general case the linear element will be set in the square-law differential
form of corresponding variables:
$\displaystyle
d\overline{s}^{2}=\sum^{n}_{i,k=1}g_{ik}d\breve{x}_{i}d\breve{x}_{k}$ (10)
where $g_{ik}=g_{ki}$ is symmetrical metrics tensor, $n=3N$.
If we have $p$ kinematics restrictions $f_{i}=f_{i}(x_{1},x_{2}...x_{n})$,
$i=1,2...p$, the motion of the system will be in $l=3N-p$ dimensional
hyperspace. In this case we have:
$d\overline{s}^{2}=\sum^{n}_{i,k=1}a_{ik}dq_{i}dq_{k}$, where $a_{ik}$ -is
known function in a new coordinates. If as kinematics conditions are potential
forces then the equation (8) will be equivalent to the motion equation of
$MP$. But for system which is a set of $SP$, the energy part is distributed by
non-potential forces. There is a question what will be an interval in this
case?
Let’s show, that for answer on this question it is necessary to present energy
of system in the form of two parts: energy of motion of the center of mass of
$SP$-$T_{N}^{tr}$, and internal energy of $SP$-$T_{N}^{ins}$. I.e. the
interval corresponding for system $SP$ also should consist of two parts. In
this case the $T_{N}^{tr}$, $T_{N}^{ins}$ expressions (7) can be written down
as:
$\displaystyle
d\overline{s}^{2}=(2T_{N}^{tr}+2T_{N}^{ins})dt^{2}=ds_{tr}^{2}+ds_{ins}^{2}=$
$\displaystyle\
N\breve{V}_{0}^{2}dt^{2}+(\sum^{N-1}_{i=1}\sum^{N}_{j=i+1}\breve{v}_{ij}^{2})dt^{2}/N$
(11)
where $\breve{V}_{0}=(\sum^{N}_{i=1}\breve{v}_{i})/N$,
$\breve{v}_{ij}=\breve{v}_{i}-\breve{v}_{j}$.
Let us transform the energy $T_{N}$ by replacement:
$\breve{v}_{i}=\breve{V}_{0}-\bar{v}_{i}$, where
$\sum^{N}_{i=1}\breve{v}_{i}=N\breve{V}_{0}$, i.e.
$\sum^{N}_{i=1}\bar{v}_{i}=0$. Then we will have:
$\displaystyle
T_{N}=N\breve{V}_{0}^{2}/2+\breve{V}_{0}\sum^{N}_{i=1}\bar{v}_{i}+\sum^{N}_{i=1}\bar{v}_{i}^{2}/2$
(12)
Because $\sum^{N}_{i=1}\bar{v}_{i}=0$, then we have
$\displaystyle\sum^{N}_{i=1}\bar{v}_{i}^{2}/2=1/(2N)(\sum^{N-1}_{i=1}\sum^{N}_{j=i+1}\breve{v}_{ij}^{2})$
(13)
As a result we obtain:
$\displaystyle
d\overline{s}^{2}=(2T_{N}^{tr}+2T_{N}^{ins})dt^{2}=ds_{tr}^{2}+ds_{ins}^{2}=$
$\displaystyle\ N\breve{V}_{0}^{2}dt^{2}+\sum^{N}_{i=1}\bar{v}_{i}^{2}dt^{2}$
(14)
Thus, the square of an interval of non-equilibrium system breaks up to the sum
of squares of two intervals. The first corresponds to the motion energy of
$SP$ center of mass and the second corresponds to the internal energy of
system. It is follows from here that the interval of the non-equilibrium
system which consists of a set of $SP$ breaks up to two independent intervals
characterizing dynamics of system: $ds_{tr}^{2}=N\breve{V}_{0}^{2}dt^{2}$ and
$ds_{ins}^{2}=\sum^{N}_{i=1}\bar{v}_{i}^{2}dt^{2}$. These intervals are
orthogonally and they correspond to adjacent of a triangle for a full interval
of system in configuration space.
The change of the $SP$ center of mass motion energy is caused by work of
potential forces $F^{tr}$. Their work is defined by expression:
$A^{tr}=\int{F^{tr}dR}$, $F^{tr}=\nabla\varphi$, where $\varphi$ is scalar
function, $dR$ is a distance of systems motion.
The forces $F^{ins}$ which change of the internal energy $SP$ are non-
potential. Their work consists from the work on change of $MP$ motion energy
relative to the center of mass, i.e.
$A^{ins}=\sum^{N}_{i=1}\int{F_{i}dr_{i}}$, where $dr_{i}$ -moving of $i$ -th
element of system relative to the center of mass. And because
$\sum^{N}_{i=1}{F_{i}}=0$ then $\int{\sum^{N}_{i=1}F_{i}dR}=0$ for any
possible way of moving of system. I.e. the potential component of the external
force $F^{tr}$ acting on $SP$ changes $s_{tr}$ but does not change $s_{ins}$.
The work of non-potential forces, $F^{ins}$ changes $s_{ins}$ but does not
change $s_{tr}$. The variables defining motion of the center of mass are
macroparameters, and the variables defining change of internal energy are
microparameters.
Thus for the description of dynamics of the non-equilibrium system it is
necessary to present this system as a set of $SP$ and then it is necessary to
represent $SP$’s energy in the form of the sum of two types of energy:
internal energy and energy of $SP$ motion. In the nature we deal with the real
bodies possessing internal energy. At their interaction the part of energy go
to their heating. This energy transforming is realized by the friction force.
So the $SP$ dynamics is determined by the two types of symmetries: the
symmetry of space in which the $SP$ motion and internal symmetry of
distributions of elements of $SP$. Thus the necessity of splitting of the
energy on two parts has under itself a real basis.
## V The equations of interaction of systems and thermodynamics
Equations (1-8) give relationship between mechanics and thermodynamics [4, 8].
According to the basic equation of thermodynamics the work of external forces
acting on the system splits into two parts. The first part corresponds to
reversible work. In our case it corresponds to the change of the motion energy
of the system as a whole. The second part of energy goes on heating. It
corresponds to the internal energy of the system.
Let us take a motionless non-equilibrium system consisting of ”$R$”
equilibrium subsystems. Each equilibrium subsystem consists of a great number
of elements $N_{L}>>1$, where $L=1,2,3...R,N=\sum\limits_{L=1}^{R}N_{L}$. Let
$dE$ be work done over the system. In thermodynamics energy $E$ is called
internal energy (in our case it is equal to the sum of all energies of
equilibrium subsystems). It is known from thermodynamics that ${dE=dQ-PdY}$
[8]. Here, according to generally accepted terminology, $E$ is the energy of
the system; $Q$ is the thermal energy; $P$ is the pressure; $Y$ is the volume.
The equation of interaction between $SP$ is also a differential of two types
of energy. It means that $dE$ in the $SP$ is redistributed in such a way that
some part of it changes energy of relative motion of the $SP$ and the other
part changes the internal energy. Thus, it follows that entropy may be
introduced into classical mechanics if it is considered as a quantity
characterizing increase in the internal energy of an $SP$ at the expense of
energy of their motion. Then the increase in entropy can be written as [3, 4]:
${{\Delta{S}}={\sum\limits_{L=1}^{R}{\\{{N_{L}}\sum\limits_{k=1}^{N_{L}}\int[{\sum\limits_{s}{{F^{L}_{ks}}v_{k}}/{E^{L}}]{dt}}\\}}}}$
(15)
Here ${E^{L}}$ is the kinetic energy of $L$-$SP$; $N_{L}$ is the number of
elements in $L$-$SP$; $L=1,2,3...R$; ${R}$ is the number of $SP$; ${s}$ is the
number of external elements which interact with ${k}$ element belonging to the
$L$-$SP$; ${F_{ks}^{L}}$ is the force acting on the $k$-element; $v_{k}$ is
the velocity of the $k$\- element.
Based on the generally accepted definition of entropy we can derive expression
for its production and define necessary conditions for stationarity of a
nonequilibrium system [4].
## VI Conclusion
The classical mechanics collides with insuperable difficulties in attempt to
describe evolution of non-equilibrium systems. The main reason is that the
process of evolution is irreversible but the classical mechanics is reversible
[9, 10]. The reversibility of classical mechanics is defined by the nature of
the second law of Newton. According to this law the acceleration of
unstructured bodies is proportional to the force acting on it. Therefore the
region of application of the second law of Newton is restricted by
unstructured bodies. It means that the second law of Newton is inapplicable
for the description of dynamics of the real bodies possessing a friction.
Hence for removal of the mentioned restrictions of classical mechanics it is
necessary to define friction forces rigorously on the basis of Newton’s second
law.
The analysis of dynamics of a hard-discs system has led to the conclusion that
in order to solve this problem it is necessary to find the motion equation of
$SP$. It has been done for a case when $SP$ represents a system of potentially
interacting $MP$, moving in the field of external forces.
During the process of search of a way which could lead to the $SP$ motions
equation and then as a result of its analysis, the following conclusions were
found out.
The motion and evolution of the system are defined by two types of symmetry:
the symmetry of space in which it is moving and its internal symmetry. In
accordance with these two types of symmetries the energy of system also breaks
up on to two types: the motion energy of system and its internal energy. In
its turn, the change of these types of energy is also defined by two types of
forces. Transformation of energy of $SP$ motion is caused by potential force.
Transformation of internal energy $SP$ is caused by work of non-potential
force. The work of the non-potential force leads to irreversibility of $SP$
dynamics.
The non-equilibrium systems in approach of the local equilibrium can be
presented as a set of the equilibrium subsystems which are in motion relative
to each other. In this case the description of dynamics of system by means of
the $SP$ motion equation can be carried out.
The state of the system as a set of $SP$ can be defined in the phase space
which consists of 6R-1 coordinates and momentums of $SP$, where R is the
number of $SP$. Location of each $SP$ is given by three coordinates and their
momentums. The phase space which is determined by coordinates and velocities
of $SP$ is compressible.
The dynamics of the non-equilibrium system composed of a set of $SP$ is
determined by the Liouville equation for equilibrium $SP$. These systems
acquires an equilibrium state when all energy of $SP$ motion transforms into
its internal energy.
The offered expansion of classical mechanics and the deterministic explanation
of irreversibility open a way to the substantiation of thermodynamics.
According to the motion equation for $SP$ the first law of thermodynamics
follows from the fact that the work of external forces changes both the energy
of particle’s motion and their internal energy. The second law of
thermodynamics follows from irreversible transformation of energy of relative
motion of system’s particles into their internal energy.
The motion equation for $SP$ also states impossibility of existence of
structureless particles in classical mechanics, which is equivalent to
infinite divisibility of matter.
Thus, the replacement of model of system in the form of set $MP$ on a model in
the form of a set of $SP$ leads to essential expansion of classical mechanics.
Such expansion allows, remaining within the frame of laws of Newton’s
mechanics, to offer the deterministic explanation of irreversibility and,
thereby, to enter the concept of entropy and evolution into the classical
mechanics. It is a bright example of that the further development of physics
is impossible without perfection of models on which basis it has been
constructed.
## References
* (1) Newton I., Mathematical principles of natural Philosophy. New York, 1846
* (2) Somsikov V.M., Equilibration of a hard-disks system. International Journal of the Bifurcation and Chaos, 2004, 14, 11, p.4027
* (3) Somsikov V.M., The restrictions of classical mechanics in the description of dynamics of nonequilibrium systems and the way to get rid of them. New Advances in Physics, Vol. 2, No 2, September 2008, pp. 125-140
* (4) Somsikov V.M., The mechanics of the systems of structured particles and irreversibility. arXiv:0908.3125v1 [physics.class-ph] 21 Aug 2009
* (5) Poincare A., About science. 1983, Nauka, Moscow
* (6) Lanczos C., The variation principles of mechanics. 1962, Univer. of Toronto press
* (7) Landau L.D., Lifshits Ye.M., Mechanics. 1958, Nauka, Moscow
* (8) Rumer Yu.B., Ryvkin M.Sh., Thermodynamics. Statistical Physics and Kinematics. 1977, Nauka, Moscow
* (9) Cohen E.G., Boltzmann and statistical mechanics, Dynamics: Models and Kinetic Methods for Nonequilibrium Many Body systems. 1998, NATO Sci. Series E: Applied Sci., 371, p. 223
* (10) Zaslavsky G.M., Chaotic dynamic and the origin of Statistical laws,1999, Physics Today, August, Part 1, p.39
|
arxiv-papers
| 2010-06-16T08:34:06 |
2024-09-04T02:49:10.943062
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V.M. Somsikov",
"submitter": "Slava Somsikov",
"url": "https://arxiv.org/abs/1006.3158"
}
|
1006.3259
|
¡html¿ ¡head¿ ¡title¿COMP5541 Winter 2010¡/title¿ ¡/head¿ ¡body¿
¡a href=”http://users.encs.concordia.ca/ mokhov”¿Serguei A. Mokhov¡/a¿¡br /¿
mokhov@cse.concordia.ca
¡h1¿TOC¡/h1¿
¡ul¿ ¡li¿¡a href=”#abstract”¿Abstract¡/a¿¡/li¿ ¡li¿¡a href=”#inital-
reqs”¿Initial Requirements¡/a¿¡/li¿ ¡li¿¡a href=”#reports”¿Reports¡/a¿¡/li¿
¡/ul¿
¡h1¿Abstract¡/h1¿ ¡a id=”abstract” name=”abstract” /¿
¡p¿This index covers the final course project reports for COMP5541 Winter 2010
at Concordia University, Montreal, Canada, Tools and Techniques for Software
Engineering by 4 teams trying to capture the requirements, provide the design
specification, configuration management, testing and quality assurance of
their partial implementation of the Unified University Inventory System (UUIS)
of an Imaginary University of Arctica (IUfA). Their results are posted here
for comparative studies and analysis. ¡/p¿
¡hr /¿
¡h1¿Initial Ambigous Requirements¡/h1¿ ¡a id=”initial-reqs” name=”initial-
reqs” /¿
¡pre¿ Client:
Imaginary University of Arctica (IUfA)
Product wanted:
Unified University Inventory System (UUIS)
Summary:
Currently 3 faculties have different subsets of inventory of their various
assets (equipment, furniture, space, software, seat assignment, etc). recorded
in various formats and forms. Need a unified inventory system for all the
Faculties of IUfA
Faculties:
\- Arts and Science \- Computer Science \- Engineering
3 Buildings, partitioned into locations (rooms, suites, cubicles, atriums,
teaching labs, research labs), whe locations can contain other locations.
Faculties are partitioned into departments, e.g. History, Religion, Visual
Arts, Math (in Arts and Science), ECE, MIE (in Engineering), SOEN, CS (in CS).
Roles of people accessing the system:
\- Inventory staff with different levels \- Common / administrative \- Per
department (e.g. a DA and part-time students) \- Per faculty (e.g. a FA and
full-time controllers) \- Full-time Faculty \- Part-time Faculty \- University
Administration \- IT Group \- Research assistants \- Research associates \-
Students are diploma, master’s thesis option, master’s course option, PhD \-
Security
Anybody can submit a request to inventory or report a problem with an
inventory item with or without a barcode, serial number, and/or a description
(level 0).
Changes are made based on the request and submitted for approval to become
permanent.
Three levels of approval: Technical staff (IT and techies) (level 1) DA (level
2), chair or director (level 3) FA, e.g. Associate Dean, Dean, Controller
(level 4)
Want: \- To be able to inventory and enter the data about: \- assets, such as
equipment and furniture, phones, etc. \- space/locations (rooms, suites,
cubicles, drawers, offices) \- grad seats (which student ID occupies a seat in
which lab) \- software, for licenses, lending \- floor plans and maps \-
reporting inventory changes \- doing the change \- toggling edit/view mode \-
ability to select any columns to show or hide \- approving the change \-
permissions per faculty and perl level; assign permissions \- auditing \- Need
to integrate the previous data like COMPID and ENGRID, ARTSID \- The items are
tracked by the unified barcode IUFAID0000000001, S/N, etc. \- Items can have
many properties or none from the base description. \- Items can be grouped
into objects and updated as group (e.g. change location). \- Problem report
form – technical or administrative \- Accessible from outside \- Powerful
search capabilities \- Bulk entry and update (e.g. from a scanner PDA) \-
Authenticate to the application with a common account \- Higher priviledged
with voice or other biometric means
Assume (need to simulate for the needed extent): \- Personal ID/data is
available for the community (profs, students, staff have all usernames,
personal info, etc.) ¡/pre¿
¡hr /¿
¡h1¿COMP5541 Winter 2010 Final Project Reports¡/h1¿ ¡a id=”reports”
name=”reports” /¿ ¡ul¿ ¡li¿Team 1’s Approach ¡ul¿ ¡!- SRS -¿ ¡li¿
LIST:arXiv:1005.0330 ¡/li¿ ¡!- SDD -¿ ¡li¿ LIST:arXiv:1005.0595 ¡/li¿ ¡/ul¿
¡/li¿ ¡li¿Team 2’s Approach ¡ul¿ ¡!- SRS -¿ ¡li¿ LIST:arXiv:1005.0783 ¡/li¿
¡!- SDD -¿ ¡li¿ LIST:arXiv:1005.0665 ¡/li¿ ¡/ul¿ ¡/li¿ ¡li¿Team 3’s Approach
¡ul¿ ¡!- SRS -¿ ¡li¿ LIST:arXiv:1005.0609 ¡/li¿ ¡!- SDD -¿ ¡li¿
LIST:arXiv:1005.0854 ¡/li¿ ¡/ul¿ ¡/li¿ ¡li¿Team 4’s Approach ¡ul¿ ¡!- SRS -¿
¡li¿ LIST:arXiv:1005.0162 ¡/li¿ ¡!- SDD -¿ ¡li¿ LIST:arXiv:1005.0169 ¡/li¿
¡/ul¿ ¡/li¿ ¡/ul¿
¡hr /¿
¡/body¿ ¡/html¿
|
arxiv-papers
| 2010-06-16T16:12:30 |
2024-09-04T02:49:10.950955
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serguei A. Mokhov",
"submitter": "Serguei Mokhov",
"url": "https://arxiv.org/abs/1006.3259"
}
|
1006.3331
|
# Generators for the Euclidean Picard Modular Groups
Tiehong zhao Institut de Mathématiques
Université Pierre et Marie Curie
4, place Jussieu
F-75252 Paris, France zhao@math.jussieu.fr
###### Abstract.
The goal of this article is to show that five explicitly given
transformations, a rotation, two screw Heisenberg rotations, a vertical
translation and an involution generate the Euclidean Picard modular groups
with coefficient in the Euclidean ring of integers of a quadratic imaginary
number field. We also obtain the relations of the isotropy subgroup by
analysis of the combinatorics of the fundamental domain in Heisenberg group.
The author is supported by the China-funded Postgraduates Studying Aboard
Program for Building Top University.
## 1\. Introduction
Let $\mathcal{K}=\mathbb{Q}(\sqrt{-d})$ be a quadratic imaginary number field.
Let $\mathcal{O}_{d}$ be the ring of algebraic integers of $\mathcal{K}$. The
Bianchi groups $PSL_{2}(\mathcal{O}_{d})$ are the simplest numerically defined
discrete groups. In number theory they have been used to study the zeta-
functions of binary Hermitian forms over the rings $\mathcal{O}_{d}$. As the
isometric groups acting on the half-upper space, they are of interest in the
theory of Fuchsian groups and the related theory of Riemann surfaces. The
Bianchi groups can be considered as the natural algebraic generalization of
the classical modular group $PSL_{2}(\mathbb{Z})$. A good general reference
for the Bianchi groups and their relation to the modular group is [3]. As a
natural generalization of the Bianchi groups, the subgroups of $PU(2,1)$ with
coefficients in $\mathcal{O}_{d}$ are called Picard modular groups, denoted by
$PU(2,1;\mathcal{O}_{d})$. These groups have attracted a great deal of
attention both for their intrinsic interest as discrete groups and also for
their applications in complex hyperbolic geometry (as the holomorphic
automorphism subgroups).
A general method to determine finite presentations for each Bianchi group
$PSL_{2}(\mathcal{O}_{d})$ was developed by Swan [17] based on geometrical
work of Bianchi, while a separate purely algebraic method was given by Cohn
[1]. In general, fundamental domains for Lie groups were studied by [11], but
the complex hyperbolic space is a particularly challenging case since no
existence of totally geodesic hypersurface. So far very few examples of
complex hyperbolic lattices have been constructed explicitly. Due to the
famous paper [12] of Mostow, other explicit constructions of fundamental
domains for lattices in $PU(2,1$) were obtained, see for example, the work of
Goldman and Parker [10], Deraux, Falbel and Paupert [2], Schwartz [15], the
survey paper of Parker [13]. In particular, the group
$PU(2,1;\mathcal{O}_{3})=PU(2,1;\mathbb{Z}[\omega]),$ where $\omega$ is a cube
root of unity was studied by Falbel and Parker in [5] and its sister was
treated recently in [19]. Analogously a fundamental domain of Gauss-Picard
group $PU(2,1;\mathbb{Z}[i])$ was described in ([6], [7], [8]) and analysis of
the combinatorics of the fundamental domain gives rise to a presentation of
the group in [6].
In this paper we give a description of generators for certain Picard modular
groups $PU(2,1;\mathcal{O}_{d})$ where the ring $\mathcal{O}_{d}$ is Euclidean
except for $d=1,3$ (these two exceptional cases have been studied in many
aspects). Among the quadratic imaginary number rings $\mathcal{O}_{d}$ only
$\mathcal{O}_{1},\mathcal{O}_{2},\mathcal{O}_{3},\mathcal{O}_{7},\mathcal{O}_{11}$
have a Euclidean algorithm, see [16], although there is a larger finite
collection of $\mathcal{O}_{d}$’s (such as $d=1,$ $2,$ $3,$ $7,$ $11,$ $19,$
$43$, $67,$ $163$, see [18]) which have class number one. For these values of
$d$ the orbifold $\mathbf{H}^{2}_{\mathbb{C}}/PU(2,1;\mathcal{O}_{d})$ has
only one cusp. The method is based on the construction of various shapes of
precisely fundamental domains for the stabilisers of infinity of
$PU(2,1;\mathcal{O}_{d})$ and then on a determination of several neighboring
isometric spheres such that the union of the boundaries of these isometric
spheres contains the fundamental domain of the stabiliser, which was used in
([5], [6], [19]). Compared with other groups, the generators of these groups
in ([5], [6], [19]) are easy to be obtained since the fundamental domain
constructed lies completely inside the boundary of the isometric sphere
centred at origin. Again this reflects the underlying number theory;
$\mathcal{O}_{1}$ and $\mathcal{O}_{3}$ have non-trivial units while the other
three do not. A simple algorithm to decompose any transformation in the Picard
group $PU(2,1;\mathcal{O}_{1})$ as a product of the generators was given in
[4], one would be interesting to extend their method to other Picard modular
groups. However, it would also be important to find the generators in terms of
geometric ways which will provide more informations that one continue to
construct a fundamental domain explicitly for each of Picard modular groups.
I would like to thank my advisor E. Falbel for his warm encouragements all
along this work and for a number of helpful comments.
## 2\. Complex hyperbolic space
### 2.1. The Siegel domain
In this section we give the necessary background material on complex
hyperbolic space. To know more details of this material we refer the reader to
[9].
Let $\mathbb{C}^{2,1}$ denote the complex vector space equipped with the
Hermitian form defined by
$\langle\mathbf{z},\mathbf{w}\rangle=z_{1}\bar{w}_{3}+z_{2}\bar{w}_{2}+z_{3}\bar{w}_{1},$
where $\mathbf{z}$ and $\mathbf{w}$ be the column vectors
$[z_{1},z_{2},z_{3}]^{t}$ and $[w_{1},w_{2},w_{3}]^{t}$ respectively. The
projective model of complex hyperbolic space $\mathbf{H}_{\mathbb{C}}^{2}$ is
defined to be the collection of negative lines in $\mathbb{C}^{2,1}$, namely,
those points $\mathbf{z}$ satisfying $\langle\mathbf{z},\mathbf{z}\rangle<0$.
We mainly take the Siegel domain $\mathfrak{S}$ as a upper half-space model
for the complex hyperbolic space, that is given by
$\mathfrak{S}=\\{(z_{1},z_{2})\in\mathbb{C}^{2}:2\Re ez_{1}+|z_{2}|^{2}<0\\}.$
The boundary of the Siegel domain $\mathfrak{S}$ is identified with the one-
point compactification of the Heisenberg group. The Heisenberg group
$\mathfrak{R}$ is $\mathbb{C}\times\mathbb{R}$ with the group law
$(\zeta_{1},t_{1})\diamond(\zeta_{2},t_{2})=(\zeta_{1}+\zeta_{2},t_{1}+t_{2}+2\Im
m(\zeta_{1}\bar{\zeta}_{2})).$
The Cygan metric on $\mathfrak{R}$ is given by
$\rho_{0}((\zeta_{1},t_{1}),(\zeta_{2},t_{2}))=\left||\zeta_{1}-\zeta_{2}|^{2}-it_{1}+it_{2}-2i\Im
m(\zeta_{1}\bar{\zeta}_{2})\right|,$
in terms of the operation of Heisenberg group, that is
$\left|(\zeta_{1},t_{1})^{-1}\diamond(\zeta_{2},t_{2})\right|$.
We can extend the Cygan metric to an incomplete metric on
$\bar{\mathfrak{S}}-\\{\infty\\}$ as follows
$\tilde{\rho}_{0}=\left||\zeta_{1}-\zeta_{2}|^{2}+|u_{1}-u_{2}|-it_{1}+it_{2}-2i\Im
m(\zeta_{1}\bar{\zeta}_{2})\right|.$
The Siegel domain $\mathfrak{S}$ is parametrised in horospherical coordinates
by
(2.1)
$(\zeta,t,u)\longrightarrow\left[\begin{array}[]{c}(-|\zeta|^{2}-u+it)/2\\\
\zeta\\\ 1\end{array}\right]$
and the point at infinity being
$q_{\infty}=\left[\begin{array}[]{c}1\\\ 0\\\ 0\end{array}\right].$
Then $\mathfrak{S}=\mathfrak{R}\times\mathbb{R}_{+}$ and
$\partial\mathfrak{S}=(\mathfrak{R}\times\\{0\\})\cup\\{q_{\infty}\\}$.
### 2.2. Complex hyperbolic isometries
Let $U(2,1)$ be the group of matrices that are unitary with respect to the
form $\langle.,.\rangle$. The group of holomorphic isometries of complex
hyperbolic space is the projective unitary group $PU(2,1)=U(2,1)/U(1)$, with a
natural identification $U(1)=\\{e^{i\theta}I,\theta\in[0,2\pi)\\}.$ We now
describe the action of the stabiliser of $q_{\infty}$ on the Heisenberg group.
The Heisenberg group acts on itself by Heisenberg translations. For
$(\tau,v)\in\mathfrak{R}$, this is
$T_{(\tau,v)}:(\zeta,t)\mapsto(\zeta+\tau,t+v+2\Im
m(\tau\bar{\zeta}))=(\tau,v)\diamond(\zeta,t).$
Heisenberg translation by $(0,v)$ for any $v\in\mathbb{R}$ is called vertical
translation by $v$.
The unitary group $U(1)$ acts on the Heisenberg group by Heisenberg rotations.
For $e^{i\theta}\in U(1)$, the rotation fixing $q_{0}=(0,0,0)$ is given by
$R_{\theta}:(\zeta,t)\mapsto(e^{i\theta}\zeta,t).$
All other Heisenberg rotations may be obtained from these by conjugating by a
Heisenberg translation.
For $\lambda\in\mathbb{R}_{+}$, Heisenberg dilation by $\lambda$ fixing
$q_{\infty}$ and $q_{0}=(0,0,0)\in\partial\textbf{H}^{2}_{\mathbb{C}}$ is
given by
$D_{\lambda}:(\zeta,t)\mapsto(\lambda\zeta,\lambda^{2}t).$
All other Heisenberg dilations fixing $q_{\infty}$ may be obtained by
conjugating by a Heisenberg translation.
The stabiliser of $q_{\infty}$ in $PU(2,1)$ is generated by all Heisenberg
translations, rotations and dilations. However, only Heisenberg translations
and rotations are isometric with respect to various natural metrics on
$\mathfrak{R}$. For this reason the group generated by all Heisenberg
translations and rotations, which is the semidirect product
$U(1)\ltimes\mathfrak{R}$, is called the Heisenberg isometry group
$Isom(\mathfrak{R})$. The nontrivial central elements of the Heisenberg
isometry group are precisely the vertical translations. In particular, each
element of $Isom(\mathfrak{R})$ preserves every horosphere.
There is a canonical projection from $\mathfrak{R}$ to $\mathbb{C}$ called
vertical projection and denoted by $\Pi$, given by
$\Pi:(\zeta,t)\longmapsto\zeta.$ Using the exact sequence
$0\longrightarrow\mathbb{R}\longrightarrow\mathfrak{R}\stackrel{{\scriptstyle\Pi}}{{\longrightarrow}}\mathbb{C}\longrightarrow
0,$
we obtain the exact sequence (see Scott [14] page 467)
(2.2) $0\longrightarrow\mathbb{R}\longrightarrow
Isom(\mathfrak{R})\stackrel{{\scriptstyle\Pi_{*}}}{{\longrightarrow}}Isom(\mathbb{C})\longrightarrow
1.$
Here $Isom(\mathbb{C})$ is the group of orientation preserving Euclidean
isometries of $\mathbb{C}$.
Observe the elements in $Isom(\mathbb{C})$ can be represented by matrices in
$GL(2,\mathbb{C})$ of the form
$\left[\begin{array}[]{cc}e^{i\theta}&\zeta_{0}\\\
0&1\end{array}\right]\left[\begin{array}[]{c}\zeta\\\
1\end{array}\right]=\left[\begin{array}[]{c}e^{i\theta}\zeta+\zeta_{0}\\\
1\end{array}\right]$
Therefore, the map $\Pi_{*}$ can be given by
(2.3)
$\Pi_{*}:\left[\begin{array}[]{ccc}1&-\bar{\zeta_{0}}e^{i\theta}&(-|\zeta_{0}|^{2}+it_{0})/2\\\
0&e^{i\theta}&\zeta_{0}\\\
0&0&1\end{array}\right]\longrightarrow\left[\begin{array}[]{cc}e^{i\theta}&\zeta_{0}\\\
0&1\end{array}\right].$
It is clear that
$Ker(\Pi_{*})=\left\\{\left[\begin{array}[]{ccc}1&0&it_{0}/2\\\ 0&1&0\\\
0&0&1\end{array}\right]:t_{0}\in\mathbb{R}\right\\}$
is the group of vertical translations fixing $q_{\infty}$.
### 2.3. Isometric spheres
Given an element $G\in PU(2,1)$ with satisfying $G(q_{\infty})\neq
q_{\infty}$, we define the isometric sphere of $G$ to be the hypersurface
$\left\\{\mathbf{z}\in\textbf{H}^{2}_{\mathbb{C}}:|\langle\mathbf{z},q_{\infty}\rangle|=|\langle\mathbf{z},G^{-1}(q_{\infty})\rangle|\right\\}.$
For example, the isometric sphere of
$I_{0}=\left[\begin{array}[]{ccc}0&0&1\\\ 0&-1&0\\\ 1&0&0\end{array}\right]$
is
(2.4)
$\mathcal{B}_{0}=\left\\{(\zeta,t,u)\in\mathfrak{S}:\left||\zeta|^{2}+u+it\right|=2\right\\}$
in horospherical coordinates or
(2.5)
$\mathcal{B}_{0}=\left\\{[z_{1},z_{2},z_{3}]\in\textbf{H}^{2}_{\mathbb{C}}:|z_{1}|=|z_{3}|\right\\}$
in homogeneous coordinates.
All other isometric spheres are images of $\mathcal{B}_{0}$ by Heisenberg
dilations, rotations and translations. Thus the isometric sphere with radius
$r$ and centre $(\zeta_{0},t_{0},0)$ is given by
$\left\\{(\zeta,t,u):\left||\zeta-\zeta_{0}|^{2}+u+it-it_{0}+2i\Im
m(\zeta\bar{\zeta}_{0})\right|=r^{2}\right\\}.$
If $G$ has the matrix form
(2.6) $\left[\begin{array}[]{ccc}a&b&c\\\ d&e&f\\\ g&h&j\end{array}\right],$
then $G(q_{\infty})\neq q_{\infty}$ if and only if $g\neq 0$. The isometric
sphere of $G$ has radius $r=\sqrt{2/|g|}$ and centre $G^{-1}(q_{\infty})$,
which in horospherical coordinates is
$(\zeta_{0},t_{0},0)=(\bar{h}/\bar{g},2\Im m(\bar{j}/\bar{g}),0).$
Isometric spheres are examples of bisectors. Mostow [12] showed that a
bisector is the preimage of a geodesic, called spine, under orthogonal
projection onto the unique complex line containing it. The fibres of this
projection are complex lines called the slices of the bisector. Goldman [9]
showed that a bisector is the union of all totally real Larangian planes
containing the spine. Such Lagrangian planes are called the meridians.
## 3\. On the structure of the stabiliser
In this section we will obtain the generators and relations of the stabiliser
of Picard modular groups by analysis of the fundamental domain in Heisenberg
group.
Let $\mathcal{O}_{d}$ be the ring of integers in the quadratic imaginary
number field $\mathbb{Q}(i\sqrt{d})$, where $d$ is a positive square-free
integer. If $d\equiv 1,2\ (mod\ 4)$, then
$\mathcal{O}_{d}=\mathbb{Z}[i\sqrt{d}]$ and if $d\equiv 3\ (mod\ 4)$, then
$\mathcal{O}_{d}=\mathbb{Z}[\omega_{d}]$, where $\omega_{d}=(1+i\sqrt{d})/2$.
The group $\Gamma_{d}=PU(2,1;\mathcal{O}_{d})$ is called Euclidean Picard
modular group if the ring $\mathcal{O}_{d}$ is Euclidean, namely, only the
rings
$\mathcal{O}_{1},\mathcal{O}_{2},\mathcal{O}_{3},\mathcal{O}_{7},\mathcal{O}_{11}$.
Further relative to amalgamation property, these five groups can be
subclassified into three groupings $\\{\Gamma_{1}\\},$ $\\{\Gamma_{3}\\},$
$\\{\Gamma_{2},\Gamma_{7},\Gamma_{11}\\}$. Since two classes
$\\{\Gamma_{1}\\},\\{\Gamma_{3}\\}$ (c.f. [5], [6]) have been studied in
detail, we mainly describe the remaining class
$\\{\Gamma_{2},\Gamma_{7},\Gamma_{11}\\}$.
### 3.1. The stabiliser of $q_{\infty}$
First we want to analyse $(\Gamma_{d})_{\infty}$ with $d=2,7,11$, the
stabiliser of $q_{\infty}$. Every element of $(\Gamma_{d})_{\infty}$ is upper
triangular and its diagonal entries are units in $\mathcal{O}_{d}$. Recall
that the units of $\mathcal{O}_{1}$ are $\pm 1,\pm i$, they are $\pm
1,\pm\omega,\pm\omega^{2}$ for $\mathcal{O}_{3}$ and they are $\pm 1$ for
others. Therefore $(\Gamma_{d})_{\infty}$ contains no dilations and so is a
subgroup of $Isom(\mathfrak{R})$ and fits into the exact sequence as
$0\longrightarrow\mathbb{R}\cap(\Gamma_{d})_{\infty}\longrightarrow(\Gamma_{d})_{\infty}\stackrel{{\scriptstyle\Pi_{*}}}{{\longrightarrow}}\Pi_{*}((\Gamma_{d})_{\infty})\longrightarrow
1.$
We can write the isometry group of the integer lattice as
$Isom(\mathcal{O}_{d})=\left\\{\left[\begin{array}[]{cc}\alpha&\beta\\\
0&1\end{array}\right]:\alpha,\beta\in\mathcal{O}_{d},\alpha\ \text{is a
unit}\right\\}.$
We now find the image and kernel in this exact sequence.
###### Proposition 3.1.
The stabiliser $(\Gamma_{d})_{\infty}$ of $q_{\infty}$ in $\Gamma_{d}$
satisfies
$0\longrightarrow
2\sqrt{d}\mathbb{Z}\longrightarrow(\Gamma_{d})_{\infty}\stackrel{{\scriptstyle\Pi_{*}}}{{\longrightarrow}}\Delta\longrightarrow
1,$
where $\Delta\subset Isom(\mathcal{O}_{d})$ is of index 2 if $d\equiv 2(mod\
4)$ and $\Delta=Isom(\mathcal{O}_{d})$ if $d\equiv 3(mod\ 4)$.
###### Proof.
Although we only consider the cases $d=2,7,11$, the ring $\mathcal{O}_{2}$
represents those for the values of $d$ with $d\equiv 2(mod\ 4)$ and the rings
$\mathcal{O}_{7},\mathcal{O}_{11}$ represent those of the values $d\equiv
3(mod\ 4)$, the remaining case is the same as $\mathcal{O}_{1}$ which has been
done in [6]. Observe that $Isom(\mathcal{O}_{d})$ is generated by the subgroup
of translations
$\left\\{\hat{T}_{\beta}=\left[\begin{array}[]{cc}1&\beta\\\
0&1\end{array}\right]:\beta\in\mathcal{O}_{d}\right\\}$
and the finite subgroup of order two
$\left\\{\hat{R}_{\alpha}=\left[\begin{array}[]{cc}\alpha&0\\\
0&1\end{array}\right]:\alpha\in\mathcal{O}_{d},\text{$\alpha$ is a
unit}\right\\}.$
Then, to understand $\Delta\subset Isom(\mathcal{O}_{d})$, it suffices to
determine which translations can be lifted. We divide into two cases to
complete the proof.
(i) The case $\mathcal{O}_{d}$ with $d\equiv 2(mod\ 4)$
Suppose that $\beta\in\mathcal{O}_{d}=\mathbb{Z}[i\sqrt{d}]$ and consider the
translation $\hat{T}_{\beta}$ by $\beta$ in $\mathbb{Z}[i\sqrt{d}]$ given
above. The preimage of $\hat{T}_{\beta}$ under $\Pi_{*}$ has the form
$T_{\beta,t}=\left[\begin{array}[]{ccc}1&-\bar{\beta}&\frac{-|\beta|^{2}+it}{2}\\\
0&1&\beta\\\ 0&0&1\end{array}\right].$
This map is in $PU(2,1;\mathbb{Z}[i\sqrt{d}])$ if and only if $|\beta|^{2}$ is
an even integer and $t\in 2\sqrt{d}\mathbb{Z}$ . Writing $\beta=m+i\sqrt{d}n$
for $m,n\in\mathbb{Z}$, then we can obtain $m\equiv 0\ (mod\ 2)$ from the
conditions $|\beta|^{2}=m^{2}+dn^{2}\in 2\mathbb{Z}$ and $d\equiv 2(mod\ 4)$.
Therefore, we conclude that $\Delta\subset Isom(\mathbb{Z}[i\sqrt{d}])$ is of
index 2. Also, the kernel of $\Pi_{*}$ is generated by
$\left[\begin{array}[]{ccc}1&0&i\sqrt{d}\\\ 0&1&0\\\ 0&0&1\end{array}\right],$
which is a vertical translation of $(0,2\sqrt{2})$.
(ii) The case $\mathcal{O}_{d}$ with $d\equiv 3(mod\ 4)$
Suppose that $\beta=m+n\frac{1+i\sqrt{d}}{2}\in\mathcal{O}_{d}$ with
$m,n\in\mathbb{Z}$ for $d\equiv 3(mod\ 4)$. By the same argument of (i), it
only suffices to determine $m,n$ such that $|\beta|^{2}$ is an integer. For
$d\equiv 3(mod\ 4)$, it is easy to show that
$|\beta|^{2}=m^{2}+mn+n^{2}(d+1)/4\in\mathbb{Z}$ for any $m,n\in\mathbb{Z}$,
which implies that $\Delta=Isom(\mathcal{O}_{d})$. Obviously, the kernel of
$\Pi_{*}$ is generated by a vertical translation of $(0,2\sqrt{d})$.
∎
### 3.2. Fundamental domain for the stabiliser
As the first step toward the construction of a fundamental domain for the
action of $(\Gamma_{d})_{\infty}$ on $\mathfrak{R}$ for $d=2,7,11$, we shall
find the suitable generators of $Isom(\mathcal{O}_{d})$ to construct a
fundamental domain in $\mathbb{C}$.
In the proof of Proposition 3.1 we saw that
$\Delta=\Pi_{*}((\Gamma_{2})_{\infty})$ is a subgroup of index 2 in
$Isom(\mathcal{O}_{2})$ consisting of elements of $GL(2,\mathcal{O}_{2})$ of
the form
$\left\\{\left[\begin{array}[]{cc}(-1)^{j}&m+i\sqrt{2}n\\\
0&1\end{array}\right]:j=0,1,m,n\in\mathbb{Z},m\equiv 0(mod\ 2)\right\\}.$
A fundamental domain for this group is the triangle in $\mathbb{C}$ with
vertices at $-1+\sqrt{2}i/2$ and $1\pm\sqrt{2}i/2$; see (a) in Figure 3.1.
Side paring maps are given by
$r^{(2)}_{1}=\left[\begin{array}[]{cc}-1&0\\\ 0&1\end{array}\right],\
r^{(2)}_{2}=\left[\begin{array}[]{cc}-1&2\\\ 0&1\end{array}\right],\
r^{(2)}_{3}=\left[\begin{array}[]{cc}-1&\sqrt{2}i\\\ 0&1\end{array}\right].$
The first of these is a rotation of order 2 fixing origin, the second is a
rotation of order 2 fixing $1/2$ and the third is a rotation of order 2 fixing
$\sqrt{2}i/2$. Indeed every element of $\Delta=GL(2,\mathcal{O}_{2})$ is
generated by $r^{(2)}_{1},r^{(2)}_{2},r^{(2)}_{3}$ as follows
$\displaystyle\left[\begin{array}[]{cc}(-1)^{j}&2m+\sqrt{2}ni\\\
0&1\end{array}\right]$ $\displaystyle=\left[\begin{array}[]{cc}1&2\\\
0&1\end{array}\right]^{m}\left[\begin{array}[]{cc}1&\sqrt{2}i\\\
0&1\end{array}\right]^{n}\left[\begin{array}[]{cc}-1&0\\\
0&1\end{array}\right]^{j}$
$\displaystyle=\left(r^{(2)}_{2}r^{(2)}_{1}\right)^{m}\left(r^{(2)}_{3}r^{(2)}_{1}\right)^{n}\left(r^{(2)}_{1}\right)^{j}.$
As the same argument, a fundamental domain for $Isom(\mathcal{O}_{d})$ with
$d=7$ or $11$ is the triangle in $\mathbb{C}$ with vertices at
$(-1+i\sqrt{d})/4$, $(1-i\sqrt{d})/4$ and $(3+i\sqrt{d})/4$; see (b) in Figure
3.1. Side paring maps are given by
$r^{(d)}_{1}=\left[\begin{array}[]{cc}-1&0\\\ 0&1\end{array}\right],\
r^{(d)}_{2}=\left[\begin{array}[]{cc}-1&1\\\ 0&1\end{array}\right],\
r^{(d)}_{3}=\left[\begin{array}[]{cc}-1&(1+i\sqrt{d})/2\\\
0&1\end{array}\right].$
All these maps are rotations by $\pi$ fixing $0,1/2$ and $(1+i\sqrt{d})/4$
respectively.
Figure 3.1. (a) Fundamental domain for a subgroup $\Delta$ of
$Isom(\mathcal{O}_{2})$ with index 2. (b) Fundamental domain for
$Isom(\mathcal{O}_{d})$ with $d=7,11$. This is also true for all the values of
$d$ with $d\equiv 3(mod\ 4)$.
In order to produce a fundamental domain for $(\Gamma_{d})_{\infty}$ we look
at all the preimages of the triangle (that is a fundamental domain of
$\Pi_{*}((\Gamma_{d})_{\infty})$) under vertical projection $\Pi$ and we
intersect this with a fundamental domain for $ker(\Pi_{*})$. The inverse of
image of the triangle under $\Pi$ is an infinite prism. The kernel of
$\Pi_{*}$ is the infinite cyclic group generated by $T$, the vertical
translation by $(0,2\sqrt{d})$.
Figure 3.2. A fundamental domain $\mathbf{\Sigma}_{2}$ for
$(\Gamma_{2})_{\infty}$ in the Heisenberg group: the map $R^{(2)}_{1}$ rotates
through $\pi$ about $\zeta=0$, the map $R^{(2)}_{2}$ is a Heisenberg rotation
through $\pi$ about $\zeta=1$ and the map $R^{(2)}_{3}$ is a Heisenberg
rotation through $\pi$ about $\zeta=\sqrt{2}i/2$.
###### Proposition 3.2.
$(\Gamma_{2})_{\infty}$ is generated by
$R^{(2)}_{1}=\left[\begin{array}[]{ccc}1&0&0\\\ 0&-1&0\\\
0&0&1\end{array}\right],R^{(2)}_{2}=\left[\begin{array}[]{ccc}1&2&-2\\\
0&-1&2\\\
0&0&1\end{array}\right],R^{(2)}_{3}=\left[\begin{array}[]{ccc}1&-i\sqrt{2}&-1\\\
0&-1&i\sqrt{2}\\\ 0&0&1\end{array}\right]$
and
$T^{(2)}=\left[\begin{array}[]{ccc}1&0&i\sqrt{2}\\\ 0&1&0\\\
0&0&1\end{array}\right].$
A presentation is given by
$(\Gamma_{2})_{\infty}=\langle
R^{(2)}_{j},T^{(2)}|{R^{(2)}_{j}}^{2}=[T^{(2)},R^{(2)}_{j}]=\left({T^{(2)}}^{2}R^{(2)}_{1}R^{(2)}_{3}R^{(2)}_{2}\right)^{2}=id\rangle.$
###### Proof.
Those matrices are constructed by lifting generators of the subgroup
$\Delta\subset Isom(\mathcal{O}_{2})$ with index 2 and also $T^{(2)}$ is a
generator of the kernel of the map $\Pi_{*}$. A fundamental domain can be
constructed with side pairings as Figure 3.2, where the vertices of the prism
are $v_{3}^{+}=(-1+\sqrt{2}i/2,\sqrt{2}),$
$v_{2}^{+}=(1+\sqrt{2}i/2,\sqrt{2}),$ $v_{1}^{+}=(1-\sqrt{2}i/2,\sqrt{2})$ for
the upper cap of the prism and $v_{3}^{-}=(-1+\sqrt{2}i/2,-\sqrt{2}),$
$v_{2}^{-}=(1+\sqrt{2}i/2,-\sqrt{2}),$ $v_{1}^{-}=(1-\sqrt{2}i/2,-\sqrt{2})$
for the base. In particular, the points $v_{4}^{\pm},$ $v_{5}^{\pm},$
$v_{6}^{\pm}$ are the middle points of the edges $(v_{1}^{\pm},v_{2}^{\pm}),$
$(v_{2}^{\pm},v_{3}^{\pm})$ and $(v_{3}^{\pm},v_{1}^{\pm})$, respectively.
The actions of side-pairing maps on $\mathfrak{R}$ are given by
$\displaystyle R^{(2)}_{1}(\zeta,t)$ $\displaystyle=(-\zeta,t),$
$\displaystyle R^{(2)}_{2}(\zeta,t)$ $\displaystyle=(-\zeta+2,t+4\Im
m{\zeta}),$ $\displaystyle R^{(2)}_{3}(\zeta,t)$
$\displaystyle=(-\zeta+i\sqrt{2},t-2\sqrt{2}\Re e{\zeta}),$ $\displaystyle
T^{(2)}(\zeta,t)$ $\displaystyle=(\zeta,t+2\sqrt{2}).$
We describe the side pairing in terms of the action on the vertice:
$\displaystyle R^{(2)}_{1}$ $\displaystyle:$
$\displaystyle(v_{6}^{+},v_{1}^{+},v_{1}^{-},v_{6}^{-})\longrightarrow(v_{6}^{+},v_{3}^{+},v_{3}^{-},v_{6}^{-}),$
$\displaystyle R^{(2)}_{2}$ $\displaystyle:$
$\displaystyle(v_{1}^{+},v_{4}^{+},v_{4}^{-})\longrightarrow(v_{2}^{-},v_{4}^{+},v_{4}^{-}),$
$\displaystyle T^{(2)}R^{(2)}_{2}$ $\displaystyle:$
$\displaystyle(v_{1}^{+},v_{1}^{-},v_{4}^{-})\longrightarrow(v_{2}^{+},v_{2}^{-},v_{4}^{+}),$
$\displaystyle R^{(2)}_{3}$ $\displaystyle:$
$\displaystyle(v_{2}^{+},v_{5}^{+},v_{5}^{-})\longrightarrow(v_{3}^{-},v_{5}^{+},v_{5}^{-}),$
$\displaystyle T^{(2)}R^{(2)}_{3}$ $\displaystyle:$
$\displaystyle(v_{2}^{+},v_{2}^{-},v_{5}^{-})\longrightarrow(v_{3}^{+},v_{3}^{+},v_{5}^{+}),$
$\displaystyle T^{(2)}$ $\displaystyle:$
$\displaystyle(v_{1}^{-},v_{4}^{-},v_{2}^{-},v_{5}^{-},v_{3}^{-},v_{6}^{-})\longrightarrow(v_{1}^{+},v_{4}^{+},v_{2}^{+},v_{5}^{+},v_{3}^{+},v_{6}^{+}).$
The presentation can be obtained following from the edge cycles of the
fundamental domain. ∎
###### Proposition 3.3.
$(\Gamma_{7})_{\infty}$ is generated by
$R^{(7)}_{1}=\left[\begin{array}[]{ccc}1&0&0\\\ 0&-1&0\\\
0&0&1\end{array}\right],R^{(7)}_{2}=\left[\begin{array}[]{ccc}1&1&-\bar{\omega}_{7}\\\
0&-1&1\\\
0&0&1\end{array}\right],R^{(7)}_{3}=\left[\begin{array}[]{ccc}1&\bar{\omega}_{7}&-1\\\
0&-1&\omega_{7}\\\ 0&0&1\end{array}\right]$
and
$T^{(7)}=\left[\begin{array}[]{ccc}1&0&i\sqrt{7}\\\ 0&1&0\\\
0&0&1\end{array}\right].$
A presentation is given by
$\displaystyle(\Gamma_{7})_{\infty}$ $\displaystyle=\langle
R^{(7)}_{j},T^{(7)}|{R^{(7)}_{1}}^{2}={R^{(7)}_{3}}^{2}=[T^{(7)},R^{(7)}_{1}]=[T^{(7)},R^{(7)}_{3}]$
$\displaystyle\hskip
113.81102pt=T^{(7)}{R^{(7)}_{2}}^{-2}=\left(R^{(7)}_{1}R^{(7)}_{3}R^{(7)}_{2}\right)^{2}=id\rangle.$
###### Proof.
Those matrices are constructed by lifting generators of
$Isom(\mathcal{O}_{7})$ and also $T^{(7)}$ is a generator of the kernel of the
map $\Pi_{*}$. A fundamental domain can be constructed with side pairings as
Figure 3.3, where the vertices of the prism are
$v_{1}^{+}=((1-i\sqrt{7})/4,\sqrt{7}),$
$v_{2}^{+}=((3+i\sqrt{7})/4,\sqrt{7}),$
$v_{4}^{+}=((-1+i\sqrt{7})/4,\sqrt{7})$ for the upper cap of the prism and
$v_{1}^{-}=((1-i\sqrt{7})/4,-\sqrt{7}),$
$v_{2}^{-}=((3+i\sqrt{7})/4,-\sqrt{7}),$
$v_{4}^{-}=((-1+i\sqrt{7})/4,-\sqrt{7})$ for the base. The points
$v_{3}^{\pm},v_{5}^{\pm}$ are the middle points of the edges
$(v_{2}^{\pm},v^{\pm}_{4})$ and $(v_{4}^{\pm},v^{\pm}_{1})$. In particular, we
introduce more three points $w_{1}^{+}=((1-i\sqrt{7})/4,\sqrt{7}/2),$
$w_{2}^{-}=((3+i\sqrt{7})/4,-\sqrt{7}/2)$ and
$w_{3}^{+}=((-1+i\sqrt{7})/4,\sqrt{7}/2)$. The actions of side-pairing maps on
$\mathfrak{R}$ are given by
$\displaystyle R^{(7)}_{1}(\zeta,t)$ $\displaystyle=(-\zeta,t),$
$\displaystyle R^{(7)}_{2}(\zeta,t)$ $\displaystyle=(-\zeta+1,t+2\Im
m{\zeta}+\sqrt{7}),$ $\displaystyle R^{(7)}_{3}(\zeta,t)$
$\displaystyle=(-\zeta+\omega_{7},t+2\Im m{(\bar{\omega}_{7}\zeta})),$
$\displaystyle T^{(7)}(\zeta,t)$ $\displaystyle=(\zeta,t+2\sqrt{7}).$
We describe the side pairing in terms of the action on the vertice:
$\displaystyle R^{(7)}_{1}$ $\displaystyle:$
$\displaystyle(v_{5}^{+},v_{1}^{+},v_{1}^{-},v_{5}^{-})\longrightarrow(v_{5}^{+},v_{4}^{+},v_{4}^{-},v_{5}^{-}),$
$\displaystyle R^{(7)}_{2}$ $\displaystyle:$
$\displaystyle(v_{1}^{-},v_{2}^{-},w_{1}^{-},w_{1}^{+})\longrightarrow(w_{1}^{-},w_{1}^{+},v_{1}^{+},v_{2}^{+}),$
$\displaystyle R^{(7)}_{3}$ $\displaystyle:$
$\displaystyle(v_{2}^{+},w_{1}^{-},v_{3}^{-},v_{3}^{+})\longrightarrow(w_{2}^{+},v_{4}^{-},v_{3}^{-},v^{+}_{3}),$
$\displaystyle T^{(7)}R^{(7)}_{3}$ $\displaystyle:$
$\displaystyle(w_{1}^{-},v_{2}^{-},v_{3}^{-})\longrightarrow(v_{4}^{+},w_{2}^{+},v_{3}^{+}),$
$\displaystyle T^{(7)}$ $\displaystyle:$
$\displaystyle(v_{1}^{-},v_{2}^{-},v_{3}^{-},v_{4}^{-},v_{5}^{-})\longrightarrow(v_{1}^{+},v_{2}^{+},v_{3}^{+},v_{4}^{+},v_{5}^{+}).$
The presentation can be obtained following from the edge cycles of the
fundamental domain. ∎
Figure 3.3. A fundamental domain $\mathbf{\Sigma}_{7}$ for
$(\Gamma_{7})_{\infty}$ in the Heisenberg group: the map $R^{(7)}_{1}$ rotates
through $\pi$ about $\zeta=0$, the action of parabolic $R_{2}^{(7)}$ is a
Heisenberg rotation through $\pi$ about $\zeta=1/2$ followed by an upward
vertical translation by $\sqrt{7}$ and the map $R_{3}^{(7)}$ is a Heisenberg
rotation through $\pi$ about $\zeta=(1+i\sqrt{7})/4$.
###### Proposition 3.4.
$(\Gamma_{11})_{\infty}$ is generated by
$R^{(11)}_{1}=\left[\begin{array}[]{ccc}1&0&0\\\ 0&-1&0\\\
0&0&1\end{array}\right],R^{(11)}_{2}=\left[\begin{array}[]{ccc}1&1&-\bar{\omega}_{11}\\\
0&-1&1\\\ 0&0&1\end{array}\right],$
$R^{(11)}_{3}=\left[\begin{array}[]{ccc}1&\bar{\omega}_{11}&-1-\bar{\omega}_{11}\\\
0&-1&\omega_{11}\\\ 0&0&1\end{array}\right]\quad\text{and}\quad
T^{(11)}=\left[\begin{array}[]{ccc}1&0&i\sqrt{11}\\\ 0&1&0\\\
0&0&1\end{array}\right].$
A presentation is given by
$\displaystyle(\Gamma_{11})_{\infty}$ $\displaystyle=\langle
R^{(11)}_{j},T^{(11)}|{R^{(11)}_{1}}^{2}=[T^{(11)},R^{(11)}_{1}]=T^{(11)}{R^{(11)}_{2}}^{-2}$
$\displaystyle\hskip
56.9055pt=T^{(11)}{R^{(11)}_{3}}^{-2}=T^{(11)}\left(R^{(11)}_{1}R^{(11)}_{3}R^{(11)}_{2}\right)^{-2}=id\rangle.$
###### Proof.
Those matrices are constructed by lifting generators of
$Isom(\mathcal{O}_{11})$ and also $T^{(11)}$ is a generator of the kernel of
the map $\Pi_{*}$. A fundamental domain can be constructed with side pairings
as Figure 3.4, where the vertices of the prism are
$v_{1}^{+}=((1-i\sqrt{11})/4,\sqrt{11}),$
$v_{2}^{+}=((3+i\sqrt{11})/4,3\sqrt{11}/2),$
$v_{3}^{+}=((-1+i\sqrt{11})/4,2\sqrt{11})$ for the upper cap of the prism and
$v_{1}^{-}=((1-i\sqrt{11})/4,-\sqrt{11}),$
$v_{2}^{-}=((3+i\sqrt{11})/4,-\sqrt{11}/2),$ $v_{3}^{-}=((-1+i\sqrt{11})/4,0)$
for the base. The points $v_{0}^{\pm}$ are the middle points of the edges
$(v_{1}^{\pm},v^{\pm}_{3})$. In particular, we introduce more three points
$w_{1}=((1-i\sqrt{11})/4,0),$ $w_{2}=((3+i\sqrt{11})/4,\sqrt{11}/2)$ and
$w_{3}=((-1+i\sqrt{11})/4,\sqrt{11})$. The actions of side-pairing maps on
$\mathfrak{R}$ are given by
$\displaystyle R^{(11)}_{1}(\zeta,t)$ $\displaystyle=(-\zeta,t),$
$\displaystyle R^{(11)}_{2}(\zeta,t)$ $\displaystyle=(-\zeta+1,t+2\Im
m{\zeta}+\sqrt{11}),$ $\displaystyle R^{(11)}_{3}(\zeta,t)$
$\displaystyle=(-\zeta+\omega_{11},t+2\Im
m{(\bar{\omega}_{11}\zeta})+\sqrt{11}),$ $\displaystyle T^{(11)}(\zeta,t)$
$\displaystyle=(\zeta,t+2\sqrt{11}).$
We describe the side pairing in terms of the action on the vertice:
$\displaystyle R^{(11)}_{1}$ $\displaystyle:$
$\displaystyle(v_{0}^{+},v_{1}^{+},w_{1},v_{0}^{-})\longrightarrow(v_{0}^{+},w_{3},v_{3}^{-},v_{0}^{-}),$
$\displaystyle T^{(11)}R^{(11)}_{1}$ $\displaystyle:$
$\displaystyle(w_{1},v_{1}^{-},v_{0}^{-})\longrightarrow(v_{3}^{+},w_{3},v_{0}^{+}),$
$\displaystyle R^{(11)}_{2}$ $\displaystyle:$
$\displaystyle(v_{1}^{+},w_{1},v_{1}^{-},v_{2}^{-})\longrightarrow(v_{2}^{+},w_{2},v_{2}^{-},v_{1}^{+}),$
$\displaystyle R^{(11)}_{3}$ $\displaystyle:$
$\displaystyle(v_{2}^{+},w_{2},v_{2}^{-},v_{3}^{-})\longrightarrow(v_{3}^{+},w_{3},v_{3}^{-},v_{2}^{+}),$
$\displaystyle T^{(11)}$ $\displaystyle:$
$\displaystyle(v_{0}^{-},v_{1}^{-},v_{2}^{-},v_{3}^{-})\longrightarrow(v_{0}^{+},v_{1}^{+},v_{2}^{+},v_{3}^{+}).$
The presentation can be obtained following from the edge cycles of the
fundamental domain. ∎
Figure 3.4. A fundamental domain $\mathbf{\Sigma}_{11}$ for
$(\Gamma_{11})_{\infty}$ in the Heisenberg group: the map $R^{(11)}_{1}$
rotates through $\pi$ about $\zeta=0$, the action of parabolic $R_{2}^{(11)}$
is a screw Heisenberg rotation through $\pi$ about $\zeta=1/2$ followed by an
upward vertical translation by $\sqrt{11}$ and the map $R_{3}^{(11)}$ is a
screw Heisenberg rotation through $\pi$ about $\zeta=(1+i\sqrt{11})/4$
followed by an upward vertical translation by $\sqrt{11}$.
## 4\. The statement of our method and results
In this section, we introduce the method used in ([5], [6]) to determine the
generators of the Euclidean Picard groups and then state our results.
Recall that the map
$I_{0}=\left[\begin{array}[]{ccc}0&0&1\\\ 0&-1&0\\\ 1&0&0\end{array}\right],$
defined in the Section 2.3. We consider the isometric sphere $\mathcal{B}_{0}$
of $I_{0}$ given by (2.4), which is a Cygan sphere centred $o=(0,0,0)$ with
radius $\sqrt{2}$. Observe that $I_{0}$ maps $\mathcal{B}_{0}$ to itself and
swaps the inside and the outside of $\mathcal{B}_{0}$. Given an element of
$\Gamma_{d}$ as the form (2.6), we know that the radius of isometric sphere is
$\sqrt{2/|g|}$. For each case $\mathcal{O}_{d}$, the radius of isometric
sphere is not greater than $\sqrt{2}$ since the absolute of $g$ is not smaller
than 1 for $g\in\mathcal{O}_{d}$. We show that the largest isometric spheres
are all the images of $\mathcal{B}_{0}$ under the elements in
$(\Gamma_{d})_{\infty}$.
###### Proposition 4.1.
An isometric sphere has the largest radius if and only if it is the image of
$\mathcal{B}_{0}$ under an element in $(\Gamma_{d})_{\infty}$.
###### Proof.
Obviously, the image of $\mathcal{B}_{0}$ under an element in
$(\Gamma_{d})_{\infty}$ has the largest radius $\sqrt{2}$. On the contrary,
given an element $G$ as the form (2.6) such that $G(q_{\infty})\neq
q_{\infty}$, then the isometric sphere of $G$ has the largest radius which
leads to $g=1$. So the centre of isometric sphere of $G$ is
$G^{-1}(\infty)=(\bar{h},2\Im m\bar{j},0)$ in horospherical coordinates. Since
$\bar{h}$ and $2\Im m\bar{j}\in\mathcal{O}_{d}$, we can take a Heisenberg
translation $T\in(\Gamma_{d})_{\infty}$ mapping the origin to $(\bar{h},2\Im
m\bar{j})$. Writing $T^{\prime}=GTI_{0}$, we know that $T^{\prime}$ fixes
$\infty$. We conclude explicitly that the isometric sphere of $G$ is
$\left\\{\mathbf{z}\in\textbf{H}^{2}_{\mathbb{C}}:|\langle\mathbf{z},q_{\infty}\rangle|=|\langle\mathbf{z},G^{-1}(q_{\infty})\rangle|=|\langle\mathbf{z},TI_{0}(q_{\infty})\rangle|\right\\},$
which is the image of $\mathcal{B}_{0}$ under $T$. ∎
Our method is based on the special feature that the orbifold
$\mathbf{H}^{2}_{\mathbb{C}}/\Gamma_{d}$ has only one cusp for $d=2,7,11$. For
these types of orbifolds, one would like to construct a fundamental domain
using the Ford domain (that is the exteriors of isometric spheres of all
elements not fixing infinity), namely, the intersection of the Ford domain and
a fundamental domain for the stabiliser of infinity. The Ford domain is
canonical, but we can choose a fundamental domain for the stabiliser freely.
As the first step toward the construction of a fundamental domain, we should
always determine the generators of the group. In the previous section, we
found suitable generators of the stabliser and constructed a fundamental
domain. We will show that adjoining $I_{0}$ to $(\Gamma_{d})_{\infty}$ gives
the Euclidean Picard modular groups $\Gamma_{d}$. The basic idea of the proof
can be described easily. Analogous to Theorem 3.5 of [5] we shall prove that
$\langle R^{(d)}_{1},R^{(d)}_{2},R^{(d)}_{3},T^{(d)},I_{0}\rangle$ has only
one cusp. The fact that $PU(2,1;\mathcal{O}_{d})$ has the same cusp and the
stabiliser of infinity as the group generated by
$R^{(d)}_{1},R^{(d)}_{2},R^{(d)}_{3},T^{(d)},I_{0}$ shows that they are the
same. The key step is to find a union of isometric spheres such that a
fundamental domain for $\Gamma_{d}$ is contained in the intersection of their
exteriors and a fundamental domain for the stabiliser, which implies that the
group $\langle R^{(d)}_{1},R^{(d)}_{2},R^{(d)}_{3},T^{(d)},I_{0}\rangle$ has
only one cusp. In other words, we want to show the union of the boundaries of
these isometric spheres in Heisenberg group contains each of the prisms we
constructed above. The problem of determining this will be discussed in the
next section.
A simple lemma will be used in the proof of our theorems many times, we state
it as follows.
###### Lemma 4.2.
(c.f. [6]) All Cygan balls are affinely convex.
Our aim is to prove the following results, we summarise them as three
theorems.
###### Theorem 4.3.
Let $\mathcal{K}=\mathbb{Q}(\sqrt{-2})$ and let
$\mathcal{O}_{2}=\mathbb{Z}[i\sqrt{2}]$. Then the group
$PU(2,1,\mathcal{O}_{2})$ is generated by the elements
$I_{0}=\left[\begin{array}[]{ccc}0&0&1\\\ 0&-1&0\\\
1&0&0\end{array}\right],R^{(2)}_{1}=\left[\begin{array}[]{ccc}1&0&0\\\
0&-1&0\\\
0&0&1\end{array}\right],R^{(2)}_{2}=\left[\begin{array}[]{ccc}1&2&-2\\\
0&-1&2\\\ 0&0&1\end{array}\right],$
$R^{(2)}_{3}=\left[\begin{array}[]{ccc}1&-i\sqrt{2}&-1\\\ 0&-1&i\sqrt{2}\\\
0&0&1\end{array}\right]\quad\text{and}\quad
T^{(2)}=\left[\begin{array}[]{ccc}1&0&i\sqrt{2}\\\ 0&1&0\\\
0&0&1\end{array}\right].$
###### Theorem 4.4.
Let $\mathcal{K}=\mathbb{Q}(\sqrt{-7})$ and let
$\mathcal{O}_{7}=\mathbb{Z}[\omega_{7}]$, where
$\omega_{7}=\frac{1}{2}(1+i\sqrt{7})$, be the ring of integers of
$\mathcal{K}$. Then the group $PU(2,1,\mathcal{O}_{7})$ is generated by the
elements
$I_{0}=\left[\begin{array}[]{ccc}0&0&1\\\ 0&-1&0\\\
1&0&0\end{array}\right],R^{(7)}_{1}=\left[\begin{array}[]{ccc}1&0&0\\\
0&-1&0\\\
0&0&1\end{array}\right],R^{(7)}_{2}=\left[\begin{array}[]{ccc}1&1&-\bar{\omega}_{7}\\\
0&-1&1\\\ 0&0&1\end{array}\right],$
$R^{(7)}_{3}=\left[\begin{array}[]{ccc}1&\bar{\omega}_{7}&-1\\\
0&-1&\omega_{7}\\\ 0&0&1\end{array}\right]\quad\text{and}\quad
T^{(7)}=\left[\begin{array}[]{ccc}1&0&i\sqrt{7}\\\ 0&1&0\\\
0&0&1\end{array}\right].$
###### Theorem 4.5.
Let $\mathcal{K}=\mathbb{Q}(\sqrt{-11})$ and let
$\mathcal{O}_{11}=\mathbb{Z}[\omega_{11}]$, where
$\omega_{11}=\frac{1}{2}(1+i\sqrt{11})$, be the ring of integers of
$\mathcal{K}$. Then the group $PU(2,1,\mathcal{O}_{11})$ is generated by the
elements
$I_{0}=\left[\begin{array}[]{ccc}0&0&1\\\ 0&-1&0\\\
1&0&0\end{array}\right],R^{(11)}_{1}=\left[\begin{array}[]{ccc}1&0&0\\\
0&-1&0\\\
0&0&1\end{array}\right],R^{(11)}_{2}=\left[\begin{array}[]{ccc}1&1&-\bar{\omega}_{11}\\\
0&-1&1\\\ 0&0&1\end{array}\right],$
$R^{(11)}_{3}=\left[\begin{array}[]{ccc}1&\bar{\omega}_{11}&-1-\bar{\omega}_{11}\\\
0&-1&\omega_{11}\\\ 0&0&1\end{array}\right]\quad\text{and}\quad
T^{(11)}=\left[\begin{array}[]{ccc}1&0&i\sqrt{11}\\\ 0&1&0\\\
0&0&1\end{array}\right].$
###### Remark.
For other values of $d$ such that $\mathcal{O}_{d}$ has class number one,
namely $d=19,$ $43,$ $67,$ $163$, we can construct the same type of
fundamental domain for $(\Gamma_{d})_{\infty}$ in Heisenberg group as
$(\Gamma_{11})_{\infty}$. Although all generators as the above types lie in
$PU(2,1;\mathcal{O}_{d})$, there is no reason why adjoining $I_{0}$ to
$(\Gamma_{d})_{\infty}$ should continue to generate the full group
$PU(2,1;\mathcal{O}_{d})$. From the point of view of geometric, the largest
radius of isometric sphere is always $\sqrt{2}$ while the shape of the
fundamental domain and the length of Heisenberg translations become large as
$d$ getting large. In contrast the radius of isometric spheres other than the
largest are going to be smaller and smaller. Consequently the mount of
isometric spheres containing the fundamental domain increases so rapidly with
the value of $d$ that it seem to be done by another way. Furthermore, the
method of [4] could not be extended to non-Euclidean Picard modular groups.
## 5\. The determination of isometric spheres
Recall that the Cygan sphere $\mathcal{B}_{0}$ is the isometric sphere of
$I_{0}$. The boundary of $\mathcal{B}_{0}$ is called a spinal sphere [9] in
Heisenberg group, we denote by $\mathcal{S}_{0}$ which is defined by
(5.1)
$\mathcal{S}_{0}=\left\\{(\zeta,t):\left||\zeta|^{2}+it\right|=2\right\\}.$
Indeed we only need to consider the boundaries of isometric spheres in
Heisenberg group because two isometric spheres have a non-empty interior
intersection if and only if the boundaries have a non-empty interior
intersection.
### 5.1. The case $\mathcal{O}_{2}$
In the cases of $PU(2,1;\mathcal{O}_{1})$ and $PU(2,1;\mathcal{O}_{3})$, all
the vertices of the fundamental domain for the stabiliser of $q_{\infty}$
acting on $\partial\textbf{H}^{2}_{\mathbb{C}}$ lie inside $\mathcal{S}_{0}$.
For the group $PU(2,1;\mathcal{O}_{2})$, it is not hard to show that six
vertices of the prism $\mathbf{\Sigma}_{2}$ lie outside $\mathcal{S}_{0}$.
Therefore we need to find more isometric spheres whose boundaries together
with $\mathcal{S}_{0}$ contain the prism $\mathbf{\Sigma}_{2}$.
We consider the map
$I_{0}R^{(2)}_{2}I_{0}=\left[\begin{array}[]{ccc}1&0&0\\\ -2&-1&0\\\
-2&-2&1\end{array}\right],$
whose isometric sphere which we denote by $\mathcal{B}_{1}$ is a Cygan sphere
centred at the point $(1,0,0)$ (in horospherical coordinates) with radius 1.
The boundary of $\mathcal{B}_{1}$ is given by
(5.2) $\mathcal{S}_{1}=\left\\{(\zeta,t):\left||\zeta-1|^{2}+it+2i\Im
m\zeta\right|=1\right\\}.$
Minimising the number of spinal spheres by the symmetry of $R^{(2)}_{1}$, it
suffice to consider $\mathcal{S}_{0}$ and several images of $\mathcal{S}_{1}$
under some suitable elements in $(\Gamma_{2})_{\infty}$, these are in
Heisenberg coordinates given by
$\displaystyle T^{(2)}(\mathcal{S}_{1})$
$\displaystyle=\left\\{(\zeta,t):\left||\zeta-1|^{2}+it-2i\sqrt{2}+2i\Im
m\zeta\right|=1\right\\},$ $\displaystyle{T^{(2)}}^{-1}(\mathcal{S}_{1})$
$\displaystyle=\left\\{(\zeta,t):\left||\zeta-1|^{2}+it+2i\sqrt{2}+2i\Im
m\zeta\right|=1\right\\},$ $\displaystyle R^{(2)}_{1}(\mathcal{S}_{1})$
$\displaystyle=\left\\{(\zeta,t):\left||\zeta+1|^{2}+it-2i\Im
m\zeta\right|=1\right\\},$
$\displaystyle{T^{(2)}}^{-1}R^{(2)}_{1}(\mathcal{S}_{1})$
$\displaystyle=\left\\{(\zeta,t):\left||\zeta+1|^{2}+it+2i\sqrt{2}-2i\Im
m\zeta\right|=1\right\\}.$
We claim that the prism $\mathbf{\Sigma}_{2}$ lies inside the union of
$\mathcal{S}_{0}$ and these images of $\mathcal{S}_{1}$, see Figure 5.1 for
viewing these spinal spheres.
Figure 5.1. (a) The shading view of neighboring spinal spheres containing the
fundamental domain for $(\Gamma_{2})_{\infty}$. (b) Another view for these
spinal spheres.
###### Proposition 5.1.
The prism $\mathbf{\Sigma}_{2}$ is contained in the union of the interiors of
the spinal spheres $\mathcal{S}_{0},$ $\mathcal{S}_{1},$
$T^{(2)}(\mathcal{S}_{1}),$ ${T^{(2)}}^{-1}(\mathcal{S}_{1}),$
$R^{(2)}_{1}(\mathcal{S}_{1})$ and
${T^{(2)}}^{-1}R^{(2)}_{1}(\mathcal{S}_{1})$.
###### Proof.
It suffices to show there exists three points $(v_{1}^{+})^{(j)}$ $(j=1,2,3)$
on the edges $(v_{1}^{+},v_{1}^{-}),$ $(v_{1}^{+},v_{2}^{+})$ and
$(v_{1}^{+},v_{3}^{+})$ which lie in the intersection of the interiors of
$\mathcal{S}_{0}$ and $\mathcal{S}_{1}$ such that the tetrahedron
$\mathbb{T}(v_{1}^{+})$ with vertices $v_{1}^{+},$ $(v_{1}^{+})^{(1)},$
$(v_{1}^{+})^{(2)},$ $(v_{1}^{+})^{(3)}$ lies inside $\mathcal{S}_{1}$. By the
same argument, we can also obtain other five tetrahedra
$\mathbb{T}(v_{2}^{+}),$ $\mathbb{T}(v_{3}^{+}),$ $\mathbb{T}(v_{1}^{-}),$
$\mathbb{T}(v_{2}^{-}),$ $\mathbb{T}(v_{3}^{-})$ with apex $v_{2}^{+},$
$v_{3}^{+},$ $v_{1}^{-},$ $v_{2}^{-},$ $v_{3}^{-}$ respectively such that
$\mathbb{T}(v_{2}^{+})\in Int(T^{(2)}(\mathcal{S}_{1})),$
$\mathbb{T}(v_{3}^{+})\in Int(R^{(2)}_{1}(\mathcal{S}_{1})),$
$\mathbb{T}(v_{1}^{-})\in Int({T^{(2)}}^{-1}(\mathcal{S}_{1})),$
$\mathbb{T}(v_{2}^{-})\in Int(\mathcal{S}_{1})$ and $\mathbb{T}(v_{3}^{-})\in
Int({T^{(2)}}^{-1}R^{(2)}_{1}(\mathcal{S}_{1}))$. Moreover, the core part
obtained by cutting off six the tetrahedra from the prism lies inside
$\mathcal{S}_{0}$.
We shall prove the existence of the tetrahedron $\mathbb{T}(v_{1}^{+})$ and
the others follow similarly. The edge joining $v_{1}^{+}$ and $v_{1}^{-}$ is
contained the complex line $\zeta=1-\sqrt{2}i/2$ which is given by points with
Heisenberg coordinates
$\zeta=1-\sqrt{2}i/2,\quad-\sqrt{2}\leq t\leq\sqrt{2}.$
The edge joining $v_{1}^{+}$ and $v_{2}^{+}$ is given by points with
Heisenberg coordinates
$\Re e\zeta=1,\quad-\sqrt{2}/2\leq\Im m\zeta\leq\sqrt{2}/2,\quad t=\sqrt{2}.$
The edge joining $v_{1}^{+}$ and $v_{3}^{+}$ is given by points with
Heisenberg coordinates
$\Re e\zeta=-\sqrt{2}\Im m\zeta,\quad t=\sqrt{2}.$
From (5.1) and (5.2), the points on the edge $(v^{+}_{1},v^{-}_{1})$ lie in
the intersection of the interiors of $\mathcal{S}_{0}$ and $\mathcal{S}_{1}$
if and only if
(5.3)
$\left|3/2+it\right|<2\quad\text{and}\quad\left|1/2-(t-\sqrt{2})i\right|<1.$
By an easy calculation, the inequalities (5.3) are equivalent to
$\sqrt{2}-\sqrt{3}/2<t<\sqrt{7}/2.$
Using the same argument as above, we obtain that the points on the edge
$(v^{+}_{1},v^{+}_{2})$ lie in the intersection of the interiors of
$\mathcal{S}_{0}$ and $\mathcal{S}_{1}$ if and only if $\Re e\zeta=1,$ and
$-\sqrt{\sqrt{2}-1}<\Im m\zeta<\delta_{1}$, where $\delta_{1}\approx-0.208$ is
the largest real root of the equation $x^{4}+4x^{2}+4\sqrt{2}x+1=0$. The
points on the edge $(v^{+}_{1},v^{+}_{3})$ lie in the intersection of the
interiors of $\mathcal{S}_{0}$ and $\mathcal{S}_{1}$ if and only if $\Re
e\zeta=-\sqrt{2}\Im m\zeta$ and $-2^{1/4}/\sqrt{3}<\Im m\zeta<\delta_{2}$,
where $\delta_{2}\approx-0.264$ is the largest real root of the equation
$9x^{4}+12\sqrt{2}x^{3}+18x^{2}+8\sqrt{2}x+2=0$.
In term of these, we choose three points as
$(v_{1}^{+})^{(1)}=(1-\sqrt{2}i/2,1)$ on the edge $(v^{+}_{1},v^{-}_{1})$,
$(v_{1}^{+})^{(2)}=(1-i/2,\sqrt{2})$ on the edge $(v^{+}_{1},v^{+}_{2})$ and
$(v_{1}^{+})^{(3)}=(\sqrt{2}/2-i/2,\sqrt{2})$ on the edge
$(v^{+}_{1},v^{+}_{3})$, which are inside the intersection of the interiors of
$\mathcal{S}_{0}$, $\mathcal{S}_{1}$ and also the vertex $v_{1}^{+}$ lies
inside $\mathcal{S}_{1}$. Therefore the tetrahedron $\mathbb{T}(v_{1}^{+})$
with the vertices $v_{1}^{+},$ $(v_{1}^{+})^{(1)},$ $(v_{1}^{+})^{(2)},$
$(v_{1}^{+})^{(3)}$ lies inside $\mathcal{S}_{1}$ by Lemma 4.1. ∎
### 5.2. The case $\mathcal{O}_{7}$
In this case, the distance between the top and base of the fundamental domain
for the stabiliser $(\Gamma_{7})_{\infty}$ is greater than the diameter of
$\mathcal{S}_{0}$, which implies that the prism $\mathbf{\Sigma}_{7}$ can not
be contained inside $\mathcal{S}_{0}$ completely. Due to increasing the length
of Heisenberg translations, only the images of $\mathcal{S}_{0}$ under the
elements in $(\Gamma_{7})_{\infty}$ could not cover the whole prism. We show
that there are also isometric spheres with Cygan radius smaller than
$\sqrt{2}$ whose centres are near to origin.
Therefore we consider the map
$Q=I_{0}R^{(7)}_{2}I_{0}=\left[\begin{array}[]{ccc}1&0&0\\\ 1&1&0\\\
\bar{\omega}_{7}&1&1\end{array}\right].$
Consider the isometric spheres of $Q$ and $Q^{-1}$, which we denote by
$\mathcal{B}_{2}$ and $\mathcal{B}_{3}$, respectively. The centre of
$\mathcal{B}_{2}$ is $Q^{-1}(\infty)$, which is the point with horospherical
coordinates $(1/4+i\sqrt{7}/4,\sqrt{7}/2,0)$ and the centre of
$\mathcal{B}_{3}$, is $Q(\infty)$ which has horospherical coordinates
$(1/4-i\sqrt{7}/4,\sqrt{7}/2,0)$. Both these isometric spheres have Cygan
radius $\sqrt{2/|\omega_{7}|}=2^{1/4}$. The boundaries of these isometric
spheres $\mathcal{B}_{2}$ and $\mathcal{B}_{3}$ are in Heisenberg coordinates
given by
(5.4) $\displaystyle\mathcal{S}_{2}$
$\displaystyle=\left\\{(\zeta,t):\left|\left|\zeta-\omega_{7}/2\right|^{2}+it+i\sqrt{7}/2+i\Im
m(\bar{\omega}_{7}\zeta)\right|=\sqrt{2}\right\\},$ (5.5)
$\displaystyle\mathcal{S}_{3}$
$\displaystyle=\left\\{(\zeta,t):\left|\left|\zeta-\bar{\omega}_{7}/2\right|^{2}+it-i\sqrt{7}/2+i\Im
m(\omega_{7}\zeta)\right|=\sqrt{2}\right\\}.$
In order to cover the prim $\mathbf{\Sigma}_{7}$ by the spinal spheres, we use
the symmetry property of $R^{(7)}_{1}$, it suffice to consider
$\mathcal{S}_{0},\mathcal{S}_{2}$ and images of $\mathcal{S}_{0}$ and
$\mathcal{S}_{3}$ under suitable elements in $(\Gamma_{7})_{\infty}$, these
are points with Heisenberg coordinates given by
$\displaystyle R^{(7)}_{2}(\mathcal{S}_{0})$
$\displaystyle=\left\\{(\zeta,t):\left||\zeta-1|^{2}+it-i\sqrt{7}+2i\Im
m\zeta\right|=4\right\\},$ $\displaystyle{R^{(7)}_{2}}^{-1}(\mathcal{S}_{0})$
$\displaystyle=\left\\{(\zeta,t):\left||\zeta-1|^{2}+it+i\sqrt{7}+2i\Im
m\zeta\right|=4\right\\},$ $\displaystyle{R^{(7)}_{2}}^{-1}(\mathcal{S}_{3})$
$\displaystyle=\left\\{(\zeta,t):\left|\left|\zeta-(1+\omega_{7})/2\right|^{2}+it+i\sqrt{7}\right.\right.$
$\displaystyle\hskip 142.26378pt\left.\left.+i\Im
m((1+\bar{\omega}_{7})\zeta)\right|=\sqrt{2}\right\\},$ $\displaystyle
R^{(7)}_{3}R^{(7)}_{2}(\mathcal{S}_{3})$
$\displaystyle=\left\\{(\zeta,t):\left|\left|\zeta+\bar{\omega}_{7}/2\right|^{2}+it-i\sqrt{7}/2-i\Im
m(\omega_{7}\zeta)\right|=\sqrt{2}\right\\},$ $\displaystyle
R^{(7)}_{1}R^{(7)}_{3}R^{(7)}_{2}(\mathcal{S}_{3})$
$\displaystyle=\left\\{(\zeta,t):\left|\left|\zeta-\bar{\omega}_{7}/2\right|^{2}+it-i\sqrt{7}/2+i\Im
m(\omega_{7}\zeta)\right|=\sqrt{2}\right\\}.$
We claim that the prism $\mathbf{\Sigma}_{7}$ lies inside the union of
$\mathcal{S}_{0},$ $\mathcal{S}_{2}$ and these images
$R^{(7)}_{2}(\mathcal{S}_{0}),$ ${R^{(7)}_{2}}^{-1}(\mathcal{S}_{0}),$
${R^{(7)}_{2}}^{-1}(\mathcal{S}_{3})$,
$R^{(7)}_{3}R^{(7)}_{2}(\mathcal{S}_{3}),$
$R^{(7)}_{1}R^{(7)}_{3}R^{(7)}_{2}(\mathcal{S}_{3})$, see Figure 5.2 for
viewing these spinal spheres.
Figure 5.2. (a) The shading view of neighboring spinal spheres containing the
fundamental domain for $(\Gamma_{7})_{\infty}$. (b) Another view for these
spinal spheres.
###### Proposition 5.2.
The prism $\mathbf{\Sigma}_{7}$ is contained in the union of the interiors of
the spinal spheres $\mathcal{S}_{0},$ $\mathcal{S}_{2},$
$R^{(7)}_{2}(\mathcal{S}_{0}),$ ${R^{(7)}_{2}}^{-1}(\mathcal{S}_{0}),$
${R^{(7)}_{2}}^{-1}(\mathcal{S}_{3}),$
$R^{(7)}_{3}R^{(7)}_{2}(\mathcal{S}_{3})$ and
$R^{(7)}_{1}R^{(7)}_{3}R^{(7)}_{2}(\mathcal{S}_{3})$.
###### Proof.
It suffice to show that the prism $\mathbf{\Sigma}_{7}$ can be decomposed into
several pieces as polyhedra such that each polyhedron lies inside a spinal
sphere which is described in the proposition and the common face of two
adjacent polyhedra lie in the intersection of the interior of two spinal
spheres which contain these two polyhedra.
We need to add sixteen points on the faces of the prism $\mathbf{\Sigma}_{7}$
in order to decompose the prim into seven polyhedra, in Heisenberg
coordinates, these are given by
$\begin{array}[]{llll}p_{1}=(1/4-i\sqrt{7}/4,3/2),&p_{2}=(0.11-i11\sqrt{7}/100,1.44+\sqrt{7}/50),\\\
p_{3}=(1/2,8/5),&p_{4}=(-1/10+i\sqrt{7}/10,\sqrt{7}),\\\
p_{5}=(-1/10+i\sqrt{7}/10,\sqrt{7}/2),&p_{6}=(3/4+i\sqrt{7}/4,1.7),\\\
p_{7}=(-1/4+i\sqrt{7}/4,1),&p_{8}=(1/4-i\sqrt{7}/4,-1),\\\
p_{9}=(1/60-i\sqrt{7}/60,-2\sqrt{7}/3),&p_{10}=(-1/20+i\sqrt{7}/20,-\sqrt{7}),\\\
p_{11}=(3/5+i\sqrt{7}/10,-2\sqrt{7}/3),&p_{12}=(7/10+i\sqrt{7}/5,-\sqrt{7}),\\\
p_{13}=(3/4+i\sqrt{7}/4,-2\sqrt{7}/3),&p_{14}=(5/12+i\sqrt{7}/4,-2\sqrt{7}/3),\\\
p_{15}=(1/4+i\sqrt{7}/4,-\sqrt{7}),&p_{16}=(-1/4+i\sqrt{7}/4,-1).\end{array}$
Figure 5.3. A view of the decomposition for the prism $\mathbf{\Sigma}_{7}$ as
several polyhedra.
We describe these polyhedra as follows:
(i) The tetrahedron $\mathbb{T}$ with the vertice $v^{+}_{1},$ $p_{1},$
$p_{2},$ $p_{3}$;
(ii) The hexahedron $\mathbb{H}_{1}$ with the vertice $v^{+}_{1},$
$v_{2}^{+},$ $p_{2},$ $p_{3},$ $p_{4},$ $p_{5},$ $p_{6}$;
(iii) The pentahedron $\mathbb{P}_{1}$ with the vertice $v_{2}^{+},$ $p_{4},$
$p_{5},$ $p_{6},$ $v^{+}_{4},$ $p_{7}$;
(iv) The pentahedron $\mathbb{P}_{2}$ with the vertice $v_{1}^{-},$ $p_{8},$
$p_{9},$ $p_{10},$ $p_{11},$ $p_{12}$;
(v) The hexahedron $\mathbb{H}_{2}$ with the vertice $p_{9},$ $p_{10},$
$p_{11},$ $p_{12},$ $p_{13},$ $v_{2}^{-},$ $p_{14},$ $p_{15}$;
(vi) The pentahedron $\mathbb{P}_{3}$ with the vertice $p_{9},$ $p_{10},$
$p_{14},$ $p_{15},$ $p_{16},$ $v_{4}^{+}$;
(vii) The octahedron $\mathbb{O}$ with the vertice $p_{1},$ $p_{2},$ $p_{3},$
$p_{5},$ $p_{6},$ $p_{7},$ $p_{8},$ $p_{9},$ $p_{11},$ $p_{13},$ $p_{14},$
$p_{16}$.
Note that the face $(p_{1},p_{2},p_{3})$ of $\mathbb{T}$ and the face
$(p_{2},p_{3},p_{5},p_{6})$ of $\mathbb{H}_{1}$ are on the face
$(p_{1},p_{5},p_{6})$ of $\mathbb{O}$; the common face
$(v^{+}_{2},p_{4},p_{5},p_{6})$ of $\mathbb{H}_{1}$ and $\mathbb{P}_{1}$ is a
vertical plane; the face $(p_{9},p_{11},p_{13},p_{14})$ of $\mathbb{H}_{2}$ is
parallel to the base of the prism. Furthermore, the faces
$(p_{9},p_{10},p_{11},p_{12})$ and $(p_{9},p_{10},p_{14},p_{15})$ are the
trapeziums since the edge $(p_{9},p_{11})$ is parallel to $(p_{10},p_{12})$
and the edge $(p_{9},p_{14})$ is parallel to $(p_{10},p_{15})$.
By examining the location of the points and applying Lemma 4.1, we conclude
that the tetrahedron $\mathbb{T}$ is inside the spinal sphere
$R^{(7)}_{1}R^{(7)}_{3}R^{(7)}_{2}(\mathcal{S}_{3})$; the hexahedron
$\mathbb{H}_{1}$ is contained inside the spinal sphere
$R^{(7)}_{2}(\mathcal{S}_{0})$; the pentahedron $\mathbb{P}_{1}$ is inside
$R^{(7)}_{3}R^{(7)}_{2}(\mathcal{S}_{3})$; the pentahedron $\mathbb{P}_{2}$ is
contained inside ${R^{(7)}_{2}}^{-1}(\mathcal{S}_{0})$; the hexahedron
$\mathbb{H}_{2}$ is contained inside ${R^{(7)}_{2}}^{-1}(\mathcal{S}_{3})$;
the pentahedron $\mathbb{P}_{3}$ is inside $\mathcal{S}_{2}$; the remaining
octahedron $\mathbb{O}$ is inside $\mathcal{S}_{0}$; see Figure 5.3 for
viewing the decomposition of the prism. ∎
### 5.3. The case $\mathcal{O}_{11}$
In this case, we know that the fundamental domain for the stabiliser
$(\Gamma_{11})_{\infty}$ cannot be still inside $\mathcal{S}_{0}$ completely.
The radius of spinal spheres other than the largest are so small that these
spinal spheres are not much contribution to covering the prism
$\mathbf{\Sigma}_{11}$. Due to the different shape of the prism
$\mathbf{\Sigma}_{11}$ with the case $\mathcal{O}_{7}$, we only need to
consider the largest spinal spheres which are the images of $\mathcal{S}_{0}$
under the elements of $(\Gamma_{11})_{\infty}$. In order to determine a union
of the spinal spheres which covering the prim $\mathbf{\Sigma}_{11}$, we
minimise their numbers by the symmetry of $R^{(11)}_{1}$, it suffice to
consider $\mathcal{S}_{0}$ and the images of $\mathcal{S}_{0}$ under suitable
elements in $(\Gamma_{11})_{\infty}$, these are in Heisenberg coordinates
given by
$\displaystyle T^{(11)}(\mathcal{S}_{0})$ $\displaystyle=$
$\displaystyle\left\\{(\zeta,t):\left||\zeta|^{2}+it-2i\sqrt{11}+2i\Im
m\zeta\right|=4\right\\},$ $\displaystyle R^{(11)}_{2}(\mathcal{S}_{0})$
$\displaystyle=$
$\displaystyle\left\\{(\zeta,t):\left||\zeta-1|^{2}+it-i\sqrt{11}+2i\Im
m\zeta\right|=4\right\\},$ $\displaystyle
R^{(11)}_{1}R^{(11)}_{2}(\mathcal{S}_{0})$ $\displaystyle=$
$\displaystyle\left\\{(\zeta,t):\left||\zeta+1|^{2}+it-i\sqrt{11}-2i\Im
m\zeta\right|=4\right\\},$ $\displaystyle{R^{(11)}_{2}}^{-1}(\mathcal{S}_{0})$
$\displaystyle=$
$\displaystyle\left\\{(\zeta,t):\left||\zeta-1|^{2}+it+i\sqrt{11}+2i\Im
m\zeta\right|=4\right\\},$ $\displaystyle R^{(11)}_{3}(\mathcal{S}_{0})$
$\displaystyle=$
$\displaystyle\left\\{(\zeta,t):\left|\left|\zeta-\omega_{11}\right|^{2}+it-i\sqrt{11}-2i\Im
m(\bar{\omega}_{11}\zeta)\right|=4\right\\},$
$\displaystyle{R^{(11)}_{3}}^{-1}(\mathcal{S}_{0})$ $\displaystyle=$
$\displaystyle\left\\{(\zeta,t):\left|\left|\zeta-\omega_{11}\right|^{2}+it+i\sqrt{11}-2i\Im
m(\bar{\omega}_{11}\zeta)\right|=4\right\\},$ $\displaystyle
R^{(11)}_{3}R^{(11)}_{2}(\mathcal{S}_{0})$ $\displaystyle=$
$\displaystyle\left\\{(\zeta,t):\left|\left|\zeta+\bar{\omega}_{11}\right|^{2}+it-i\sqrt{11}-2i\Im
m(\omega_{11}\zeta)\right|=4\right\\},$ $\displaystyle
R^{(11)}_{1}R^{(11)}_{3}R^{(11)}_{2}(\mathcal{S}_{0})$ $\displaystyle=$
$\displaystyle\left\\{(\zeta,t):\left||\zeta-\bar{\omega}_{11}|^{2}+it-i\sqrt{11}+2i\Im
m(\omega_{11}\zeta)\right|=4\right\\},$ $\displaystyle
R^{(11)}_{1}{R^{(11)}_{3}}^{-1}R^{(11)}_{2}(\mathcal{S}_{0})$ $\displaystyle=$
$\displaystyle\left\\{(\zeta,t):\left||\zeta-\bar{\omega}_{11}|^{2}+it+i\sqrt{11}+2i\Im
m(\omega_{11}\zeta)\right|=4\right\\}.$
###### Definition 5.3.
Let $X$ be a closed polygonal chain (not necessarily in a plane), then a
topological disk defined by the cone over $X$ with apex $v$ is called a cone-
polygon, denoted by $\mathbb{D}_{v}(X)$.
Note that a polygon in traditional sense can be interpreted as a cone-polygon,
in that case, the boundary of cone-polygon and the apex lie in the same plane
and moreover the apex is in the interior of the boundary. We claim that the
prism $\mathbf{\Sigma}_{11}$ lies inside the union of $\mathcal{S}_{0}$ and
its images as above, see Figure 5.4 for viewing these spinal spheres.
Figure 5.4. (a) The shading view of neighboring spinal spheres containing the
fundamental domain for $(\Gamma_{11})_{\infty}$. (b) Another view for these
spinal spheres.
###### Proposition 5.4.
The prism $\mathbf{\Sigma}_{11}$ is contained in the union of the interiors of
the spinal spheres $\mathcal{S}_{0},$ $T^{(11)}(\mathcal{S}_{0}),$
$R^{(11)}_{2}(\mathcal{S}_{0}),$ ${R^{(11)}_{2}}^{-1}(\mathcal{S}_{0}),$
$R^{(11)}_{3}(\mathcal{S}_{0}),$ ${R^{(11)}_{3}}^{-1}(\mathcal{S}_{0}),$
$R^{(11)}_{1}{R^{(11)}_{3}}^{-1}R^{(11)}_{2}(\mathcal{S}_{0})$,
$R^{(11)}_{1}R^{(11)}_{3}R^{(11)}_{2}(\mathcal{S}_{0})$,
$R^{(11)}_{1}R^{(11)}_{2}(\mathcal{S}_{0})$ and
$R^{(11)}_{3}R^{(11)}_{2}(\mathcal{S}_{0})$.
###### Proof.
Using the same argument of Proposition 5.2, we want to decompose the prism
$\mathbf{\Sigma}_{11}$ into several polyhedral cells. The difference is the
complicated intersection of the spinal spheres, which leads that the prism is
difficultly decomposed into several polyhedral cells each of which is
contained in one spinal sphere. Observe that a union of interiors of several
spinal spheres is a star-convex set if they have a non-empty interior
intersection. We shall show that the collection of these spinal spheres can be
separated into several parts such that each part contains certain polyhedral
cell. All these polyhedral cells are defined by the star-disk as its boundary.
We first define a tetrahedron $\mathbb{T}$ with vertices
$v_{1}^{-},q_{1},q_{2},q_{3}$, where
$\begin{array}[]{l}q_{1}=\left(1/4-i\sqrt{11}/4,-2\sqrt{11}/3\right),\\\
q_{2}=\left(3/20-3i\sqrt{11}/20,-4\sqrt{11}/5\right),\\\
q_{3}=\left(7/20-3i\sqrt{11}/20,-9\sqrt{11}/10\right).\end{array}$
Observe that the points $q_{1},q_{2},q_{3}$ lie on the edges
$(v^{+}_{1},v^{-}_{1}),$ $(v^{-}_{1},v_{3}^{-})$ and $(v_{1}^{-},v_{2}^{-})$,
respectively. An easy calculation shows that this tetrahedron is contained
inside $R^{(11)}_{1}{R^{(11)}_{3}}^{-1}R^{(11)}_{2}(\mathcal{S}_{0})$ by Lemma
4.1.
Next, we define a hexahedron $\mathbb{H}_{1}$ with vertices
$q_{1},q_{2},q_{3},q_{4},q_{5},q_{6},q_{7},v^{+}_{0}$ and another hexahedron
$\mathbb{H}_{2}$ with vertices $v_{2}^{-},q_{5},q_{6},q_{7},q_{8},q_{9}$,
where
$\begin{array}[]{llll}q_{4}=\left(1/4-i\sqrt{11}/4,-1/2\right),&q_{5}=\left(0.42+0.26i,-0.71\sqrt{11}+0.39\right),\\\
q_{6}=\left(0.6+i\sqrt{11}/10,-0.65\sqrt{11}\right),&q_{7}=\left(0.58+2i\sqrt{11}/25,-1.92\right),\\\
q_{8}=\left(3/4+i\sqrt{11}/4,0\right),&q_{9}=(0.55+i\sqrt{11}/4,-2\sqrt{11}/5).\end{array}$
Observe that the points $q_{4},q_{6},q_{8},q_{9}$ lie on the edges
$(v^{+}_{1},v^{-}_{1}),$ $(v^{-}_{1},v_{2}^{-}),$ $(v_{2}^{+},v_{2}^{-})$ and
$(v^{-}_{2},v^{-}_{3})$, respectively. The points $q_{5}$ lies on the interior
of the base of the prism and $q_{7}$ lies on the interior of the face
$(v^{+}_{1},v^{-}_{1},v^{-}_{2},v^{+}_{2})$. Then we know the hexahedron
$\mathbb{H}_{1}$ has the faces $(q_{1},q_{2},q_{3}),$
$(q_{1},q_{3},q_{6},q_{7},q_{4}),$ $(q_{1},q_{2},v^{+}_{0},q_{4}),$
$(q_{4},q_{5},v^{-}_{0}),$ $(q_{4},q_{5},q_{7})$ and $(q_{5},q_{6},q_{7})$ and
the hexahedron $\mathbb{H}_{2}$ has the faces $(q_{5},q_{6},q_{7}),$
$(q_{5},q_{7},q_{8}),$ $(v^{-}_{2},q_{8},q_{9}),$ $(q_{5},q_{8},q_{9})$
$(q_{6},q_{7},q_{8},v^{-}_{2})$ and $(q_{5},q_{6},v^{-}_{2},q_{9})$. By
examining the location of these points and Lemma 4.1, we conclude that the
hexahedron $\mathbb{H}_{1}$ is contained inside
${R^{(11)}_{2}}^{-1}(\mathcal{S}_{0})$ and the hexahedron $\mathbb{H}_{2}$ is
lied inside ${R^{(11)}_{3}}^{-1}(\mathcal{S}_{0})$.
We focus on describing other polyhedral cells in the decomposition of the
prism $\mathbf{\Sigma}_{11}$. Let $\mathcal{U}_{1}$ denote the union of
$R^{(11)}_{2}(\mathcal{S}_{0}),$ $R^{(11)}_{1}R^{(11)}_{2}(\mathcal{S}_{0})$
and $R^{(11)}_{1}R^{(11)}_{3}R^{(11)}_{2}(\mathcal{S}_{0})$. We verify that
$q_{10}=(0.2-0.4i,2.4)$ is in the intersection of the interiors of these three
spinal spheres, which implies that $\mathcal{U}_{1}$ is a star-convex set
about $q_{11}$. Analogously, we know $\mathcal{U}_{2}$, denoted by the union
of $T^{(11)}(\mathcal{S}_{0}),$ $R^{(11)}_{3}(\mathcal{S}_{0}),$
$R^{(11)}_{1}R^{(11)}_{2}(\mathcal{S}_{0})$ and
$R^{(11)}_{3}R^{(11)}_{2}(\mathcal{S}_{0})$, is a star-convex set about
$q_{11}=(0.18+0.72i,4.8)$. This can be verified by examining the location of
$q_{12}$ which is in the intersection of the interiors of these four spinal
spheres. We need to add the following points on the faces of the prism
$\mathbf{\Sigma}_{11}$, each of which is in the intersection of the interiors
of at least two spinal spheres.
$\begin{array}[]{ll}q_{12}=(1/4-i\sqrt{11}/4,\sqrt{11}/2),&q_{13}=(0.21-0.21i\sqrt{11},\sqrt{11}/2),\\\
q_{14}=(0,\sqrt{11}/2),&q_{15}=(-0.21+0.21i\sqrt{11},\sqrt{11}/2),\\\
q_{16}=(i\sqrt{11}/4,1),&q_{17}=(3/4+i\sqrt{11}/4,1),\\\
q_{18}=(0.42-2i\sqrt{11}/25,1.95),&q_{19}=(3/4+i\sqrt{11}/4,\sqrt{11}),\\\
q_{20}=(0.6+i\sqrt{11}/10,27\sqrt{11}/20),&q_{21}=(0.42+0.26i,1.29\sqrt{11}+0.39),\\\
q_{22}=(-1.4+1.4i\sqrt{11},4\sqrt{11}/5),&q_{23}=(-1/4+i\sqrt{11}/4,\sqrt{11}/2).\end{array}$
Observe that the points $q_{12},q_{20},q_{23}$ lie on the edges
$(v^{+}_{1},v^{-}_{1}),$ $(v^{+}_{1},v^{+}_{2})$ and $(v^{+}_{3},v^{-}_{3})$
respectively and the points $q_{17},q_{19}$ lie on the edge
$(v^{+}_{2},v^{-}_{2})$. Moreover, the points $q_{13},$ $q_{14},$ $q_{15},$
$q_{22}$ lie on the interior of the face
$(v^{+}_{1},v^{-}_{1},v^{-}_{3},v^{+}_{3})$, the point $q_{16}$ lies on the
interior of the face $(v^{+}_{2},v^{-}_{2},v^{-}_{3},v^{+}_{3})$, the point
$q_{18}$ lies on the interior of the face
$(v^{+}_{1},v^{-}_{1},v^{-}_{2},v^{+}_{2})$ and the points $q_{21}$ lies on
the interior of the top $(v^{+}_{1},v^{+}_{2},v^{+}_{3})$. We need to add
other three points in the interior of the prism $\mathbf{\Sigma}_{11}$ which
are used to define the cone-polygon,
$\begin{array}[]{l}q_{24}=(-0.16+0.74i,1.4),\\\ q_{25}=(0.328-0.28i,1.99),\\\
q_{26}=(0.325+0.29i,4.652).\end{array}$
We verify the location of all these points as follows:
$\bullet$ The point $q_{12}$ is in the intersection of the interiors of
$R^{(11)}_{1}R^{(11)}_{3}R^{(11)}_{2}(\mathcal{S}_{0})$ and $\mathcal{S}_{0}$;
$\bullet$ The point $q_{13}$ is in the intersection of the interiors of
$\mathcal{S}_{0}$, $R^{(11)}_{1}R^{(11)}_{2}(\mathcal{S}_{0})$ and
$R^{(11)}_{1}R^{(11)}_{3}R^{(11)}_{2}(\mathcal{S}_{0})$;
$\bullet$ The point $q_{14}$ is in the intersection of the interiors of
$\mathcal{S}_{0}$, $R^{(11)}_{2}(\mathcal{S}_{0})$ and
$R^{(11)}_{1}R^{(11)}_{2}(\mathcal{S}_{0})$;
$\bullet$ The point $q_{15}$ is in the intersection of the interiors of
$\mathcal{S}_{0}$, $R^{(11)}_{2}(\mathcal{S}_{0})$,
$R^{(11)}_{3}(\mathcal{S}_{0})$ and
$R^{(11)}_{3}R^{(11)}_{2}(\mathcal{S}_{0})$;
$\bullet$ The points $q_{16},q_{19},q_{20}$ are in the intersection of the
interiors of $R^{(11)}_{2}(\mathcal{S}_{0})$ and
$R^{(11)}_{3}(\mathcal{S}_{0})$;
$\bullet$ The point $q_{17}$ is in the intersection of the interiors of
$\mathcal{S}_{0}$ and $R^{(11)}_{2}(\mathcal{S}_{0})$;
$\bullet$ The point $q_{18}$ is in the intersection of the interiors of
$\mathcal{S}_{0}$, $R^{(11)}_{2}(\mathcal{S}_{0})$ and
$R^{(11)}_{1}R^{(11)}_{3}R^{(11)}_{2}(\mathcal{S}_{0})$;
$\bullet$ The point $q_{21}$ is in the intersection of the interiors of
$T^{(11)}(\mathcal{S}_{0})$, $R^{(11)}_{2}(\mathcal{S}_{0})$ and
$R^{(11)}_{3}(\mathcal{S}_{0})$;
$\bullet$ The point $q_{22}$ is in the intersection of the interiors of
$R^{(11)}_{1}R^{(11)}_{2}(\mathcal{S}_{0})$, $R^{(11)}_{2}(\mathcal{S}_{0})$
and $R^{(11)}_{3}(\mathcal{S}_{0})$;
$\bullet$ The point $v^{+}_{0}$ is in the intersection of the interiors of
$R^{(11)}_{1}R^{(11)}_{2}(\mathcal{S}_{0})$, $T^{(11)}(\mathcal{S}_{0})$ and
$R^{(11)}_{2}(\mathcal{S}_{0})$;
$\bullet$ The point $q_{23}$ is in the intersection of the interiors of
$\mathcal{S}_{0}$, $R^{(11)}_{3}(\mathcal{S}_{0})$ and
$R^{(11)}_{3}R^{(11)}_{2}(\mathcal{S}_{0})$;
$\bullet$ The point $q_{24}$ is in the intersection of the interiors of
$\mathcal{S}_{0},$ $R^{(11)}_{2}(\mathcal{S}_{0}),$
$R^{(11)}_{3}(\mathcal{S}_{0})$ and
$R^{(11)}_{3}R^{(11)}_{2}(\mathcal{S}_{0})$;
$\bullet$ The point $q_{25}$ is in the intersection of the interiors of
$\mathcal{S}_{0},$ $R^{(11)}_{2}(\mathcal{S}_{0}),$
$R^{(11)}_{1}R^{(11)}_{2}(\mathcal{S}_{0})$ and
$R^{(11)}_{1}R^{(11)}_{3}R^{(11)}_{2}(\mathcal{S}_{0})$;
$\bullet$ The point $q_{26}$ is in the intersection of the interiors of
$T^{(11)}(\mathcal{S}_{0}),$ $R^{(11)}_{2}(\mathcal{S}_{0}),$
$R^{(11)}_{3}(\mathcal{S}_{0})$ and
$R^{(11)}_{1}R^{(11)}_{2}(\mathcal{S}_{0})$.
In term of these, we denote by $X_{1}$ a closed polygonal chain joining in
order with the points $p_{12},$ $p_{13},$ $p_{14},$ $p_{15},$ $p_{16},$
$p_{17},$ $p_{18}$ and denote by $X_{2}$ a closed polygonal chain joining in
order with the points $p_{16},$ $p_{19},$ $p_{20},$ $p_{21},$ $v^{+}_{0}$,
$p_{22},$ $p_{24}$. So then we can define two cone-polygons
$\mathbb{D}_{q_{25}}(X_{1})$ and $\mathbb{D}_{q_{26}}(X_{2})$. By examining
the locations of these points, we show that $\mathbb{D}_{q_{25}}(X_{1})$ is in
the intersection of the interiors of $\mathcal{S}_{0}$, $\mathcal{U}_{1}$ and
$\mathbb{D}_{q_{26}}(X_{2})$ is in the intersection of the interiors of
$R^{(11)}_{2}(\mathcal{S}_{0})$ and $T^{(11)}(\mathcal{S}_{0}),$
$R^{(11)}_{1}R^{(11)}_{2}(\mathcal{S}_{0}),$ $R^{(11)}_{3}(\mathcal{S}_{0})$,
namely, the intersection of the interiors of $\mathcal{U}_{1}$ and
$\mathcal{U}_{2}$. The remaining faces can be easily verified which are
contained inside $\mathcal{S}_{0},\mathcal{U}_{1}$ or $\mathcal{U}_{2}$ .
Finally, we define three polyhedral cells as follows:
(i) The polyhedral cell $\mathbb{P}_{1}$ is defined by the faces
$\mathbb{D}_{q_{25}}(X_{1}),$ $\mathbb{D}_{q_{26}}(X_{2}),$
$(v^{+}_{1},q_{12},q_{18},q_{17},q_{19},q_{20})$,
$(v_{1}^{+},q_{12},q_{13},q_{14},q_{15},q_{22},v^{+}_{0})$ and
$(v^{+}_{1},q_{20},q_{21},v^{+}_{0})$ as its boundary;
(ii) The polyhedral cell $\mathbb{P}_{2}$ is defined by the faces
$\mathbb{D}_{q_{26}}(X_{2}),$ $(q_{15},q_{23},q_{24}),$
$(v^{+}_{1},q_{20},q_{21},v^{+}_{0})$
$(v^{+}_{1},q_{12},q_{18},q_{17},q_{19},q_{20})$,
$(v_{1}^{+},q_{12},q_{13},q_{14},q_{15},q_{22},v^{+}_{0})$ and
$(q_{23},q_{16},q_{24})$ as its boundary;
(iii) The polyhedral cell $\mathbb{P}_{3}$ is defined by the faces
$\mathbb{D}_{q_{25}}(X_{1}),$ $(q_{4},q_{5},q_{7}),$ $(q_{5},q_{7},q_{8}),$
$(q_{4},q_{5},v^{-}_{0}),$ $(q_{15},q_{23},q_{24}),$
$(q_{8},q_{9},v^{-}_{3},q_{23},q_{16},q_{17})$, $(q_{23},q_{16},q_{24})$,
$(q_{4},q_{7},q_{8},q_{17},q_{18},q_{12})$,
$(v^{-}_{3},v^{-}_{0},q_{5},q_{9})$ and
$(q_{12},q_{13},q_{14},q_{15},q_{23},v^{-}_{3},v^{-}_{0},q_{4})$ as its
boundary;
By Lemma 4.1 and the properties of star-convex of $\mathcal{U}_{1}$ and
$\mathcal{U}_{2}$, we conclude that the polyhedral cell $\mathbb{P}_{1}$
contained inside $\mathcal{U}_{1}$; the polyhedral cell $\mathbb{P}_{2}$
contained inside $\mathcal{U}_{2}$; the polyhedral cell $\mathbb{P}_{3}$ is
contained inside $\mathcal{S}_{0}$. This completes the proof. ∎
## References
* [1] P. M. Cohn, A presentation for $SL_{2}$ for Euclidean quadratic imaginary number fields, _Mathematika_. 15(1968), 156¨C163.
* [2] M. Deraux, E. Falbel and J. Paupert. New constructions of fundamental polyhedra in complex hyperbolic space. _Acta Math._ 194(2005), no. 2, 155-201.
* [3] B. Fine, Algebraic theory of the Bianchi groups, _Marcel Dekker Inc._ (1989).
* [4] E. Falbel, G. Francsics, P. Lax and J. R. Parker. Generators of a Picard modular group in two complex dimensions. _to appear in proc. AMS._
* [5] E. Falbel and J. R. Parker. The Geometry of Eisenstein-Picard Modular Group. _Duke Math. J._ 131(2006), no. 2, 249-289.
* [6] E. Falbel, G. Francsics and J. R. Parker. The geometry of the Gauss-picard modular group. _Math Annalen_. (pubilished online: 04 May 2010).
* [7] G. Francsics and P. Lax. A semi-explicit fundamental domain for the Picard Modular Group in complex hyperbolic space. _Contemporary Mathematics._ 368(2005), 211-226.
* [8] G. Francsics and P. Lax. An explicit fundamental domain for the Picard Modular Group in two complex dimensions. (Preprint 2005).
* [9] W. M. Goldman. _Complex Hyperbolic Geometry._ (Oxford Mathematical Monographs, Oxford University Press 1999).
* [10] W. M. Goldman and J. R. Parker, Complex hyperbolic ideal triangle groups, _J. reine angewandte Math._ 425(1992), 71-86.
* [11] H. Garland and M. S. Raghunathan. Fundamental domains for lattices in ($\mathbf{R}$-)rank 1 semisimple Lie groups. _Ann. of Math._ 92(1970), no.2, 279-326..
* [12] G. D. Mostow. On a remarkable class of polyhedra in complex hyperbolic space. _Pacific J. Mathematics_. 86(1980), 171-276.
* [13] J .R. Parker. Complex hyperbolic lattices. _Contemporary Mathematics._ 501(2009), 1-42.
* [14] G. P. Scott. The geometries of 3-manifolds. _Bull. London Math. Soc._ 15(1983), 401-487.
* [15] R. E. Schwartz. Complex hyperbolic triangle groups. _Proc. of the intern. Congress of Mathematicians_ : Invited Lectures 1(2002), 339-350.
* [16] I. N. Stewart and D. O. Tall, Algebraic number theory._Chapman and Hall Ltd._ (1979).
* [17] R.G. Swan, Generators and relations for certain special linear groups, _Adv. in Math._ 6(1971), 1-77.
* [18] T. Zink. Üer die Anzahl der Spitzen einiger arithmetischer Untergruppen unitäer Gruppen. _Math. Nachr._ 89 (1979), 315-320.
* [19] T. Zhao. A minimal volume arithmetic cusped complex hyperbolic orbifold. (Preprint).
|
arxiv-papers
| 2010-06-16T21:40:42 |
2024-09-04T02:49:10.957094
|
{
"license": "Public Domain",
"authors": "Tiehong Zhao",
"submitter": "Tiehong Zhao",
"url": "https://arxiv.org/abs/1006.3331"
}
|
1006.3464
|
[labelstyle=]
# Grothendieck rings of universal quantum groups
Alexandru Chirvăsitu University of California, Berkeley, 970 Evans Hall #3480,
Berkeley CA, 94720-3840, USA chirvasitua@gmail.com
###### Abstract.
We determine the Grothendieck ring of finite-dimensional comodules for the
free Hopf algebra on a matrix coalgebra, and similarly for the free Hopf
algebra with bijective antipode and other related universal quantum groups.
The results turn out to be parallel to those for Wang and Van Daele’s deformed
universal compact quantum groups and Bichon’s generalization of those results
to universal cosovereign Hopf algebras: in all cases the rings are isomorphic
to those of non-commutative polynomials over certain sets, these sets varying
from case to case. In most cases we are able to give more precise information
about the multiplication table of the Grothendieck ring.
###### Key words and phrases:
free Hopf algebra, cosovereign Hopf algebra, matrix coalgebra, Grothendieck
ring of comodules, corepresentation
###### 2010 Mathematics Subject Classification:
16T05, 16T15, 16T20, 20G42
## Introduction
The representation theory of quantum groups has played an important role in
mathematics during the past several decades. Several approaches can be
identified, which yield interesting different, but often related families of
Hopf algebras. One has, for example, Drinfeld and Jimbo’s deformed universal
enveloping algebras ([Dr1, Dr2, Ji]), the compact matrix groups of Woronowicz
([Wo1, Wo2]), or various “quantum automorphism groups”, such as those of Manin
([Ma]), the quantum group of a bilinear form ([DVL]), that of a measured
algebra ([Bi1]), etc.
The “universal quantum groups” in the title are Hopf algebras which enjoy
certain universality properties; they are described in more detail below. We
are interested in their finite-dimensional comodules, so they are to be
regarded as quantum groups of the “function algebra” flavor.
One class of Hopf algebras which will be relevant to our discussion and will
provide the motivation for what follows is that of universal or free
cosovereign Hopf algebras. These were introduced by Bichon in [Bi2], and are
defined essentially as follows: given an invertible $n\times n$ matrix $F$,
the universal cosovereign Hopf algebra $H(F)$ is the free Hopf algebra
generated by an $n\times n$ matrix coalgebra $u=(u_{ij})$ with the provision
that the squared antipode acts on $u$ as conjugation by $F$ (see [Bi2] for
more details).
The main objects of study here are the following:
(1) The free Hopf algebra $H(n)$ on the matrix coalgebra $M_{n}(k)^{*}$ (for
some field $k$ and $n\geq 2$). It was shown in [Ta] that the forgetful functor
from Hopf algebras to coalgebras (always over some fixed base field $k$) has a
left adjoint. $H(n)$ is precisely the image of the matrix coalgebra
$M_{n}(k)*$ through this adjoint.
(2) $H_{\infty}(n)$, the free Hopf algebra with bijective antipode on the same
matrix coalgebra $M_{n}(k)^{*}$. As in (1), it is shown in [Sc] that the
forgetful functor from Hopf algebras with bijective antipode to that of
coalgebras has a left adjoint. Just as before, $H_{\infty}(n)$ denotes here
the image of the matrix coalgebra through that adjoint.
(3) We introduce an object denoted by $H_{d}(F)$. Here $d$ is a positive
integer, while $F$ is an invertible $n\times n$ matrix over $k$. With this
data, $H_{d}(F)$ is the free Hopf algebra generated by a matrix coalgebra
$u=(u_{ij})$ such that the $2d$’th power of the antipode acts on $u$ as
conjugation by $F$. We chose to consider these objects because they generalize
at the same time the universal cosovereign Hopf algebras discussed above
($H(F)$ from [Bi2] would be $H_{1}(F)$ here), and the free Hopf algebra with
antipode of order $2d$ on a matrix coalgebra, used in [Ch]
($H_{d}(M_{n}(k)^{*})$ from that paper is $H_{d}(I_{n})$ here, where $I_{n}\in
M_{n}(k)$ is the identity matrix).
Finally, we reserve the notation $\tilde{H}$ or $\tilde{H}(n)$ as a
placeholder for any of the above; the $n$ indicates that we are considering
either $H(n)$, or $H_{\infty}(n)$, or $H_{d}(F)$ for some $n\times n$ matrix
$F\in GL(n,k)$. We will be concerned primarily with determining the
Grothendieck rings of finite-dimensional comodules for the various
$\tilde{H}(n)$’s.
It turns out that when the base field is $\mathbb{C}$ and the matrix $F$ used
in the definition of $H(F)$ is positive definite, the $H(F)$ are precisely the
CQG algebras (in the sense of [DK], for example) associated to Wang and Van
Daele’s compact quantum matrix groups $A_{u}(Q)$ ([VDW]). The
corepresentations of the latter were determined by Bănică in [Ba], and the
results were later generalized by Bichon ([Bi4]) to include all cosemisimple
$H(F)$’s in characteristic zero. The corepresentations of $A_{u}(Q)$ (and by
extension those of $H(F)$) are of interest because collectively, the
$A_{u}(Q)$ play the role of the unitary group $U(n)$ (see [Ba]). We will
recall the relevant results in the next section.
This discussion provides part of the motivation for our problem: the
combinatorics of the multiplication table for the Grothendieck rings under
consideration turns out to mimic the results obtained in [Ba] and [Bi4] quite
closely, and seems interesting in its own right. Essentially, our results say
that at least for $\tilde{H}(n)$ excluding $H_{1}(F)$, the Grothendieck ring
is “as free of relations” as one can expect (see the next section for precise
statements).
Further motivation comes from the desire to obtain more information on the
free Hopf algebras $H(n)$ (and their relatives). Ever since the introduction
of $H(n)$ (and in fact of the free Hopf algebra on any coalgebra) by Takeuchi
in [Ta], where they were used to give the first examples of Hopf algebras with
non-bijective antipode, they have appeared in several other papers, also as
the basis for counterexamples: in [Ni], Nichols constructs a basis for $H(n)$,
proves that its antipode is injective, and then constructs a quotient
bialgebra of $H(2)$ which is not a Hopf algebra. In a similar vein, in [Sc],
Schauenburg introduces $H_{\infty}(n)$ and constructs a quotient Hopf algebra
of $H_{\infty}(4)$ whose antipode is not injective, thus giving the first
example of a non-injective surjective antipode. In view of their universal
properties, objects such as $H(n)$ and $H_{\infty}(n)$ are well-suited to be
starting points for the construction of counterexamples (as seen above), so it
seems worthwhile to gather more information about their structure.
The paper is organized as follows:
In Section 1 we set up the notations, introduce some preliminary results
needed later on, and state our main theorems.
In Section 2 Bergman’s diamond lemma ([Be]) is used to find bases for the
objects of interest $\tilde{H}(n)$, $n\geq 2$. These bases are somewhat
different from those which have appeared in the literature ([Ni, Sc]), and
will prove more convenient for our goals.
In Section 3 we prove that the Grothendieck rings of finite-dimensional
comodules of the Hopf algebras $\tilde{H}$ are non-commutative polynomial
rings.
Section 4 contains the main results of this paper, determining the
multiplication table of the Grothendieck (semi)ring of $\tilde{H}$ for all
cases except for the $H_{1}(F)$’s, and recovering the known results on the
latter assuming cosemisimplicity.
## 1\. Preliminaries
We begin by introducing the main conventions and some of the notation, and
recalling some generalities on the Hopf algebras alluded to in the previous
section.
We will be working over a fixed base field $k$, which will henceforth be
assumed to be algebraically closed. This assumption will simplify things by
ensuring, for example, that all simple coalgebras are actually matrix
coalgebras. Here, by matrix coalgebra we mean the dual $M_{n}(k)^{*}$ of the
usual algebra $M_{n}(k)$ of $n\times n$ matrices over $k$. $M_{n}(k)^{*}$ has
a basis $\displaystyle(x_{ij})_{i,j=1}^{n}$ with the coalgebra structure being
defined by
$\Delta(x_{ij})=\sum_{k=1}^{n}x_{ik}\otimes x_{kj},\
\varepsilon(x_{ij})=\delta_{ij},$ (1.1)
where $\Delta,\varepsilon$ stand, as usual, for the comultiplication and
counit respectively, and $\delta_{ij}$ is the Kronecker symbol. The
terminology “matrix coalgebra” always refers to $M_{n}(k)^{*}$ in this paper.
A collection of not necessarily linearly independent elements $x_{ij}$ in a
coalgebra (bialgebra, Hopf algebra) satisfying (1.1) will be referred to as a
multiplicative matrix (following [Ma]). Note that the linear span of a
multiplicative matrix is a coalgebra.
We assume familiarity with Hopf algebra theory as appearing, for example, in
[Sw, A, Mo]. We also use the standard notations: $\Delta,\varepsilon,S$ for
comultiplication, counit and antipode respectively. The words ‘comodule’ and
‘corepresentation’ are used interchangeably, and unless specified otherwise,
all comodules are right and finite-dimensional.
For a Hopf algebra $H$, $\mathcal{M}^{H}$ denotes the category of (finite-
dimensional, right) $H$-comodules. The Grothendieck ring of such comodules
will be denoted by $K(H)$. Sometimes, when there is no danger of confusion, we
might denote a comodule and its representative in the Grothendieck ring by the
same symbol. As the category of comodules is left rigid, we have an anti-
endomorphism $*$ on $K(H)$, sending the representative of a comodule to the
representative of its (left) dual. We might denote the map either by $u\mapsto
u^{*}$ or by $u\mapsto*(u)$. The trivial $H$-comodule will be denoted by $1$;
it is the multiplicative identity of the ring $K(H)$.
In fact, we will also be concerned with the Grothendieck semiring $K_{+}(H)$,
by which we mean the sub-semiring of $K(H)$ generated by the representatives
of the comodules. $K_{+}(H)$ is, of course, invariant under $*$. It is well
known that $K(H)$ has a basis (as an abelian group) formed by the set
$\mathcal{S}=\mathcal{S}(H)$ of (isomorphism classes of) simple comodules.
There is a natural order on $K$, for which $K_{+}$ is the positive cone. With
this order, $K(H)$ is also a lattice; $\vee$ will denote the supremum
operation on this lattice.
Note that there is a bijection between $\mathcal{S}(H)$ and the set of matrix
subcoalgebras of $H$, the simple comodule $M$ corresponding to the smallest
subcoalgebra $C$ such that the comodule structure map of $M$ factors as
$\rho:M\to M\otimes C\to M\otimes H$
(the last map being induced by the inclusion $C\to H$). $C$ is precisely the
linear span of the $x_{ij}$, which are uniquely determined by
$\rho(e_{j})=\sum_{i=1}^{n}e_{i}\otimes x_{ij},\ j=\overline{1,n}.$
More generally, the same construction for an $n$-dimensional (not necessarily
simple) comodule $M$ yields an $n\times n$ multiplicative matrix in $H$ as
soon as we fix a basis $(e_{i})_{i=1}^{n}$ for $M$. In this context, we write
$C$ as $C(M)$ and refer to $C$ as the coalgebra corresponding to the comodule
$M$.
The Hopf algebras of interest have already been introduced in the preceding
section: they are $H(n)$, the free Hopf algebra on an $n\times n$ matrix
coalgebra, $H_{\infty}(n)$, the free Hopf algebra with bijective antipode on
an $n\times n$ matrix coalgebra, and $H_{d}(F)$, where $d$ is a positive
integer and $F\in GL(n,k)$ is an invertible $n\times n$ matrix. $n\geq 2$ will
always be assumed, and as stated in the introduction, we use $\tilde{H}$ (or
$\tilde{H}(n)$ if we want to be more precise) as a generic symbol for any of
these Hopf algebras.
Recall ([Ta, Ni]) that $\tilde{H}=H(n)$ is defined as follows: one has a
multiplicative matrix $\displaystyle X^{r}=(x^{r}_{ij})_{i,j}$ for each non-
negative integer $r$, satisfying the relations
$\sum_{k=1}^{n}x^{r}_{ik}x^{r+1}_{jk}=\delta_{ij}=\sum_{k=1}^{n}x^{r+1}_{ki}x^{r}_{kj},\
\forall i,j,r.$ (1.2)
In other words, the transpose $\displaystyle\left(X^{r+1}\right)^{t}$ is the
inverse (in $M_{n}\left(\tilde{H}\right)$) of $X^{r}$. The antipode sends
$X^{r}$ to this transpose, i.e. acts by $S(x^{r}_{ij})=x^{r+1}_{ji}$. An
entirely analogous presentation can be given for $H_{\infty}(n)$, except that
this time, $r$ runs through the integers instead of the non-negative integers
(see [Sc]).
As for $\tilde{H}=H_{d}(F)$, we again have multiplicative matrices $X^{r}$ as
above, but this time $r$ runs through $\mathbb{Z}/2d$, the integers modulo
$2d$, and the relations (1.2) hold as stated for $r=\overline{0,2d-2}$. For
$r=2d-1$ we have instead (in compressed form, using the matrices $X$)
$(X^{2d-1})^{-1}=F(X^{0})^{t}F^{-1}.$ (1.3)
That is, instead of making the transpose $(X^{0})^{t}$ the inverse of
$X^{2d}$, we “twist” by $F$.
Notice that all the $\tilde{H}(n)$ have a distinguished $n$-dimensional
corepresentation, corresponding to the multiplicative matrix $X^{0}$: it is a
vector space with basis $e_{i},\ i=\overline{1,n}$ on which $\tilde{H}$ acts
by
$e_{j}\mapsto\sum_{i=1}^{n}e_{i}\otimes x^{0}_{ij}.$
We refer to this as the fundamental corepresentation of $\tilde{H}$, and we
will usually denote its representative in $K_{+}(\tilde{H})$ by $f$.
Finally, whenever we discuss one of the Hopf algebras $\tilde{H}$,
$R=R(\tilde{H})$ stands for the set over which the $r$ in the notation $X^{r}$
used above range: $R=\mathbb{N}$, the set of non-negative integers for
$\tilde{H}=H(n)$, $R=\mathbb{Z}$ for $\tilde{H}=H_{\infty}(n)$, and
$R=\mathbb{Z}/2d$ when $\tilde{H}=H_{d}(F)$.
We can now state the theorems proven in the paper. First, we explain the
weaker results, but which hold in greater generality, to be proven in Section
3.
Suppose we are working with $\tilde{H}$. Consider the free monoid $A_{R}$ on
$R$, with generators $\alpha_{r},\ r\in R$, and endow it with the unique anti-
endomorphism $*$ sending $\alpha_{r}$ to $\alpha_{r+1}$ for all $r\in R$. We
will refer to the elements of $A_{R}$ as words in the $\alpha_{r}$’s, as
usual, and for convenience, $\alpha_{r}$ and $r$ might be identified when
there is no danger of confusion. We have a partial order on $A_{R}$, given by
the length of the words.
There is a unique monoid map $\phi:A_{R}\to K=K(\tilde{H})$ which intertwines
the anti- endomorphisms $*$ and sends $\alpha_{0}$ (for $0\in R$) to the
fundamental corepresentation $f$. Now write
$\phi(x)=\sum_{s\in\mathcal{S}^{\prime}}n_{s}s+\sum_{s\in\mathcal{S}^{\prime\prime}}n_{s}s,$
(1.4)
where $n_{s}$ are positive integers, and $\mathcal{S}^{\prime\prime}$ is the
set of those $s$ which appear in a similar expansion for $\phi(y)$, $y<x$
(i.e. $y\in A_{R}$ is shorter than $x$). Denote the first sum in the right
hand side of (1.4) by $u_{x}$. Our first theorem is then the following:
###### Theorem 1.1.
With $\tilde{H}$ as above, the map $x\mapsto u_{x}$ induces a bijection
between $A_{R}$ and $\mathcal{S}(\tilde{H})$.
In other words, the simple comodules of $\tilde{H}$ can be labeled in a very
natural manner by the elements of the free monoid $A_{R}$. We will also see in
Section 3 that this easily implies the following:
###### Corollary 1.2.
The Grothendieck ring $K(\tilde{H})$ is isomorphic to the free unital algebra
$\mathbb{Z}[A_{R}]$ on $R$.
###### Remark 1.3.
The corollary implies that $K(H(n))$ is isomorphic to $K(H_{\infty}(m))$, of
course ($m,n\geq 2$), since in these two cases we have $R=\mathbb{N}$ and
$R=\mathbb{Z}$. However, the isomorphism appearing in the proof of the
corollary will make specific use of these sets $R$, and not just of their
cardinality.
Section 4 is concerned with a stronger version of Theorem 1.1, but which does
not hold for all $\tilde{H}$. In order to state it, we need to introduce more
notations.
Let $x\in A_{R}$. We keep the notation introduced before the statement of
Theorem 1.1. Write
$x=r_{1}r_{2}\ldots r_{n},$
where each $r_{i}$ is one of the letters $\alpha_{r}$, $r\in R$. Denote by
$I(x)$ the set of those $i\in\overline{1,n-1}$ for which $r_{i}r_{i+1}$ is
either of the form $\alpha_{r}\alpha_{r+1}$ or $\alpha_{r+1}\alpha_{r}$. For
each $i\in I(x)$, denote
$x_{i}=r_{1}r_{2}\ldots r_{i-1}r_{i+2}\ldots r_{n}.$
$\phi$ sends $\alpha_{r}\alpha_{r+1}$ and $\alpha_{r+1}\alpha_{r}$ to modules
of the form $uu^{*}$ and respectively $u^{*}u$ for $u\in K(\tilde{H})$, and
both of these are $\geq 1$ in $K(\tilde{H})$. In conclusion, we get
$1\leq\phi(r_{i}r_{i+1})$, and hence $\phi(x_{i})\leq\phi(x)$ for every $i\in
I(x)$. Denote
$u^{\prime}_{x}=\phi(x)-\bigvee_{i\in I(x)}\phi(x_{i}).$
It’s clear that $u^{\prime}_{x}\geq u_{x}$. Our result is the following:
###### Theorem 1.4.
(a) Suppose $\tilde{H}$ is not of the form $H_{1}(F)$. Then, with the
notations used above, we have $u^{\prime}_{x}=u_{x}$ for every $x\in A_{R}$,
and hence $x\mapsto u^{\prime}_{x}$ is a bijection between $A_{R}$ and
$\mathcal{S}(\tilde{H})$.
(b) For $\tilde{H}=H_{1}(F)$, the statement in (a) is true if and only if
$\tilde{H}$ is cosemisimple.
We now take a moment to recall the situation in the literature for the free
cosovereign Hopf algebras $H_{1}(F)$, and make the connection between those
results and the theorems stated above.
In [Ba] the free monoid $A$ on two generators $\alpha,\beta$ is considered,
with the involution $*$ used above in the more general situation; here, this
involution simply interchanges $\alpha$ and $\beta$. Bănică then introduces a
new product $\odot$ on the monoid ring $\mathbb{Z}[A]$:
$x\odot y=\sum_{x=ag,y=g^{*}b}ab,\ x,y\in A.$ (1.5)
It is shown that this is indeed an associative product, and moreover,
$(\mathbb{Z}[A],\odot)$ is again the free ring generated by $\alpha,\beta$.
The results in [Bi4] which are relevant here can be rephrased and summarized
as follows ([Bi4, Theorem 1.1,(iii)]):
###### Theorem 1.5.
Assume $k$ has characteristic zero and $\tilde{H}=H_{1}(F)$ is cosemisimple.
Then, the map $(\mathbb{Z}[A],\odot)\to K(\tilde{H})$ defined by sending
$\alpha$ and $\beta$ to $f$ and $f^{*}$ respectively is an isomorphism of
rings with involution, and induces a bijection of $A$ with the set of
isomorphism classes of irreducible corepresentations.
Note that this generalizes [Ba, Théorème 1 (i)], and so includes the
corepresentation theory of Wang and Van Daele’s universal compact quantum
groups mentioned in the introduction. Bichon actually determines exactly when
a universal cosovereign Hopf algebra is cosemisimple in characteristic zero,
but we do not make use of that result here.
It is not difficult to see that part (b) of Theorem 1.4 (in characteristic
zero) is, in fact, another way of stating Theorem 1.5.
For $\tilde{H}=H(n)$, Theorem 1.4 says, essentially, that the Grothendieck
ring $K(\tilde{H})$ is generated as a ring with anti-endomorphism by the
fundamental corepresentation $f$, and the relations satisfied by the
generators $f,f^{*},f^{**}$, etc. are precisely those imposed by the fact that
$\mathcal{M}^{H}$ is a left rigid monoidal category, and nothing more. In
other words, $K(\tilde{H})$ is “as free as possible” on the dual iterates
$f,f^{*},f^{**}$, etc. of $f$. We refer to this situation as “maximal
freeness”, hence the title of Section 4.
The meaning of Theorem 1.4 for $\tilde{H}=H_{\infty}(n)$ or
$\tilde{H}=H_{d}(F)$ is similar: in the first case $K(\tilde{H})$ is maximally
free on the iterates $*^{r}(f)$, $r\in R=\mathbb{Z}$ under the constraints
that $\mathcal{M}^{H}$ be a rigid (both left and right) monoidal category,
while for $\tilde{H}=H_{d}(F)$, in the good cases (i.e. when either $d>1$ or
$d=1$ and $H_{1}(F)$ is cosemisimple), $K$ is maximally free on the dual
iterates of $f$ under the constraint that $\mathcal{M}^{H}$ be a rigid
monoidal category for which the $2d$’th power of the dual is naturally
isomorphic to the identity functor.
## 2\. Putting the diamond lemma to good use
As announced in the introduction, in this section we will look at the Hopf
algebras $\tilde{H}$ in more detail, and bases over $k$ will be constructed
for them using Bergman’s diamond lemma. We use the results and language in
[Be] freely, and refer to that paper for the necessary background and
terminology.
Typically, we won’t go through the actual verification of the fact that the
ambiguities we get ([Be]) are resolvable. Instead, for the more formidable
ambiguities, we give an argument which simplifies the situation considerably
and makes the verification itelf more or less trivial.
A basis for $H(n)$ was constructed by Nichols in [Ni], and the technique was
adapted to $H_{\infty}(n)$ in [Sc]. We stated in [Ch] that an analogous
approach works for what here would be called $H_{d}(I_{n})$. Because the
result will be different here, we recall only that the bases used in these
papers consisted of all words in the generators $x^{r}_{ij}$ (introduced in
the previous section) which contain no subwords of either one of the forms
$x^{r}_{in}x^{r+1}_{jn},\quad x^{r+1}_{ni}x^{r}_{nj},\quad
x^{r}_{in}x^{r+1}_{jn-1}x^{r+2}_{kn-1},\quad
x^{r+2}_{ni}x^{r+1}_{n-1j}x^{r}_{n-1k},$
for $r$ ranging through $R=R(\tilde{H})$.
Let us now look at $\tilde{H}=H(n)$, $H_{\infty}(n)$, or $H_{d}(F)$, with
$F\in GL(n,k)$. The following notation will be useful: bold symbols such as
${\bf r}=(r_{1},\ldots,r_{k})$ and ${\bf i}=(i_{1},\ldots,i_{k})$ denote
vectors of elements $r_{j}\in R$ and $i_{j}\in\overline{1,n}$ respectively.
The length of the vector ${\bf r}$ will be denoted by $|{\bf r}|$. $x^{\bf
r}_{\bf ij}$ denotes the product $x^{r_{1}}_{i_{1}j_{1}}\ldots
x^{r_{k}}_{i_{k}j_{k}}$; $x^{\bf r}_{\bf ij}$ will also occasionally be
referred to as a monomial of type ${\bf r}$.
In order to apply the diamond lemma, we need a collection of reductions, and a
semigroup partial order on the monoid $\langle\mathcal{X}\rangle$ freely
generated by the set $\mathcal{X}$ of symbols $x^{r}_{ij}$, $r\in R$ and
$i,j\in\overline{1,n}$. We take care of the ordering later; the reductions are
as follows:
$x^{r}_{in}x^{r+1}_{jn}\to\delta_{ij}-\sum_{a<n}x^{r}_{ia}x^{r+1}_{ja},\quad\mbox{r
even}$ (2.1)
$x^{r}_{i1}x^{r+1}_{j1}\to\delta_{ij}-\sum_{a>1}x^{r}_{ia}x^{r+1}_{ja},\quad\mbox{r
odd}$ (2.2)
$x^{r+1}_{ni}x^{r}_{nj}\to\delta_{ij}-\sum_{a<n}x^{r+1}_{ai}x^{r}_{aj},\quad\mbox{r
odd}$ (2.3)
$x^{r+1}_{1i}x^{r}_{1j}\to\delta_{ij}-\sum_{a>1}x^{r+1}_{ai}x^{r}_{aj},\quad\mbox{r
even}$ (2.4)
Here $\delta_{ij}$ is the Kronecker delta, and since $R$ is one of the sets
$\mathbb{N}$, $\mathbb{Z}$ or $\mathbb{Z}/2d$, it makes sense to talk about
even and odd elements $r\in R$.
These reductions, with $r$ ranging through the whole set $R$, account for all
the relations defining the algebras $H(n)$ and $H_{\infty}(n)$ (and even
$H_{d}(I_{n})$). So by the diamond lemma, in order to conclude that the
monomials which contain no subwords as in the left hand sides of (2.1) - (2.4)
form a basis in these cases, it suffices to prove (once the semigroup partial
order with the descending chain condition and compatible with the reductions
has been found) that all resulting overlap and inclusion ambiguities are
resolvable.
The advantage of this choice of reductions over those in [Ni, Sc, Ch] is the
fact that now there is essentially only one ambiguity to resolve
(“essentially” meaning up to interchanging $1$ and $n$, a translation of $R$,
etc.). This essentially unique (overlap) ambiguity is
$x^{r}_{in}x^{r+1}_{1n}x^{r}_{1j}$ for even $r$, and one sees easily that it
is indeed resolvable. Hence, we now have a basis for $H(n)$ and
$H_{\infty}(n)$.
In order to treat $H=H_{d}(F)$, the arbitrary invertible matrix $F$ must be
brought into the picture. Recall ((1.2)) that as an algebra, $H$ is generated
by the elements $x^{r}_{ij}$ for $r\in\mathbb{Z}/2d=\overline{0,2d-1}$, and
$i,j\in\overline{1,n}$, subject to the relations
$(X^{r+1})^{t}=(X^{r})^{-1},\ \forall r\in\overline{0,2d-2},$
$F(X^{0})^{t}F^{-1}=(X^{2d-1})^{-1}.$
Here, $X^{r}$ is the matrix $(x^{r}_{ij})_{i,j}\in M_{n}(H)$, and the
superscript t denotes the transpose of an $n\times n$ matrix.
To get reductions which account for all of this, we first make the observation
that it suffices to consider the case when $F$ is upper triangular. More
precisely, we have an isomorphism $H_{d}(F)\cong H_{d}(PFP^{-1})$ for any
$P\in GL(n,k)$, and any matrix can be made upper triangular by conjugation
(the field is algebraically closed!).
The claim about the isomorphism is proven in [Bi2] for $d=1$, i.e. for the
free cosovereign Hopf algebras. It suffices to send $X^{0}$ from
$H_{d}(PFP^{-1})$ to $(P^{t})^{-1}X^{0}P^{t}$ from $H_{d}(F)$, and this is
easily seen to extend to a Hopf algebra isomorphism for the Hopf algebra
structures described in the previous section. Hence, from now on, whenever
$H_{d}(F)$ comes up, we assume that $F$ is upper triangular. With this
assumption in place, we keep the reductions (2.1) - (2.4) for
$r=\overline{0,2d-2}$, and add the two reductions
$x^{2d-1}_{i1}x^{0}_{j1}\to
F_{11}^{-1}F_{jj}\left(\delta_{ij}-\sum_{(l,p,u)\neq(1,1,j)}F_{lp}(F^{-1})_{uj}x^{2d-1}_{il}x^{0}_{up}\right)$
(2.5)
$x^{0}_{ni}x^{2d-1}_{nj}\to
F_{ii}^{-1}F_{nn}\left(\delta_{ij}-\sum_{(p,u,l)\neq(i,n,n)}x^{0}_{up}x^{2d-1}_{lj}\right)$
(2.6)
We have postponed tackling the issue of the semigroup partial order on
$\langle\mathcal{X}\rangle$ until now because we would like to find such an
order which is compatible with all of our reductions (2.1) - (2.6) at once (in
addition to having the descending chain condition). For our purposes, the
following works.
First, words in the $x^{r}_{ij}$ are ordered according to their length (that
is, shorter words are smaller). Then, among words of the same length, we only
compare pairs of the form $x^{\bf r}_{\bf ij}$, $x^{\bf r}_{\bf
i^{\prime}j^{\prime}}$ (i.e. with the same vector ${\bf r}$). So consider such
a pair, say
$x^{\bf r}_{\bf ij}=x^{r_{1}}_{i_{1}j_{1}}\ldots x^{r_{k}}_{i_{k}j_{k}},\quad
x^{\bf r}_{\bf
i^{\prime}j^{\prime}}=x^{r_{1}}_{i^{\prime}_{1}j^{\prime}_{1}}\ldots
x^{r_{k}}_{i^{\prime}_{k}j^{\prime}_{k}}.$
Let $\ell$ be the smallest index for which the pairs $(i_{\ell},j_{\ell})$ and
$(i^{\prime}_{\ell},j^{\prime}_{\ell})$ are different. Then, the order between
our monomials $x^{\bf r}_{\bf ij}$ and $x^{\bf r}_{\bf i^{\prime}j^{\prime}}$
is the same as the order between the two-term monomials $x^{\bf s}_{\bf uv}$
and $x^{\bf s}_{\bf u^{\prime}v^{\prime}}$ respectively, where
${\bf s}=(r_{\ell},r_{\ell+1}),$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad{\bf u}$
$\displaystyle=(i_{\ell},i_{\ell+1}),$ $\displaystyle{\bf v}$
$\displaystyle=(j_{\ell},j_{\ell+1}),\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad{\bf u^{\prime}}$
$\displaystyle=(i^{\prime}_{\ell},i^{\prime}_{\ell+1}),$ $\displaystyle{\bf
v^{\prime}}$
$\displaystyle=(j^{\prime}_{\ell},j^{\prime}_{\ell+1}).\qquad\qquad\qquad\qquad\qquad\qquad$
The order is undefined if $\ell=k$, i.e. the monomials are incomparable in our
partial order in this case.
The above is clearly a semigroup partial order for any partial order
whatsoever on the two-term monomials, so it suffices to describe that. We
simply make the two-term monomials on the left hand side of each of (2.1) -
(2.6) greater than any two-term monomial in the right hand side of the same
reduction; it is not difficult to see that this can be extended to a partial
order on the two-term monomials.
For example, if ${\bf r}=(r,r\pm 1)$ and $r$ is even, then the order can be
defined as follows:
$x^{\bf r}_{\bf ij}>x^{\bf r}_{\bf
i^{\prime}j^{\prime}}\quad\mbox{if}\quad{\bf i}=(n,n)\neq{\bf i^{\prime}},$
$x^{\bf r}_{\bf ij}>x^{\bf r}_{\bf
i^{\prime}j^{\prime}}\quad\mbox{if}\quad{\bf i}=(n,n)={\bf
i^{\prime}},\quad{\bf j^{\prime}}\neq(n,n),\quad{\bf ij}<{\bf
i^{\prime}j^{\prime}}\ \mbox{lexicographically},$ $x^{r}_{in}x^{r\pm
1}_{jn}>x^{r}_{ia}x^{r\pm 1}_{ja},\ \forall a<n,\ i,\ j.$
Here, ${\bf ij}$ is simply the concatenation of the vectors ${\bf i}$ and
${\bf j}$. In checking that this works, one must make use of the fact that our
matrix $F$ is now assumed to be upper triangular. A similar arrangement works
for ${\bf r}=(r,r\pm 1)$ with odd $r$, and this is enough for our purposes.
Apart from the ambiguities resulting from the reductions (2.1) - (2.4) (for
$r=\overline{0,2d-2}$), which are easily checked to be resolvable, we must
also consider the ambiguities of the form $x^{0}_{nj}x^{2d-1}_{n1}x^{0}_{i1}$
and $x^{2d-1}_{i1}x^{0}_{n1}x^{2d-1}_{nj}$. Because of the complicated form of
the reductions (2.5), (2.6), it is much more cumbersome to check the
resolvability of these. We will make use of a trick to reduce (2.5) and (2.6)
to the case when $F$ is diagonal; this simplifies the task of checking the
resolvability significantly, and we leave that task to the reader.
The trick alluded to in the previous paragraph is of the following nature: (1)
first, we would like to conclude that the desired resolvability depends only
on the conjugacy class of $F$ in the group $T(n,k)$ of upper triangular
$n\times n$ matrices; (2) next, we observe that it suffices to prove the
resolvability only for $F$ in a Zariski dense subset of $T(n,k)$. These two
steps would indeed reduce the checking to the case when $F$ is diagonal,
because we can take our Zariski dense set to be that of diagonalizable upper
triangular matrices.
To prove step (1), notice that by the diamond lemma, the resolvability can be
regarded as a statement about the dimension of the span of the $x^{\bf r}_{\bf
ij}$ in $H_{d}(F)$, where ${\bf r}$ is either $(0,2d-1,0)$ or $(2d-1,0,2d-1)$.
But by the argument used to prove the isomorphism $H_{d}(F)\cong
H_{d}(PFP^{-1})$, this dimension depends only on the conjugacy class of $F$ in
$T(n,k)$.
For step (2), let us focus on resolving $x^{0}_{nj}x^{2d-1}_{n1}x^{0}_{i1}$
(the other ambiguity being essentially the same). We can either apply (2.6) to
the first two factors and then (2.5) to every term in the resulting sum for
which it applies, or apply (2.5) to the last two factors and then (2.6) to all
the terms to which it applies in the resulting sum. The aim is to prove that
if for a Zariski dense subset of $T(n,k)$ the resulting expressions are
identical, then they are identical for all $F$. But this is clear: the
resulting expressions are linear combinations of terms of the form $x^{\bf
r}_{\bf ij}$ for ${\bf r}=(0,2d-1,0)$, and the coefficients of each such term
are regular functions defined on the algebraic variety $T(n,k)$; if these
coefficients coincide on a Zariski dense subset of $T(n,k)$, they coincide
everywhere by continuity.
We now summarize the conslusions of this section:
###### Proposition 2.1.
(a) For $\tilde{H}=H(n)$ or $H_{\infty}(n)$, the diamond lemma is applicable
to the reductions (2.1) - (2.4) (for $r\in R(\tilde{H})$), so the words in
$x^{r}_{ij}$ containing no subwords as in the left hand sides of those
reductions form a basis for $\tilde{H}$.
(b) Let $F\in T(n,k)$. For $\tilde{H}=H_{d}(F)$, the same conclusion as in (a)
holds, with the reductions (2.1) - (2.4), $r=\overline{0,2d-2}$ and (2.5),
(2.6).
The expansion of an element of $\tilde{H}$ as a linear combination of the
basis given here will be referred to as the standard form of the element.
Similarly, the standard form of an element of $\tilde{H}\otimes\tilde{H}$ is
its expansion as a linear combination of tensor products of reduced monomials.
The terms reducible/irreducible for monomials $x^{\bf r}_{\bf ij}$ as above
always refer to the reductions (2.1) - (2.6).
Finally, note that $\tilde{H}$ is filtered by the non-negative integers, with
$\tilde{H}_{k}$ being the span of the monomials $x^{\bf r}_{\bf ij}$ for
$|{\bf r}|\leq k$.
## 3\. Freeness
In this section we prove Theorem 1.1 and its consequence, Corollary 1.2. Let
us take care of the corollary first, assuming the theorem is proven.
We introduce some more notation first: given $r\in R=R(\tilde{H})$, $f_{r}\in
K=K(\tilde{H})$ denotes the comodule of $\tilde{H}$ corresponding to the
matrix coalgebra $X^{r}$. Similarly, given a vector ${\bf
r}=(r_{1},\ldots,r_{k})$ with entries in $R$, $f_{\bf r}$ denotes the product
$f_{r_{1}}\ldots f_{r_{k}}$. Similarly, $X^{\bf r}$ denotes the product of the
coalgebras $X^{r_{i}}$; it is the coalgebra $C(f_{r_{i}})$ corresponding to
the tensor product of the comodules $f_{r_{i}}$ (in the same order
$r_{1},r_{2},\ldots$).
Since the words $x\in A_{R}$ are clearly in one-to-one correspondence with the
vectors ${\bf r}$ with entries in $R$, we may denote the elements
$u_{x},u^{\prime}_{x}$ introduced in Section 1 by $u_{\bf r}$ and
$u^{\prime}_{\bf r}$ respectively (for the vector ${\bf r}$ corresponding to
$x$).
###### Proof of Corollary 1.2.
Recall the morphism $\phi:\mathbb{Z}[A_{R}]\to K=K(\tilde{H})$ of rings
endowed with an anti-endomorphism introduced in Section 1. Both the free
unital ring $\mathbb{Z}[A_{R}]$ on $R$ and the Grothendieck ring $K$ are
filtered: the former by the length of the words on $R$, and the latter by
setting, $K_{n}$ equal to the linear combination of those simple comodules
which are $\leq f_{\bf r}$ for some vector $\bf r\subset R$ of length $\leq n$
for each non-negative integer $n$ (remember that there is an order on $K$,
with $K_{+}$ as a positive cone).
The map $\phi$ from Section 1 preserves the filtration, and Theorem 1.1 says
precisely that the induced graded map between associated graded rings is an
isomorphism. But this implies that $\phi$ itself is bijective, and we are
done. ∎
###### Remark 3.1.
The corollary generalizes [Bi4, Corollary 5.5], which consists of the
corresponding statement for the cosemisimple universal cosovereign Hopf
algebras $H_{1}(F)$ in characteristic zero.
Before going into the proof of the theorem, we make several preliminary
observations on the problem. One of these is the following reformulation:
###### Lemma 3.2.
Theorem 1.1 is equivalent to the fact that the elements $u_{\bf r}\in
K(\tilde{H})$ appearing in its statement are simple.
###### Proof.
That the $u_{\bf r}$ are simple is part of the statement of Theorem 1.1, so we
only need the opposite implication. Hence, we now assume that all $u_{\bf r}$
are simple.
Since the Hopf algebra $\tilde{H}$ is the sum of the subcoalgebras $X^{\bf r}$
(for vectors ${\bf r}$ with entries in $R$), it follows that its comodules are
subcomodules of the tensor products (represented by) the $f_{\bf r}$. Now
consider (the representative of) a simple comodule $u\in K=K(\tilde{H})$. We
have just noticed that we must have $u\leq f_{\bf r}$ in $K$ for some vector
${\bf r}$; choose such an ${\bf r}$ of the smallest length possible. It then
follows from the definition of the $u_{\bf s}$’s that $u=u_{\bf r}$;
consequently, $\phi$ is a surjection of $A_{R}$ on $\mathcal{S}(\tilde{H})$.
On the other hand, again from the definition of $u_{\bf r}$, it follows that
the elements of the corresponding matrix subcoalgebra of $\tilde{H}$, in their
standard form, contain reduced monomials of type ${\bf r}$ (apart from those
of type ${\bf s}$ for $|\bf s|<|{\bf r}|$). But this immediately implies that
the $u_{\bf r}$ are all different, so $\phi$ is also injective. ∎
The previous lemma allows us to focus on proving that $u_{\bf r}$ are all
simple. In order to state the next preliminary result, we introduce the
following terminology: a vector ${\bf r}=(r_{1},\ldots,r_{k})\subset R$ is
said to be a 1-step vector if $r_{i+1}=r_{i}\pm 1$ for all $i$. The claim is
now the following:
###### Lemma 3.3.
If $u_{\bf r}$ is simple for every 1-step vector ${\bf r}\subset R$, then all
$u_{\bf r}$ are simple.
###### Proof.
We prove (under the hypothesis of the lemma) that all $u_{\bf r}$ are simple
by induction on the length of ${\bf r}$. Vectors of length $1$ (or $0$, i.e.
the empty vector) are by definition 1-step, so the base case of the induction
is taken care of. Now fix a vector ${\bf r}$, and assume the statement is
proven for all shorter vectors.
If ${\bf r}$ is 1-step, there is nothing to prove. Otherwise, we can write
${\bf r}$ as a concatenation ${\bf r}_{1}{\bf r}_{2}$, where ${\bf r}_{1}$ and
${\bf r}_{2}$ are vectors such that the last entry $r_{1}$ of ${\bf r}_{1}$
and the first entry $r_{2}$ of ${\bf r}_{2}$ satisfy $r_{2}\neq r_{1}\pm 1$.
By the induction hypothesis, the coalgebras $C_{i}$, $i=1,2$ corresponding
respectively to $u_{{\bf r}_{i}}$ are matrix coalgebras; since the
intersection of $C_{i}$ with the matrix coalgebra $X^{\bf s}$ for ${\bf s}$
shorter than ${\bf r}_{i}$ is trivial, the projection of $C_{i}$ on the span
of the monomials of type ${\bf r}_{i}$ (respectively) obtained by sending all
other monomials to zero is injective. But the form of the basis in Proposition
2.1 makes it clear that the product of two irreducible monomials of types
${\bf r}_{1}$ and respectively ${\bf r}_{2}$ is again irreducible. This,
together with the previous observation, implies that the multiplication map
from the tensor product $C_{1}\otimes C_{2}$ to the product $C=C_{1}C_{2}$
inside $\tilde{H}$ is an isomorphism, and hence that (a) $u_{\bf r}=u_{{\bf
r}_{1}}u_{{\bf r}_{2}}$, and (b) $u_{\bf r}$ is simple, with matrix coalgebra
$C$. This completes the induction step. ∎
In the proof of Theorem 1.1, we will deal separately with the universal
cosovereign Hopf algebras $H_{1}(F)$. For the other cases, $\tilde{H}=H(n)$,
$H_{\infty}(n)$ or $H_{d}(F)$ for some $d>1$, the following observation will
be useful:
###### Lemma 3.4.
If Theorem 1.1 holds for $\tilde{H}=H(n)$, then it holds for
$\tilde{H}=H_{\infty}(n)$ or $\tilde{H}=H_{d}(F)$, $d>1$.
###### Proof.
By the two previous lemmas, it is enough to check that $u_{\bf r}$ is simple
for any 1-step vector ${\bf r}$.
Assume first that $\tilde{H}=H_{\infty}(n)$. In this case, by applying a high
enough power of the antipode (which is bijective), we may as well assume that
integer entries of ${\bf r}$ are, in fact, non-negative. But the bases for our
Hopf algebras given by Proposition 2.1 make it clear that the map $H(n)\to
H_{\infty}(n)$ sending $x^{0}_{ij}$ in $H(n)$ to $x^{0}_{ij}$ in
$H_{\infty}(n)$ induces an isomorphism of $K(H(n))$ onto the subring of
$K(H_{\infty}(n))$ generated by the subcomodules of the $f_{\bf r}$’s for non-
negative vectors ${\bf r}$.
Now take $\tilde{H}=H_{d}(F)$ for some $d>1$ and $F\in GL(n,k)$. We have a
surjective Hopf algebra map $H(n)\to H_{d}(F)$, sending $x^{r}_{ij}$ in $H(n)$
to $x^{\bar{r}}_{ij}$ in $H_{d}(F)$, where $r\mapsto\bar{r}$ is the obvious
surjection $\mathbb{N}\to\mathbb{Z}/2d$. If we prove that the matrix coalgebra
$C_{\bf r}$ corresponding to $u_{\bf r}\in K(H(n))$ gets mapped to a matrix
coalgebra, then we are done.
It is clear from the reductions (2.1) - (2.6) that whenever ${\bf
r}\subset\mathbb{N}$ is a 1-step vector, a reduced monomial of type ${\bf r}$
in $H(n)$ is mapped onto a reduced word of type
${\bf\bar{r}}\subset\mathbb{Z}/2d$ in $H_{d}(F)$ as long as $d>1$. In other
words, the span of the reduced words of type ${\bf r}$ is mapped injectively
into $H_{d}(F)$. In view of the fact (also noted in the previous proof) that
the projection onto the span of the words of type ${\bf r}$ obtained by
sending all other monomials to zero is injective on the matrix coalgebra
$C_{\bf r}$, this concludes the proof. ∎
For $H_{1}(F)$ we will have to make use of Bichon’s results on Hopf-Galois
systems ([Bi3], [Bi4, Proposition 2.1, 2.4]): what is relevant for us here is
that if $F$ is upper triangular with diagonal $D$, then there is an
equivalence of monoidal categories between $H_{1}(F)$ and $H_{1}(D)$ matching
up the fundamental corepresentations. Hence, when dealing with $H_{1}(F)$ in
the proof, we can (and will) assume that $F$ is diagonal. With this assumption
in place, the proof below will take care of all the possibilities for
$\tilde{H}$ at once.
###### Proof of Theorem 1.1.
The following argument applies to $\tilde{H}=H(n)$ or $H_{1}(F)$ for some
diagonal invertible matrix $F\in GL(n,k)$ (see the comments above). Recall
that $n\geq 2$. Lemma 3.4 says that we will then get the cases
$\tilde{H}=H_{\infty}(n)$ or $H_{d}(F)$, $d>1$ for free, so this suffices to
prove the theorem. Furthermore, by Lemma 3.2, we only have to prove that the
comodules $u_{\bf r}$ are simple.
Fix an $R$-vector ${\bf r}=(r_{1},\ldots,r_{k})$. Let $C$ be a simple (hence
matrix) subcoalgebra of $C_{\bf r}=C(u_{\bf r})$. Denote by ${\bf\ell}$ the
alternating vector $(1,n,1,n,\ldots)$, of length $|{\bf r}|$ (we could have
used any two different elements of $\overline{1,n}$ instead of $1$ and $n$). I
claim that $C$ necessarily contains an element $x$ whose standard form
contains the monomial $x^{\bf r}_{\bf\ell\ell}$.
Assuming the claim for now, the proof continues as follows. Consider the Hopf
algebra $H$, obtained as a quotient of $H(n)$ by sending all off-diagonal
generators $x^{0}_{ij}$, $i\neq j$ to zero. $H$ is nothing but the group
algebra of the free group $F_{n}$ on the $n$ generators $x_{i}=x^{0}_{ii}$,
$i=\overline{1,n}$. Because in this proof $\tilde{H}$ is $H(n)$ or $H_{1}(F)$
for a diagonal matrix $F$, the surjection $H(n)\to H$ factors through
$\tilde{H}$. Hence, we now have a surjection $\psi:\tilde{H}\to H$, obtained
by sending all off-diagonal generators $x^{0}_{ij}$, $i\neq j$ to zero. The
induced map on Grothendieck rings will also be denoted by $\psi$.
Because $x^{\bf r}_{\bf\ell\ell}$ has non-zero coefficient in $x\in C$, it
follows that the simple $\tilde{H}$-comodule corresponding to $C$, when
regarded as an $H$-comodule by “scalar corestriction” via $\psi$, contains the
$1$-dimensonal $H$-comodule $v$ corresponding to
$x_{1}^{\varepsilon_{1}}x_{n}^{\varepsilon_{2}}x_{1}^{\varepsilon_{3}}\ldots$
as a summand, where the expression contains $|{\bf r}|$ factors, and
$\varepsilon_{i}=1$ if $r_{i}$ is even and $-1$ otherwise. $C$ was an
arbitrary matrix subcoalgebra; unless $u_{\bf r}$ is simple, this means that
$2v\leq\psi(u_{\bf r})$ (in the usual order on the Grothendieck ring $K(H)$).
This, however, is plainly false: on the one hand we have $u_{\bf r}\leq f_{\bf
r}$ in $K(\tilde{H})$ (recall that $f_{\bf r}=f_{r_{1}}\ldots f_{r_{2}}$), and
on the other hand, $\psi(x^{\bf r}_{\bf ij})$ is equal to
$x_{1}^{\varepsilon_{1}}x_{n}^{\varepsilon_{2}}x_{1}^{\varepsilon_{3}}\ldots$
for precisely one (reducible or irreducible) monomial $x^{\bf r}_{\bf ij}$ of
type ${\bf r}$, which means that $2v\not\leq\psi(f_{\bf r})$ in $K(H)$.
It remains to prove the claim that $x^{\bf r}_{\bf\ell\ell}$ has non-zero
coefficient in the standard form of some element of $C$. The following
technique was used in the proof of [Ch, Proposition 2.6], as well as several
other results in that paper.
Consider any non-zero element $x$ of $C$. Because $C\subset X^{\bf r}$ and the
intersection of $C$ with any coalgebra of the form $X^{\bf s}$, $|{\bf
s}|<|{\bf r}|$ is trivial, the standard form of $x$ must contain some reduced
monomial $x^{\bf r}_{\bf ij}$. Using the comultiplication
$\Delta(x^{r}_{ij})=\sum_{a=1}^{n}x^{r}_{ia}\otimes x^{r}_{aj},$
we conclude that the standard form of $\Delta(x)$ contains $x^{\bf r}_{\bf
i\ell}\otimes x^{\bf r}_{\bf\ell j}$ (one sees easily that both $x^{\bf
r}_{\bf i\ell}$ and $x^{\bf r}_{\bf\ell j}$ must be reduced if $x^{\bf r}_{\bf
ij}$ is). But that the standard form of some element of $C$ (which we may as
well assume is our $x$) contains $x^{\bf r}_{\bf i\ell}$. Now simply repeat
the argument to conclude that $x^{\bf r}_{\bf\ell\ell}$ is indeed contained in
the standard form of some element of $C$. ∎
## 4\. Maximal freeness
The goal in this section is to prove Theorem 1.4. We begin by noticing that
the lemmas in the previous section have analogues which apply here almost word
for word.
The first observation is that since we now know that $u_{\bf r}$ are simple
and it we remarked in Section 1 that $u_{\bf r}\leq u^{\prime}_{\bf r}$ in
$K(\tilde{H})$, the result that $u^{\prime}_{\bf r}=u_{\bf r}$, which is what
we’re after in Theorem 1.4, is equivalent to saying that $u^{\prime}_{\bf r}$
being simple. This is an analogue of Lemma 3.2. In each particular case, we
use whichever formulation seems more convenient.
Lemma 3.3 can also be adapted to $u^{\prime}_{\bf r}$:
###### Lemma 4.1.
Let $\tilde{H}$ be one of our Hopf algebras, and $R=R(\tilde{H})$, as usual.
If $u^{\prime}_{\bf r}=u_{\bf r}$ for every 1-step $R$-vector ${\bf r}$, then
the same holds for all vectors ${\bf r}$.
###### Proof.
We will adapt the proof of Lemma 3.3, using induction on $|{\bf r}|$ again. If
${\bf r}$ is not 1-step, then write it as a concatenation ${\bf r}_{1}{\bf
r}_{2}$, as in that proof. By the induction hypothesis we know that
$u^{\prime}_{{\bf r}_{i}}=u_{{\bf r}_{i}}$, $i=1,2$, so the argument used in
the proof of Lemma 3.3 shows that the tensor product $u^{\prime}_{{\bf
r}_{1}}u^{\prime}_{{\bf r}_{2}}$ is simple. Since it’s easy to see from the
definition of the $u^{\prime}_{\bf s}$’s that $u^{\prime}_{\bf r}\leq
u^{\prime}_{{\bf r}_{1}}u^{\prime}_{{\bf r}_{2}}$, we get the desired result
that $u^{\prime}_{\bf r}$ is simple. ∎
The following analogue of Lemma 3.4 will come in handy in the proof of Theorem
1.4, (a). Once more, the proof of Lemma 3.4 can be adapted immediately to the
present situation.
###### Lemma 4.2.
If $u^{\prime}_{\bf r}=u_{\bf r}$ for $\tilde{H}=H(n)$ and all
$R(\tilde{H})$-vectors ${\bf r}$, then the same is true for
$\tilde{H}=H_{\infty}(n)$ or $H_{d}(F)$, $d>1$.
Theorem 1.4 (a) has now been reduced to the case $\tilde{H}=H(n)$. We reduce
it further to $\tilde{H}=H(2)$ by the following observation: it was shown in
[Bi3, Corollary 5.3] that there is a monoidal equivalence between the
categories of comodules of $H(n)$ and $H(2)$ for every $n\geq 2$. Furthermore,
it follows from the discussions in that paper that this equivalence matches up
the fundamental corepresentations. Since the statement of Theorem 1.4 clearly
depends only on the Grothendieck ring (as a ring endowed with an anti-
endomorphism) and the choice of a distinguished element of that ring (the
fundamental corepresentation), we can indeed work only with $H(2)$.
We now need to go into the combinatorics of the multiplication in
$K(\tilde{H})$ in more detail, and this requires yet more new terminology and
notations. It will be very useful to know the dimensions of (the comodules
represented by) the $u^{\prime}_{\bf r}$’s, so we begin by introducing the
notations necessary to state that result.
Fix our Hopf algebra $\tilde{H}=H(n)$, $H_{\infty}(n)$, or $H_{d}(F)$ for some
$F\in GL(n,k)$. Let
${\bf r}=(r_{1},\ldots,r_{k})$
be a vector with entries in $R=R(\tilde{H})$, as usual. Now consider sequences
$n_{1},\ldots,n_{k}$ of positive integers in the range $\overline{1,n}$ with
the properties that (a) if $r_{i}$ is even and $r_{i+1}=r_{i}\pm 1$, then the
pair $(n_{i},n_{i+1})$ is different from $(n,n)$, and (b) if $r_{i}$ is odd
and $r_{i+1}=r_{i}\pm 1$, then $(n_{i},n_{i+1})\neq(1,1)$. Denote by
${\mathcal{O}}_{\bf r}$ the collection of such vectors, and by $n_{\bf r}$ the
cardinality of ${\mathcal{O}}_{\bf r}$.
###### Remark 4.3.
A quick look at the reduction formulas (2.1) - (2.6) shows that when
$R=\mathbb{Z}/2$ (i.e. $\tilde{H}$ is one of the universal cosovereign Hopf
algebras $H_{1}(F)$), the number of irreducible monomials of type ${\bf r}$ is
precisely $n_{\bf r}^{2}$. This observation will be crucial in the proof of
Theorem 1.4.
It will be seen below (Corollary 4.7) that the dimension of $u^{\prime}_{\bf
r}$ is precisely $n_{\bf r}$, and at the same time, we will see how the basic
tensor products $f_{\bf r}=f_{r_{1}}\ldots f_{r_{k}}$ decompose as sums of
$u^{\prime}_{\bf s}$’s. The following setup is relevant for the latter
purpose.
For a vector ${\bf r}=(r_{i},\ i=\overline{1,k})$ as above, we introduce the
following notion:
###### Definition 4.4.
An ${\bf r}$-configuration is a sequence of length $k=|{\bf r}|$ of symbols,
with each symbol being either empty (i.e. no symbol at all) or one of the
parantheses ‘$($’, ‘$)$’, according to the following rules:
(a) the sequence of symbols is grammatically correct as a sequence of
parantheses;
(b) if $|{\bf r}|=0,1$, then the only ${\bf r}$-configuration is the empty one
(only the empty symbol, or in other words, no symbols at all);
(c) if we have a $($ at position $i$ and its pair $)$ at $j>i$, then
$r_{j}=r_{i}\pm 1$;
(d) if we have a $($ at $i$ and its pair $)$ at $j>i$, then all positions
between $i$ and $j$ are filled up completely with paired up parantheses (in
particular, it follows that $j-i$ is odd).
The collection of all ${\bf r}$-configurations will be denoted by ${\rm
Conf}_{\bf r}$, with $\emptyset$ standing for the empty configuration. We give
some examples to help clarify the definition. The parantheses appear above
their positions, with nothing appearing over the positions corresponding to
the empty symbol.
Suppose ${\bf r}=(1,2,1)$. Apart from the empty configuration, we have two
more, namely
$\displaystyle($ $\displaystyle)$ $\displaystyle($ $\displaystyle)$
$\displaystyle 1$ $\displaystyle 2$ $\displaystyle 3$ $\displaystyle{\rm
and}\qquad\qquad$ $\displaystyle 1$ $\displaystyle 2$ $\displaystyle 3$ .
Similarly, if ${\bf r}=(1,2,1,2)$, then there are five non-empty ${\bf
r}$-configurations. Those with only one pair of parantheses are
$\displaystyle($ $\displaystyle)$ $\displaystyle($ $\displaystyle)$
$\displaystyle($ $\displaystyle)$ $\displaystyle 1$ $\displaystyle 2$
$\displaystyle 3$ $\displaystyle 4$ $\displaystyle,\qquad$ $\displaystyle 1$
$\displaystyle 2$ $\displaystyle 3$ $\displaystyle 4$ $\displaystyle,\qquad$
$\displaystyle 1$ $\displaystyle 2$ $\displaystyle 3$ $\displaystyle 4$ ,
while those with two pairs of parantheses are
$\displaystyle($ $\displaystyle)$ $\displaystyle($ $\displaystyle)$
$\displaystyle($ $\displaystyle($ $\displaystyle)$ $\displaystyle)$
$\displaystyle 1$ $\displaystyle 2$ $\displaystyle 3$ $\displaystyle 4$
$\displaystyle{\rm and}\qquad\quad$ $\displaystyle 1$ $\displaystyle 2$
$\displaystyle 3$ $\displaystyle 4$ .
Given a vector ${\bf r}$ and an ${\bf r}$-configuration $c\in{\rm Conf}_{\bf
r}$, we denote by ${\bf r}_{c}$ the vector obtained from ${\bf r}$ by removing
the entries whose positions hold parantheses in $c$.
###### Remark 4.5.
Note that ${\bf r}$-configurations enjoy is a certain “transitivity” property:
suppose $($ occupies position $i$ in $c$, while its pair $)$ occupies position
$i+1$. Let $d$ be the ${\bf r}$-configuration consisting of only these two
parantheses at $i$ and $i+1$, and let $d^{\prime}$ be the ${\bf
r}_{d}$-configuration consisting of all the symbols left after striking out
the two parantheses at $i$ and $i+1$. Then, we have ${\bf r}_{c}=({\bf
r}_{d})_{d^{\prime}}$.
We have now made the combinatorial preparations necessary to describe the
“multiplication table” of $K(\tilde{H})$ in terms of the $u^{\prime}_{\bf
r}$’s. Both in the proposition and in the corollary following it, it is
understood that we are working with $\tilde{H}=\tilde{H}(n)$, as usual; this
is where the $n$ necessary in the definition of $n_{\bf r}$ comes from.
###### Proposition 4.6.
For an $R=R(\tilde{H})$-vector ${\bf r}$, the formula
$f_{\bf r}=\sum_{c\in{\rm Conf}_{\bf r}}u^{\prime}_{{\bf r}_{c}}$ (4.1)
holds in $K(\tilde{H})$.
###### Proof.
This is more or less a tautology, once we translate the definition of
$u^{\prime}_{\bf r}$ given in Section 1 using the notations employed here.
Recall that we used the notation $u^{\prime}_{x}$, $x\in A_{R}$ in Section 1,
and then renamed that to $u^{\prime}_{\bf r}$ by identifying the elements of
the free monoid $A_{R}$ on $R$ with the $R$-vectors ${\bf r}$. The definition
now reads
$u^{\prime}_{\bf r}=f_{\bf r}-\bigvee f_{{\bf r}_{c}},$ (4.2)
the supremum ranging over those ${\bf r}$-configurations $c$ with only two
parantheses (necessarily, these would have to be a $($ at some position $i$,
and its pair $)$ at position $i+1$).
The proposition now follows by induction on the length of the vector ${\bf
r}$, by applying the induction hypothesis to the vectors ${\bf r}_{c}$ and
using the remark made above on the transitivity of configurations (Remark
4.5). ∎
We also record the following consequence, as announced above:
###### Corollary 4.7.
The dimension of the comodule represented by $u^{\prime}_{\bf r}$ is $n_{\bf
r}$.
###### Proof.
With Proposition 4.6 at our disposal, the proof is a simple counting argument
plus induction by the length of ${\bf r}$, the base case of the induction
($|{\bf r}|=0,1$) being trivial.
Fix ${\bf r}=(r_{1},\ldots,r_{k})$, and assume the statement is proven for
shorter $R$-vectors. We then know that it holds for all ${\bf r}_{c}$,
$c\in{\rm Conf}_{\bf r}$, except for $c=\emptyset$. Hence, by formula (4.1)
(and since $\dim(f_{\bf r})=n^{k}$), it suffices to show that
$n^{|{\bf r}|}=\sum_{c\in{\rm Conf}_{\bf r}}n_{{\bf r}_{c}}.$
To see how this comes about, remember that $n_{\bf r}$ is the cardinality of
the set ${\mathcal{O}}_{\bf r}$, which is a certain collection of length
$|{\bf r}|$ sequences with entries in $\overline{1,n}$; we will exhibit a
bijection between the disjoint union of the sets ${\mathcal{O}}_{{\bf r}_{c}}$
and the set ${\overline{1},n}^{|{\bf r}|}$ of all such sequences.
Fix an ${\bf r}$-configuration $c$, and consider the set ${\mathcal{O}}_{\bf
r}^{c}$ of sequences in $\overline{1,n}^{k}$ defined by the following rules:
(a) if $i,i+1$ correspond to the empty symbol in $c$, then the same rules
apply as for ${\mathcal{O}}_{\bf r}$, i.e. $(n_{i},n_{i+1})\neq(n,n)$ if
$r_{i+1}=r_{i}\pm 1$, $r_{i}$ even, and $(n_{i},n_{i+1})\neq(1,1)$ if
$r_{i+1}=r_{i}\pm 1$, $r_{i}$ odd;
(b) if $i<j$ hold parantheses $($ and respectively $)$ in $c$, then
$n_{i},n_{j}$ are both $n$ or both $1$, according to whether $r_{i}$ is even
or odd, respectively.
Given a sequence $n_{1},\ldots,n_{k}$ in ${\mathcal{O}}_{\bf r}^{c}$, by
simply deleting the $n_{i}$’s in the sequence for those $i$ which hold a
paranthesis, we get a subsequence belonging to the set ${\mathcal{O}}_{{\bf
r}_{c}}$. The opposite map from ${\mathcal{O}}_{{\bf r}_{c}}$ to
${\mathcal{O}}_{\bf r}^{c}$ is easily constructed by simply inserting the
missing terms $n_{i}$ according to rule (b) above, so we have a bijection
between the two sets. On the other hand, the set $\overline{1,n}^{k}$ of all
length $k$ sequences with terms in the range $\overline{1,n}$ is clearly
partitioned by the sets ${\mathcal{O}}_{\bf r}^{c}$, so we get the desired
result. ∎
We can now take care of part (b) of the theorem.
###### Proof of Theorem 1.4 (b).
“$\Leftarrow$” Suppose $H_{1}(F)$ is cosemisimple, and fix an
$R=\mathbb{Z}/2$-vector ${\bf r}$. By Corollary 4.7, $u^{\prime}_{\bf r}$ is a
direct sum of simple comodules of total dimension $n_{\bf r}$. By Theorem 1.1,
one of these comodules is $u_{\bf r}$.
By the very definition of $u_{\bf r}$, the only matrix subcoalgebra of $X^{\bf
r}$ which does not appear as a summand of $X^{\bf s}$ for some shorter vector
$|{\bf s}|<|{\bf r}|$ is the one denoted above by $C_{\bf r}$, corresponding
to the simple comodule $u_{\bf r}$. This means that $\dim(C_{\bf r})$ is
precisely the number of irreducible monomials of type ${\bf r}$, i.e. $n_{\bf
r}^{2}$ (see Remark 4.3). But this then implies that the dimension of $u_{\bf
r}$ is $n_{\bf r}$, so $u_{\bf r}$ accounts for the entire $u^{\prime}_{\bf
r}$.
“$\Rightarrow$” We want to prove that if $u^{\prime}_{\bf r}=u_{\bf r}$ for
all $\mathbb{Z}/2$-vectors ${\bf r}$, then $\tilde{H}=H_{1}(F)$ is the sum of
its matrix subcoalgebras $C_{\bf r}$ (corresponding respectively to the simple
comodules $u_{\bf r}$).
Consider an element
$x=\sum a^{\bf s}_{\bf ij}x^{\bf s}_{\bf ij}\in\tilde{H}$ (4.3)
in its standard form, where $a^{\bf s}_{\bf ij}$ are coefficients in the field
$k$. If $t$ is a non-negative integer, denote
$x_{t}=\sum_{|{\bf s}|=t}a^{\bf s}_{\bf ij}x^{\bf s}_{\bf ij}.$
In other words, we are “truncating” $x$ to its portion of length $t$.
Typically, we will choose $t$ to be the top length of a monomial appearing in
(4.3).
Now fix a $\mathbb{Z}/2$-vector ${\bf r}$. By hypothesis, $u^{\prime}_{\bf
r}=u_{\bf r}$ is simple; according to Corollary 4.7, its dimension is $n_{\bf
r}$, so the dimension of its corresponding matrix coalgebra $C_{\bf r}$ is
$n_{\bf r}^{2}$. But by Remark 4.3, this is precisely the number of
irreducible monomials of type ${\bf r}$.
It has been noticed before that the map sending $x\in C_{\bf r}$ to $x_{|{\bf
r}|}$ is an injection into the span of irreducible monomials of type ${\bf
r}$. By the dimension count in the previous paragraph, $x\mapsto x_{|{\bf
r}|}$ is an isomorphism of $C_{\bf r}$ onto this span. By induction on the
length of the vectors, $x-x_{|{\bf r}|}$ is contained in the sum of all
coalgebras $C_{{\bf s}}$, $|{\bf s}|<|{\bf r}|$, so finally, every irreducible
monomial is contained in the sum of the subcoalgebras $C_{\bf r}$. ∎
In the proof of Theorem 1.4 we will make use of known facts about the
corepresentations of the quantized function algebra on $SL(2)$, which we
denote here by $SL_{q}(2)$. As $SL_{q}(2)$ is one of the most well studied
quantum groups, we do not recall the definition here; it can be found in
numerous sources in the literature. The reference we will be making use of for
the very basic results on its corepresentations that will actually come up
here is [KP]. Recall only that $q\in k^{*}$ is an invertible scalar. One
usually considers it over fields $k$ of characteristic zero (typically
$\mathbb{C}$), and furthermore, the corepresentations behave well (i.e. there
is an isomorphism between the Grothendieck rings of $SL_{q}(2)$ and the usual
$SL(2)$) when $q$ is not a root of unity. However, all the usual proofs go
through in positive characteristic, even in the bad case when $q$ is a root of
unity, as soon as its order is coprime to the characteristic; we invite the
reader to check this as an exercise, going through the proofs in [KP], for
example.
$SL_{q}(2)$ has a fundamental $2\times 2$ matrix subcoalgebra denoted in [KP]
by
$m=\begin{pmatrix}\alpha&\beta\\\ \gamma&\delta\end{pmatrix}$
which generates $SL_{q}(2)$ as an algebra. We also denote $m$ by $m_{1}$, and
we use the same notation for the corresponding $2$-dimensional comodule, and
its class in the Grothendieck ring; our $m$ is denoted by $u^{\frac{1}{2}}$ in
[KP]. One has, for small enough positive integers $t$, simple
corepresentations $m_{t}$ which satisfy the Clebsch-Gordan multiplication
table:
$m_{t}\otimes m\cong m_{t+1}\oplus m_{t-1},$ (4.4)
where $m_{0}$ stands for the trivial corepresentation. It follows that the
dimension of $m_{t}$ is $t+1$. Here, $t$ less than half the order of $q$ minus
1 is “small enough” in case $q$ is a root of unity. All of these
corepresentations are self-dual. Only these partial results on the
corepresentation theory of $SL_{q}(2)$ are important here; they follow
immediately from the more detailed versions stated briefly at the end of [KP,
Section 0] and proven in that paper.
###### Proof of Theorem 1.4 (b).
In the remarks immediately after Lemma 4.2 we observed that it suffices to
consider $\tilde{H}=H(2)$. Furthermore, by Lemma 4.1, it suffices to prove the
statement of the theorem for 1-step $R=\mathbb{N}$-vectors ${\bf r}$.
Now fix a 1-step $\mathbb{N}$-vector ${\bf r}$. We know from Proposition 4.6
that $f_{\bf r}$ can be broken up as the sum of all $u^{\prime}_{{\bf
r}_{c}}$’s, as $c$ ranges through all the ${\bf r}$-configurations. Moreover,
Corollary 4.7 says that the dimension of $u^{\prime}_{{\bf r}}$ is $n_{{\bf
r}}$. Since here the $n$ used in the calculation of $n_{{\bf r}_{c}}$ is $2$,
it is a simple matter to compute $n_{\bf r}=|{\bf r}|+1$ (the fact that ${\bf
r}$ is 1-step is crucial here).
The plan of the proof is as follows:
Let $H$ be a Hopf algebra with a multiplicative matrix $m$ (we denote the
corresponding $2$-dimensional comodule and its class in the Grothendieck ring
by $m$ again). Let $\psi:H(2)\to H$ be the map sending $X^{0}$ to $m$, and
denote the induced map on Grothendieck rings by the same symbol. If
$\psi(f_{\bf r})$ contains some simple composition factor $m^{\prime}$ of
dimension $|{\bf r}|+1$ which does not appear as a composition factor in
$\psi(f_{{\bf r}_{c}})$ for any non-empty ${\bf r}$-configuration $c$, then we
must have
$m^{\prime}=\psi(u^{\prime}_{\bf r}),$
and hence $u^{\prime}_{\bf r}$ must be simple.
Hence, it suffices to find $H,m$ as above, and this is where the $q$-analogues
of $SL(2)$ come in. We take $H=SL_{q}(2)$ for some adequate $q$ (either not a
root of unity, or, if the field $k$ is the algebraic closure of a finite field
and we have no choice, a root of unity of order greater than $2|{\bf r}|+1$).
$m$ will be the $m_{1}$ introduced above. Since $m$ is self-dual, it follows
that $\psi(f_{\bf r})$ is precisely the $|{\bf r}|$’th tensor power of $m$.
Finally, (4.4) shows that $m^{\prime}=m_{|{\bf r}|}$ has the desired
properties. ∎
We end by recasting the results obtained here in a form that is similar to
Theorem 1.5. The main observation is that given Theorem 1.4, Proposition 4.6
gives the formulas for the multiplication in the Grothendieck ring in terms of
the basis $u^{\prime}_{\bf r}=u_{\bf r}$ (in the cases covered by the
theorem). In order to get explicit formulas (i.e. express the product $u_{\bf
r}u_{\bf s}$ as a linear combination of the $u$’s), we need to introduce an
operation on the monoid ring $\mathbb{Z}[A_{R}]$, similar to Bănică’s $\odot$
mentioned in the introduction ((1.5)).
Recall that we defined an anti-endomorphism $*$ on the free monoid $A_{R}$
generated by $R$, given by sending the generator $\alpha_{r}$, $r\in R$ to
$\alpha_{r+1}$, which extends by linearity to the monoid ring. In the
discussion below, we identify words on $R$ (i.e. elements of $A_{R}$) with
$R$-vectors ${\bf r}$ is the obvious way; the multiplication in the monoid
$A_{R}$ is expressed in terms of vectors as concatenation (and written ${\bf
rs}$ for vectors ${\bf r}$, ${\bf s}$), and the $*$ operation is given by
$(r_{1},r_{2},\ldots,r_{k})^{*}=(r_{k}+1,\ldots,r_{2}+1,r_{1}+1).$
We call two vectors ${\bf r},{\bf s}\in A_{R}$ linked and write ${\bf
r}\sim{\bf s}$ if ${\bf r}^{*}={\bf s}$ or ${\bf s}^{*}={\bf r}$ (note that
this is equivalent to ${\bf r}^{*}={\bf s}$ for $R=\mathbb{Z}/2$, which is the
case treated in [Ba]). Now consider the binary operation on
$\mathbb{Z}[A_{R}]$, extended by linearity from the formula
${\bf r}\odot{\bf s}=\sum{\bf ab},\ {\bf r},{\bf s}\in A_{R},$ (4.5)
where the sum ranges over all possible ways of writing ${\bf r}={\bf at}$,
${\bf s}={\bf t^{\prime}b}$ with ${\bf t}\sim{\bf t^{\prime}}$. This operation
is actually associative, and has the same unit as the usual multiplication in
$\mathbb{Z}[A_{R}]$; all of this is easily checked.
We extend the notation $u^{\prime}_{\bf r}$ to $u^{\prime}_{a}$ for any
$a\in\mathbb{Z}[A_{R}]$ by linearity in $a$. In this setting, I claim that
Proposition 4.6 can be reformulated as follows:
* Proposition4.6 bis
Let $\tilde{H}=H(n)$, $H_{\infty}(n)$, or $H_{d}(F)$. Then, the formula
$u^{\prime}_{\bf r}u^{\prime}_{\bf s}=u^{\prime}_{{\bf r}\odot{\bf s}},\
\forall{\bf r},{\bf s}\in A_{R}$ (4.6)
holds in the Grothendieck ring $K(\tilde{H})$.
###### Proof.
This is proven by induction on $|{\bf r}|+|{\bf s}|$, the base case when ${\bf
r}$ and ${\bf s}$ are both empty (i.e. of length zero) being trivial. Now fix
${\bf r}$, ${\bf s}$, and assume the statement is proven for smaller combined
lengths of the two vectors.
Proposition 4.6 says that we have
$f_{\bf r}=\sum_{c\in{\rm Conf_{\bf r}}}u^{\prime}_{{\bf r}_{c}},$ (4.7)
$f_{\bf s}=\sum_{d\in{\rm Conf_{{\bf s}}}}u^{\prime}_{{\bf s}_{d}},$ (4.8)
and
$f_{{\bf rs}}=\sum_{e\in{\rm Conf_{{\bf rs}}}}u^{\prime}_{({\bf rs})_{e}}.$
(4.9)
Since $f_{{\bf r}}f_{{\bf s}}=f_{{\bf rs}}$, we multiply (4.7) and (4.8) and
compare the result to the right hand side of (4.9). Apply the induction
hypothesis to express all products $u^{\prime}_{{\bf r}_{c}}u^{\prime}_{{\bf
s}_{d}}$ with $c\neq\emptyset$ or $d\neq\emptyset$ as a sum of $u^{\prime}$
terms. This gives us some of the terms $u^{\prime}_{({\bf rs})_{e}}$ in (4.9),
and the sum of the ones we do not get in this way will be exactly
$u^{\prime}_{{\bf r}}u^{\prime}_{{\bf s}}$.
It now remains to observe that the ${\bf rs}$-configurations $e$ which do not
arise from products of the form $u^{\prime}_{{\bf r}_{c}}u^{\prime}_{{\bf
s}_{d}}$ with $c,d$ not both empty are precisely those consisting of an
unbroken string of ‘$($’ symbols at the end of ${\bf r}$, followed by an
unbroken string (necessarily of the same length) of ‘$)$’ symbols at the
beginning of ${\bf s}$. On the other hand, it’s clear from our definitions
that the $u^{\prime}_{({\bf rs})_{e}}$ for such configurations $e$ are
precisely the terms appearing in the definition (4.5) of the product $\odot$.
∎
As promised above, we now have a complete, explicit description of the
multiplication in $K=K(\tilde{H})$ in the cases covered by Theorem 1.4, when
the $u^{\prime}_{\bf r}$ form a basis for $K$ as a free abelian group: the
multiplication table is described by (4.6).
## References
* [A] Abe, E. - Hopf algebras, Cambridge University Press 1980
* [Ba] Bănică, T. - Le groupe quantique compact libre $U(n)$, Comm. Math. Phys. 190 (1997), pp. 143 - 172
* [Be] Bergman, G. - The diamond lemma for ring theory, Adv. Math. 29 (1978), pp. 178 - 218
* [Bi1] Bichon, J. - Galois reconstruction of finite quantum groups, J. Algebra 230 (2000), pp. 683 - 693
* [Bi2] \- Cosovereign Hopf algebras, J. Pure Appl. Algebra 157 (2001), pp. 121 - 133
* [Bi3] \- Hopf-galois systems, J. Algebra 264 (2003), pp. 565 - 581
* [Bi4] \- Corepresentation theory of universal cosovereign Hopf algebras, J. London Math. Soc. 75 (2007), pp. 83 - 98
* [Ch] Chirvăsitu, A. - Subcoalgebras and endomorphisms of free Hopf algebras, preprint available online, to appear in J. Pure Appl. Algebra
* [DK] Dijkhuizen, M. S. and Koorwinder, T. H. - CQG algebras: a direct algebraic approach to compact quantum groups, Lett. Math. Phys 32 (1994), pp. 315 - 330
* [Dr1] Drinfeld, V. G. - Hopf algebras and the Yang-baxter equations, soviet Math. Dokl. 32 (1985), pp. 254 - 258
* [Dr2] \- Quantum groups, In Proc. of the ICM-1986, Berkeley, Vol. I, Providence , R. I., Am. Math. Soc. 1987, pp. 798 - 820
* [DVL] Dubois-Violette, M. and Launer, G. - The quantum group of a non-degenerate bilinear form, Phys. Lett. B 245 (1990), pp. 175 - 177
* [Ji] Jimbo, M. - A $q$-difference analogue of ${\mathcal{U}}({\mathfrak{g}})$ and the Yang-Baxter equations, Lett. Math. Phys. 10 (1985), pp. 63 - 69
* [KP] Kondratowicz, P. and Podleś, P. - On representation theory of quantum $SL_{q}(2)$ at roots of unity,
* [Ma] Manin, Y. - Quantum Groups and Noncommutative Geometry, Publications du CRM 1561, Univ. de Montreal 1988
* [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
* [VDW] Van Daele, A. and Wang, S. Z. - Universal quantum groups, International J. of Math. 7 (1996), pp. 255 - 264
* [Wo1] Woronowicz, S. L. - compact matrix pseudogroups, Commun. Math. Phys. 111 (1987), pp. 613 - 665
* [Wo2] \- Tannaka-Krein duality for compact matrix pseudogroups. Twisted $SU(N)$ groups., Invent. Math. 93 (1988), pp. 35 - 76
|
arxiv-papers
| 2010-06-17T13:37:54 |
2024-09-04T02:49:10.969686
|
{
"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/1006.3464"
}
|
1006.3483
|
# Exact epidemic dynamics for generally clustered, complex networks
Thomas House
###### Abstract
The last few years have seen remarkably fast progress in the understanding of
statistics and epidemic dynamics of various clustered networks. This paper
considers a class of networks based around a new concept (the locale) that
allow exact results to be derived for epidemic dynamics. While there is no
restriction on the motifs that can be found in such graphs, each node must be
uniquely assigned to a generally clustered subgraph in this construction.
## 1 Introduction
Recent progress on exact analytic approaches to epidemics on clustered
networks has been extremely fast. Models have been proposed based on
households [17, 3, 4], and the more general concept of local-global networks
[1, 2]. Another recent innovation has come from generalisations of random
graph theory [13, 10, 8], and at the same time, general methods have been
proposed for manipulation of master equations [16, 15]. These complement the
traditional epidemiological approach to clustering based on moment closure [9]
that has recently been applied graphs with more general motif structure [7].
This paper draws on much of this recent activity, making three main
contributions. Firstly, a set of networks is defined using the new concept of
a locale (which is distinct from the recently introduced concept of a role
[8]) that have no restriction on the motifs that can be present. Secondly,
exact epidemic dynamics are derived for these networks—the first time that
manifestly exact results for transient epidemic dynamics of an infinite
clustered network with non-homogeneous mixing outside the clusters have been
derived. Finally, techniques are presented for practical efficient calculation
of quantities of interest.
## 2 General Theory
### 2.1 Network generation
We start with the definition of a network (or graph—we use the terms
interchangeably) $G$ of size $N$ as a set of nodes (vertices)
$V\cong\mathbb{Z}_{N}$, which are indexed by $i,j,\dots\in\mathbb{Z}_{N}$, and
a set of links (edges) $E\subseteq V\times V$. The information contained in a
network can be encoded in an adjacency matrix $\mathbf{A}=(A_{ij})$, whose
elements are given by
$A_{ij}=\begin{cases}1&\text{ if }\left(i,j\right)\in E\text{ ,}\\\ 0&\text{
otherwise.}\end{cases}$ (1)
Here we consider symmetric, non-weighted networks without self-links and so
$A_{ii}=0$, $A_{ij}=A_{ji}$.
We now present a model for network creation that is both more general than
previous work, and also allows significant analytic progress to be made. This
starts by defining a set of objects we call stubby subnets, which are indexed
by type $\sigma$. A stubby subnet of type $\sigma$ and size $n_{\sigma}$
consists of three elements:
1. 1.
A set of nodes $v^{\sigma}\cong\mathbb{Z}_{n_{\sigma}}$;
2. 2.
A set of within-subnet links, $e^{\sigma}\subseteq v^{\sigma}\times
v^{\sigma}$, with a within-subnet adjacency matrix $\mathbf{a}^{\sigma}$
defined as for $\mathbf{A}$ above;
3. 3.
A vector of ‘stubs’ $\mathbf{s}^{\sigma}$, such that $\forall i\in
v^{\sigma},s_{i}^{\sigma}\in\mathbb{Z}$.
A full network is then constructed in the following way. Firstly, we take a
number $M_{\sigma}\gg 1$ of each stubby subnet type, such that the network
size and nodes are given respectively by
$N=\sum_{\sigma}M_{\sigma}n_{\sigma}\text{ ,}\qquad
V=\bigoplus_{\sigma}\bigoplus_{m=1}^{M_{\sigma}}v^{\sigma}\text{ .}$ (2)
Here we use tensor sums $\oplus$ to represent the aggregation of subnet nodes
without the removal of ‘duplicates’ that would be implicit in set-theoretic
union. We can also apply this concept to the within-subnet links, providing
one part of the full link set,
$E_{1}=\bigoplus_{\sigma}\bigoplus_{m=1}^{M_{\sigma}}e^{\sigma}\text{ .}$ (3)
The remainder of links are then provided by constructing a full vector of
‘stubs’ and connecting these using the standard Configuration Model [11].
$\mathbf{S}=\bigoplus_{\sigma}\bigoplus_{m=1}^{M_{\sigma}}\mathbf{s}^{\sigma}\text{
,}\quad E_{2}=\mathrm{ConfigurationModel}(V,\mathbf{S})\text{ ,}\quad
E=E_{1}\cup E_{2}\text{ .}$ (4)
In the limit where the network is sufficiently large, no duplicate links will
be produced through the union of $E_{1}$ and $E_{2}$, however for explicit
generation of finite-size networks, the removal of duplicates implicit in (4)
is commonly used.
Having defined such a network, it is straightforward to calculate degree
distributions and clustering coefficients, since a node $i$ from a stubby
subnet of type $\sigma$ has degree and clustering coefficient
$d_{i}=s_{i}^{\sigma}+\sum_{j}a_{ij}^{\sigma}\text{ ,
and}\quad\phi_{i}=\frac{{(\mathbf{a}^{\sigma})^{3}}_{ii}}{d_{i}(d_{i}-1)}\text{
.}$ (5)
From consideration of the standard configuration model, a giant component
emerges within a network of this kind provided
$\sum_{\sigma}M_{\sigma}D^{\sigma}(D^{\sigma}-2)>0\text{
,}\quad\text{where}\quad
D^{\sigma}:=\sum_{i=1}^{n_{\sigma}}s_{i}^{\sigma}\text{ .}$ (6)
Note that we have implicitly assumed that all stubby subnets are internally
connected.
### 2.2 Invasion and final size
We now introduce a framework for the determination of whether a network of the
kind considered can support the invasion of a species obeying SIR dynamics. To
do this, we define the concept of a locale, which is a stubby subnet of type
$\sigma$, together with an ‘origin’ node $o\in v^{\sigma}$. Clearly, there are
at least $\sum_{\sigma}n_{\sigma}$ such locales to consider, although
symmetries may reduce the effective number of these. Locale types are denoted
using indices like $\lambda=\left(\sigma,o\right)$.
Invasibility of a network of the type under consideration (i.e. one
constructed from stubby subnets) can therefore be considered by constructing a
branching process on locales. If we define a ‘locale next generation’ matrix
as the number of secondary locales infected by an initially infected locale
early in the epidemic, then we can use the dominant eigenvalue of such a
matrix to define a threshold parameter.
In order to do this, we need to define two dynamical quantities. The first of
these is $T$, the probability that infection eventually passes across a
network link where one node starts infectious and the other susceptible. The
second is $P_{\sigma}(j|o)$, which is the probability that within the locale
$\left(\sigma,o\right)$, where infection is first introduced to node $o$, that
infection eventually reaches node $j\in v^{\sigma}$. The calculation of these
two quantities depends on the precise dynamical system underneath the
transmission process, but once they have been determined, the locale next
generation matrix (interpreted as the expected number of locales of type
$\bar{\lambda}=\left(\bar{\sigma},\bar{o}\right)$ created by a locale of type
$\lambda=\left(\sigma,o\right)$ early in the epidemic) is given by
$\mathcal{K}^{L}_{\bar{\lambda}\lambda}=T\frac{M_{\bar{\sigma}}s^{\bar{\sigma}}_{\bar{o}}}{s_{\mathrm{tot}}}\left(\left(s_{o}-1\right)+\sum_{j\in
v_{\sigma}\ominus o}P_{\sigma}(j|o)s_{j}\right)\text{ ,}$ (7)
where the total number of stubs in the network is
$s_{\mathrm{tot}}=\sum_{\sigma}M_{\sigma}\sum_{i\in
v_{\sigma}}s_{i}^{\sigma}\text{ .}$ (8)
The locale basic reproduction number, which is different from the standard
basic reproductive number $R_{0}$, is then the dominant eigenvalue of this
matrix
$R_{L}:=\left|\left|\mathcal{K}^{L}\right|\right|\text{ .}$ (9)
By using a ‘susceptibility sets’ argument as in [3, 4], the final size of an
epidemic can also be calculated using the following set of transcendental
equations:
$\displaystyle R_{\infty}$
$\displaystyle=1-\frac{\sum_{\sigma}M_{\sigma}\sum_{i\in
v_{\sigma}}x_{i}^{\sigma}}{\sum_{\sigma}M_{\sigma}n_{\sigma}}\text{ ,}$ (10)
$\displaystyle x_{i}^{\sigma}$ $\displaystyle=\pi_{i}^{\sigma}\prod_{j\in
v_{\sigma}\ominus
i}\left(\left(1-P(i|j)\right)+P(i|j)\pi_{j}^{\sigma}\right)\text{ ,}$
$\displaystyle\pi_{i}^{\sigma}$
$\displaystyle=\left((1-T)+T\sum_{\lambda^{\prime}}\frac{M_{\sigma^{\prime}}s_{o^{\prime}}^{\sigma^{\prime}}}{s_{\mathrm{tot}}}\tilde{x}_{o^{\prime}}^{\sigma^{\prime}}\right)^{s_{i}^{\sigma}}\text{
,}$ $\displaystyle\tilde{x}_{i}^{\sigma}$
$\displaystyle=\tilde{\pi}_{i}^{\sigma}\prod_{j\in v_{\sigma}\ominus
i}\left(\left(1-P(i|j)\right)+P(i|j)\pi_{j}^{\sigma}\right)\text{ ,}$
$\displaystyle\tilde{\pi}_{i}^{\sigma}$
$\displaystyle=\left((1-T)+T\sum_{\lambda^{\prime}}\frac{M_{\sigma^{\prime}}s_{o^{\prime}}^{\sigma^{\prime}}}{s_{\mathrm{tot}}}\tilde{x}_{o^{\prime}}^{\sigma^{\prime}}\right)^{s_{i}^{\sigma}-1}\text{
.}$
Here $R_{\infty}$ is the proportion of the population that is ultimately
infected by the epidemic, $x_{i}^{\sigma}$ is the probability that the $i$-th
node in a stubby subnet $\sigma$ avoids infection during the epidemic and
$\pi_{i}^{\sigma}$ is the corresponding probability for avoidance of global
infection. Variables marked with a tilde represent secondary locales in the
susceptibility-set branching process, and other quantities are as defined
above.
### 2.3 Full Dynamics
In order to consider full transient dynamics for the system, we assume that
transmission of infection across a link is a one-step Poisson process,
happening at rate $\tau$, and that recovery is Markovian with rate $\gamma$.
Our methodology is straightforwardly extended to the case where shedding
happens at a variable rate during an individual’s infectious period or the
case of non-exponentially distributed recovery times through the method of
stages (and other compartmental methods). In the Markovian case,
$T=\tau/(\tau+\gamma)$, but to calculate $P(i|j)$ we must consider internal
dynamics for a subnet of size $n$ with adjacency matrix $\mathbf{a}$ and
infection starting on node $o$. Since the general dynamics in this case are
rather hard to write down, we make use of Dirac notation, using the
appropriate links to Markov chains [6], to simplify notation.
#### 2.3.1 Within-subnet dynamics
Our starting point is a node-level state space
$\mathcal{S}=\left\\{\left|S\right>,\left|I\right>,\left|R\right>\right\\}\text{
,}$ (11)
Defined such that, where we use letters $A,B,\ldots$ to represent generic
states
$\left<A|B\right>=\delta_{A,B}\text{ .}$ (12)
We then define five abstract operators: three that return the appropriate
infection state
$\displaystyle\hat{S}\left|S\right>$ $\displaystyle=\left|S\right>,$
$\displaystyle\hat{S}\left|I\right>$ $\displaystyle=0,$
$\displaystyle\hat{S}\left|R\right>$ $\displaystyle=0,$
$\displaystyle\hat{I}\left|S\right>$ $\displaystyle=0,$
$\displaystyle\hat{I}\left|I\right>$ $\displaystyle=\left|I\right>,$
$\displaystyle\hat{I}\left|R\right>$ $\displaystyle=0,$
$\displaystyle\hat{R}\left|S\right>$ $\displaystyle=0,$
$\displaystyle\hat{R}\left|I\right>$ $\displaystyle=0,$
$\displaystyle\hat{R}\left|R\right>$ $\displaystyle=\left|R\right>;$ (13)
and two that correspond to transmission and recovery
$\displaystyle\hat{t}\left|S\right>$ $\displaystyle=\left|I\right>,$
$\displaystyle\hat{t}\left|I\right>$ $\displaystyle=0,$
$\displaystyle\hat{t}\left|R\right>$ $\displaystyle=0,$
$\displaystyle\hat{r}\left|S\right>$ $\displaystyle=0,$
$\displaystyle\hat{r}\left|I\right>$ $\displaystyle=\left|R\right>,$
$\displaystyle\hat{r}\left|R\right>$ $\displaystyle=0.$ (14)
So a general state under consideration obeys
$\left|p\right>\in\mathcal{S}^{\otimes n}\text{
,}\quad\left<1|p\right>=1\text{ , where
}\left|1\right>:=\left(\left|S\right>+\left|I\right>+\left|R\right>\right)^{\otimes
n}\text{ .}$ (15)
This is in contrast to normalisation in quantum mechanics—where states obey
$\left<\psi|\psi\right>=1$—and the ‘ket’ $\left|1\right>$ is henceforth used
without explicit definition to stand for an unweighted sum over basis states.
Where $\hat{\mathcal{O}}$ is an operator defined to act on elements of
$\mathcal{S}$, we define an operator acting on the complete state space using
subscripting so that
$\hat{\mathcal{O}}_{i}:=\mathbb{1}\otimes\cdots\underbrace{\otimes\hat{\mathcal{O}}\otimes}_{i\text{th
place}}\cdots\mathbb{1}\text{ .}$ (16)
Having set up this machinery, we can now write the system’s dynamics in an
extremely compact form:
$\frac{d}{dt}\left|p\right>=\hat{Q}\left|p\right>\text{ , where
}\hat{Q}=\tau\sum_{i}(\hat{t}_{i}-\hat{S}_{i})\sum_{j}a_{ij}\hat{I}_{j}+\gamma\sum_{i}(\hat{r}_{i}-\hat{I}_{i})\text{
.}$ (17)
Despite this compact expression, the actual dimensionality of the system above
grows extremely quickly with network size for numerical and analytical work.
There are two general methods available for increasing the tractability of
these equations, particularly for final outcomes.
#### Path integrals for Markov chains
The outcome probabilities for local subnets can be written in terms of the
following integral
$P(j|o)=\int_{0}^{\infty}\left<p_{j}\right|e^{\hat{Q}t}\left|o\right>dt\text{
,}$ (18)
where we have defined two new states
$\left|o\right>:=\hat{I}_{o}\prod_{j\neq o}\hat{S}_{j}\left|1\right>\text{
,}\qquad\left|p_{j}\right>:=\gamma\hat{I}_{j}\left|1\right>\text{ .}$ (19)
In order to evaluate (18) efficiently, we can make use of the general theory
of path integrals for Markov chains [14]. To do this, we need first to
decompose the state space into an absorbing set $\mathcal{A}$ and a non-
absorbing set $\mathcal{C}$:
$\mathcal{S}^{\otimes n}=\mathcal{A}\cup\mathcal{C}\text{ ,}$ (20)
which can be done through the definition of projection operators
$\displaystyle\hat{P}_{\mathcal{A}}$
$\displaystyle=\sum_{\\{A_{i}\\}_{i=1}^{n}\in\\{S,R\\}^{\otimes
n}}\left|A_{1}\right>\otimes\cdots\otimes\left|A_{n}\right>\left<A_{1}\right|\otimes\cdots\otimes\left<A_{n}\right|\text{
,}$ (21) $\displaystyle\hat{P}_{\mathcal{C}}$
$\displaystyle=\mathbb{1}-\hat{P}_{\mathcal{A}}\text{ .}$
Two further definitions are needed. Firstly, the time evolution operator
restricted to the non-absorbing states is given by
$\hat{Q}_{\mathcal{C}}:=\hat{Q}\circ\hat{P}_{\mathcal{C}}\text{ .}$ (22)
Secondly, in contrast with quantum mechanics, operators are not Hermitian, and
so ‘transposed’ operators that act on the adjoint space of ‘bra’ states are
denoted using the dagger $\dagger$ and are not identical to the un-daggered
operators on ‘ket’ states. Using these definitions, is is possible to write
final outcome probabilities for the epidemic process in a particularly compact
form:
$P(j|o)=\left<p_{j}\right|((\hat{Q}_{\mathcal{C}})^{\dagger})^{-1}\left|o\right>\text{
.}$ (23)
This method of path integrals was applied to household epidemic models in
[15]. In practice, the inverse operator in (23) need not be calculated in
full—for SIR dynamics, a matrix representation will exist in which $Q$ is
triangular, and so quantities of interest can be calculated by solving a
system of triangular linear equations, which is relatively numerically
efficient.
#### Automorphism-driven lumping
Recently, the technique of automorphism-driven lumping has been applied to
epidemic dynamics on networks [16] and percolation [8]. This approach reduces
the complexity of network problems by making systematic use of discrete
symmetries of the network. In particular, the automorphism group of a graph
$G$ of size $n$ with adjacency matrix $\mathbf{a}$ is a subset of the
permutation group: $\mathrm{Aut}(G)\subseteq\mathrm{S}_{n}$. The elements of
the automorphism group leave the adjacency matrix invariant:
$\mathbf{M}\in\mathrm{Aut}(G)\quad\Leftrightarrow\quad\mathbf{a}=\mathbf{M}\mathbf{a}\mathbf{M}^{\mathrm{T}}\text{
.}$ (24)
The use of this insight to lump epidemic equations requires some care in the
labelling of dynamical variables [16]. Using the notation above, we relabel a
generic dynamical state of the system
$\left|A_{1}\right>\otimes\cdots\otimes\left|A_{n}\right>\equiv\left|\\{(A_{1},1),\ldots,(A_{n},n)\\}\right>\text{
,}$ (25)
i.e. we go from an ordered set of states to an unordered set of pairs of
states and node numbers. ‘Lumped’ basis states for the dynamical system (17)
can then be defined according to the orbits of the automorphism group—this
means that states like the above are lumped together into classes like
$L(A_{1},\ldots,A_{n})=\\{\ \\{(A_{1},M(1)),\ldots,(A_{n},M(n))\\}\ |\
\mathbf{M}\in\mathrm{Aut}(G)\ \\}\text{ ,}$ (26)
where $M(i)$ is the index of the non-zero component of the $i$-th row of the
permutation matrix $\mathbf{M}$. The dynamical equivalence of these states can
be seen by repeated substitution of
$\mathbf{a}\rightarrow\mathbf{M}\mathbf{a}\mathbf{M}^{\mathrm{T}}$ into (17).
Clearly, lumping classes must contain states that all have the same
eigenvalues of $\hat{S}$ and $\hat{I}$; and in the limiting case of a fully
connected graph such that $\mathrm{Aut}(G)=\mathrm{S}_{n}$, only these
aggregate eigenvalues are required to describe the system [16].
#### 2.3.2 Global dynamics
Recently, a set of dynamics was presented that are a manifestly exact
description of the mean behaviour of an SIR epidemic on a configuration-model
network [2] (equivalent to a stubby subnet model where all subnets have one
node). We now re-write this in Dirac notation, so that this approach may be
readily combined with the within-subnet dynamics above to define exact global
dynamics.
Our starting point is a set of states that represent a number of ‘remaining
half-links’
$\mathcal{S}=\left\\{\left|l\right>\right\\}_{l=0}^{k_{\mathrm{max}}}\text{ ,
such that }\left<l^{\prime}|l\right>=\delta_{l,l^{\prime}}\text{ ,}$ (27)
where $k_{\mathrm{max}}$ is the maximum node degree (or more generally maximum
number of stubs). We define two operators on such states: a link number
operator, and a link-number lowering operator:
$\hat{l}\left|l\right>=l\left|l\right>\text{
,}\qquad\hat{l}^{-}\left|l\right>=\begin{cases}\left|l-1\right>&\text{ if
}l\geq 1\text{ ,}\\\ 0&\text{ otherwise.}\end{cases}$ (28)
We now consider how remaining half-links interact with disease state. These
are taken as a tensor product,
$\left|A,l\right>=\left|A\right>\otimes\left|l\right>\text{ , so that
}\left<B,l^{\prime}|A,l\right>=\delta_{A,B}\delta_{l,l^{\prime}}\text{ .}$
(29)
By construction, however, recovered individuals lose all their half-links, so
the state space for this system is
$\mathcal{S}=\left\\{\left|S,l\right>,\left|I,l\right>,\left|R,0\right>\right\\}_{l=0}^{k_{\mathrm{max}}}\text{
.}$ (30)
We then define four operators on this space, which we present in terms of
their non-trivial action
$\displaystyle\hat{t}\left|S,l\right>$
$\displaystyle:=\left(\hat{t}\left|S\right>\right)\otimes\left|l\right>=\left|I,l\right>\text{
,}$ (31) $\displaystyle\hat{b}\left|S,l\right>$
$\displaystyle:=\left(\hat{t}\left|S\right>\right)\otimes\left(\hat{l}^{-}\left|l\right>\right)=\left|I,l-1\right>\text{
,}$ $\displaystyle\hat{l}^{-}\left|A,l\right>$
$\displaystyle:=\left|A\right>\otimes\left(\hat{l}^{-}\left|l\right>\right)=\left|A,l-1\right>\text{
,}$ $\displaystyle\hat{r}\left|I,l\right>$
$\displaystyle:=\left|R,0\right>\text{ .}$
Three of these operators are simple uplifts, but the operator $\hat{b}$ for
global infection is new. To define the dynamics of this system, we start with
a general state
$\left|p\right>=\sum_{l}\left(x_{l}(t)\left|S,l\right>+y_{l}(t)\left|I,l\right>\right)+z(t)\left|R,0\right>\text{
,}$ (32)
which obeys
$\left<1|p\right>=1\text{ , for
}\left|1\right>:=\sum_{l}\left(\left|S,l\right>+\left|I,l\right>\right)+\left|R,0\right>\text{
.}$ (33)
There is also a non-linear term for the density of infection amongst free
half-links that appears in the system,
$\rho[p]:=\frac{\left<1\right|\hat{I}\hat{l}\left|p\right>}{\left<1\right|\hat{l}\left|p\right>}\text{
.}$ (34)
Then an exact representation of expected SIR dynamics on a configuration-model
network is given by
$\displaystyle\hat{Q}[p]$
$\displaystyle:=\gamma\left(\hat{r}-\hat{I}\right)+\tau\left(\hat{l}^{-}-\mathbb{1}\right)\hat{l}\hat{I}+\rho[p]\left(\gamma+\tau\right)\left(\hat{l}^{-}-\mathbb{1}\right)\hat{l}+\rho[p]\tau\left(\hat{b}-\hat{S}\right)\hat{l}\text{
,}$ (35) $\displaystyle\frac{d}{dt}\left|p\right>$
$\displaystyle=\hat{Q}[p]\left|p\right>\text{ .}$
The significance of these dynamics is that they do not grow in size with
network size; in fact, they are exact in the infinite-size limit, which is
inaccessible through simulation or direct integration of (17).
#### 2.3.3 Full system dynamics
For a network made up of stubby subnets, it is possible to a make the same
construction as above, where global links are made along with the epidemic
process. In this case, a general state can be written
$\left|p\right>=\sum_{\sigma,\left\\{A_{i},l_{i}\right\\}_{i=1}^{n_{\sigma}}}{{p_{\sigma}}^{A_{1}\ldots
A_{n}}}_{l_{1}\ldots
l_{n}}(t)\left|\sigma\right>\otimes\left|A_{1},l_{1}\right>\otimes\cdots\otimes\left|A_{n_{\sigma}},l_{n_{\sigma}}\right>\text{
,}$ (36)
where $\left<\bar{\sigma}|\sigma\right>=\delta_{\bar{\sigma},\sigma}$ as would
be expected. Clearly, any attempt to write down differential equations for the
tensor representation of this system, ${{p_{\sigma}}^{A_{1}\ldots
A_{n}}}_{l_{1}\ldots l_{n}}(t)$, will involve extremely complex expressions.
By contrast, using the formalism of Dirac notation and operators that we have
developed above, we can write the exact dynamics for this system as
$\displaystyle\hat{P}_{\sigma}$
$\displaystyle:=\sum_{\left\\{A_{i},l_{i}\right\\}_{i=1}^{n_{\sigma}}}\left|\sigma\right>\otimes\left|A_{1},l_{1}\right>\otimes\cdots\otimes\left|A_{n_{\sigma}},l_{n_{\sigma}}\right>\left<A_{n_{\sigma}},l_{n_{\sigma}}\right|\otimes\cdots\otimes\left<A_{1},l_{1}\right|\otimes\left<\sigma\right|$
(37) $\displaystyle\rho[p]$
$\displaystyle:=\frac{\left<1\right|\sum_{\sigma}\sum_{i=1}^{n_{\sigma}}\hat{I}_{i}\hat{l}_{i}\hat{P}_{\sigma}\left|p\right>}{\left<1\right|\sum_{\sigma}\sum_{i=1}^{n_{\sigma}}\hat{l}_{i}\hat{P}_{\sigma}\left|p\right>}\text{
,}$ $\displaystyle\hat{Q}[p]$
$\displaystyle:=\gamma\sum_{i}\left(\hat{r}_{i}-\hat{I}_{i}\right)+\tau\sum_{i}\left(\hat{l}^{-}_{i}-\mathbb{1}\right)\hat{l}_{i}\hat{I}_{i}+\tau\sum_{i}\left(\hat{t}_{i}-\hat{S}_{i}\right)\sum_{\sigma,j}a^{\sigma}_{ij}\hat{I}_{j}\hat{P}_{\sigma}$
$\displaystyle\quad+\rho[p]\left(\gamma+\tau\right)\sum_{i}\left(\hat{l}^{-}_{i}-\mathbb{1}\right)\hat{l}_{i}+\rho[p]\tau\sum_{i}\left(\hat{b}_{i}-\hat{S}_{i}\right)\hat{l}_{i}\text{
,}$ $\displaystyle\frac{d}{dt}\left|p\right>$
$\displaystyle=\hat{Q}[p]\left|p\right>\text{ .}$
These equations have the same significance as above: the exact expected
epidemic dynamics of a class of clustered dynamics can be calculated for the
infinite-size limit of a network.
## 3 Examples
We now turn to some examples of the methodology presented above to specific
networks. Throughout this section we work in natural units such that the
recovery rate $\gamma=1$.
### 3.1 Invasion and final size
We consider invasion on the two locales shown in Figure 1. These networks are
constructed from the envelope / diamond motif as shown, so that every
individual has exactly $n$ links. This means that all differences between this
model and an $n$-regular random graph derive from the presence and structure
of short loops in the network and not heterogeneity in node degree. The locale
basic reproductive ratio is given by:
$R_{L}=\big{(}\tau\big{(}2(n-3)^{2}+(n(25n-142)+204)\tau+(n(133n-716)+982)\tau^{2}\\\
+(n(377n-1948)+2570)\tau^{3}+(n(563n-2846)+3672)\tau^{4}\\\
+2(n(193n-968)+1239)\tau^{5}+12(8(n-5)n+51)\tau^{6}\big{)}\big{)}\\\
/\big{(}(2n-5)(1+\tau)^{4}(1+2\tau)^{2}(1+3\tau)\big{)}\text{ .}$ (38)
Final sizes are calculated using (10). In Panes (c) and (d) of Figure 1, to
compare the asymptotically exact results (blue line) with finite-size
networks, $10^{6}$ Simulations were run for envelope-based networks of size
100 and 1000, with $n=4$, over a range of transmission parameter values. For
comparison, theoretical curves were also plotted for a configuration model
where every node has four links (red line), and a household model for
households of size 4 where every individual has one stub (green line). Final
sizes for these two comparators are a special case of the analysis in [3, 4].
Clearly, all of these comparator networks also have the property that
neighbourhood sizes are uniformly equal to four, and so all outcome
differences are due to local clustering structure.
The results shown in Panes (c) and (d) of Figure 1 show firstly that this
local clustering structure does have a significant impact on epidemic
outcomes, and also that even for relatively small networks the results of
simulation demonstrate this difference and agree well with the asymptotic
result. The blue line representing final outcomes also has two interesting
features: there is a short plateau of small but finite final sizes above the
invasion threshold; and for very fast transmission, the predicted final sizes
are larger than for the unclustered regular graph.
### 3.2 Full Dynamics
While invasion thresholds are of practical interest, transient dynamical
features of epidemics are also important, and are not always simply determined
by consideration of thresholds. Figure 2 shows the exact transient behaviour
for two special graphs, both of which give all nodes degree three: (a) a
configuration-model network where each node has 3 stubs; (b) a stubby-subnet
graph composed of triangles with each node having one stub. The dynamics as
defined above give the epidemic curves shown in (c) for the CM network and (d)
for the triangle-based network respectively.
These show the interesting feature that once we are in a region of much faster
transmission than is required for invasion, the clustered network exhibits
later but higher peaks—an analogue of the lager final sizes seen for clustered
networks at very large $\tau$ above.
## 4 Other solvable networks
It has been clear for some time that a network (or otherwise structured
population) with a local-global distinction will admit a solution to an
epidemic on that network [1]. As a practical adjunct to this, both the local
and global features of the network must individually admit solution. The
stubby-subnet networks here propose one such distinction: each node can be
uniquely assigned to a local unit of clustered structure; and global mixing
happens through a configuration model network.
We now consider three other versions of this concept, firstly by introducing
assortative mixing outside the subnet, secondly using the recently defined
role-based networks, and finally to weighted networks.
### 4.1 Assortativity
In [12], a generalisation of the configuration model was developed to
incorporate the notion of assortativity. Such assortativity (or even
disassortativity) is a mainstay of epidemiology, and much theoretical effort
has been expended to model its effects [5]. To describe assortativity, we
introduce a correlation matrix ${C}_{\bar{\lambda},\lambda}$ (analogous to the
$e_{kl}$ of [12]) that multiplies the probabilities that two locales are
linked globally compared to the configuration model. For such a network, the
locale next generation matrix is
$\mathcal{K}^{L}_{\bar{\lambda}\lambda}=T\frac{M_{\bar{\sigma}}s^{\bar{\sigma}}_{\bar{o}}}{s_{\mathrm{tot}}}\left(\left(s_{o}-1\right)C_{\bar{\lambda},\lambda}+\sum_{j\in
v_{\sigma}\ominus
o}P_{\sigma}(j|o)s_{j}C_{\bar{\lambda},(\sigma,j)}\right)\text{ ,}$ (39)
and an appropriate threshold parameter will be given by the dominant
eigenvalue of this matrix. Exact transient dynamics for such a system should
also be straightforward to write down: in addition to indexing a node with its
effective remaining half-links and disease state, each node should also be
indexed by locale. Instead of having homogeneous transmission on the basis of
pairing half-links at rate $\tau$, the rate should then be multiplied by
${C}_{\bar{\lambda},\lambda}$. Of course, this yields equations that are at
least quadratic rather than linear in maximum node degree, making numerical
integration correspondingly more difficult.
### 4.2 Role-based networks
Role-based networks as considered in [13, 10, 8] involve a different
definition of local and global. In these networks, it is links that can be
uniquely assigned to a local unit of clustered structure, meaning that nodes
can be attached to many different clustered subgraphs. This clearly allows a
next-generation matrix to be established by indexing cases by the unit of
structure through which they acquired infection, as in [10]. The definition of
manifestly exact dynamics is less clear in this case, however dynamical
approaches such as [18] that are in extremely good numerical agreement with
simulation, and may turn out to be exact through further work, can clearly be
extended to role-based networks. The primary differences between stubby-subnet
and role-based networks are that the former can specify an exact structure of
stubs for each node in a clustered motif, while the latter can involve each
node in several motifs. As such, these are best seen as complementary
approaches to the fast-moving field of solvable clustered networks.
### 4.3 Weighted networks
While all networks discussed above have been topological (i.e. links are
either present or not) all of the analysis above carries through exactly if
within-subnet links are weighted, so $a^{\sigma}_{ij}\in\mathbb{R}$. It is
also possible to stratify global links into multiple contexts, each with a
given strength (i.e. different values of $T$) although this latter
modification does increase the system dimensionality, while weighting within-
subnet dynamics does this only if the weighting breaks a discrete symmetry of
the topological network.
## Acknowledgements
Work funded by the UK Engineering and Physical Sciences Research Council
(Grant Number EP/H016139/1). The author would like to thank Matt Keeling and
Josh Ross for helpful discussions and comments on this work.
## References
* [1] F. Ball and P. Neal, A general model for stochastic SIR epidemics with two levels of mixing, Mathematical Biosciences, 180 (2002), pp. 73–102.
* [2] , Network epidemic models with two levels of mixing, Mathematical Biosciences, 212 (2008), pp. 69–87.
* [3] F. Ball, D. Sirl, and P. Trapman, Threshold behaviour and final outcome of an epidemic on a random network with household structure, Advances in Applied Probability, 41 (2009), pp. 765–796.
* [4] , Analysis of a stochastic SIR epidemic on a random network incorporating household structure, Mathematical Biosciences, 224 (2010), pp. 53–73.
* [5] O. Dieckmann and J. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, J Wiley, 2000\.
* [6] P. J. Dodd and N. M. Ferguson, A many-body field theory approach to stochastic models in population biology, PLoS ONE, 4 (2009), p. e6855.
* [7] T. House, G. Davies, L. Danon, and M. J. Keeling, A motif-based approach to network epidemics, Bulletin of Mathematical Biology, 71 (2009), pp. 1693–1706.
* [8] B. Karrer and M. E. J. Newman, Random graphs containing arbitrary distributions of subgraphs, arXiv:1005.1659v1, (2010).
* [9] M. J. Keeling, The effects of local spatial structure on epidemiological invasions, Proc Biol Sci, 266 (1999), pp. 859–67.
* [10] J. Miller, Percolation and epidemics in random clustered networks, Physical Review E, 80 (2009), pp. 1–4.
* [11] M. Molloy and B. Reed, A critical point for random graphs with a given degree sequence, Random Struct. Algorithms, 6 (1995), pp. 161–179.
* [12] M. Newman, Assortative mixing in networks, Physical Review Letters, 89 (2002), p. 208701.
* [13] M. Newman, Random graphs with clustering, Physical Review Letters, 103 (2009), pp. 1–4.
* [14] P. K. Pollett and V. E. Stefanov, Path integrals for continuous-time Markov chains, J. Appl. Prob., 39 (2002), pp. 901–904.
* [15] J. V. Ross, T. House, and M. J. Keeling, Calculation of disease dynamics in a population of households, PLoS ONE, 5 (2010), p. e9666.
* [16] P. L. Simon, M. Taylor, and I. Z. Kiss, Exact epidemic models on graphs using graph automorphism driven lumping. Sussex Maths Preprint: SMRR-2010-02. Published online at the Journal of Mathematical Biology ahead of print, 2010.
* [17] P. Trapman, On analytical approaches to epidemics on networks, Theoretical Population Biology, 71 (2007), pp. 160–173.
* [18] E. M. Volz, Dynamics of infectious disease in clustered networks with arbitrary degree distributions, arXiv:1006.0970v1, (2010).
(a)
(b)
(c)
(d)
Figure 1: Epidemics on envelope / diamond motif-based networks. (a) and (b)
show the two locales involved. Bottom panes show final sizes for $10^{6}$
simulations on networks of size (c) 100 and (d) 1000. Each translucent dot
represents a realisation; blue lines are asymptotic predictions for a regular
graph of degree 4; red lines are the asymptotic predictions for the envelope
network with $n=4$; and green lines are asymptotic predictions for four-
cliques with one global link per node.
(a)
(b)
(c)
(d)
Figure 2: Exact transient epidemic dynamics for two special networks. (a)
shows a typical location in the unclustered graph, and (b) shows a typical
location in the clustered graph. Epidemic curves (grey) for different
parameter values are shown in (c), (d) respectively. Peak times (blue) and
peak heights (red) are projected onto the appropriate axes.
|
arxiv-papers
| 2010-06-17T14:41:26 |
2024-09-04T02:49:10.981145
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Thomas House",
"submitter": "Thomas House",
"url": "https://arxiv.org/abs/1006.3483"
}
|
1006.3484
|
# Mass discrepancy in galaxy clusters as a result of the offset between dark
matter and baryon distributions
HuanYuan Shan1,2, Bo Qin2,⋆, and HongSheng Zhao2,3
1Department of Astronomy, School of Physics, Peking University, Beijing,
100871, China
2National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China
3SUPA, University of St Andrews, KY16 9SS, UK E-mail: shanhuany@gmail.com,
qinbo@bao.ac.cn
(Accepted …. Received …; in original form …)
###### Abstract
Recent studies of lensing clusters reveal that it might be fairly common for a
galaxy cluster that the X-ray center has an obvious offset from its
gravitational center which is measured by strong lensing. We argue that if
these offsets exist, then X-rays and lensing are indeed measuring different
regions of a cluster, and may thus naturally result in a discrepancy in the
measured gravitational masses by the two different methods. Here we
investigate theoretically the dynamical effects of such lensing-X-ray offsets,
and compare with observational data. We find that for typical values, the
offset alone can give rise to a factor of two difference between the lensing
and X-ray determined masses for the core regions of a cluster, suggesting that
such “offset effect” may play an important role and should not be ignored in
our dynamical measurements of clusters.
###### keywords:
dark matter-gravitational lensing-X-rays: galaxies: clusters
### 1 Introduction
Galaxy clusters, the largest gravitationally-bound structures in the universe,
are ideal cosmological tools. Accurate measurements of their masses provide a
crucial observational constraint on cosmological models. Several dynamical
methods have been available to estimate cluster masses, such as (1) optical
measurements of the velocity dispersions of cluster galaxies, (2) measurements
of the X-ray emitting gas, and (3) gravitational lensing. Good agreements
between these methods have been found on scales larger than cluster cores.
However, joint measurements of lensing and X-rays often identify large
discrepancies in the gravitational masses within the central regions of
clusters by the two methods, and the lensing mass has always been found to be
$2-4$ times higher than the X-ray determined mass. This is the so-called “Mass
Discrepancy Problem” (Allen 1998; Wu 2000). Many plausible explanations have
been suggested, e.g., the triaxiality of galaxy clusters (Morandi et al.
2010), the oversimplification of the strong lensing model for the central mass
distributions of clusters (Bartelmann & Steinmetz 1996), the inappropriate
application of the hydrostatic equilibrium hypothesis for the central regions
of clusters (Wu 1994; Wu & Fang 1997), or the magnetic fields in clusters
(Loeb & Mao 1994).
Recently Richard et al. (2010) present a sample of $20$ strong lensing
clusters taken from the Local Cluster Substructure Survey (LoCuSS), among
which $18$ clusters have X-ray data from Chandra observations (Sanderson et
al. 2009). They show that the X-ray/lensing mass discrepancy is $1.3$ at
$3\sigma$ significance — clusters with larger substructure fractions show
greater mass discrepancies, and thus greater departures from hydrostatic
equilibrium.
On the other hand, lensing observations of the bullet cluster 1E0657-56 (Clowe
et al. 2006), combined with earlier X-ray measurements (Markevitch et al.
2006), clearly indicate that the gravitational center of the cluster has an
obvious offset from its baryonic center. Furthermore, recent studies (Shan et
al. 2010) of lensing galaxy clusters reveal that offset between the lensing
center and X-ray center appears to be quite common, especially for unrelaxed
clusters. Among the recent sample of 38 clusters of Shan et al. (2010), $45\%$
have been found to have offsets greater than $10^{\prime\prime}$, and $5$
clusters even have offsets greater than $40^{\prime\prime}$. Motivated by such
observations, we propose to investigate galaxy cluster models where the center
of the dark matter (DM) halo does not coincide with the center of the X-ray
gas (See Figure 1).
Figure 1: Offset between the dark matter center and the X-ray center in a
galaxy cluster.
If the X-ray center of a cluster has an offset from its lensing
(gravitational) center, then the X-rays and lensing are indeed measuring
different regions of the cluster. Given the same radius, the lensing is
measuring the DM halo centered at the gravitational center (shown by the dark
blue sphere in Figure 1 ), while the X-rays are measuring the sphere of the
halo that is offset from the true gravitational center (shown by the red
circle in Figure 1). In this case, there will always be a natural discrepancy
between the lensing and X-ray measured masses — or specifically, the X-ray
mass will always be lower than the lensing mass, just as the long-standing
“mass discrepancy problem” has indicated.
In this paper, we investigate the lensing-X-ray mass discrepancy caused by the
offsets between DM and X-ray gas. To check our predictions, we compile a
sample of $27$ clusters with good lensing and X-ray measurements. We conclude
that such “offset” effect should not be ignored in our dynamical measurements
of galaxy clusters. A flat $\Lambda$CDM cosmology is assumed throughout this
paper, where $\Omega_{m}$=0.3, $\Omega_{\Lambda}$=0.7, and $\rm
H_{0}=70\,km\,s^{-1}Mpc^{-1}$.
### 2 Mass discrepancy as a result of the dark matter-baryon offset
We model our galaxy cluster with a fiducial model as the following: (1) the DM
halo is modeled by the Navarro-Frenk-White (NFW) profile (Navarro et al. 1997)
with concentration $c=4.32$ and scaled radius $r_{s}=516\,{\rm kpc}$, (2) the
gas distribution is modeled by a $\beta$ model with $\beta=0.65$, the cluster
core radius $r_{c}=200\,{\rm kpc}$, and the gas fraction $f_{\rm gas}=12\%$,
(3) the mass density of the BCG is described by a Singular Isothermal Sphere
(SIS) with a velocity dispersion of $300\,\rm km/s$.
The projected mass within a sphere of radius $R_{x}$ is
$\displaystyle m(R_{x},d)$ $\displaystyle=$
$\displaystyle\int_{0}^{2\pi}\int_{0}^{R_{x}}\left[\Sigma_{\rm
NFW}(R^{\prime})+\Sigma_{\rm gas}(R)\right.$ $\displaystyle\left.+\Sigma_{\rm
BCG}(R^{\prime})\right]R^{\prime}\,dR^{\prime}\,d{\theta},$
where $R^{\prime}=\sqrt{d^{2}+R^{2}+2dR\cos\theta}$ is the 2-D radius from the
halo center, $R$ is the 2-D radius from the X-ray gas center, $d$ is the 2-D
offset between the halo center and X-ray center, and $\Sigma_{\rm NFW}$,
$\Sigma_{\rm gas}$, and $\Sigma_{\rm BCG}$ are the projected mass densities of
the DM halo, the gas and the BCG, respectively. For a given radius $R_{x}$,
the gravitational mass measured by lensing $m_{\rm lens}$ can be given by
$m(R_{x},0)$ (as shown by the dark blue sphere in Figure 1), while the
projected mass measured by X-rays $m_{\rm xray}$ is described by $m(R_{x},d)$
(the mass within the red circle in Figure 1). We now calculate the mass ratio
$m(R_{x},d)/m(R_{x},0)$, or equivalently, $m_{\rm lens}/m_{\rm xray}$.
Figure 2 shows the mass ratio as a function of the 2-D offset $d$, for a
typical rich cluster. The solid curves are the mass ratio with the fiducial
model, the dashed and dotted curves are the mass ratio with the NFW
concentration $c=4.04$ and $5.13$ (top left), the cluster core radius
$r_{c}=150\,{\rm kpc}$ and $400\,{\rm kpc}$ (top right), the $\beta$ index
$\beta=0.6$ and $0.9$ (bottom left), the gas fraction $f_{\rm gas}=0.1$ and
$0.2$, respectively. For these cases, the three curves from top to bottom are
for the three measuring radii $R_{x}=50\,{\rm kpc},100\,{\rm kpc},200\,{\rm
kpc}$, respectively. From Figure 2 we have the following conclusions:
(1) The lensing measured mass $m_{\rm lens}$ is always higher than the X-ray
measured mass $m_{\rm xray}$. For typical values of offset $d=100\,{\rm kpc}$
and $R_{x}=100\,{\rm kpc}$, $m_{\rm lens}/m_{\rm xray}\sim 2$, comparable to
the ratio found in early studies (Allen 1998; Wu 2000; Richard et al. 2010).
(2) The “Offset Effect” we are reporting here should contribute significantly
to the long-standing “Mass Discrepancy Problem”.
(3) The ratio of $m_{\rm lens}/m_{\rm xray}$ increases with offset $d$.
(4) $m_{\rm lens}/m_{\rm xray}$ depends very strongly on $R_{x}$. Here $R_{x}$
acts like the arc radius $r_{\rm arc}$ in strong lensing, i.e., we only
measure the enclosed mass within a small region of $R\leq R_{x}$. When $R_{x}$
is very small, the offset effect is most prominent and gives large $m_{\rm
lens}/m_{\rm xray}$. Increasing $R_{x}$ will reduce $m_{\rm lens}/m_{\rm
xray}$. When $R_{x}$ is very large (compared with $d$), the offset effect will
be “smeared out”, and the $m_{\rm lens}$-$m_{\rm xray}$ discrepancy introduced
by the offset will vanish.
(5) The mass ratio is very sensitive to the NFW concentration, and it
increases dramatically with $c$.
(6) The mass ratio increases with the core radius, and decreases with $\beta$
index and gas fraction. However, the mass ratio is not very sensitive to the
gas model.
Figure 2: Ratio of projected gravitational masses, as a function of the 2-D
offset $d$ and the measuring radius $R_{x}$. The solid curves are the mass
ratio for the fiducial model with $c=4.32$, $r_{s}=516\rm kpc$, $\beta=0.65$,
$r_{c}=200\rm kpc$, and $f_{\rm gas}=0.12$. The dashed and dotted curves are
for the NFW concentration $c=4.04$ and $5.13$ (top left), the cluster core
radius $r_{c}=150\,{\rm kpc}$ and $400\,{\rm kpc}$ (top right), the $\beta$
index $\beta=0.6$ and $0.9$ (bottom left), the gas fraction $f_{\rm gas}=0.1$
and $0.2$, respectively. The three dotted (dashed, solid as well) curves from
top to bottom in one panel correspond to $R_{x}=50,100,200\,\rm kpc$,
respectively.
### 3 Comparison with Observational Data
To compare with our theoretical predictions, we compile a sample of $27$
clusters with $48$ arc-like images, which have both strong lensing and X-ray
measurements. The clusters and their lensing and X-ray data are listed in
Table 1. For the $22$ arcs that have no redshift information, we estimate
their lensing masses $m_{\rm lens}$ by assuming the mean redshifts of
$\left<z_{d}\right>=0.8$ and $2.0$, respectively. The X-ray data are taken
from Tucker et al. (1998), Wu (2000), Bonamente et al. (2006), and references
therein. The offsets between lensing and X-ray centers are taken from Shan et
al. (2010). The clusters in our table are classified as relaxed (with cooling
flow) and unrelaxed (which are dynamically unmature), from their X-ray
morphologies. The definition has been used in the literature by Allen (1998),
Wu (2000), Baldi et al. (2007), and Dunn & Fabian (2008).
Mass from strong lensing. Assuming a spherical matter distribution, one can
calculate the gravitational mass of a galaxy cluster projected within a radius
of $r_{\rm arc}$ on the cluster plane as
$m_{\rm lens}(<r_{\rm arc})=\pi r_{\rm arc}^{2}\Sigma_{\rm crit},$ (1)
where $\Sigma_{\rm crit}=\frac{c^{2}}{4\pi G}\frac{D_{s}}{D_{l}D_{ls}}$ is the
critical surface mass density, $D_{l}$, $D_{s}$ and $D_{ls}$ are the angular
diameter distances to the cluster, to the background galaxy, and from the
cluster to the galaxy, respectively. The above equation is actually the
lensing equation for a cluster lens of spherical mass distribution with a
negligible small alignment parameter for the distant galaxy within $r_{\rm
arc}$. The values of $m_{\rm lens}$ within the arc radius $r_{\rm arc}$ are
listed in Table 1.
Allen (1998) pointed out that the use of more realistic, elliptical mass
models can reduce the masses within the arc radii by up to $40\%$, though a
value of $20\%$ is more typical. However, such corrections are still not very
significant compared with the large discrepancies between the lensing and
X-ray determined masses. We will discuss it in more detail in the next
section.
Mass from X-rays. Assuming that the intra-cluster gas is isothermal and in
hydrostatic equilibrium, the cluster mass $m(r)$ enclosed within a radius $r$
can be easily calculated from
$-\frac{Gm(r)}{r^{2}}=\frac{kT}{\mu m_{p}}\frac{d{\,\rm ln}n_{\rm
gas}(r)}{dr},$ (2)
where $T$ is the gas temperature, $n_{\rm gas}$ the gas number density,
$m_{p}$ the proton mass, and $\mu=0.585$ the mean molecular weight. Here we
assume that the gas follows the conventional $\beta$ model, i.e., $n_{\rm
gas}(r)=n_{\rm gas}(0)(1+r^{2}/r_{c}^{2})^{-\frac{3\beta}{2}}$. In order to
compare the mass measured by X-rays with the lensing result, we need to
convert this $m(r)$ (i.e., 3-D) into the projected mass $m_{\rm xray}$ (see
e.g. Wu 1994):
$m_{\rm xray}=1.13\times 10^{13}\beta\bar{f}\left({R\over
r_{c}}\right)\left(\frac{r_{c}}{0.1{\rm Mpc}}\right)\left(\frac{kT}{1{\rm
keV}}\right)M_{\odot},$ (3)
where
$\bar{f}(y)=\frac{\pi y^{2}}{2(1+y^{2})^{1/2}},$ (4)
the mass ratio $m_{\rm lens}/m_{\rm xray}$ are listed in Table 1.
Figure 3 shows the relation between the mass ratios $m_{\rm lens}/m_{\rm
xray}$ and the (scaled) offsets for our sample of $27$ clusters ($48$ arc
images). It should be pointed out that the $27$ clusters in our sample have
quite different sizes and masses. This can be seen from the wide range of the
cluster temperatures — from $4~{}\rm keV$ to $14~{}\rm keV$. Therefore, it is
useful to compare the offsets of the clusters on the same scale. We realize
that the M-T relation of clusters scales as${\rm M}\sim{\rm T}^{3/2}$ (e.g.,
Nevalainen et al. 2000; Xu et al. 2001), and that ${\rm M}\sim{\rm R}^{3}$,
where R is the size of the cluster. Therefore, in Figure 3, instead of using
the physical offset $d$, we use a scaled offset which is characterized by
${\rm d_{\rm kpc}}/{\rm T_{\rm keV}}^{1/2}$.
From Figure 3, the mass ratios $m_{\rm lens}/m_{\rm xray}$ exhibits large
dispersions — roughly ranging from $2$ to $4$. Many clusters have large error
bars. It appears that relax clusters (marked by crosses) have smaller $m_{\rm
lens}/m_{\rm xray}$ ratios. The fact that $m_{\rm lens}>m_{\rm xray}$ is
consistent with our theoretical predictions, and the ratio of $m_{\rm
lens}/m_{\rm xray}\sim 2-4$ is also roughly consistent with our predictions as
plotted in Figure 2.
However, no strong correlation has been found between the offset and mass
discrepancies. We notice that many clusters in the sample have very small
offset values — smaller than the errors in lensing and X-ray measurements
which are typically a few arcseconds. So these offset values are not robustly
measured themselves, and we thus remove these data points and only focus on
clusters with large offsets of $d>10^{\prime\prime}$, as has been suggested in
Shan et al. (2010). This leaves a sub-sample of only $24$ arc images. The
dashed line in Figure 3 shows a $\chi^{2}$ fit to this sub-sample, which
satisfies $m_{\rm lens}/m_{\rm xray}\sim 3.24(\frac{d/100\,{\rm
kpc}}{\sqrt{kT/{\rm keV}}})^{0.20}$ with a reasonable $\chi^{2}=0.75$. We can
find $m_{lens}/m_{xray}$ increasing slightly with $d$.
Figure 3: The ratio of lensing and X-ray determined masses for our sample of
$27$ clusters ($48$ arc images). The x-label show the scaled offset between DM
and baryons. The squares denote unrelaxed clusters, and crosses the relaxed
clusters. The dashed line shows a $\chi^{2}$ fit satisfying $m_{\rm
lens}/m_{\rm xray}\sim 3.24(\frac{d/100\,{\rm kpc}}{\sqrt{kT/{\rm
keV}}})^{0.20}$ with $\chi^{2}=0.75$ for the clusters with offset larger than
$10^{\prime\prime}$.
### 4 Discussion and Conclusions
As has been reported by Shan et al. (2010), it might be fairly common in
galaxy clusters that the X-ray center has an obvious offset from the
gravitational center. We have explored the dynamical consequences of this
lensing-X-ray offset and tried to attribute such an effect to the long-
standing “Mass Discrepancy Problem” in galaxy clusters. Our theoretical model
predicts that such an offset effect will always result in a larger $m_{\rm
lens}$ than $m_{\rm xray}$, with a typical mass ratio $m_{\rm lens}/m_{\rm
xray}\sim 2$, which is consistent with observations.
To test our model, we have compiled a sample of $27$ clusters, and studied in
detail their lensing and X-ray properties and obtained their lensing and X-ray
masses, $m_{\rm lens}$ and $m_{\rm xray}$. The lack of strong correlation
between $m_{\rm lens}/m_{\rm xray}$ and the offset $d$ suggests that the
problem is more complicated. As we have found in Section 2, $m_{\rm
lens}/m_{\rm xray}$ is not only a function of $d$, but also depends very
strongly on $R_{x}$ (or the arc radius $r_{\rm arc}$). Apparently, each
cluster in our sample has quite different $r_{\rm arc}$.
Probably, other mechanisms than the offset effect should play important roles,
and the lensing-X-ray mass discrepancy may not be just from one mechanism, but
a combination of many effects:
(1) The central regions of clusters may be still undergoing dynamical
relaxation, and the X-ray gas may not be in good hydrostatic equilibrium.
Therefore, large errors could be induced in the X-ray measurement of cluster
cores, especially for unrelaxed clusters.
(2) The spherical models are too simple to reflect the real mass distribution
of clusters. The use of more realistic mass model could reduce the lens mass
within the arc radius by up to $40\%$, though values of $\sim 20\%$ are more
typical (Bartelmann 1995; Allen 1998).
(3) The presence of substructures may complicate our simple spherical lens
model, and hence could be a main source of uncertainties in $m_{\rm lens}$.
The absence of the secondary arc-like images in most arc-cluster systems may
indicate the limitations of the spherical mass distribution in the central
regions of clusters.
It should be noted that the mass ratios we obtained here are slightly higher
than Allen (1998) and Wu (2000) because they unfortunately used a Hubble
constant of $\rm H_{0}=50\,km\,s^{-1}Mpc^{-1}$. The use of $\rm
H_{0}=70\,km\,s^{-1}Mpc^{-1}$ here will of course make the mass discrepancy
problem more pronounced.
It should be noted that the gas represents only a $10\%$ perturbation due to
the small ratio of gas-to-DM in the central region, likewise the offset of the
gas is only a small perturbation (less than $10\%$) to the otherwise
concentric matter density or potential. It is unlikely to create a factor of
two difference in the lensing-derived enclosed masses within an arc.
To illustrate the lensing effect of the offset perturbation and triaxiality,
we show the critical curves in Figure 4. The solid curves indicate the
critical curve of circular NFW plus $\beta$ model without offset, the dotted
curves indicate the critical curve of elliptical NFW plus $\beta$ model with
offset $d=10^{\prime\prime}$. The square and cross denote the center of dark
matter and the hot gas, respectively. For the NFW profile, $c=4.3,r_{s}=516\rm
kpc$; for the $\beta$ model, $\beta=0.65,r_{c}=150\rm kpc$. We also introduce
the triaxiality with the ellipticity $e=0.15$ and position angle
$\theta=30^{\circ}$. We also assume the lens and source redshifts $z_{l}=0.3$,
$z_{s}=1$. We can see that the predicted critical curves (dotted lines) have
very similar sizes as the predicted critical curves for a benchmark model
(solid lines) with the same mass DM and gas mass but in concentric spheres.
Figure 4: The effects of offset and triaxiality on the critical curves. The
solid curves indicate the critical curve of circular NFW & $\beta$ model
without offset. The dotted curves indicate the critical curve of elliptical
NFW & $\beta$ model with offset $d=10^{\prime\prime}$. The square and cross
denote the center of dark matter and the X-ray gas, respectively. For the NFW
profile, $c=4.3,r_{s}=516\rm kpc$; for the $\beta$ model,
$\beta=0.65,r_{c}=150\rm kpc$. The ellipticity and position angle are $e=0.15$
and $\theta=30^{\circ}$. The lens and source redshifts are $z_{l}=0.3$,
$z_{s}=1$.
Early studies have suggested that statistically unrelaxed clusters have larger
mass discrepancies than relaxed clusters (Allen 1998; Wu 2000; Richard et al.
2010). As Shan et al. (2010) have reported, the clusters with large offset of
$d>10^{\prime\prime}$ are all unrelaxed clusters. If such offsets exist and
are big, then they must come into play in our dynamical studies of galaxy
cluster, and should not be ignored, especially for unrelaxed clusters.
## Chapter Acknowledgments
We thank Bernard Fort, Charling Tao, and Xiang-Ping Wu for discussions, and an
anonymous referee for helpful suggestions. HYS and BQ are grateful to the CPPM
for hospitality. This work was supported by the National Basic Research
Program of China (973 Program) under grant No. 2009CB24901, and CAS grants
KJCX3-SYW-N2 and KJCX2-YW-N32.
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Table 1: The X-ray and lensing mass discrepancies of $27$ clusters. For the
$22$ arcs that have no redshift information, we estimate the mean redshifts of
$\left<z_{d}\right>=0.8$ and $2.0$, respectively. Refs A and B give the
references of the lensing and X-ray data, respectively. The last column shows
the classification of the clusters: “R/U” means relaxed/unrelaxed.
Cluster | $z_{\rm cluster}$ | Offset | $z_{\rm arc}$ | $r_{\rm arc}$ | $m_{\rm lens}$ | Ref. Ad | kT | $\beta$ | $r_{c}$ | $m_{\rm xray}$ | $m_{\rm lens}/m_{\rm xray}$ | Classc | Ref.Bd
---|---|---|---|---|---|---|---|---|---|---|---|---|---
| | (arcsec) | (kpc) | | (Mpc) | $\times 10^{14}M_{\odot}$ | | (keV) | | (Mpc) | $\times 10^{14}M_{\odot}$ | | |
1E0657-56 | 0.296 | 47.4 | 209.2 | 3.24 | 0.25 | 4.37 | 3,4 | $14.1^{+0.2}_{-0.2}$ | $0.62^{+0.07}_{-0.07}$ | $0.36^{+0.05}_{-0.05}$ | $2.15^{+0.46}_{-0.46}$ | $2.03^{+0.44}_{-0.44}$ | U | 12
A68a | 0.255 | 14.3 | 56.7 | 1.60 | 0.04 | 0.13 | 11 | $10.0^{+1.1}_{-0.9}$ | $0.72^{+0.04}_{-0.03}$ | $0.25^{+0.02}_{-0.02}$ | $0.08^{+0.019}_{-0.016}$ | $1.66^{+0.39}_{-0.34}$ | U | 2
| | | | 1.60 | 0.10 | 0.80 | | | | | $0.47^{+0.11}_{-0.091}$ | $1.77^{+0.40}_{-0.34}$ | |
| | | | 2.63 | 0.11 | 0.94 | | | | | $0.56^{+0.13}_{-0.11}$ | $1.67^{+0.38}_{-0.32}$ | |
| | | | … | 0.211 | 4.49(3.54)b | | | | | $1.69^{+0.35}_{-0.29}$ | $2.64^{+0.55}_{-0.46}(2.08^{+0.43}_{-0.36})$ | |
| | | | 0.86 | 0.28 | 7.49 | | | | | $2.61^{+0.51}_{-0.43}$ | $2.95^{+0.58}_{-0.48}$ | |
| | | | … | 0.27 | 7.26(5.72)b | | | | | $2.47^{+0.49}_{-0.41}$ | $2.98^{+0.59}_{-0.49}(2.35^{+0.47}_{-0.39})$ | |
| | | | 1.27 | 0.32 | 8.66 | | | | | $3.14^{+0.60}_{-0.50}$ | $2.82^{+0.54}_{-0.45}$ | |
| | | | … | 0.12 | 1.54(1.22)b | | | | | $0.65^{+0.15}_{-0.12}$ | $2.23^{+0.50}_{-0.42}(1.76^{+0.39}_{-0.33})$ | |
A267 | 0.230 | 9.62 | 35.3 | … | 0.12 | 1.48(1.20)b | 11 | $6.0^{+0.6}_{-0.5}$ | $0.71^{+0.03}_{-0.03}$ | $0.19^{+0.01}_{-0.01}$ | $0.39^{+0.08}_{-0.07}$ | $3.86^{+0.78}_{-0.71}(3.13^{+0.63}_{-0.58})$ | U | 2
A370a | 0.375 | 19.9 | 102.7 | 1.30 | 0.41 | 13.1 | 7 | $7.13^{+1.05}_{-1.05}$ | $0.95^{+0.75}_{-0.35}$ | $0.56^{+0.44}_{-0.26}$ | $4.34^{+4.10}_{-2.27}$ | $3.00^{+2.83}_{-1.57}$ | U | 13
| | | | 0.72 | 0.19 | 4.09 | | | | | $0.71^{+1.15}_{-0.65}$ | $5.84^{+9.54}_{-5.42}$ | |
A697 | 0.282 | 3.07 | 13.1 | … | 0.12 | 1.51(1.15)b | 10 | $9.9^{+0.6}_{-0.6}$ | $0.61^{+0.01}_{-0.01}$ | $0.24^{+0.01}_{-0.01}$ | $0.26^{+0.21}_{-0.13}$ | $5.53^{+4.52}_{-2.84}(4.21^{+3.44}_{-2.16})$ | U | 2
A773a | 0.217 | 6.43 | 22.6 | 0.65 | 0.11 | 1.39 | 11 | $7.6^{+0.5}_{-0.4}$ | $0.61^{+0.01}_{-0.01}$ | $0.19^{+0.06}_{-0.06}$ | $0.37^{+0.042}_{-0.037}$ | $3.75^{+0.42}_{-0.38}$ | U | 2
| | | | 0.40 | 0.21 | 7.08 | | | | | $1.26^{+0.26}_{-0.24}$ | $5.85^{+1.19}_{-1.11}$ | |
| | | | … | 0.25 | 6.50(5.36)b | | | | | $1.61^{+0.29}_{-0.26}$ | $4.10^{+0.73}_{-0.68}(3.38^{+0.60}_{-0.56})$ | |
| | | | … | 0.23 | 5.41(4.45)b | | | | | $1.43^{+0.27}_{-0.25}$ | $3.89^{+0.74}_{-0.69}(3.21^{+0.61}_{-0.57})$ | |
| | | | 1.11 | 0.213 | 4.34 | | | | | $1.26^{+0.26}_{-0.24}$ | $3.35^{+0.68}_{-0.64}$ | |
| | | | 0.40 | 0.16 | 4.14 | | | | | $0.84^{+0.20}_{-0.19}$ | $5.12^{+1.23}_{-1.17}$ | |
| | | | … | 0.04 | 0.18(0.15)b | | | | | $0.07^{+0.023}_{-0.022}$ | $2.51^{+0.86}_{-0.83}(2.07^{+0.71}_{-0.69})$ | |
| | | | 0.49 | 0.23 | 7.42 | | | | | $1.43^{+0.27}_{-0.25}$ | $5.07^{+0.96}_{-0.90}$ | |
A963a | 0.206 | 7.10 | 24.0 | … | 0.057 | 0.35(0.29)b | 11 | $6.13^{+0.45}_{-0.30}$ | $0.51^{+0.04}_{-0.04}$ | $0.11^{+0.02}_{-0.02}$ | $0.14^{+0.038}_{-0.034}$ | $2.41^{+0.63}_{-0.57}(2.01^{+0.53}_{-0.48})$ | R | 13
| | | | 0.71 | 0.09 | 0.87 | | | | | $0.31^{+0.073}_{-0.066}$ | $2.89^{+0.68}_{-0.61}$ | |
A1689 | 0.183 | 0.60 | 1.85 | … | 0.20 | 4.5(3.8)b | 8 | $9.02^{+0.40}_{-0.30}$ | $0.65^{+0.04}_{-0.02}$ | $0.14^{+0.02}_{-0.02}$ | $1.68^{+0.24}_{-0.17}$ | $2.68^{+0.38}_{-0.27}(2.29^{+0.32}_{-0.23})$ | R | 13
A1835 | 0.252 | 1.61 | 6.33 | … | 0.17 | 2.82(2.23)b | 11 | $9.8^{+1.4}_{-1.4}$ | $0.65^{+0.04}_{-0.04}$ | $0.08^{+0.01}_{-0.01}$ | $1.72^{+0.36}_{-0.36}$ | $1.70^{+0.36}_{-0.36}(1.35^{+0.28}_{-0.28})$ | R | 13
A1914 | 0.171 | 11.3 | 32.9 | … | 0.10 | 1.16(1.01)b | 10 | $9.9^{+0.3}_{-0.3}$ | $0.90^{+0.01}_{-0.01}$ | $0.200^{+0.003}_{-0.003}$ | $0.70^{+0.036}_{-0.036}$ | $1.67^{+0.09}_{-0.09}(1.44^{+0.07}_{-0.07})$ | U | 2
A2204a | 0.151 | 1.20 | 3.15 | … | 0.025 | 0.08(0.07)b | 10 | $6.5^{+0.2}_{-0.2}$ | $0.48^{+0.002}_{-0.002}$ | $0.02^{+0.0003}_{-0.0003}$ | $0.11^{+0.0040}_{-0.0040}$ | $0.71^{+0.03}_{-0.03}(0.63^{+0.02}_{-0.02})$ | R | 2
| | | | … | 0.01 | 0.013(0.012)b | | | | | $0.03^{+0.0009}_{-0.0009}$ | $0.49^{0.02}_{-0.02}(0.43^{+0.02}_{-0.02})$ | |
A2163 | 0.203 | 44.0 | 146.9 | 0.73 | 0.07 | 0.58 | 1 | $14.6^{+0.85}_{-0.85}$ | $0.62^{+0.02}_{-0.02}$ | $0.33^{+0.02}_{-0.02}$ | $0.23^{+0.033}_{-0.033}$ | $2.39^{+0.35}_{-0.35}$ | U | 13
A2218a | 0.176 | 19.1 | 56.9 | 1.03 | 0.28 | 8.60 | 11 | $7.1^{+0.2}_{-0.2}$ | $0.65^{+0.08}_{-0.05}$ | $0.25^{+0.09}_{-0.05}$ | $1.67^{+0.49}_{-0.31}$ | $5.07^{+1.49}_{-0.93}$ | U | 13
| | | | 0.70 | 0.09 | 0.89 | | | | | $0.25^{+0.11}_{-0.065}$ | $3.94^{+1.73}_{-1.04}$ | |
| | | | 2.52 | 0.09 | 0.82 | | | | | $0.25^{+0.11}_{-0.065}$ | $3.20^{+1.40}_{-0.85}$ | |
A2219a | 0.228 | 11.3 | 41.2 | … | 0.09 | 0.79(0.64)b | 11 | $12.4^{+0.5}_{-0.5}$ | $0.40^{+0.07}_{-0.07}$ | $0.16^{+0.08}_{-0.08}$ | $0.38^{+0.21}_{-0.21}$ | $2.19^{+1.17}_{-1.17}(1.78^{+0.95}_{-0.95})$ | U | 13
| | | | … | 0.12 | 1.54(1.26)b | | | | | $0.63^{+0.30}_{-0.30}$ | $2.39^{+1.15}_{-1.15}(1.94^{+0.94}_{-0.94})$ | |
A2259 | 0.164 | 16.3 | 45.9 | 1.48 | 0.04 | 0.13 | 10 | $5.6^{+0.3}_{-0.3}$ | $0.58^{+0.02}_{-0.02}$ | $0.14^{+0.01}_{-0.01}$ | $0.06^{+0.0089}_{-0.0089}$ | $2.71^{+0.38}_{-0.38}$ | U | 2
A2261a | 0.224 | 1.31 | 4.72 | … | 0.12 | 1.49(1.22)b | 10 | $7.2^{+0.4}_{-0.4}$ | $0.56^{+0.01}_{-0.01}$ | $0.08^{+0.004}_{-0.003}$ | $0.71^{+0.058}_{-0.056}$ | $2.12^{+0.17}_{-0.17}(1.73^{+0.14}_{-0.14})$ | R | 2
| | | | … | 0.11 | 1.26(1.03)b | | | | | $0.63^{+0.053}_{-0.051}$ | $2.00^{+0.17}_{-0.16}(1.63^{+0.14}_{-0.13})$ | |
A2390 | 0.228 | 6.00 | 21.9 | 0.91 | 0.20 | 3.8 | 10 | $11.1^{+1.0}_{-1.0}$ | $0.59^{+0.02}_{-0.02}$ | $0.16^{+0.01}_{-0.01}$ | $1.79^{+0.26}_{-0.26}$ | $2.22^{+0.32}_{-0.32}$ | U | 13
CL0024 | 0.395 | 13.2 | 70.4 | 1.68 | 0.26 | 4.7 | 6 | $5.7^{+4.9}_{-2.1}$ | $0.48^{+0.08}_{-0.05}$ | $0.08^{+0.05}_{-0.03}$ | $1.19^{+1.22}_{-0.56}$ | $4.03^{+4.14}_{-1.91}$ | U | 13
MS0440 | 0.190 | 1.50 | 4.89 | 0.53 | 0.10 | 1.23 | 5,10 | $5.30^{+1.27}_{-0.85}$ | $0.45^{+0.03}_{-0.03}$ | $0.03^{+0.01}_{-0.01}$ | $0.40^{+0.15}_{-0.12}$ | $3.26^{+1.19}_{-0.94}$ | R | 13
MS0451 | 0.550 | 12.1 | 76.8 | … | 0.23 | 7.6(3.5)b | 10 | $10.17^{+1.55}_{-1.26}$ | $0.68^{+0.13}_{-0.09}$ | $0.31^{+0.09}_{-0.06}$ | $1.64^{+0.85}_{-0.61}$ | $4.81^{+2.49}_{-1.79}(2.11^{+1.09}_{-0.78})$ | U | 13
MS1008 | 0.360 | 5.43 | 27.3 | … | 0.30 | 9.2(6.2)b | 1 | $7.29^{+2.45}_{-1.52}$ | $0.63^{+0.11}_{-0.07}$ | $0.23^{+0.07}_{-0.05}$ | $1.89^{+1.15}_{-0.73}$ | $4.84^{+2.95}_{-1.89}(3.29^{+2.00}_{-1.28})$ | R | 13
MS1358 | 0.329 | 2.79 | 13.2 | 4.92 | 0.14 | 1.24 | 10 | $7.5^{+4.3}_{-4.3}$ | $0.47^{+0.02}_{-0.02}$ | $0.05^{+0.02}_{-0.01}$ | $0.82^{+0.53}_{-0.51}$ | $1.54^{+0.99}_{-0.97}$ | R | 13
MS1455 | 0.258 | 2.77 | 11.1 | … | 0.11 | 1.22(0.96)b | 9,10 | $5.45^{+0.29}_{-0.28}$ | $0.64^{+0.04}_{-0.03}$ | $0.07^{+0.01}_{0.01}$ | $0.57^{+0.077}_{-0.067}$ | $2.14^{+0.29}_{-0.25}(1.68^{+0.23}_{-0.20})$ | U | 13
MS2053 | 0.580 | 10.5 | 69.1 | 3.15 | 0.16 | 1.41 | 10 | $4.7^{+0.5}_{-0.4}$ | $0.64^{+0.04}_{-0.03}$ | $0.16^{+0.02}_{-0.01}$ | $0.60^{+0.13}_{-0.094}$ | $2.47^{+0.54}_{-0.39}$ | U | 2
MS2137 | 0.313 | 5.70 | 26.1 | … | 0.10 | 0.99(0.72)b | 9,10 | $4.37^{+0.38}_{-0.72}$ | $0.63^{+0.04}_{-0.03}$ | $0.05^{+0.01}_{-0.01}$ | $0.44^{+0.067}_{-0.094}$ | $2.26^{+0.34}_{-0.48}(1.65^{+0.25}_{-0.35})$ | R | 13
PKS0745 | 0.103 | 6.82 | 12.9 | 0.43 | 0.05 | 0.42 | 10 | $8.7^{+1.6}_{-1.2}$ | $0.59^{+0.01}_{-0.01}$ | $0.06^{+0.01}_{-0.01}$ | $0.29^{+0.075}_{-0.061}$ | $1.55^{+0.40}_{-0.33}$ | R | 13
RXJ1347 | 0.451 | 2.81 | 16.2 | 0.81 | 0.28 | 8.9 | 1 | $11.37^{+1.10}_{-0.92}$ | $0.57^{+0.04}_{-0.014}$ | $0.07^{+0.01}_{-0.01}$ | $3.07^{+0.54}_{-0.35}$ | $2.90^{+0.51}_{-0.33}$ | R | 13
a Multiple-arc system.
b Arc-like image is assumed at $z_{s}=0.8$ $(z_{s}=2)$.
c R: Relaxed, U: Unrelaxed.
d References: (1) Allen 1998; (2) Bonamente et al. 2006; (3) Bradac et al.
2006; (4) Clowe et al. 2006; (5) Gioia et al. 1998; (6) Jee et al. 2007; (7)
Kneib et al. 1993; (8) Limousin et al. 2007; (9) Newbury & Fahlman 1999; (10)
Sand et al. 2005; (11) Smith et al. 2005; (12) Tucker et al. 1998; (13) Wu
2000
|
arxiv-papers
| 2010-06-17T14:44:02 |
2024-09-04T02:49:10.988506
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "HuanYuan Shan, Bo Qin, HongSheng Zhao",
"submitter": "HuanYuan Shan",
"url": "https://arxiv.org/abs/1006.3484"
}
|
1006.3553
|
# The effect of radiation pressure on emission line profiles and black hole
mass determination in active galactic nuclei
Hagai Netzer11affiliation: School of Physics and Astronomy and the Wise
Observatory, The Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-
Aviv University, Tel-Aviv 69978, Israel , Paola Marziani22affiliation: INAF,
Osservatorio Astronomico di Padova, Vicolo dell’ Osservatorio 5, IT35122
Padova, Italy
###### Abstract
We present a new analysis of the motion of pressure-confined, broad line
region (BLR) clouds in active galactic nuclei (AGNs) taking into account the
combined influence of gravity and radiation pressure. We calculate cloud
orbits under a large range of conditions and include the effect of column
density variation as a function of location. The dependence of radiation
pressure force on the level of ionization and the column density are
accurately computed. The main results are: a. The mean cloud locations
($r_{BLR}$) and line widths (FWHMs) are combined in such a way that the simple
virial mass estimate, $r_{BLR}FWHM^{2}/G$, gives a reasonable approximation to
$M_{\rm BH}$ even when radiation pressure force is important. The reason is
that $L/M$ rather than $L$ is the main parameter affecting the planar cloud
motion. b. Reproducing the mean observed $r_{BLR}$, FWHM and line intensity of
H$\beta$ and C iv $\lambda 1549$ requires at least two different populations
of clouds. c. The cloud location is a function of both $L^{1/2}$ and $L/M$.
Given this, we suggest a new approximation for $r_{BLR}$ which, when inserted
into the BH mass equation, results in a new approximation for $M_{\rm BH}$.
The new expression involves $L^{1/2}$, FWHM and two constants that are
obtained from a comparison with available $M-{\sigma*}$ mass estimates. It
deviates only slightly from the old mass estimate at all luminosities. d. The
quality of the present black hole mass estimators depends, critically, on the
way the present $M-{\sigma*}$ AGN sample (29 objects) represents the overall
population, in particular the distribution of $L/L_{\rm Edd}$.
###### Subject headings:
Galaxies: Active – Galaxies: Black holes – Galaxies: Nuclei – Galaxies:
Quasars: Emission Lines
## 1\. Introduction
The profiles of the broad emission lines in the spectrum of active galactic
nuclei (AGNs) are the main source of information about the motion of the high
density gas in the broad line region (BLR). Detailed studies of such profiles
have been the focus of intense investigation for many years (see Netzer 1990
for a review of older work and Marziani et al. 1996 and Richards et al. 2002
for more recent publications). Unfortunately, several rather different
geometries can conspire to result in similar line profiles and today, there is
no way to infer, directly, the global BLR motion from line profile fitting.
A less ambitious goal is to use a measure of the observed line width, e.g. the
line FWHM, or the line dispersion (see Peterson et al. 2004 for definitions)
as indicators of the mean emissivity-weighted velocity of the BLR gas. Such
measurements are crucial for deducing black hole (BH) mass ($M_{\rm BH}$) in
cases where the emissivity-weighted radius, $r_{BLR}$, is measured directly
from reverberation mapping (RM) experiments, or estimated from from
L-$r_{BLR}$ relationships that are based on such studies (see Kaspi et al.
2000; Kaspi et al 2005; Vestergaard and Peterson 2006 for reviews). A typical
expression of this type is
$r_{BLR}=a\left[\frac{L_{5100}}{10^{46}\,{\rm
erg\,s^{-1}}}\right]^{\gamma}\,\,pc,$ (1)
where $L_{5100}$ is the continuum luminosity ($\lambda L_{\lambda}$) at 5100Å
and $\gamma=0.6\pm 0.1$. The constant $a$ depends on the line in question. For
H$\beta$, $a\simeq 0.4$ pc (e,g, Bentz et al. 2009) and for C iv $\lambda
1549$, $a\simeq 0.13$ pc (Kaspi et al. 2007 after assuming
$L_{1350}=2L_{5100}$). For a virialized BLR, the above $r_{BLR}$ can be
combined with a measure of the FWHM, or the line dispersion, to obtain the BH
mass,
$M_{BH}=fr_{BLR}FWHM^{2}/G\,$ (2)
where the constant $f$ is a geometrical correction factor of order unity that
takes into accounts the (unknown) gas distribution and dynamics. Various
possible values of $f$ have been computed by Collin et al. (2006) for various
possible geometries. However, the only empirical way to determine $f$ is to
compare the results of eqn. 2 with independent measurements of $M_{\rm BH}$,
like those available in the the case where the central BH resides in a bulge
and $M_{\rm BH}$ can be estimates from the $M-{\sigma*}$ relationship (e.g.
Tremaine et al. 2002). Such comparisons by Onken et al (2004) and by Woo et
al. (2010), using the H$\beta$ RM data base, suggest $f=1\pm 0.1$.
In a recent paper, Marconi et al. (2008; hereafter M08) investigated the role
of radiation pressure force and its effect on the motion of the BLR gas and
the required modification to the BH mass estimate. According to M08, radiation
pressure plays an important role in affecting the cloud motion provided the
column density (${N_{\rm col}}$) of most BLR clouds is smaller than about
$10^{23}$ cm-2. According to M08, in such a case, there is a need to add a
second term to eq. 2. This term depends on the source luminosity and ${N_{\rm
col}}$. The modified form suggested in M08 is
$M_{BH}=f_{1}r_{BLR}FWHM^{2}/G+f_{g}L/N_{col}\,$ (3)
where$f_{1}$ replaces $f$ in eqn. 2 and $f_{g}$ is a second constant. If
$L=L_{5100}/10^{44}\,{\rm erg\,s^{-1}}$ and ${N_{\rm col}}$ is measured in
units of $10^{23}\,{\rm cm^{-2}}$, $f_{g}\simeq 10^{7.7}$$M_{\odot}$.̇
According to M08, failing to account for the second term results in the
underestimation of $M_{\rm BH}$. Obviously, the inclusion of such a term
results in $f_{1}<f$. M08 repeated the analysis of Onken et al. (2004) and
Vestergaard and Peterson (2006), taking into account the new term and solving
for $f_{1}$ and ${N_{\rm col}}$. This resulted in $f_{1}\simeq 0.56$ and
${N_{\rm col}}$$\simeq 10^{23}$ cm-2.
The M08 suggestion can be tested by comparing low redshift samples of type-I
and type-II AGNs since the estimate of $M_{\rm BH}$ in the latter does not
involve the source luminosity and gas dynamics. Netzer (2009) carried out such
a comparison and found that radiation pressure force plays only a marginal
role in such sources. The conclusion is that, in many AGNs, the mean column
density of the BLR clouds exceeds $\sim 10^{23}$cm-2. In a later work, Marconi
et al (2009; hereafter M09) argued that firm conclusions regarding the role of
radiation pressure force are difficult to obtain since the column density in
some BLRs can be different than in others and there is no simple way to
evaluate the overall effect of such a column density distribution. The
treatment of a certain type of cloud in all sources, or even in a single BLR,
is of course highly simplified and eqns. 2 and 3 must be treated as crude
first approximations.
The critical and detailed evaluation of the role of radiation pressure force
in “real” BLRs is the subject of the present paper. In §2 we present our basic
equations and in §3 we use them to calculate various expected broad emission
line profiles and mass normalization factors, $f$. §4 deals with the
evaluation of present day $M_{\rm BH}$ estimates and suggests a new way to
estimate $M_{\rm BH}$ and $r_{BLR}$ which is consistent with our calculations.
## 2\. Cloud motion in the BLR
In this work we focus on the “cloud model” of the BLR. The general framework
of this model is explained in Netzer (1990) and in Kaspi and Netzer (1999) and
a major empirical justification is obtained from the recent X-ray detected
single blobs, or clouds, moving in a region which is typical, in terms of
velocity and dimension, of the BLR (Risaliti et al. 2010; Maiolino et al.
(2010). We do not consider the locally optimally-emitting cloud (LOC) model
(Baldwin et al. 1995; Korista & Goad 2000) where, at every location, there is
a large range in cloud properties. The dynamics of the BLR gas in this model
has never been treated and is far more complicated than the one considered
here. Another possibility that has been discussed, extensively, is that wind
from the inner disk plays an important role in feeding and driving the BLR
gas. Possible evidence for this scenario comes from radio observations (e.g.
Vestergaard, Wilkes, & Barthel 2000; Jarvis & McLure 2006) and
spectropolarimetry (Smith et al. 2004; Young et al. 2007). Theoretical
considerations are discussed in Bottorff et al. (1997), Murray & Chiang
(1997), Proga, Stone, & Kallman (2000), Everett (2003), Young et al. (2007),
and several other papers. While our calculations apply to any cloud, even
those created and driven by such winds, the specific examples given below are
more applicable to bound clouds where inward and outward motions are both
allowed.
### 2.1. The equation of motion of BLR clouds
The basic equation of motion, ignoring drag force, is
$a(r)=\frac{\sigma_{T}L_{bol}}{\mu m_{H}c4\pi
r^{2}}[M(r)-1/\Gamma]-\frac{1}{\rho}\frac{dP_{g}}{dr}\,\,,$ (4)
where $M(r)$ is the force multiplier, $L_{bol}$ is the bolometric luminosity,
$\mu$ is the average number of nucleons per electron, and $\Gamma=$$L/L_{\rm
Edd}$. The force multiplier depends on the gas composition and its level of
ionization. An interesting case is a Compton thin neutral cloud that absorbs
all the ionizing radiation (a Compton thin “block”). In this case
$M(r)\simeq\alpha(r)/(\sigma_{T}N_{col}$) where $N_{col}$ is the hydrogen
column density and $\alpha(r)$ is the fraction of the bolometric luminosity
which is absorbed by the gas. For such a “block”, $\alpha(r)=L_{ion}/L_{bol}$
but in general $\alpha(r)$ is radius dependent because of the changing column
density and level of ionization of the gas (see below).
Ignoring thermal pressure we obtain
$a(r)=\frac{L_{bol}}{r^{2}}\left[\frac{1.14\times
10^{-11}\alpha(r)}{N_{23}}-\frac{8.8\times 10^{-13}}{\Gamma}\right]$ (5)
where $N_{23}=N_{col}/10^{23}$. Thus, radiation pressure is the dominant force
when
$\Gamma\geq 7.7\times 10^{-2}\frac{N_{23}}{\alpha(r)}\,\,.$ (6)
The above expressions, including the one for the limiting $\Gamma$, include
only radial terms and assume a pure radially dependent radiation pressure
force. The calculations of real orbits, and the conditions for cloud escape,
require their integration and will thus include the standard constants of
motion (energy and angular momentum). Obviously, the conditions for escape
depend on the cloud azimuthal velocity, $v_{\theta}$, and can differ
substantially from what is obtained by using eqn. 6.
M08 and M09 derived similar expressions for the case of completely opaque
clouds. According to them, radiation pressure dominates the cloud motion if
$\Gamma\geq 1.27\times 10^{-2}b_{5100}N_{23}$ (7)
where $b_{5100}=L_{bol}/L_{5100}$. The two expressions provide the same
limiting $\Gamma$ when
$\alpha(r)b_{5100}\simeq 6.1\,\,.$ (8)
A recent paper by Ferland et al. (2009), where mostly neutral, infalling
clouds are considered, reaches basically identical conclusions.
### 2.2. Confined clouds
BLR clouds are likely to be confined. The confining mechanism is not known but
high temperature gas and magnetic confinement have been proposed. The approach
chosen here is consistent with the idea of magnetic confinement and some
justifications for it are given in Rees, Netzer and Ferland (1989). We adopt a
simple model of numerous individual clouds that are moving under the combined
influence of the BH gravity and radiation pressure force. Following Netzer
(1990), and Kaspi and Netzer (1999), we assume that clouds retain their mass
as they move in or out and the gas density changes with radius in a way that
depends on the radial changes of the confining pressure.
Assume the external pressure and hence the gas density in individual clouds
are proportional to the radial coordinate, $n_{H}\propto r^{-s}$. A reasonable
guess that agrees with observations is $1\leq s\leq 5/2$ (Rees, Netzer and
Ferland 1989). This results in a radial dependence of the ionization parameter
(the ratio of ionizing photon density to gas density), $U\propto r^{s-2}$. For
spherical clouds, $N_{col}\propto r^{-2s/3}$, $R_{c}\propto r^{s/3}$ and
$A_{c}\propto r^{2s/3}$, where $R_{c}$ is the cloud radius and $A_{c}$ its
geometrical cross section. The line intensity contributed by a single cloud,
$\epsilon(r)$, depends on its covering factor and the line emissivity $j(r)$
which depends on the conditions in the gas,
$\epsilon(r)\propto j(r)A_{c}/r^{2}\propto j(r)r^{2s/3-2}\,\,.$ (9)
In the real calculations we ignore factors of order unity relating the mean
cloud “size” and its mean column density since this is not known and require
different type of calculations.
The above considerations suggest that the importance of radiation pressure
increases with distance because of the dependence of $N_{col}$ on $r$, i.e.
$\Gamma_{lim}\propto r^{-2s/3}/\alpha(r)\,\,.$ (10)
Since $\Gamma$ depends on the global accretion rate which has little to do
with cloud properties, the more physical approach is to consider the case of a
certain $\Gamma$ and follow the cloud motion. The examples discussed below
follow this approach.
In this work we consider three types of clouds:
1. 1.
Very large column density clouds where radiation pressure force is negligible
at all distances. Here virial cloud motion is a good approximation (the
Ferland et al 2009 infalling clouds belong to this category).
2. 2.
Cloud for which radiation pressure is very important somewhere inside the
“classical BLR” (e.g. inside the RM radius). Such clouds will escape the
system on dynamical time scales and their contribution to the line profiles is
small except for times immediately after a large increase in $L_{bol}$.
3. 3.
Clouds for which radiation pressure is non-negligible but is not strong enough
to allow escape. Such clouds are the ones discussed by M08 albeit without the
radial dependence of ${N_{\rm col}}$ considered here. This case is the one
most relevant to real BLRs and we discuss it in detail in the following
section.
### 2.3. Modified equation of motion
The modified equation of motion is obtained from eqn. 5 by including the
radial dependence of $N_{col}$. We define $r_{23}$ to be the distance where
$N_{23}=1$. This gives
$a(r)=\frac{L_{bol}}{r^{2}}\left[\frac{1.14\times
10^{-11}\alpha(r)}{(r/r_{23})^{-2s/3}}-\frac{8.8\times
10^{-13}}{\Gamma}\right]\,\,.$ (11)
The column density dependent critical distance where radiation force dominates
the cloud motion is,
$\frac{r}{r_{23}}\geq\left[\frac{7.7\times
10^{-2}}{\alpha(r)\Gamma}\right]^{3/2s}\,\,.$ (12)
For example, the case of $s=1$ and $\alpha(r)=0.5$ gives a critical radius of
$r=0.06\Gamma^{-1.5}r_{23}$ for radially moving clouds. This dependence of $r$
on $\Gamma$ is the main motivation to suggest a new method for evaluating
$M_{\rm BH}$ and $r_{BLR}$ (§4). As explained, the critical radius should not
be confused with the point of escape from the system. Non-radial velocity
components ($v_{\theta}$), that reflect the energy and angular momentum of the
system will act to reduce this radius (see examples below).
The motion of BLR clouds with the above properties involves an acceleration
term of the form,
$a(r)=\frac{c_{1}\alpha(r)}{r^{2-2s/3}}-\frac{c_{2}}{r^{2}}\,\,,$ (13)
where
$c_{1}=1.14\times 10^{-11}L_{bol}r_{23}^{-2s/3}$ (14)
and
$c_{2}=8.8\times 10^{-13}L_{bol}/\Gamma\,\,.$ (15)
The radial potential is,
$\Phi(r)=-\int_{r}^{r*}a(r)dr\,\,,$ (16)
where $r*$ is the radius where $\Phi(r)=0$. Below we use this potential to
calculate cloud orbits and line profiles. The energy and angular momentum
terms that result from the above integration, are included in the calculation
by fixing the initial conditions, $r$ and the two velocity components at this
location.
## 3\. Line profile calculations
### 3.1. Method
We carried out a series of calculations under a variety of conditions
considered to be typical of different BLRs. Every model is calculated for
assumed $M_{\rm BH}$ and $\Gamma$. This specify $L_{bol}$ and thus the
potential $\Phi(r)$. The additional model parameters are:
1. 1.
The radial parameter $s$.
2. 2.
The cloud column density normalization factor $r_{23}$.
3. 3.
The initial radius $r_{0}$ and the initial velocity $v_{0}=v_{\theta}(r_{0})$.
We assume that the orbits of clouds with very large $N_{col}$ are ellipses of
given eccentricities. $r_{0}$ is chosen to be the apogee of the orbit and
$v_{0}$ (given below in units of the Keplerian velocity, $v_{Kepler}$) is
determined from these conditions. This a simple way to specify the angular
momentum. In the examples below, we focus on those cases where the resulting
FWHMs are consistent with the observations of the broad H$\beta$ and C iv
$\lambda 1549$ emission lines but give details for several others.
4. 4.
The initial ionization parameter, $U(r_{0})$. We note that the exact value of
the gas density, $n_{H}(r)$, is less important. In the following we assume
$n_{H}(r_{0})=10^{10}$ cm-2 for all cases.
5. 5.
The three-dimensional distribution of orbits. This is done in two steps. First
we calculate the motion of numerous identical clouds in a plane and then
distribute many such planes in a spherical geometry specified by the
inclinations of the planes to the line of sight. The profiles given below are
only those for a line of sight which is perpendicular to the central plane of
motion (if such a plane exists). All calculations assume a large enough number
of clouds such that the predicted profiles are smooth (see Bottorff and
Ferland 2000 for discussion and earlier references on this issue).
Physical properties that are not included in the present calculations are non-
isotropic central radiation field, non-isotropic line emission, the
photoionization of gas with a range of density and metallicity, different
inclinations of the line of sight to the central plane of motion, large cloud
covering factors in a specific direction, and central obscuration, e.g. by the
accretion disk. Several of those are likely to be important in real BLRs but
are beyond the scope of the present work.
Fig. 1 illustrates the orbits of three $s=1.2$, $r_{0}=10^{17}$ cm,
$\Gamma=0.1$ and $v_{0}=0.5v_{Kepler}$ clouds moving under the influence of a
$10^{8}$ $M_{\odot}$ BH. The first is an ellipse typical of a cloud which is
not affected by radiation pressure (e.g. $r_{23}=1000r_{0}$). This is shown by
a thick solid line. Formally speaking, such clouds are Compton thick but this
is of no practical implications since the only intention is to show a simple,
gravity dominated orbit. The second is a case where $r_{23}=10r_{0}$. Here,
radiation pressure force is significant and acts to constantly changing the
direction of motion of the cloud. This results in a rotating planar orbit. The
third orbit (dashed line) follows the trajectory of a smaller column density
cloud ($r_{23}=0.82r_{0}$) that escapes the system. Increasing $\Gamma$ will
result in similar type orbits for the rotating orbit second cloud except that
the angle between two successive revolutions will increase.
Figure 1.— Planar orbits of three clouds with $\Gamma=0.1$ and different
column densities. The large column density cloud (thick line,
$r_{23}=1000r_{0}$) moves in a closed elliptical orbit. A smaller column
density cloud (thin line, $r_{23}=10r_{0}$) moves in a closed rotating orbit
and a marginal column density cloud (dashed line, $r_{23}=0.82r_{0}$) escapes
the system.
We calculated various line profiles for the case of $M_{\rm
BH}$=$10^{8}$$M_{\odot}$, $r_{23}=10r_{0}$, $v_{0}=0.5$ and $\Gamma$ in the
range of 0.05 (negligible radiation pressure force) to 0.735 (just below
escape). The bolometric luminosity in each of those is obtained from the
combination of $M_{\rm BH}$ and $\Gamma$. We assume $\alpha(r)=0.5$ at all
radii and $\epsilon(r)$ which takes into account only geometrical factors
(i.e. constant $j(r)$, see eqn. 9) and isotropic line emission. In terms of
total line emission, this is a reasonable approximation for lines like
H$\beta$ that reprocess roughly a constant fraction of the ionizing continuum
radiation. Obviously, a large optical depth in H$\beta$ will result in line
emission anisotropy which is not considered here. It is not appropriate for
lines like C iv $\lambda 1549$ whose intensity is more sensitive to the level
of ionization and the gas temperature. At this stage we specifically avoid the
use of a varying $\alpha(r)$ since the effect on the orbit can be significant
even for small changes in this parameter (see below). The resulting profiles,
assuming a complete spherical atmosphere (the entire $\pm\pi/2$ radians range
relative to the central plane), are shown in Fig. 2. As expected, the profile
becomes narrower with the increasing $\Gamma$ reflecting the fact that, as the
luminosity increases, the cloud spend less and less time at small radii.
Figure 2.— Line profiles for spherical s=1.2 atmospheres around a
$10^{8}$$M_{\odot}$ BH and a range of $\Gamma$ as marked. All clouds start at
$r_{0}=10^{17}$ cm with $v_{r}=0$ and $v_{\theta}=0.5\,v_{Kepler}$. The column
densities are changing as $(r/r_{23})^{-2s/3}$ with $r_{23}=10^{18}$ cm
($N_{col}\approx 6.3\cdot 10^{23}$ cm-2 at $r_{0}$). The FWHM of the profile
decreases with the increasing $\Gamma$ due to the increasing importance of
radiation pressure force. The profile parameters are listed in Table 1.
The top part of Table 1 provides additional information about the
calculations. For each profile we give the FWHM in units of
$v_{Kepler}(r_{0})$, the mean emissivity weighted radius, $<r>/r_{0}$, and the
mass correction factor $f$ (eqn. 2). The calculation of $<r>$ is obtained by
weighting the emissivity of the cloud and the time it spends at each radius.
This is roughly equivalent to the observed RM radius. The mass correction
factor is obtained by requiring $fFWHM^{2}<r>/G=M_{BH}$. We also show (in
parenthesis) the values of FWHM and $f$ obtained for the case of a thick
central disk which represents only a part of a spherical distribution. Here
the cloud distribution correspond to a width, relative to the central plane,
of $\pm\pi/4$ radians. The reduction in FWHM relative to the complete sphere
is about a factor of 0.6 and there is a corresponding increase in $f$.
Computed line profiles that are typical of this and similar geometries are
shown in Fig. 4.
Table 1Line widths, mass conversion factor $f$, and emissivity-weighted radii
for various modelsaafootnotemark:
$\Gamma$ | FWHM/$v_{Kepler}(r_{0})$ | $<r>/r_{0}$ | $f$
---|---|---|---
$s=1.2$ | $r_{23}=10r_{0}$ | $v_{0}=0.5$ |
0.05 | 1.58 (0.93) | 0.54 | 0.75 (2.18)
0.1 | 1.55 (0.92) | 0.54 | 0.77 (2.21)
0.3 | 1.45 (0.87) | 0.56 | 0.85 (2.37)
0.5 | 1.34 (0.81) | 0.59 | 0.94 (2.56)
0.7 | 1.15 (0.72) | 0.68 | 1.11 (2.78)
0.735 | 1.06 (0.68) | 0.78 | 1.13 (2.76)
$s=1.2$ | $r_{23}=10r_{0}$ | $v_{0}=0.25$ |
0.05 | 1.04 | 0.45 | 2.05
0.1 | 1.02 | 0.45 | 2.10
0.3 | 0.95 | 0.47 | 2.39
0.5 | 0.87 | 0.49 | 2.74
0.7 | 0.76 | 0.52 | 3.31
0.91 | 0.59 | 0.67 | 4.32
$s=1.2$ | $r_{23}=r_{0}$ | $v_{0}=0.5$ |
0.01 | 1.57 | 0.54 | 0.76
0.03 | 1.51 | 0.55 | 0.80
0.1 | 1.23 | 0.64 | 1.03
0.116 | 1.055 | 0.79 | 1.13
aAssuming the line emissivity is strictly proportional to the cloud cross
section and $\alpha(r)=0.5$. In all cases $v_{0}=v_{\theta}(r_{0})$. Numbers
for $f$ assume spherical BLRs (numbers in brackets assume a $\pm\pi/4$ radians
thick disk).
To explore models with different initial conditions, we computed two cases of
planar orbits with the same orbital energy and different angular momentum. One
such example is shown in Fig. 3. The less eccentric case in the diagram
corresponds to the orbit labeled with 2) in Fig. 1 for which $v_{0}=0.5$. The
more eccentric one assumes $v_{0}=0.25$ but with a non-zero radial velocity of
$v_{r}=0.433$. This results in a much narrower profile. In the middle part of
Table 1, we report other cases where $v_{0}=0.25$ and $v_{r}=0$. Such orbits
are again very eccentric and the profiles are, indeed, much narrower. The
corresponding values of $f$ are now larger by a factor of 2-3 than those
observed. Additional models (not shown here) with larger initial angular
momentum, give larger FWHM and smaller $f$. Obviously, some combination of all
those is required to explain real observations.
Figure 3.— Top: Planar orbits of two clouds with the same orbital energy and
different angular momentum. The model parameters are $\Gamma=0.1$, $s$ = 1.2
$M_{\rm BH}$= 108 M⊙ and other parameters as in case 2) of Fig. 1. The less
eccentric case ((1), thin line) corresponds to maximum angular momentum at
$r=r_{0}$ with $v_{0}=0.5$ and no initial radial velocity ($v_{r,0}=0$). The
more eccentric case ((2), thick line) assumes $v_{0}=0.25$ and
$v_{r,0}=0.433v_{Kepler}$. Bottom: Line profiles for the two cases (same
notation as in Fig. 2). The narrower profile corresponds to orbit 2).
The bottom part of Table 1 shows the results of a set of line profile
calculations carried out for smaller column density clouds. We chose
$r_{23}=r_{0}$ which corresponds to a factor of 6.3 decrease in ${N_{\rm
col}}$ relative to the case shown at the top of the table. The scaling of FWHM
between the two cases is simply by the corresponding factor in $\Gamma$ (i.e.
the same FWHM for $\Gamma$ smaller by a factor of 6.3). This illustrates the
fact that in an atmosphere with a large range of column densities, there are
always clouds that are close to being ejected from the system at large
distances.
The changes in $<r>$ for a given $r_{23}$ shown in Table 1 are due to the fact
that as $\Gamma$ increases, and radiation pressure is more important, the
clouds spend more and more time away from the BH. This is noticeable for the
case of $r_{23}=10r_{0}$ when $\Gamma$ approaches 0.73 and for the case of
$r_{23}=r_{0}$ when $\Gamma$ approaches 0.1.
As noted in §1, RM campaigns show that $r_{BLR}$(H$\beta$)$\propto
L_{bol}^{0.6\pm 0.1}$. It is interesting to note that this behavior is not
very different from what is calculated here for the changes in $<r>$ if we
compare values over the range where $\Gamma$ approaches its limiting value.
However, it is not the case when $\Gamma$ changes by similar factors close to
the lower range shown in the table, where radiation pressure is negligible.
The values of $f$ computed here should be compared with those determined
observationally for selected AGN samples with measured $\sigma*$, in
particular the Onken et al. (2004) and the Woo et al. (2010) AGN samples. The
simulations illustrate how this factor depends on the BLR geometry, the
distribution of $\Gamma$ among objects in the sample and the distribution of
${N_{\rm col}}$ in individual BLRs.
An important point of the new calculation is the relatively little change in
the value of FWHM listed in Table 1, only a factor of $\sim 1.5$ over most of
the range of $\Gamma$ except very close to the limiting value. The changes in
$f$ are also small, only a factor of $\sim 1.3$ over the same range in
$\Gamma$. This seems to be in contradiction to the naive expectation that, for
cases of increasing $L$, the term $<r>FWHM^{2}/G$ will deviate more and more
from $M_{\rm BH}$ (e.g. eqn. 3). There are two reasons for this behavior.
First, for realistic cases where ${N_{\rm col}}$ depends on the cloud
location, the mean emissivity distance and the velocity depend on $L/M$ rather
than on $L$. This suggests that very low and very high luminosity AGNs with
similar $\Gamma$ will react to radiation pressure force in a similar way.
Second, for a planar motion, the changes in the radial potential $\Phi(r)$ do
not affect the cloud velocity in a linear way. In fact, the mean orbital
changes in $v_{\theta}$ are small enough such that the overall FWHM is very
far from zero even for marginally escaping clouds. Moreover, the mean cloud
location, $<r>$, is increasing in reaction to the increasing radiation
pressure term. The end results is that the product $f<r>FWHM^{2}/G$, with a
constant value of $f$, is always a reasonable approximation for $M_{\rm BH}$
with little dependence on the relative importance of gravity and radiation
pressure force. We return to this issue in §4 where we suggest a new way to
evaluate $M_{\rm BH}$ taking into account radiation pressure acceleration.
Finally, we note that while radiation pressure is negligible for very small
values of $\Gamma$, the $s$-dependence of the cloud properties is still very
important. For example, an $s=0$ atmosphere gives constant column density
clouds (similar to what was assumed in M08) yet, the mean emissivity radius,
the FWHM of the emission lines and the mass correction factor $f$ in this case
are always different from those of the $s=1.2$ case, regardless of the column
density. The reason is the dependence of the cloud cross section on $s$. For
example, in the case of $\Gamma=0.01$ (first entry in the bottom part of Table
1), the $s=0$ case gives $<r>/r_{0}=0.39$ (compared with 0.53 for $s=1.2$) and
FWHM$/v_{Kepler}=2.45$ (compared with 1.51). The resulting $f$ is therefore
much smaller (0.42 compared with 0.76). Thus, the radial dependence of the
cloud properties are important for all $\Gamma$.
Figure 4.— Same initial conditions as in Fig. 2 for $\Gamma=0.5$. The various
profiles represent motion in different spherically shaped atmospheres. The
narrowest profile (dashed line) represents a sphere where clouds occupy only
the section between -0.3 and +0.3 radians relative to the mid-plane (which is
perpendicular to the line of sight). The other cases are for wider coverage
with clouds between -0.9 and +0.9 rad (dotted line) and -1.5 to +1.5 rad
(solid line). The double peak profile illustrates the case of two polar caps
where the clouds occupy a sphere whose mid-section, between -1.2 and +1.2 rad,
has been removed.
### 3.2. Applications to spectroscopic observations of AGNs
The examples discussed above were normalized to give a typical
$r_{BLR}$(H$\beta$) for AGNs with $M_{\rm BH}$=$10^{8}$$M_{\odot}$ and
$\Gamma=0.1$. However, the computed line profiles cannot be directly compared
with the observations of such sources for several reasons. First, we only
consider a situation involving one type of clouds and neglect the possibility
of different populations under different physical conditions in the same
source. This applies to the distributions of both ${N_{\rm col}}$ and $U(r)$.
For example, eqn. 1 and the constants given in §1 suggest that, in general,
$r_{BLR}(H_{\beta})$/$r_{BLR}(Civ\lambda 1549)\simeq 3$. The question is
whether cloud distributions like those considered in Table 1 can reproduce
this ratio. Second, we did not take into account changes in $\alpha(r)$, the
fraction of $L_{bol}$ which is absorbed by clouds at various distances. This
can be an important factor close to the BH where clouds become partly
transparent. In this case, much of the Lyman continuum radiation is not
absorbed and radiation pressure force is reduced. It can also affect medium to
large column density clouds at large distances where $\alpha(r)$ approaches
unity. For example, assuming $\alpha(r)=0.75$ instead of $\alpha(r)=0.5$ in
the calculations of Table 1 results in a limiting value of $\Gamma$ which is
about 0.4 compared with $\Gamma=0.735$ listed in the table.
To illustrate these effects, and to provide more realistic line profiles, we
computed two grids of photoionization models for a range of column density and
ionization parameter using the code ION (Netzer 2006). The first grid supplies
calculated line intensities for H$\beta$ and C iv $\lambda 1549$ over a large
range in $U(r)$. Given $r$ from the cloud motion simulation, we use the grid
to compute $j(r)$ and thus a more realistic $\epsilon(r)$. The second grid
supplies the absorbed fraction, $\alpha(r)$, as a function of $U(r)$ and
${N_{\rm col}}$. Fig. 5 shows part of the $\alpha(r)$ grid to illustrate the
expected range in this parameter. We have not included the changes in gas
density since they do not play a major role over the range of conditions
considered here. We have also not considered anisotropy of the line emission
which is bound to have an effect on the FWHM of some lines. Such modifications
will be included in a forthcoming paper that is intended to present a
comparison with observed line profiles.
Figure 5.— Part of the $\alpha(r)$ grid (fraction of the total continuum flux
absorbed by the clouds) used in the present calculations. Numbers along the
contour lines are $log\,\,\alpha$.
We tested a large number of single-zone models using the above grids of $U(r)$
and $\alpha(r)$. The models cover a large range in angular momentum and BLR
geometries. We have specifically investigated three cases of different
eccentricity, defined by three values of $v_{0}(r_{0})$, 0.25, 0.5 and 0.75.
These were calculated with different $\Gamma$ and $r_{23}$. In general, it is
easy to reproduce the observed I(C iv $\lambda 1549$)/I(H$\beta$) but
difficult to account, at the same time, for the emissivity weighted radii of
the two lines (eqn. 1) and the line width ratio. For example, the best case of
the three, with $v_{0}(r_{0})=0.25$, gives I(C iv $\lambda
1549$)/I(H$\beta$)=4.4, $<r>$(C iv $\lambda 1549$)/$<r>$(H$\beta$)=0.67 and
FWHM(C iv $\lambda 1549$)/FWHM(H$\beta$)=1.43. The conclusion is that, within
the range of parameters assumed here, there is no obvious way to explain all
those properties when keeping with the idea of a single column density
distribution (i.e. a single $r_{23}$).
We also tested a case of $M_{\rm BH}$$=10^{8}$$M_{\odot}$, $\Gamma=0.1$ and
two distinct cloud populations in inner and outer zones with some overlap
between the two. In this case, the initial conditions for the two populations
are decoupled from each other but the changes in density, column density and
ionization parameter follow the same pattern with the same $s=1.2$ density
law. The FWHM of both emission lines were calculated under the assumption of a
thick central spherical sector with clouds occupying a region of $\pm\pi/4$
radians relative to the central plane. The inner zone clouds have
$r_{0}=5\times 10^{16}$ cm and $U(r_{0})=10^{-1}$ and the outer-zone clouds
$r_{0}=3\times 10^{17}$ cm and $U(r_{0})=10^{-2.5}$. The starting velocity in
both zones is $v_{0}=0.5$ at the appropriate $r_{0}$. In both zones
$r_{23}=3r_{0}$. We followed the cloud motion and calculated, in each zone,
the line intensity ratio, I(C iv $\lambda 1549$)/I(H$\beta$), the line FWHMs,
and the emissivity weighted radii. These numbers are listed in Table 2 where
we also show the properties of the combined spectrum which is calculated under
the assumption of equal contributions to H$\beta$ from both zones. The
emissivity weighted radii for the two zones are given in units of the RM-radii
of the two lines (eqn. 1). This very simple two-zone model gives results that
are in good agreement with the observations of many low-to-intermediate
luminosity AGNs. Fig. 6 is a graphical summary of these results. The left and
central panels show H$\beta$ and C iv $\lambda 1549$ profiles for the inner
and outer zones, again assuming isotropic line emission, and the right panel
shows the combined two-zone profiles.
Table 2Properties of the two-zone model with $v_{0}(r_{0})=0.5$. Zone | FWHM(H$\beta$) | FWHM($Civ$) | $\frac{r({\rm H}\beta)}{r(RM,{\rm H}\beta)}$ | $\frac{r(Civ)}{r(RM,Civ)}$ | $\frac{I(Civ)}{I({\rm H}\beta)}$
---|---|---|---|---|---
| (km s-1) | (km s-1) | | |
Inner | 3160 | 3450 | 0.32 | 0.88 | 9.55
Outer | 1390 | 2580 | 1.64 | 3.2 | 1.45
Combined | 2060 | 3390 | 0.98 | 1.1 | 5.5
Figure 6.— Calculated H$\beta$ (solid line) and C iv $\lambda 1549$ (dashed
line) profiles for a two zone model. Left: line profile for the inner zone.
Middle: line profiles for the outer zone. Right: The combined line profile.
For FWHMs and general normalization see Table 2.
In conclusion, the simple single zone models explored here cannot reproduce
all the observed properties: line intensity ratio, mean emissivity radii and
FWHM ratio. The main reason is that the starting conditions fix the cloud
orbit, and hence the line emissivity and FWHM. Simple two-zone models like the
ones presented here can account for most observed properties of the H$\beta$
and C iv $\lambda 1549$ lines. In particular, they can account for the mean
line ratio, the mean emissivity weighted radii and the mean relative FWHM of
the H$\beta$ and C iv $\lambda 1549$ lines measured in various RM samples.
Obviously, such simple models do not intend to explain all the observed line
profile properties that can differ from one object to the next and contain
additional components (see some obvious examples for complex C iv $\lambda
1549$ profiles in Richards et al. 2002 and Sulentic et al. 2007). Fitting
those is deferred to a forthcoming paper.
## 4\. Discussion
### 4.1. General considerations
The above calculations allow us to investigate the intensity, the width and
the shape of the broad emission lines and to evaluate various methods used to
estimate $M_{\rm BH}$. We defer the discussion of specific observed line
profiles to a future paper.
Assume a system of clouds with a given total amount of gas and a large range
of column densities. Such a system will eventually break into three:
virialized clouds, non-virialized bound clouds and escaping clouds. The third
group will not contribute significantly to the observed line emission for more
than several dynamical times. The relative contribution of the first and
second groups to the line emission depend on the cloud mass distribution. A
sudden increase in $L_{bol}$ will increase the importance of radiation
pressure and will remove more gas from the system. A decrease in $L_{bol}$
will drive the system closer to virial equilibrium. A new gas supply, e.g.
from a disk-wind, will produce bound as well as unbound clouds. All aspects of
this general scenario must be considered when evaluating the observed line
profiles and the various methods developed to use them in estimating $M_{\rm
BH}$.
A major objective of the present paper is to evaluate the accuracy and the
normalization of various $M_{\rm BH}$ estimators in type-I AGNs. The results
presented in Tables 1 & 2 suggest the following:
1. 1.
Every AGN is likely to contain a large number of clouds with a large range in
${N_{\rm col}}$. This can be the result of a broad cloud mass distribution
and/or due to cloud motion in a radial-pressure dependent environment with a
positive value of $s$. A given $\Gamma$ results in a lower limit on ${N_{\rm
col}}$ at a given location for a given orbit eccentricity. Under such
conditions, there are always some clouds, e.g. those that are very close to
the BH, for which radiation pressure is negligible. For others, radiation
pressure can be very important.
2. 2.
For a small enough ${N_{\rm col}}$, the effective $r_{BLR}$ depends on both
$\Gamma$ and ${N_{\rm col}}$. Under these conditions, the BH mass itself is an
important factor in determining $r_{BLR}$. To illustrate this, consider two
AGNs with identical SED, $L_{bol}$, BLR geometry, ${N_{\rm col}}$ distribution
and inclination to the line of sight. The effective $r_{BLR}$ in the two is
the same provided they harbor identical BHs. Different $r_{BLR}$ will be
measured if the two BHs have different masses despite of the fact that
$L_{bol}$ is the same in both. This is the result of the larger $\Gamma$ in
the smaller BH AGN. The effect may not be recognized in a large sample of
sources and can, in fact, be attributed to a large intrinsic scatter in the
$L_{bol}-r_{BLR}$ relationship. Any derived $L_{bol}-r_{BLR}$ relationship
will depend on the properties of the sources in the chosen RM sample, in
particular on the distribution of $\Gamma$.
3. 3.
Assuming a range in ${N_{\rm col}}$ in every AGN, the M08 suggestion to
include a luminosity dependent term in the calculation of $M_{\rm BH}$ (eqn.
3) is not in accord with our calculation that indicate that $r_{BLR}$ and FWHM
depend on $L/M$ and not on $L$.
The multi-year RM campaign of NGC 5548 is the best example to test some of
these ideas in a specific source. The campaign has been described and analyzed
in numerous papers and the ones most relevant to the present study are
Peterson et al. (1999) and Gilbert and Peterson (2003).
Fig. 7 shows the variations in $L_{5100}$ and time lag (in this case the
centroid of the CCF) in NGC 5548. Each point represents a full observing
season which is typically $\sim 300$ days long. The data are taken from the
recent compilation of Bentz et al. (2009) which provides the best galaxy
subtracted flux at 5100Å. The uncertainty on $L_{5100}$ is basically the range
of this quantity over the observing season. This is of the same order as the
variation from one season to the next. As clearly seen from the diagram,
$r_{BLR}$(H$\beta$) lags the continuum in such a way that more luminous phases
are associated with longer lags. This has been noted in earlier publications,
e.g. Gilbert and Peterson (2003). Fig. 8 shows t(lag) vs. $L_{5100}$ for the
same data set. While the uncertainties are large, some correlation, with a
slope of 0.5-1, is evident. An earlier version of the diagram, with fewer
points, is shown in Peterson et al. (1999).
Figure 7.— Changes in continuum luminosity ($L_{5100}$) and time lag for NGC
5548 (data from Bentz et al. 2009). Error bars on $L_{5100}$ were omitted for
clarity. Figure 8.— The correlation of $L_{5100}$ vs. t(lag) for NGC 5548.
Data as in fig. 7. The dashed line has a slope of 0.5.
For NGC 5548, $M_{\rm BH}$$\simeq 10^{8}$ $M_{\odot}$ and
$r_{BLR}$(H$\beta$)$\simeq 20$ l.d.. Thus, the dynamical time is of order 6
years and the time it takes to change $r_{BLR}$ by 50% (e.g. Table 1) is
approximately 3 years. This seems to be compatible with the changes in
$L_{5100}$ and t(lag) in fig. 7, thus some adjustment of $r_{BLR}$(H$\beta$)
due to the effects discussed in this work are possible. The measured
$L_{5100}$, with a bolometric correction factor of about 10, indicates a mean
$\Gamma$ of about 0.02. The bottom part of Table 1 provides approximate
parameters for such a case. Any successful model of NGC 5548 should account
for the behavior shown in Fig. 7, as well as for the observed FWHMs and
luminosities of both H$\beta$ and C iv $\lambda 1549$. While the full
investigation is deferred to a future paper, we consider here the predicted
lags for $\Gamma$ =0.005, 0.01, 0.02 assuming a BH mass of 108 M⊙ and two
families of clouds: one with $r_{23}=10r_{0}$ and $v_{0}=0.5$, and one with
$r_{23}=0.093r_{0}$ and $v_{0}=0.25$. The second assumed family of clouds
results in pseudo-orbits of higher eccentricity that, as explained earlier,
are more strongly affected by radiation pressure. In both cases, the predicted
lags for $\Gamma=0.02$ are consistent with the observed values ($\log t$(lag)
$\approx$ 1.34 at $\log L_{5100}\approx 43.48$). However, the calculated slope
of $\log t$(lag) vs. $L_{5100}$ is flatter than observed.
As argued earlier, a single family of BLR clouds cannot provide a full
explanation to the observed spectrum of many AGNs. This must applied to NGC
5548 (to appreciate the complexity of this case see the various components
considered by Kaspi and Netzer (1999) to explain only the variable line
intensities). The simple examples considered here suggests that dynamical
scaling of the BLR in NGC 5548, due to radiation pressure force, is an
additional, physically-motivated mechanism that must be added to any cloud
model when attempting to explain the observed variations in $t$(lag).
### 4.2. Evaluation of present $M_{\rm BH}$ estimators
Current BH mass estimates utilize RM-based measurements of $r_{BLR}$, measured
FWHMs (or an equivalent velocity estimator) of certain broad emission lines,
and eqn. 2. The normalization constant $f$ is obtained by a comparing $M_{\rm
BH}$ obtained in this way with the mass obtained from the $M-{\sigma*}$
method. Having examined a large range of cloud orbits and line profiles under
various conditions, and the corresponding values of the effective $r_{BLR}$,
we are now in a position to evaluate the merits of this method.
We consider three general possibilities. The first is the case where all AGNs
contain BLR clouds with a wide column density distribution. A randomly chosen
object will have in its BLR some clouds that are affected by radiation
pressure force and others that are not. This is the case for any $\Gamma$. The
cloud dynamics and the observed line profiles reflect the (unknown) column
density distribution. Our calculations suggest that an RM sample drawn
randomly from such an AGN population can be safely used to determine the best
value of $f$ by comparing the derived $M_{\rm BH}$ with the $M-{\sigma*}$
method. This is justified by the fact that $M_{BH}\propto<r>FWHM^{2}$ even if
radiation pressure force is important (see Table 1). The observed FWHMs are,
indeed, smaller than the ones that would have been observed if all clouds had
extremely large column densities. This, however, has no practical implication
since the column densities are not known and $f$ is simply a normalization
factor that serves to bring two completely different methods of estimating
$M_{\rm BH}$ into agreement. Mass estimates obtained in this ways are reliable
provided the properties of the RM sample represent well the population
properties.
The second case reflects a situation where the cloud column density
distribution is, again, very broad but part of the population is under-
represented in the RM sample. For example, the RM sample may contain mostly
sources with $\Gamma\sim 0.1$ while the overall distribution of $\Gamma$ is
much wider. In this case, the normalization factor $f$ will reflect only the
properties of the measured sources and its use will provide poor mass
estimates for cases with much larger or much smaller accretion rates. This may
well be the case in the RM sample which is most commonly used (Kaspi et al.
2000; Bentz et al, 2009) that contains only very few AGNs with $\Gamma>0.3$.
The numbers in Table 1 enable us to evaluate the resulting deviations in the
estimated $M_{\rm BH}$. For example, if we use the first part of the table and
assume a source with a certain $L_{bol}$ and $\Gamma=0.1$, we find that the
mass of a similar $L_{bol}$ source with $\Gamma=0.7$ will be under-estimated
by a factor of 1.11/0.75.
Regarding the second case, it is important to note that under-estimates and
over-estimates of $M_{\rm BH}$ are equally likely. Consider again an RM sample
where, for most sources, $\Gamma=0.1$. This results in a certain value of $f$
which takes into account the effect of radiation pressure force in some of
these sources (see bottom part of Table 1). Assume a second, randomly selected
AGN sample with a similar BH mass distribution but a typical $\Gamma$ which is
much smaller than 0.1. Most measured FWHMs in this sample are broader than
those in the RM sample because radiation pressure force is not as effective in
reducing the cloud velocity. Using the value of $f$ derived for the RM sample
will result in over estimating $M_{\rm BH}$ in the second sample. The lower
part of Table 1 gives some idea about the magnitude of this effect, e.g. an
over estimate by a factor of 1.01/0.76.
The third case is similar to the first one except that large luminosity
variations, on time scales that are not too different from the BLR dynamical
time, are occurring in most sources, including those selected for RM
monitoring. Table 1 shows that, like the first case, the deduced $f$
represents well the population because $<r>$ follows the variations in
$L_{bol}$. The mean $M_{\rm BH}$ in such a sample is recovered albeit with a
larger uncertainty.
### 4.3. Alternative $M_{\rm BH}$ estimators
Given the above considerations, we now investigate an alternative method to
calculate $M_{\rm BH}$. The method takes into account the effect of radiation
pressure force on the cloud motion and the results will be compared to those
obtained by the old method (eqn. 2) and by the M08 method.
Our new calculations indicate that the emissivity weighted $r_{BLR}$ depends
both on the (large) range in $L$ across the entire AGN population, as well as
on short time scale changes in $r_{BLR}$ in individual sources. The first of
those depends roughly on $L^{1/2}$ and is a manifestation of the observational
fact that the ionization parameter, $U(r)$, and the spectral energy
distribution (SED), are not changing much with source luminosity. The second
reflects changes in the BLR structure due to the reaction of various column
density clouds to the (changing) radiation pressure force. This depends on
both $L_{bol}$ and $M_{\rm BH}$. This is seen for example in eqn. 12 for the
critical radius where clouds can escape the system and also in the
calculations of Table 1. It is therefore reasonable to assume that $r_{BLR}$
is given by an expression of the form,
$r_{BLR}=a_{1}L^{\gamma}+a_{2}(L/M)^{\delta}\,\,,$ (17)
where $a_{1}$ and $a_{2}$ are constants and $L$ is a measure of the source
luminosity, e.g. $L_{5100}$ if $r_{BLR}$=$r_{BLR}$(H$\beta$). Obviously, the
above approximation is not unique and one can assume other dependences that
are consistent with the line profile calculations, e.g. a dependence of FWHM
on $L/M$.
The idea of introducing a second, luminosity dependent term into the
calculation of $M_{\rm BH}$ is not new. In particular, M08 suggested an
expression for $M_{\rm BH}$ which depends on both $L^{1/2}$ and $L/N_{col}$
(eqn. 3). Assuming all AGNs obey the same relationship, and ${N_{\rm col}}$ is
the same in all, the M08 expression leads to extremely large values of $M_{\rm
BH}$ for the most luminous AGNs. The reason is the linear dependence of
$M_{\rm BH}$ on $L$ at very high luminosities combined with the calibration of
the relationship at small $L$, typical of the $M-{\sigma*}$ sample of Onken et
al. (2004). The additional consequence of this approach is an upper limit of
$\Gamma\sim 0.1$ in many high luminosity, large BH mass sources. In their
later work, M09 considered the possibility that ${N_{\rm col}}$ can differ
from one source to another but is still constant for all clouds in a given
BLR. This would result in smaller $M_{\rm BH}$ and larger $\Gamma$ in some
high luminosity sources since in some BLRs, ${N_{\rm col}}$ can exceed
$10^{23}$ cm-2 by a large factor thus reducing the importance of radiation
pressure force.
The limitation of the M08 mass estimate is the detachment of $L$ from $M_{\rm
BH}$. As shown here, this is not the case in more realistic BLRs, especially
those where the masses of the clouds are conserved. In such cases, the
location of the outer clouds that still contribute to the line profiles
depends on $L/M$ and the 3D-velocities of the marginally bound clouds are such
that the product $r_{BLR}$$FWHM^{2}$ is not very different from what is found
in pure gravity dominated systems. Moreover, for pressure confined clouds, the
dependence on ${N_{\rm col}}$ is likely to be different in different parts of
the BLR. Thus, we are looking for an expression that will reflect, properly,
all these effects and will allow for the possibility of a range of column
densities in every source. We also want to avoid biasing in the derivation of
$M_{\rm BH}$ in the limits of very large or very small $L$ and to retain the
experimental results that $r_{BLR}\propto L^{\gamma}$ with $\gamma=0.6\pm
0.1$.
All the above can be achieved by assuming that $r_{BLR}$ is given by eqn. 17
and requiring that $M_{BH}\propto r_{BLR}FWHM^{2}$. For the sake of
simplicity, we assume $\gamma=0.5$ and $\delta=1$ and substitute eqn. 17 into
the mass expression. This leads to a simple quadratic equation in $M_{\rm BH}$
with the following solution,
$M_{BH}=\frac{1}{2}a_{1}L^{1/2}FWHM^{2}\left[1+\sqrt{(}1+\frac{4a_{2}}{a_{1}^{2}FWHM^{2}})\right]\,\,,$
(18)
where $a_{1}$ and $a_{2}$ are the same ones used in eqn. 17 except for a
common multiplicative constant which depends on the units of $r_{BLR}$,
$L_{5100}$ and $M_{\rm BH}$. For example, using the measure parameters for the
H$\beta$ line, $L=$$L_{5100}$, FWHM=FWHM(H$\beta$), then the constant
multiplying $a_{1}$ and $a_{2}$ in eqn. 17 is 1016.123 when $M_{\rm BH}$ is
measured in $M_{\odot}$, $L_{5100}$ in units of $10^{44}$ ergs s-1 and
$r_{BLR}$ in cm.
We used eqn. 18 and the Woo et al. (2010) sample to find $a_{1}$ and $a_{2}$
for 29 AGNs with measured $\sigma*$. The list is an extension of the one used
by Onken et al. (2004) that contains only 16 sources. We have supplemented the
data in Woo et al. by data from Bentz et al. (2009) on $r_{BLR}$ and
$L_{5100}$ where this information was missing. First, we performed a
$\chi^{2}$ analysis on $M_{\rm BH}$(RM) vs. $M-{\sigma*}$ using the parameters
recommended by Gültekin et al. (2009). This gave $f=1.0$ which is consistent
with the values found by Onken et al. (2004) and Woo et al. (2010)111Onken et
al. (2004) and Woo et al. (2010) carried the analysis using the H$\beta$ line
dispersion rather than FWHM(H$\beta$). For the sample in question, this line-
width measure is smaller than the FWHM(H$\beta$) by a factor of approximately
1.9 leading to a corresponding increase in the mean $f$ by a factor of about
$1.9^{2}$. All these numbers are sensitive to the error estimate in $\sigma*$
and in the virial product.
Next we carried out a $\chi^{2}$ minimization to solve for $a_{1}$ and $a_{2}$
in eqn. 18. Since the minimization involves the error estimate on $L_{5100}$,
and since this error is not very well defined given the combination of
observational uncertainly and the intrinsic scatter in $L_{5100}$ over several
long RM campaigns, we decided to adopt a uniform value of $\Delta
L_{5100}/L_{5100}=0.3$. We also assume a minimum of 0.1 to
$\Delta(\sigma*)/\sigma*$ and a minimum of 0.05 on $\Delta(FWHM)/FWHM$. Our
results depend slightly on these assumptions.
The best values obtained in this procedure are $a_{1}=4.1$, $a_{2}=7.1\times
10^{7}$ and $\chi^{2}/{\nu}=1.73$. Extensive tests show that the $\chi^{2}$
changes very little if $a_{1}$ or $a_{2}$ are changing by up to 10%. This is
the result of some degeneracy between $a_{1}$ and $a_{2}$ (see eqn. 18). The
average deviation between the new mass estimates and those obtained by the
$M-\sigma*$ method is 0.31 dex. There is a weak dependence of the deviation on
the line width (larger deviation for larger FWHM(H$\beta$)) which is marginal
given the small number of sources in the sample. The corresponding number for
the deviation of masses obtained directly from the RM measurements and the
above value of $f$ is 0.36 dex. Thus the new method is, indeed, superior in
this respect. Obviously it is not surprising to find such an improvement when
adding a new free parameter to the model.
To compare the various mass estimates more thoroughly, we calculated $M_{\rm
BH}$ in three different ways: the old method (eqn. 2) with $f=1.0$, the M08
method (eqn. 3) with $f_{1}=0.56$ and $f_{g}=10^{7.7}$ (as in M08), and the
new method (eqn. 18) with the above $a_{1}$ and $a_{2}$. For the M08 method,
we followed the M09 recommendation and assumed a log normal distribution of
${N_{\rm col}}$ with a mean of $10^{23}$ cm-2 and a large standard deviation
of 0.5 dex. We also calculated $r_{BLR}$ in the old (eqn. 1) and new (eqn. 17)
ways.
Fig. 9 compares two mass ratios, $M_{\rm BH}$(new)/$M_{\rm BH}$(old) (red
points) and $M_{\rm BH}$(new)/$M_{\rm BH}$(M08) (black points), in a large
simulated AGN sample. The sample covers, uniformly, the luminosity range
$L_{5100}$=$10^{43}-10^{47}$ ergs s-1 and the simulations assume a Gaussian,
luminosity independent distribution of FWHM(H$\beta$) with a mean of 4,500 km
s-1 and a variance of 1,500 km s-1. The diagram shows that the new and old
estimates are similar at all $L_{bol}$ but $M_{\rm BH}$(M08) deviates from
both, by a large factor, at both low and high luminosities. Moreover, the
slight deviation between the new and old methods at the very high luminosity
end, by up to about 0.2 dex in $M_{\rm BH}$, is most likely due to the fact
that the procedure used to obtain $a_{1}$ and $a_{2}$ is based on a sample of
29 mostly low-to-intermediate luminosity AGNs while the simulations reach a
much larger value of $L_{bol}$. A comparison of the estimated $r_{BLR}$ (eq. 1
and 17) leads to similar conclusions.
Figure 9.— Comparison of the various methods for calculating $M_{\rm BH}$. The
ratio of the new-to-old (red) and new-to-M08 (black) methods are shown as a
function of $L_{bol}$ for the simulated sample described in the text. Note the
good agreement between the old and the new methods and the large deviation
from the method described in M08 for very large and very small values of
$L_{bol}$.
We also made a similar test on the Netzer and Trakhtenbrot (2007) sample using
all three methods. The luminosity range in this case is smaller but the
FWHM(H$\beta$) distribution more typical of observed AGNs. The results (not
shown here) are very similar to those of the simulated sample.
In conclusion, the new method for estimating $M_{\rm BH}$ gives results that
do not deviate much from the old method which is based on a single constant
$f$. This is true at both high and low luminosities and over a large range in
FWHM. Obviously, the range of parameters tested here ($s$, orbit eccentricity,
several types of cloud distributions, etc.) is rather limited and more
extensive modeling is required to confirm these results. However, it is our
opinion that the main limitation of the $M_{\rm BH}$ determination methods
remains observational and is related to the fact that the present AGN
$M-{\sigma*}$ sample is small (29 sources) and cannot possibly represent the
entire range of properties, mostly $\Gamma$, observed in AGNs.
## 5\. Conclusions
We have investigated the motion of BLR clouds with time-independent mass under
a range of conditions defined by a radial-dependent confining pressure. These
conditions enforce a range of ${N_{\rm col}}$ in every BLR, even if the
intrinsic mass distribution of the cloud is narrow. We calculated cloud orbits
under a central potential that includes a radiation pressure term. The orbits
were then combined to predict emission line profiles in several simple
situations. We only considered uniformly emitted emission lines and the
preliminary comparison with with actual observations used realistic emissivity
and column density distributions but was limited to the H$\beta$ and C iv
$\lambda 1549$ lines and at most two different cloud distributions. We found
significant changes in cloud locations and velocities for those cases where
the column densities are small enough to allow a significant contribution due
to radiation pressure. This can be important in both high and low $\Gamma$
sources. However, while cloud orbits are strongly influenced by the radiation
pressure force, there is a relatively small change in the mean
$r_{BLR}$$FWHM^{2}$ and hence no large underestimation or overestimation of
$M_{\rm BH}$. We illustrate this behavior in several cases but note that other
cloud distributions, with different mass, location and velocity distributions,
may lead to somewhat different conclusions. We used the new results to suggest
a novel method for calculating $r_{BLR}$ and $M_{\rm BH}$ by applying two new
constants that were calculated by a comparison of the H$\beta$ and $L_{5100}$
observations and the $M-\sigma*$ AGN sample of Woo et al. (2010). We applied
the method to several large observed and simulated AGN samples and
demonstrated good agreement between the new and the old, pure gravity based
methods. The comparison with the M08 methods shows large deviations in the
estimates of $M_{\rm BH}$.
We acknowledge useful comments by an anonymous referee and a detection of a
typo in Table 1 by J.M. Wang. Funding for this work has been provided by the
Israel Science Foundation grant 364/07 and by the Jack Adler Chair for
Extragalactic Astronomy. HN thanks the hospitality of Imperial College London
and University College London where part of this work has been done. PM is
grateful for the hospitality and support of the school of Physics and
Astronomy at Tel Aviv University.
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|
arxiv-papers
| 2010-06-17T19:34:53 |
2024-09-04T02:49:10.996582
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hagai Netzer and Paola Marziani",
"submitter": "Paola Marziani",
"url": "https://arxiv.org/abs/1006.3553"
}
|
1006.3582
|
# Sonic Gradient Index Lens for Aqueous Applications
Theodore P. Martin Acoustics Division, Naval Research Laboratory, Washington,
DC 20375, USA Michael Nicholas Acoustics Division, Naval Research
Laboratory, Washington, DC 20375, USA Gregory J. Orris Acoustics Division,
Naval Research Laboratory, Washington, DC 20375, USA Liang-Wu Cai Department
of Mechanical and Nuclear Engineering, Kansas State University, Manhattan,
Kansas 66506, USA Daniel Torrent Wave Phenomena Group, Department of
Electronic Engineering, Universidad Politecnica de Valencia, C/ Camino de Vera
s7n, E-46022 Valencia, Spain José Sánchez-Dehesa Wave Phenomena Group,
Department of Electronic Engineering, Universidad Politecnica de Valencia, C/
Camino de Vera s7n, E-46022 Valencia, Spain
###### Abstract
We study the acoustic scattering properties of a phononic crystal designed to
behave as a gradient index lens in water, both experimentally and
theoretically. The gradient index lens is designed using a square lattice of
stainless-steel cylinders based on a multiple scattering approach in the
homogenization limit. We experimentally demonstrate that the lens follows the
graded index equations derived for optics by mapping the pressure intensity
generated from a spherical source at 20 kHz. We find good agreement between
the experimental result and theoretical modeling based on multiple scattering
theory.
###### pacs:
43.20.Fn, 43.58.Ls,43.20.Dk
††preprint: NRL-TMartin-v1.4
Composed of ordered arrays of scatterers similar to atoms in a conventional
solid, phononic crystals (PnC) are a class of metamaterial designed to control
acoustic wave propagation in a medium. PnCs have been proposed for a broad
range of applications in wave acoustics, with acoustic lensing Cervera _et
al._ (2002); Hu and Chan (2005); Yang _et al._ (2004); Torrent and Sánchez-
Dehesa (2007); Cai _et al._ (2007); Li _et al._ (2009); Qiu _et al._
(2005); Deng _et al._ (2009); Zhang _et al._ (2009); Lin _et al._ (2009);
Ke _et al._ (2007) featuring prominently in the literature due in part to the
ease with which focusing can be achieved by altering a crystal’s gemoetric
shape Cervera _et al._ (2002); Hu and Chan (2005) or compositional structure.
Cai _et al._ (2007); Torrent and Sánchez-Dehesa (2007); Li _et al._ (2009)
Although negative index lenses have received much attention due to their
potential for near-field imaging, Qiu _et al._ (2005); Deng _et al._ (2009);
Zhang _et al._ (2009) some positive index solutions such as the acoustic
analogue of the optical graded index lens Torrent and Sánchez-Dehesa (2007);
Lin _et al._ (2009) have not yet been explored experimentally. In addition,
the majority of PnC experiments have been performed in air, Cervera _et al._
(2002); Li _et al._ (2009); Torrent _et al._ (2006); Ke _et al._ (2007)
where the large density contrast with respect to the constituent scatterers in
the crystal (typically metals) allows the scatterers to be treated as rigid.
We demonstrate below that despite the physical limitation in impedance
contrasts between an aqueous medium and the scattering elements, it is
possible to design a PnC that behaves as an ideal graded index lens (GIL) in
water based on a fully elastic multiple scattering theory (MST). Torrent _et
al._ (2006); Torrent and Sánchez-Dehesa (2007); Krokhin _et al._ (2003)
Figure 1(a) shows a plan view schematic of the GIL design. The axes of Fig.
1(a) and throughout the paper are oriented with the lens center at position
$(x,y)=(0,0)$. The GIL is made up of 75 stainless steel cylinders (T-316) that
are 75 cm in length and arranged in a square lattice with spacing $a=1.8$ cm
and dimensions $5a\times 15a$. Figure 1(b) plots the cylinder radii $R(y)$,
which are stepped toward zero at each successive layer above and below the
central axis ($y=0$) of the GIL. In the homogenization limit (propagation
wavelength $\lambda\gtrsim 4a$) each stratified layer in Fig. 1(a) can be
treated as an effective medium. MST Torrent and Sánchez-Dehesa (2007) is used
to calculate each layer’s effective sound speed $c_{eff}$, which is inversely
proportional to the filling fraction of the cylinders. The layers will have an
effective refractive index $n_{eff}=c_{b}/c_{eff}$ ($c_{b}=1470$ m/s is the
sound speed in water) that is maximal at the center of the GIL and decreases
to that of water at the edges. Our choice of $R(y)$ in Fig. 1(b) produces a
graded $n_{eff}$ that obeys the same relation as an optical GIL, Smith (2000)
$n_{eff}=n_{0}(1-\alpha^{2}y^{2})^{1/2}$, where $n_{0}$ is the refractive
index at the central layer. Our design results in $n_{0}=1.2$ and
$\alpha=0.04$ cm-1.
Figure 1(c) shows an image of the GIL submerged in a $6\times 6\times 4$ m3
isolation tank. The cylinders are mounted between reinforced Plexiglas plates
to provide stability. Acoustic waves are produced by a 10 cm-diameter
spherical source at 20 kHz ($\lambda\simeq 4a$). The wave propagation is
measured in the time-domain using monitoring hydrophones at a sampling rate of
1 MHz. Hydrophones are mounted to the source and onto a translational 3-axis
Velmex VXM® positioning system. The transmission intensity is measured by
averaging over a 10-cycle pulse from the source; this pulse is long enough to
approximate a continuous wave measurement, while being short enough to prevent
contamination from reflections off the surfaces of the tank. We have
experimentally verified that the intensity $P_{0}$ produced by the source in
the absence of the GIL drops radially in proportion to $1/r^{2}$.
Figure 2(a) shows the normalized pressure amplitude $P/P_{0}$ measured after
transmission through the GIL on the side opposite the source ($x>0$). The
source is located at $(x,y)=(-196,0)$ cm, and both the source and the
translational hydrophone are positioned in the plane bisecting the axial
center of the cylinders. The GIL is shown schematically to scale and at its
proper location in each figure throughout the paper. The data in Fig. 2(a) is
measured 2.144 ms after the initial cycle began to leave the source. This time
gives a snapshot when the pulse is centered on an enhancement in signal
amplitude observed in the vicinity of $x\simeq 80$ cm.
Figure 2(b) shows the normalized intensity averaged over the 10-cycle pulse
and obtained from the same data set shown in Fig. 2(a). As in Fig. 2(a), a
clear focusing peak is observed centered close to $x\simeq 80$ cm. In Fig.
2(c) we show a two-dimensional MST calculation Torrent _et al._ (2006);
Torrent and Sánchez-Dehesa (2007) of the total pressure intensity (incident
plus scattered) derived by placing a continuous-wave cylindrical source at
$(x,y)=(-196,0)$ cm. The calculation assumes the cylinders to be a penetrable
elastic. Torrent and Sánchez-Dehesa (2007); Krokhin _et al._ (2003) As with
the experimental data, the simulated pressure intensity is normalized to that
of the source in the absence of the GIL. The source amplitude is a Hankel
function $P_{0}=H^{(1)}_{0}(kr)$ with wavevector $k=\pi/2a$. The MST
simulation also shows a clear focusing peak, but with two important
differences: (1) the measured intensity is $\sim 2$ times larger than the
simulation, and (2) the simulated focusing peak is slightly farther from the
GIL and decays more slowly.
To quantify whether our GIL design behaves as an ideal lens, in Fig. 3(a) we
present measurements of the focusing peak along the central axis of the lens
($y=0$) for different source positions $d_{s}$. For each $d_{s}$, a large-
amplitude peak is observed above $x\gtrsim 60$ cm, while smaller peaks are
also observed closer to the GIL. As the source is moved closer to the GIL, the
large-amplitude peak moves away in qualitative agreement with the expected
behavior of a lens. In Fig. 3(b) we show MST calculations along $y=0$ for
source positions similar to those in the experiment. On initial inspection it
appears that the theory shows slowly decaying focusing peaks that change very
little with $d_{s}$. However, expansion of the region around the focusing
peaks [Fig. 3(b) inset] reveals that the peak positions move away in a manner
similar to the experiment.
We now analyze the experimental data in Fig. 3(a) above $x>62$ cm to determine
whether the focusing positions in this region follow the ideal lens equations.
For an ideal lens, the focusing peak positions $d_{p}$ should scale with
$d_{s}$ as $1/d_{p}=1/f-1/d_{s}$. The focal length $f$ of a GIL can be
approximated as, Smith (2000)
$\displaystyle f\approx\frac{1}{n_{0}\alpha\sin\alpha t}$ (1)
where $t=5a$ is the thickness of the GIL. Equation (1) gives an estimate of
$f=58.9$ cm using values of $n_{0}$ and $\alpha$ calculated in the effective
medium approximation. 111Although we are technically measuring the _back focal
length_ of the GIL, $bfl=f\cos\alpha t$, the difference between our estimate
of $f$ and the sum $bfl+t/2$ is less than $1$ cm, which is less than both our
experimental error and our data resolution ($=1$ cm). Thus we ignore this
subtlety and compare to $f$ directly. See Ref. 14 for details.
A close inspection of Fig. 3(a) reveals that the data above $x>62$ cm is
actually composed of two superimposed peaks that both move to larger $x$ as a
function of $d_{s}$. Figure 3(c) shows two examples of a double-gaussian fit
to the data in this region using a standard unconstrained, nonlinear
optimization routine. The gaussians resulting from the fit are shown
individually (blue and red) in addition to the combined fit (black). Although
the fit equation
$\gamma_{1}e^{-\beta_{1}(x-d_{p1})^{2}}+\gamma_{2}e^{-\beta_{2}(x-d_{p2})^{2}}$
contains six free parameters, the purpose of the fit is to obtain an estimate
of the peak positions $d_{p1,2}$ and their relative amplitudes $\gamma_{1,2}$.
Fig. 3(c) demonstrates that the data is well described by the double-gaussian,
with a low-amplitude peak (Peak 1) closer to the GIL and a larger-amplitude
peak (Peak 2) farther away. In both cases the amplitude of Peak 2 is about
three times larger than Peak 1, suggesting that Peak 2 is the main focusing
peak of the GIL. The relative amplitudes of Peaks 1 and 2 are observed to
follow the same qualitative behavior for all the source positions in Fig.
3(a).
Figure 3(d) plots the inverse positions $1/d_{p1,2}$ extracted from the
gaussian fits as a function of $1/d_{s}$. An ideal lens will produce a linear
trend with a slope of $-1$ and an intercept of $1/f$. Although the trends for
both peaks are linear and have intercepts that yield similar focal lengths,
the slope of Peak 1 is much less than that of an ideal lens. However, the
trend for Peak 2 results in a slope of $-1$ and its focal length $f=59.3\pm
1.5$ cm agrees with the estimate of $f$ calculated using Eqn. (1). The dashed
line in Fig. 3(d) plots the peak locations obtained from the MST calculations
in Fig. 3(b). The theory produces a slope of $-1$ and focal length $f=61.1$ cm
that closely match both the measured data and the ideal lens equations.
We propose that Peak 1 and the other low-amplitude peaks in Fig. 3(a) are the
result of constructive interference between waves scattered from the support
structure of the lens. Low-amplitude, circular interference fringes can be
observed in Fig. 2(a) emanating from above and below the plot area centered at
$x\simeq 25$ cm. These fringes are the result of scattering off of stabilizing
pillars at the corners of the GIL support structure. While averaging over a
few initial pulse cycles will reduce the interference, a small number of
cycles gives a poor approximation to a continuous wave measurement and limits
the number of multiple scattering events that contribute to the focusing peak.
Therefore we have chosen to average over many cycles and rely on the gaussian
fitting routine to remove the spurious interference.
Figure 4 demonstrates that our GIL design acts as a lens with the source off
the central axis. Figure 4(a) shows $(P/P_{0})^{2}$ measured with the
spherical source located at a $14.7\,^{\circ}$ angle with respect to the
origin. Figure 4(b) shows the MST calculation for the same source location.
Thin white lines in Figs. 4(a,b) indicate a $14.7\,^{\circ}$ angle with
respect to the $x$-axis and extend to the expected focusing positions of an
ideal lens with $f\approx 60$ cm. Both the measured data and the MST
calculation demonstrate a strong focusing peak at the expected location.
Interference fringes from the support pillar can also be observed superimposed
on the focusing peak in Fig. 4(a).
In summary, we have designed and constructed a gradient index lens that
operates in water at sonic frequencies. Our transmission measurements
demonstrate that our GIL design focuses as an ideal lens based on the optical
GIL equations. Our measurements are also consistent with the focusing
positions obtained from two-dimensional models using multiple scattering
theory. We emphasize that our GIL behaves as an ideal lens at the limit of
homogenization ($\lambda\simeq 4a$) and with a thickness on the order of a
wavelength ($t=5\lambda/4$). Such performance at the limit of homogenization
theory demonstrates the versatility of phononic crystals designed using
multiple scattering theory.
This work was supported by the U.S. Office of Naval Research.
## References
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* Hu and Chan (2005) X. Hu and C. T. Chan, Phys. Rev. Lett., 95, 154501 (2005).
* Yang _et al._ (2004) S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan, and P. Sheng, Phys. Rev. Lett., 93, 024301 (2004).
* Torrent and Sánchez-Dehesa (2007) D. Torrent and J. Sánchez-Dehesa, New J. Phys., 9, 323 (2007).
* Cai _et al._ (2007) F. Cai, F. Liu, Z. He, and Z. Liu, Appl. Phys. Lett., 91, 203515 (2007).
* Li _et al._ (2009) J. Li, L. Fok, X. Yin, G. Bartal, and X. Zhang, Nature Mater., 8, 931 (2009).
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* Zhang _et al._ (2009) S. Zhang, L. Yin, and N. Fang, Phys. Rev. Lett., 102, 194301 (2009).
* Lin _et al._ (2009) S.-C. S. Lin, T. J. Huang, J.-H. Sun, and T.-T. Wu, Phys. Rev. B, 79, 094302 (2009).
* Ke _et al._ (2007) M. Ke, Z. Liu, P. Pang, C. Qiu, D. Zhao, S. Peng, and J. Shi, Appl. Phys. Lett., 90, 083509 (2007).
* Torrent _et al._ (2006) D. Torrent, A. Hakansson, F. Cervera, and J. Sánchez-Dehesa, Phys. Rev. Lett., 96, 204302 (2006).
* Krokhin _et al._ (2003) A. A. Krokhin, J. Arriaga, and L. N. Gumen, Phys. Rev. Lett., 91, 264302 (2003).
* Smith (2000) W. J. Smith, _Modern Optical Engineering_ (McGraw-Hill, Inc., 2000).
* Note (1) Although we are technically measuring the _back focal length_ of the GIL, $bfl=f\mathop{cos}\nolimits\alpha t$, the difference between our estimate of $f$ and the sum $bfl+t/2$ is less than $1$ cm, which is less than both our experimental error and our data resolution ($=1$ cm). Thus we ignore this subtlety and compare to $f$ directly. See Ref. 14 for details.
Figure 1: (a) Plan schematic of the gradient index lens. (b) Cylinder radius
$R$ plotted vs position along the $y$-axis. (c) Digital photograph of the GIL
in the isolation tank.
Figure 2: (a) Normalized pressure amplitude $P/P_{0}$ plotted vs $x$ and $y$,
measured 2.144 ms after the initial pulse leaves the source. (b) Measured,
normalized pressure intensity $(P/P_{0})^{2}$ plotted vs $x$ and $y$ after
averaging over a 10-cycle pulse. (c) Normalized pressure intensity
$(P/P_{0})^{2}$ calculated using MST. The focusing peak maximum is marked by a
$+$.
Figure 3: (a) Measured, normalized pressure intensity vs $x$ for source
positions $d_{s}=196$, $185.8$, $175.7$, $165.5$, $155.4$, $145.2$, $135$,
$124.9$, and $114.7$ cm. (b) Normalized pressure intensity calculated using
MST for source positions $d_{s}=109a$, $103a$, $98a$, $92a$, $86a$, $81a$, and
$75a$. Inset: expanded region of the $y$-axis showing the focusing peak
positions. (c) Two focusing peaks from panel (a) are replotted as circles
(upper region offset for clarity). Black lines indicate a double-gaussian fit,
with blue (Peak 1) and red (Peak 2) lines showing the component gaussians
individually. (d) Inverse peak positions $1/d_{p1,2}$ plotted vs inverse
source positions $1/d_{s}$. Blue and red lines are fits to the trends of Peaks
1 and 2 respectively. The dashed line plots the peak positions of the
simulated data in panel (b).
Figure 4: (a) Measured, normalized pressure intensity $(P/P_{0})^{2}$ plotted
vs $x$ and $y$ with the source at a $14.7\,^{\circ}$ angle with respect to the
$x$-axis. (b) Normalized pressure intensity $(P/P_{0})^{2}$ calculated using
multiple scattering theory with the source at a $15\,^{\circ}$ angle. White
lines indicate the off-axis angle; positions of the expected focusing peaks
are marked with a $+$.
|
arxiv-papers
| 2010-06-18T00:33:00 |
2024-09-04T02:49:11.007320
|
{
"license": "Public Domain",
"authors": "Theodore P. Martin, Michael Nicholas, Gregory J. Orris, Liang-Wu Cai,\n Daniel Torrent and Jose Sanchez-Dehesa",
"submitter": "Theodore Martin",
"url": "https://arxiv.org/abs/1006.3582"
}
|
1006.3594
|
11institutetext: Key Laboratory of Optical Astronomy, National Astronomical
Observatories, CAS, 20A Datun Road, Chaoyang District, 100012, Beijing, China
22institutetext: Graduate University of the Chinese Academy of Sciences, 19A
Yuquan Road, Shijingshan District, 100049, Beijing, China
33institutetext: Sydney Institute for Astronomy (SIfA), School of Physics, The
University of Sydney, NSW 2006, Australia
44institutetext: Zentrum für Astronomie der Universität Heidelberg,
Landessternwarte, Königstuhl 12, D-69117 Heidelberg, Germany
55institutetext: Division of Astronomy and Space Physics, Department of
Physics and Astronomy, Uppsala University, Box 516, 75120 Uppsala, Sweden
# The Hamburg/ESO R-process Enhanced Star survey (HERES) ††thanks: Based on
observations collected at the European Southern Observatory, Paranal, Chile
(Proposal numbers 170.D-0010, and 280.D-5011).
VI. The Galactic Chemical Evolution of Silicon
L. Zhang 1122 T. Karlsson 33 N. Christlieb 44 A. J. Korn 55 P. S. Barklem 55
G. Zhao 11
We determined the silicon abundances of 253 metal-poor stars in the
metallicity range $-4<\mathrm{[Fe/H]}<-1.5$, based on non-local thermodynamic
equilibrium (NLTE) line formation calculations of neutral silicon and high-
resolution spectra obtained with VLT-UT2/UVES. The $T_{\mathrm{eff}}$
dependence of [Si/Fe] noticed in previous investigation is diminished in our
abundance analysis due to the inclusion of NLTE effects. An increasing slope
of [Si/Fe] towards decreasing metallicity is present in our results, in
agreement with Galactic chemical evolution models. The small intrinsic scatter
of [Si/Fe] in our sample may imply that these stars formed in a region where
the yields of type II supernovae were mixed into a large volume, or that the
formation of these stars was strongly clustered, even if the ISM was enriched
by single SNa II in a small mixing volume. We identified two dwarfs with
$\mathrm{[Si/Fe]}\sim+1.0$: HE 0131$-$3953, and HE 1430$-$1123\. These main-
sequence turnoff stars are also carbon-enhanced. They might have been pre-
enriched by sub-luminous supernovae.
###### Key Words.:
line: formation – line: profiles – stars: abundances – stars: Population II –
Galaxy: abundances – Galaxy: halo
## 1 Introduction
Studying the detailed elemental abundances of metal-deficient stars in the
Galactic halo is a standard approach to probe the origin of our Galaxy and its
early evolution, as many of these stars have formed from the local
counterparts to high-redshift gas clouds during the early chemo-dynamical
evolution of the Galaxy (e.g. Beers & Christlieb 2005, and reference therein).
While abundance ratios as a function of [Fe/H]111 [A/B] =
$\log(N_{\rm{A}}/N_{\rm{B}})-\log(N_{\rm{A}}/N_{\rm{B}})_{\odot}$ provide
information about the chemical enrichment history of the Galaxy, the scatter
of these ratios allow to study mixing processes of the interstellar medium
(ISM) in the early phases of the formation of the Galaxy (e.g. Argast et al.
2000; Karlsson & Gustafsson 2005; Karlsson 2005).
In investigations of the enrichment of the ISM, the $\alpha$-elements (e.g.,
Mg, Si, Ca, and Ti) are often used as tracer elements, because their yields
depend on the mass and the explosion energy of the SN and the amount of
fallback (Karlsson 2005). Silicon, which is produced by explosive oxygen
burning, belongs to the most abundant metals, and it can be detected over a
wide metallicity range. Besides, some extreme examples are found, which
challenge the enrichment model of SNe II. For instance, HE 1424$-$0241, an
extreme metal-poor star with $\mathrm{[Fe/H]}=-4.0$, has a very low Si
abundance (i.e., $\mathrm{[Si/Fe]}\sim-1.0$ dex, Cohen et al. 2007).
Therefore, Si is an element to probe the enrichment of the ISM.
Previous studies of silicon abundances in metal-poor stars yielded a range of
scatter in [Si/Fe]; typically from $\sim 0.06$ dex to 0.4 dex (e.g. Ryan et
al. 1996; Cayrel et al. 2004; Cohen et al. 2004; Honda et al. 2004; Aoki et
al. 2005; Preston et al. 2006; Lai et al. 2008; Shi et al. 2009). However,
these dispersions can not be simply considered as cosmic scatter reflecting
the ISM mixing process. This is mainly due to three reasons: (1) the small
sample size of analysis stars in most of the above-mentioned studies; (2) when
several analyses from the literature are combined, systematic offsets in the
Si abundances due to different methods of stellar parameter determination and
different structure of model atmospheres may arise, which artificially
increases the scatter in the combined sample; (3) the Si abundance derived
from the Si I line at 3905 Å, which is the only line that can be reliably
measured in stars at $\mathrm{[Fe/H]}<-2.5$ may not represent the true value,
because this line may be contaminated by CH lines (Cayrel et al. 2004) and the
abundance determined from this line shows an abnormal dependence on effective
temperature ($T_{\mathrm{eff}}$)(Preston et al. 2006; Lai et al. 2008). All
these may conceal the “real” cosmic scatter. Thus, Si abundances determined in
a careful and homogeneous way for a large sample of metal-poor stars are
needed.
Very recently, an NLTE analysis of silicon abundances of metal-poor stars has
been carried out by Shi et al. (2009), who discuss the NLTE effects of the
strong Si I lines at 3905 Å and 4103 Å. A strong correlation between the
difference of [Si/Fe] calculated under NLTE and LTE assumptions of these two
lines and the stellar parameters in their sample was noticed. This confirms
the suggestion of Preston et al. (2006) that Si abundances determined from the
Si I line at 3905 Å without NLTE corrections for metal-deficient star may not
be considered as the true values at $T_{\mathrm{eff}}$ warmer than 5800 K.
From these results, the anomalous $T_{\mathrm{eff}}$ dependence of [Si/Fe]
(Preston et al. 2006; Lai et al. 2008) can be partially explained. Hence NLTE
has to be taken into account when studying the chemical evolution of Si and
the scatter of [Si/Fe] as a function of [Fe/H].
The aim of this work is thus to obtain detailed silicon abundances of metal-
poor stars, so that the correlation between the abundance ratios and the
stellar parameters and the chemical enrichment of the ISM are explored. This
work is based on spectra of the Hamburg/ESO R-process Enhanced Star survey
(HERES), as described in Section 2. The method and the procedures of the
abundance analysis are described in Section 3. The results are presented in 4
and discussed in Section 5.
## 2 Observations and stellar parameters
The present work is based on the spectra of 253 HERES stars. The sample
selection and observations are described in Christlieb et al. (2004). For the
convenience of the reader, we repeat here that the spectra were obtained with
the Ultraviolet-Visual Échelle Spectrograph (UVES, Dekker et al. 2000) mounted
on the 8 m Unit Telescope 2 (Kueyen) of the Very Large Telescope (VLT). The
pipeline-reduced spectra cover the wavelength range from 3769 Å to 4980 Å at a
minimum seeing-limited resolving power of $R=20,000$. The coordinates and
barycentric radial velocities of the stars are listed in Table 1 of Barklem et
al. (2005) (heareafter B05).
We adopt the stellar parameters of B05 in our analysis. In the work of B05,
photometric $T_{\mathrm{eff}}$, metallicity estimated from the calibration of
the Ca II K-line index along with $B-V$ color (Beers et al. 1999), $\log g$
estimated from $\log g-T_{\mathrm{eff}}$ correlation (Honda et al. 2004),
$\xi=1.8$ km s-1, and $v_{\mathrm{macro}}=1.5$ km s-1 were set as initial
guess, and then were refined in an automated analysis which is based on the
Spectroscopy Made Easy (SME) package by Valenti & Piskunov (1996). The details
are described in Sections 2 and 3 of B05.
## 3 Abundance analysis
In our analysis, the one-dimensional line-blanked local thermodynamic
equilibrium (LTE) model atmospheres MAFAGS (Fuhrmann et al. 1997), with
opacity distribution functions (ODF) of Kurucz (1992) are employed. For
consistency, solar abundances are the same as B05, i.e., C is taken from
Allende Prieto et al. (2002) and other elements are those of Grevesse & Sauval
(1998). During the computation of model atmospheres at $\mathrm{[Fe/H]}<-0.6$,
an $\alpha$-element enhancement of 0.4 dex is adopted. A convective efficiency
of $\alpha_{\mathrm{mlt}}=0.5$ is used. For more details on the model
atmospheres, we refer the reader to Grupp (2004).
### 3.1 Line synthesis
The silicon abundances were determined by spectrum synthesis of the Si I lines
at 3905.53 Å and 4102.93 Å, using the Spectrum Investigation Utility (SIU) of
Reetz (1991), which computes line formation under both LTE and NLTE
conditions. Continuum scattering is considered in the computation of the
source function.
Shi et al. (2009) studied the silicon abundance discrepancy between NLTE and
LTE analyses for the two lines adopted in our analysis, and they suggested
that this departure is correlated with the strength of lines and stellar
parameters. The main characteristics are: the NLTE effects of weak lines is
small; the NLTE corrections of these two lines increase for extremely metal-
poor warm stars, and the values can reach more than 0.15 dex for the 3905 Å
line and 0.25 dex for the 4103 Å line. Thus, the NLTE effects of these two
lines are considered in the present analysis. The silicon model atom and the
NLTE calculation method are described in detail in Shi et al. (2008, 2009).
Another factor which may affect the determination of the silicon abundance is
contamination with CH lines. Cohen et al. (2004) suggested that the Si I line
at 3905.53 Å is probably blended with the B-X bandhead, which is located
approximately at $\lambda=3900$ Å. Preston et al. (2006) noticed that the
blend effect of this CH band is weak in their sample of red horizontal-branch
stars. However, the [C/Fe] ratio of most of their sample stars is less than
0.0 dex. Therefore, in order to get reasonable results for our metal-deficient
sample stars including giants and main-sequence stars, the CH B-X lines are
included in our line synthesis.
Although B05 have already derived the carbon abundance, in order to keep the
consistency of the abundance analysis technique, the abundance determination
for A-X system of CH near 4310 Å were independently performed with the
analysis code. The oxygen abundance was adopted to be $\mathrm{[O/Fe]}=0.6$
dex.
The atomic line data of Si I lines are listed in Table 1. The oscillator
strengths ($\log gf$) are adopted from the experimental results of Garz
(1973), and van der Waals interaction constants ($\log\rm C_{6}$, in the unit
of $\rm s^{-1}\rm cm^{6}$, frequency definition) are calculated according to
the interpolation tables of Anstee & O’Mara (1991, 1995). The molecule line
data of the CH A-X system are taken from B05, and reference therein. The line
positions and $\log gf$ values of the CH lines around 3900 Å are selected from
the database of Kurucz (1993). They are listed in Table 2. For stars in which
neither of the Si lines can be detected clearly, the feature which is on the
position of theoretical silicon line was fitted, and the maximum value for Si
that could fit the spectrum is considered as the upper limits for the Si
abundance. Synthetic spectra for six representative stars of our sample are
shown in Fig. 1.
Figure 1: Examples of spectral synthesis for six representative stars. The dots are the observational spectra, the solid lines are the best-fitting profile, and the dotted lines are the synthetic spectra with Si abundances of $\pm 0.15$ dex relative to the best fit, corresponding to less/larger than 5% in the continuum. The listed parameters are $T_{\rm eff}$, $\log g$, [Fe/H], and $\xi_{t}$, respectively. Table 1: Atomic data of the Si I lines used in our analysis. $\lambda$ [Å] | Transition | $E_{\mathrm{low}}$ [eV] | $\log gf$ | $\log C_{6}$
---|---|---|---|---
3905.53 | 3p1S0 – 4s1P${}^{0}_{1}$ | $-$1.909 | $-$1.09 | $-$30.917
4102.93 | 3p1S0 – 4s3P${}^{0}_{1}$ | $-$1.909 | $-$3.14 | $-$30.972
Table 2: Molecular line data for B-X system of the CH molecule near 3905 Å from Kurucz (1993) $\lambda$ [Å] | $E_{\mathrm{low}}$ [eV] | $\log gf$ | $\log C_{6}$
---|---|---|---
3905.675 | 0.124 | $-$1.178 | $-$32.521
3905.716 | 0.124 | $-$3.862 | $-$32.521
### 3.2 Abundance uncertainties
The main uncertainties in the abundances are caused by (1) uncertainties in
the analysis of individual lines, including random errors of atomic data and
fitting uncertainties; (2) errors in the continuum rectification; (3)
uncertainties of the stellar parameters.
The errors of $\log gf$ given in Garz (1973) were adopted as the perturbation
which was added to change the abundance. The variances of the silicon
abundance were taken as the uncertainties affected by $\log gf$, and they are
around 0.02 dex. It results in an error of 0.02 dex on average. After getting
the best fitting profile of a certain silicon line, the abundance was changed
until the profile deviates from the best one. This abundance change is adopted
as the fitting uncertainty. Typically, this value is 0.03 dex, which is close
to the noise. Finally, the random error is estimated by summing the estimated
error on the adopted $\log gf$ value and the fitting uncertainty in
quadrature. This result is around 0.04 dex.
The continuum around the silicon line at $\lambda=3905.53$ Å is affected by
the wings of H$\epsilon$ and Ca II K lines if the effective temperature
exceeds 5500 K in our analysis. It is difficult to get the accurate continuum
location for this wavelength range in this case, which has a direct effect on
abundance determination for the dwarfs. The situation is similar for the
4102.93 Å line, which is located in the wing of the H$\delta$ line. In the
worst case, the error in continuum rectification was estimated to be five
percent, which results in a change of the Si abundance of up to 0.11 dex.
From the determination of atmospheric parameters described in B05, 100 K, 0.25
dex, 0.1 dex, and 0.15 km s-1 are the average uncertainties of
$T_{\mathrm{eff}}$, $\log g$, metallicity, and micro-turbulent velocity,
respectively. These uncertainties typically result in abundance changes of
0.06 dex, 0.03ḋex, 0.01 dex, and 0.1 dex, respectively. The overall
uncertainty from errors in the atmospheric parameters is estimated by summing
these four abundance changes in quadrature.
Finally, the quadratic sum of the uncertainties from these three sources is
adopted as the total abundance error.
## 4 Results
### 4.1 Carbon
The abundance results are listed in Table 3, and a comparison with the
abundances derived by B05 is shown in Fig. 2. The carbon abundances agree well
with each other:
$\log\varepsilon(\rm C)_{\rm B05}=-0.05(\pm 0.07)+0.99(\pm
0.01)\times\log\varepsilon(\rm C)_{\rm ThisWork}$
We note that the $\log\varepsilon(\rm C)$ values derived by us are
systematically higher by about 0.10 dex. This difference can be explained by
the difference of the model atmospheres. The theoretical continuum computed by
the MAFAGS is higher than that calculated by MARCS (used in B05), which
results in a higher carbon abundance.
Figure 2: Comparison of the carbon abundance determined in this work with
those of B05. The open circles refer to giants, while the filled circles
represent subgiants and dwarfs. The solid line is the one-to-one correlation
and the dotted line represents a linear fit of the data.
The carbon abundance ratio as a function of $T_{\mathrm{eff}}$ is shown in
Fig. 3. The decreasing [C/Fe] towards decreasing $T_{\rm eff}$ for stars whose
$T_{\rm eff}$ are below 5000 K is expected, because the surface abundance of
carbon of evolved giants may be deficient due to the mixing processes
including first dredge-up and extra-mixing (Cayrel et al. 2004; Lucatello et
al. 2006; Aoki et al. 2007). For the giants with $T_{\rm eff}$ lower than 5000
K, contamination of Si I 3905 Å by CH B-X band can be neglected. Excluding
these low temperature giants and the carbon enhanced stars ([C/Fe] $>$ 1.0 dex
(see Lucatello et al. 2006), the $<$[C/Fe]$>$ = $0.33\pm 0.24$ dex. If the
carbon enhanced stars are accounted in, the average value is changed to
0.42$\pm$0.44 dex, with larger dispersion. These values imply that the CH B-X
band may affect the line profile of Si I 3905 Å for most of our sample stars,
thus it is necessary to add CH B-X band in our line fitting.
Figure 3: [C/Fe] as a function of $T_{\mathrm{eff}}$. The symbols are the same
as in Fig. 2.
### 4.2 Silicon
Our silicon abundance results are also listed in Table 3. The average value
and standard deviation of the abundance ratios derived by these two lines are
as follows: $<[\rm Si/Fe]_{3905}>=0.44\pm 0.39$ (247 stars) and $<[\rm
Si/Fe]_{4103}>=0.41\pm 0.42$ (199 stars). Note that the stars for which only
upper limits are available are not considered in these calculations.
The abundance discrepancy between Si3905 and Si4103 as a function of the C
abundance and the stellar parameters is shown in Fig. 4. In this figure, only
the stars that had both lines measured are used to make a comparison. The
dashed lines in Fig. 4 present the average difference (0.06 dex) and 1
$\sigma$ scatter $(\pm 0.09$ dex). In the upper panel one can notice that
there is no trend in $\Delta$ ($=\log\varepsilon(\rm
Si)_{3905}-\log\varepsilon(\rm Si)_{4103}$) vs. $\log\varepsilon(\rm C)$. It
reflects the fact that the contamination with the CH B-X band has been
eliminated in our final results.
There is a small offset between the results derived from 3905 Å and those
derived from the 4102.93 Å line. According to Shi et al. (2008), the 4102.93 Å
line should give a higher abundance if the $\log gf$ values of Garz (1973) are
adopted (see Shi et al. 2008, Fig.7). Our results show the contrary. As
discussed above, the blend with CH lines is unlikely to be the reason.
Moreover, most of the sample stars are very metal-poor, thus blends of other
metal components can be neglected. From the panels of Fig. 4, the distribution
of the difference shows a concentration around giants. This phenomenon may be
explained by two reasons:
(1) Lai et al. (2008) raised the hypothesis that strong lines would lead to
larger abundance values than weak ones, especially in giants, if the $T-\tau$
relationship of the adopted model atmosphere is shallower than that of true
one. For most of the giants in our sample, the equivalent width (EW) of the
line at 3905 Å ($\mathrm{EW}>150$ mÅ) is much larger than that of the 4103 Å
line ($\mathrm{EW}<120$ mÅ), thus the larger derived abundance from the 3905 Å
line and a slight increase of the difference towards decreasing $T_{\rm eff}$
(see the second panel of Fig. 4) are reasonable.
(2) The strong lines are sensitive to the micro-turbulence velocity. Twenty
stars were used as a test: if the $\xi_{t}$ value is increased 0.15 km/s, the
$\log\varepsilon(\rm Si_{3905})$ will decrease by 0.11 dex, while the
$\log\varepsilon(\rm Si_{4103})$ only decreases by 0.04 dex. Hence, the
determination of $\xi_{t}$ may cause higher silicon abundances for the 3905 Å
line.
It can also be seen in Fig. 4 that $\Delta$ decreases with increasing
metallicity. This s probably an artifact caused by the fact that the 4103 Å
line is difficult to detected at low metallicity. In these comparison, stars
in which only $\rm Si_{3905}$ can be detected are unavailable in such a low
metallicity range.
The average of the Si abundance determined from Si3905 and Si4103 are taken to
represent the final abundance. If only an upper limit can be derived from one
line, we adopt the value derived from the other line. The average Si abundance
ratio and its standard deviation are $<[\rm Si/Fe]>=0.46\pm 0.20$ (253 stars).
This value is closed to the prediction in Goswami & Prantzos (2000) ( about
0.5 dex in the low metallicity regime), while the value is 0.53–0.68 dex in
the calculation of Kobayashi et al. (2006). The higher theoretical value is
primarily due to the adopted IMF in the models, because [$\alpha$/Fe] is
higher for larger stellar masses.
Considering the mixing effect in low temperature giants and the accretion from
a companion for the carbon enhanced metal-poor (CEMP) star, an average [Si/C]
of $0.13\pm 0.21$ in the range of $0<\mathrm{[C/Fe]}<1$ and $T_{\rm eff}>$
5000 K was estimated. In the predictions of (Woosley & Weaver 1995; Heger &
Woosley 2002), [Si/C] is about 0.15 dex if the initial mass of the progenitor
star was about 12–40 M⨀.
In the upper panel of Fig. 6, we show our results along with the results of
previous LTE silicon abundance analyses. Most of these studies presented large
scatters in [Si/Fe]. For instance, Ryan et al. (1996) showed that the star-to-
star scatter increases towards decreasing [Fe/H], that is 0.11 for [Fe/H]
$>-1.5$, 0.14 for $-2.5<$ [Fe/H] $<-1.5$, and 0.32 for [Fe/H] $\leq-2.5$.
Preston et al. (2006) gave a star-to-star scatter of 0.22 for 24 giants([Fe/H]
$<-2.0$). In our NLTE results, the scatter of dwarfs is smaller ($\sim 0.13$).
Also, for the whole sample, the star-to-star scatter is close with the
estimated uncertainties ($\sim$ 0.16), that is, 0.23 dex, 0.18 dex, and 0.16
dex in the metallicity ranges of [$-4$,$-3$], [$-3$,$-2$], and [$-2$,$-1$],
respectively. In the lower panel of the same figure, our result shows stronger
correlation between [Si/Fe] and [Fe/H]. The slope of [Si/Fe] versus [Fe/H]
found in our NLTE analysis is $-0.14$ ([Si/Fe] = $0.15(\pm 0.07)-0.14(\pm
0.03)\times$ [Fe/H]), which is larger to the values found by most LTE results
(e.g., –0.03 in McWilliam et al. (1995), –0.07 in Ryan et al. (1996), 0.03 in
Honda et al. (2004), –0.06 in Preston et al. (2006), and so on). More details
are discussed in Sec. 5.
Figure 4: Difference between the abundances of Si determined by the Si I 3905 and 4103 Å lines as a function of the C abundance and stellar parameters. The symbols are the same as in Fig. 2. The dashed lines show the average difference between these two lines and $1\sigma$ scatter. Table 3: Abundance results of carbon and silicon. The entire table is available only electronically. A portion is shown here for guidance regarding its form and content. The last column is the average of [Si/Fe] from two Si I lines. If only upper limit can be got from one line, taking the value of the other line represents the average value. | | | | | $\log\varepsilon(\rm Si)_{\rm NLTE}$ | [Si/H]NLTE | … | … | …
---|---|---|---|---|---|---|---|---|---
star | [Fe/H] | $\log\varepsilon(\rm C)$ | [C/H] | [C/Fe] | 3905 | 4103 | 3905 | 4103 | … | … | …
CS22175-007 | –2.81 | 5.80 | –2.72$\pm$0.14 | 0.09$\pm$0.16 | 5.16 | $<$5.19 | –2.39$\pm$0.13 | $<$–2.36$\pm$0.15 | … | … | …
CS22186-023 | –2.72 | 6.00 | –2.52$\pm$0.10 | 0.20$\pm$0.12 | 5.26 | 5.17 | –2.29$\pm$0.09 | –2.38$\pm$0.11 | … | … | …
CS22186-025 | –2.87 | 5.35 | –3.17$\pm$0.15 | –0.30$\pm$0.17 | 5.22 | 5.28 | –2.33$\pm$0.14 | –2.27$\pm$0.16 | … | … | …
CS22886-042 | –2.68 | 5.71 | –2.81$\pm$0.11 | –0.13$\pm$0.13 | 5.46 | 5.22 | –2.09$\pm$0.10 | –2.33$\pm$0.12 | … | … | …
CS22892-052 | –2.95 | 6.35 | –2.17$\pm$0.11 | 0.78$\pm$0.13 | 5.31 | 5.13 | –2.24$\pm$0.10 | –2.42$\pm$0.12 | … | … | …
. | . | . | . | . | . | . | . | . | . | . | .
. | . | . | . | . | . | . | . | . | . | . | .
. | . | . | . | . | . | . | . | . | . | . | .
HE2338-1618 | –2.65 | 6.31 | –2.21$\pm$0.10 | 0.44$\pm$0.12 | 5.41 | 5.25 | –2.14$\pm$0.09 | –2.30$\pm$0.11 | … | … | …
HE2345-1919 | –2.46 | 6.40 | –2.12$\pm$0.10 | 0.34$\pm$0.12 | 5.58 | 5.60 | –1.97$\pm$0.09 | –1.95$\pm$0.11 | … | … | …
HE2347-1254 | –1.83 | 7.02 | –1.50$\pm$0.14 | 0.33$\pm$0.16 | 6.07 | 6.11 | –1.48$\pm$0.13 | –1.44$\pm$0.15 | … | … | …
HE2347-1334 | –2.55 | 5.20 | –3.32$\pm$0.13 | –0.77$\pm$0.15 | 5.36 | 5.26 | –2.19$\pm$0.12 | –2.29$\pm$0.14 | … | … | …
HE2347-1448 | –2.31 | 6.84 | –1.68$\pm$0.11 | 0.63$\pm$0.13 | 5.79 | $<$5.74 | –1.76$\pm$0.10 | $<$–1.81$\pm$0.12 | … | … | …
## 5 Discussion and conclusions
### 5.1 Abundance correlations with stellar parameters
In Figs. 5 and 6, we show [Si/Fe] as a function of the stellar parameters. The
abundance correlation with stellar parameters is discussed below.
Figure 5: Si abundance ratio as a function of stellar parameters. The arrows
refer to upper limits; otherwise, the symbols are the same as in Fig. 2. The
average error bar is shown in the lower right corner of each panel. The
crosses are the results of Preston et al. (2006), the while the open triangles
are the ones of Lai et al. (2008). Besides, in the upper panel, dashed line,
short dashed line, and dot dashed line represent the least square fits of the
results of our observed data, Preston et al. (2006), and Lai et al. (2008),
respectively. Figure 6: Same as Fig. 5, but for [Si/Fe] vs. [Fe/H]. In the
upper panel, we compare our results with those of previous LTE analyses:
McWilliam et al. (1995, pluses); Ryan et al. (1996, asteriks); Cayrel et al.
(2004, open square); Honda et al. (2004, open diamonds); Aoki et al. (2005,
filled square). In the lower panel, thick solid lines are 1-$\sigma$ scatter,
short dashed lines crossing data points represents the fitting slope, and
dashed-dotted lines are the fitting uncertainties.
Previous LTE silicon abundance analyses of metal-poor stars reported a
correlation of [Si/Fe] with $T_{\mathrm{eff}}$ (e.g. Preston et al. 2006; Lai
et al. 2008), i.e., [Si/Fe] decreases with increasing temperature. In our
results with NLTE correction, the phenomenon is not obvious. The slopes of
these three data sources are listed below:
(1) This work:
[Si/Fe] = $0.29(\pm 0.13)+0.33(\pm 0.23)\times T^{\prime}_{\rm eff}$
(2) Preston et al. (2006):
[Si/Fe] = $4.16(\pm 0.39)-6.74(\pm 0.68)\times T^{\prime}_{\rm eff}$
(3) Lai et al. (2008):
[Si/Fe] = $1.28(\pm 0.48)-1.74(\pm 0.83)\times T^{\prime}_{\rm eff}$
Note that $T_{\rm eff}=T^{\prime}_{\rm eff}\times 10^{4}$.
These relationships can be also seen in the upper panel of Fig. 5, in which
our results are plotted along with previous LTE abundance analyses. The steep
slope in [Si/Fe] versus $T_{\mathrm{eff}}$ in previous studies is mainly
caused by the low [Si/Fe] stars hotter than $\sim 5500$ K. The NLTE correction
decreases with decreasing temperature. At higher $T_{\mathrm{eff}}$, the
results with NLTE correction will become larger, which causes a higher silicon
abundance than those of LTE and makes this slope much smaller. Therefore, our
results support the conclusion of Shi et al. (2009) that NLTE effects can
explain the temperature dependency of [Si/Fe]. Therefore, the increasing trend
of [Si/Fe] with the declined $T_{\mathrm{eff}}$ is diminished, if NLTE is
considered in the abundance analysis of silicon.
Preston et al. (2006) concluded that there was no correlation between [Si/Fe]
and $\log g$, and our NLTE results also confirm this conclusion.
In the lower panel of Fig. 6, an increase of [Si/Fe] with decreasing [Fe/H]
can be seen. Although Fe I is affected by significant NLTE effects for giants
and very metal-poor stars (e.g. $T\mathrm{eff}=5000$ K, $\log g=2.00$, and
$\mathrm{[Fe/H]}=-3.00$, Mashonkina et al. 2010), the NLTE correction of Fe I
leads only to small changes in our final [Si/Fe] results and the slope of
[Si/Fe] vs. [Fe/H]. In the worst case, we find a NLTE correction for [Fe/H] of
$+0.25$ dex, corresponding to a change in [Si/Fe] of $+0.03$ dex. Applying the
corrections to our 22 very metal-poor giants ($\mathrm{[Fe/H]}<-3.0$ dex)
would lead a change of $+0.02$ in the slope of [Si/Fe] vs. [Fe/H]. In
addition, the corrections for stellar granulation for Si and Fe are small
(i.e., $<0.1$ dex), and significant only for high-excitation potential lines
in metal-deficient stars (Asplund 2005). Therefore, we conclude that the
observed slope in Fig. 6 may not be the result of NLTE/3D effects.
Magnesium is also used as the tracer to discuss the metallicity dependence. In
Fig. 7, [Si/Mg] against [Mg/H] is plotted, where the magnesium abundances are
taken from B05. A slope of [Si/Mg] vs. [Mg/H] can be noticed: [Si/Fe] =
$0.02(\pm 0.06)-0.07(\pm 0.03)\times$ [Mg/H]. The NLTE effect of Mg may not be
the reason which causes this tendency. This is because in the very recent NLTE
study of Mg of Andrievsky et al. (2010), the NLTE results of Mg have the same
evolution behavior as the LTE ones, and the NLTE correction of Mg just
enhances the abundance.
More discussion about the trends will be presented in 5.3.
Figure 7: [Si/Mg] as a function of [Mg/H]. The symbols are the same as in Fig.
5. The big triangles are marked as the Si-enhancement stars. Open ones are
giants, while filled ones are dwarfs.
### 5.2 The outliers in our sample
We did not find any stars with a deficiency of Si (such as HE 1424$-$0241,
Cohen et al. 2007). This star is at [Fe/H]$\sim-4$ with an unusually low Si
abundance such that [Si/Fe]$=-1.01$ and[Si/Mg]$=-1.45$. Cohen et al. (2008)
speculated that this phenomenon may be the result of a chemically
inhomogeneous ISM and that the star probably was enriched by a single SN. If
so, our results imply that our sample stars may not be formed in the gas which
was contributed by ejecta from only one SN. This will be discussed further in
Sec. 5.3.1.
On the other hand, we noticed five candidates with large overabundance of
silicon, [Si/Fe] are 1.47 dex, 0.99 dex, 1.10 dex, 1.01 dex, and 1.03 dex for
HE 0308$-$1154, HE 1246$-$1344, HE 2314$-$1554, HE 0131$-$3953, and HE
1430$-$1123, respectively. The first three are giants and the other two are
dwarfs. Only HE 0308$-$1154 whose [Si/Fe] is outside of the 3$\sigma$ limit
can be clearly considered as Si-enhancement (in our observed sample, [Si/Fe]
is in Gaussian distribution, that is $\\#=253.,\mu=0.46,\sigma=0.20$). To
probe the nature of these stars, we investigate the abundance patterns of
these stars, as derived by B05, and discuss them below.
Giants:
Two additional metal-deficient giants with large Si-enhancement are known:
(1) CS29498$-$043 [Fe/H]=$-$3.75 dex, [C/Fe]=1.90 dex, [Mg/Fe]=1.81 dex,
[Si/Fe]=1.07 dex (Aoki et al. 2002)
(2) CS22949$-$037 [Fe/H]=$-$3.79 dex, [C/Fe]=1.05 dex, [Mg/Fe]-1.22 dex,
[Si/Fe]=1.04 dex (Norris et al. 2001).
Both of them are CEMP stars with a large excess of $\alpha$-elements.
However, in our study, the giants HE 0308$-$1154, HE 1246$-$1344, and HE
2314$-$1554 have otherwise “normal” abundance ratios. We checked the EW of two
Si I lines of these three stars, and found that both of the EWs of these lines
are larger than 100 mÅ, and the differences of derived abundance between Si I
3905 and 4103 are small. The incorrect ”T-$\tau$” relationship in model
atmosphere (Lai et al. 2008) can results in an offset of 0.2 dex. This
phenomena can be partially interpreted by the following hypothesis.
Dwarfs:
Previously, large excesses of Si were rarely found in dwarfs. The [Si/Fe]
value of metal-deficient dwarfs determined by using Si I transitions in the
red spectral region which are not affected by NLTE effects, are seldom higher
than 0.6 dex (e.g. Stephens & Boesgaard 2002; Shi et al. 2009; Zhang et al.
2009), but these lines are difficult to detected at $\mathrm{[Fe/H]}<-2.0$
dex. Even assuming a NLTE correction of $+0.2$ dex for the [Si/Fe] values
determined by Preston et al. (2006); Lai et al. (2008), where the Si abundance
is derived from the 3905.93 Å line, none of the stars in their sample would be
Si-enhanced by more than 0.75 dex.
The two Si-enhanced dwarfs, HE 0131$-$3953 and HE 1430$-$1123, are Ba-enhanced
CEMP stars. Furthermore, HE 0131$-$3953 was identified as an s-II star 222this
kind of star is also called $r+s$ star (Jonsell et al. 2006) by B05, and HE
1430$-$1123 has rather low [Sr/Ba] value of $-1.58$ dex, which is thought to
be associated with the s-II stars. This star can not be identified as a s-II
star because of lacking abundance information for Eu (see more details in
B05). Although mass transfer from a formerly more massive companion during its
AGB phase might have caused the enhancements of C and Ba seen in these stars,
this scenario does not provide an explanation for the Si-enhancements.
Tsujimoto & Shigeyama (2003) suggested that it might be due to pre-enrichment
by subluminous SNe experiencing mixing and fallback. The fallback which
occurred inside the Si layer in subluminous SNe can result in smaller
abundances of elements heavier than Si and the enhancement of Si in these CEMP
stars relative to iron and the abundance ratio in the Sun.
### 5.3 Star-to-star scatters and mixing of the interstellar medium
The dispersion in the abundance ratios of metal-poor stars provides a measure
of the chemical inhomogeneities in the star-forming gas, and hence of the
mixing processes in the ISM. Audouze & Silk (1995) argued that increasing
inhomogeneity is to be expected with decreasing metallicity, as a result of
the small number statistics of enriching events (i.e., SN II). This was also
observed for a number of element ratios (Ryan et al. 1996; McWilliam 1997).
In the wake of these findings, Argast et al. (2000) derived the expected
scatter for several abundance ratios, including [Si/Fe], as a function of
metallicity. They predict a star-to-star scatter of $\sim 0.4$ dex in [Si/Fe]
in the range of $-4<\mathrm{[Fe/H]}<-3$, at which the model ISM was
essentially unmixed. The scatter reduces to $\sim 0.25$ dex in the range
$-3<\mathrm{[Fe/H]}<-2$ due to a gradually increased mixing. At
$\mathrm{[Fe/H]}>-2.0$, the scatter is around $0.2$ dex, reaching typical
levels of the observational uncertainties depending on the data quality.
In contrast, more recent studies have reported on a number of elements for
which the scatter in the abundance ratios, like [Mg/Fe], are consistent with
the observational uncertainties, all the way down to [Fe/H] $\sim-3.5$ (e.g.,
B05; Cohen et al. 2004; Arnone et al. 2005; Lai et al. 2008; Bonifacio et al.
2009). In the present study, the 1-$\sigma$ scatter in [Si/Fe] is 0.23 dex,
0.18 dex, and 0.16 dex in the metallicity range [$-4$,$-3$], [$-3$,$-2$], and
[$-2$,$-1$], respectively. Because the halo ISM should be well mixed at
metallicities higher than $-2.0$ dex, as suggested by minimal mixing models
like the one by Argast et al. (2000), the scatter of 0.16 dex can be
considered as the observational error. If so, the cosmic scatter is less than
0.15 dex in the full range $-4<\mathrm{[Fe/H]}<-2$, which is considerably
smaller than what was predicted by Argast et al. (2000). It therefore seems
that also Si belongs to the class of elements that show very little cosmic
scatter. However, extreme outliers do exist also in [Si/Fe] (see Cohen et al.
2007). It is not entirely known which role such outliers play. Have they been
formed out of gas enriched by SNe in a specific mass range or are they “freak
objects” formed under very particular circumstances? In the latter case, the
measured surface abundances may not uniquely reflect common SN
nucleosynthesis. We shall further discuss these issues in the next sections.
#### 5.3.1 Stochastic modelling of the chemical evolution of Si
In order to investigate the enrichment and amount of mixing in the early ISM,
our large, homogeneous sample is compared with a stochastic model of the
chemical evolution of Si. The statistics discussed here are based on a model
originally developed by Karlsson (2005, 2006) and Karlsson et al. (2008). In
this model, stars are assumed to form randomly within the system. They enrich
their surroundings locally, by ejecting heavy elements such as Si and Fe. The
Fe yields used to calculate the metallicity distribution function (MDF)
depicted in Fig. 8, are taken from Umeda & Nomoto (2002), which are nearly
identical to the Fe-yields presented in Nomoto et al. (2006). The turbulent
mixing of the ISM is modeled as a diffusion process such that each individual
SN remnant continues to grow in time as
$V_{\mathrm{mix}}(t)=\frac{4\pi}{3}(6D_{\mathrm{turb}}t+\sigma_{E})^{3/2},$
(1)
where $V_{\mathrm{mix}}$ is the mixing volume and $D_{\mathrm{turb}}=1.2\times
10^{-4}$ kpc Myr-1 is the turbulent diffusion coefficient. Here, $\sigma_{E}$,
which is a measure of the initial size of the SN remnant as it merges with the
ambient medium, is set to zero.The model used to calculate the MDF is nearly
identical to model A in Karlsson (2005).
Figure 8: The logarithm of the predicted metallicity distribution function
(MDF). The quantity f is the fraction of stars that fall within each [Fe/H]
bin (1 dex). The black, solid line shows the metal-poor tail of the predicted
MDF of the Galactic halo while the black, dashed line shows the predicted MDF
of our observational sample. The red, solid and dashed lines denote the
distribution of stars enriched by a single SN for the Galactic halo and the
current sample, respectively. Below [Fe/H]$\sim-3.8$, the number of stars
quickly goes to zero.
The large number of stars in the present sample enables us to discuss outlier
statistics. For example, what is the probability of finding an extreme Si
abundance star, similar to HE $1424-0241$ (Cohen et al. 2007), in our sample?
We shall make the simplifying assumption that stars with such extreme [Si/Fe]
ratios can only occur if they were enriched by a single SN (Cohen et al. 2008)
within a certain range of masses. Theoretically, the low Si-star may be
enriched by two, or more SNe, all within that same mass range but this
probability quickly goes to zero if the fraction of SNe within this range is
$\lesssim 30\%$, or so. About $16\%$ of all Galactic halo stars are found to
have a metallicity below [Fe/H]$=-2.5$ (Carney et al. 1996). Assuming that
stars enriched by one SN predominantly are found in this metallicity regime
(see Fig. 8), the probability of finding a star enriched by a single SN in the
Galactic halo is thus estimated to $p_{1,\mathrm{halo}}=9\times 10^{-3}$,
given the simulated metallicity distribution function (MDF) in Fig. 8.
As our sample is biased against stars above [Fe/H]$\sim-2.5$, this must be
accounted for if we seek to directly compare the observations with the model.
A selection function of $B-V=0.7$ was adopted (see Schörck et al. 2009, their
Table. 12). While stars enriched by one SN are hardly affected at all by this
bias (ı.e., they are mostly found below [Fe/H]$=-2.5$), the number of stars
enriched by more than one SN is significantly smaller, by a factor of $\sim
7$. Consequently, the fraction of stars enriched by single SNe in the present
observational sample is higher, as compared to the corresponding fraction of
the Galactic halo (see Fig. 8). The biased fraction is estimated to
$p_{1,\mathrm{bias}}=6.1\times 10^{-2}$.
The probability of finding exactly $k$ stars with similarly extreme abundances
like HE $1424-0241$, in a sample of $n$ stars is given by the Binomial
statistics $B(n,k)=C(n,k)p^{k}q^{n-k}$, where $p$ is the probability of
success, $q=1-p$ and $C(n,k)=n!/k!(n-k)!$. Given that only a fraction,
$f_{\mathrm{xtrm}}$, of the stars enriched by a single SN may show an extreme
abundance, the probability of finding such a star is therefore
$p_{\mathrm{xtrm}}=f_{\mathrm{xtrm}}\,p_{1,\mathrm{bias}}$. The fraction
$f_{\mathrm{xtrm}}$ depends critically on the stellar yields and the IMF. Both
parameters are uncertain, in particular in this extremely metal-poor regime.
#### 5.3.2 Abundance ranges, dispersions and outlier statistics
Including the low Si-star HE $1424-0241$, the observed range in [Si/Fe]
between this star and the mean of the sample is $\sim 1.5$ dex. The lowest
$33\%$ of this range, will still keep us below [Si/Fe]$=-0.5$ (i.e., outside
$\sim 5\sigma$ of the current sample), which is $\geq 0.5$ dex below the next
lowest observed [Si/Fe] ratio at $\sim 0$. From current observations, we are
unable to estimate how big $f_{\mathrm{xtrm}}$ is in this lower range.
However, even though the theoretical yields do not predict such low values in
[Si/Fe], we can estimate $f_{\mathrm{xtrm}}$ by calculating the fraction of
stars that falls within the lowest $33\%$ of the corresponding theoretical
range. This range, as predicted by the yield calculations of Heger & Woosley
(2008), is reached by $7.5\%$ of the massive stars within
$10-40\leavevmode\nobreak\ M_{\odot}$, for a Salpeter IMF. The corresponding
fraction using the yields by Nomoto et al. (2006) is $41.5\%$, in the mass
range $13-40\leavevmode\nobreak\ M_{\odot}$. We will adopt a fiducial value of
$f_{\mathrm{xtrm}}=0.15$, and allow for a range of $0.05\leq
f_{\mathrm{xtrm}}\leq 0.45$.
The probability of finding one or more stars ($k\geq 1$) with a low [Si/Fe] in
a sample of $n=253$ stars can be expressed as $B(n=253,k\geq
1)=1-(1-p_{\mathrm{xtrm}})^{n}=90.2\%$, in the case of
$f_{\mathrm{xtrm}}=0.15$ and $p_{1,\mathrm{bias}}=6.1\times 10^{-2}$. Within
the range $f_{\mathrm{xtrm}}=0.05-0.45$, the chance is $B=53.8-99.9\%$, with
increasing B for increasing $f_{\mathrm{xtrm}}$. This is high, irrespectively
of the value of $f_{\mathrm{xtrm}}$. For $f_{\mathrm{xtrm}}=0.075$, the chance
is $B=68.7\simeq 70\%$. Hence, the probability is high that at least one star
with an extremely low [Si/Fe] would have been detected in the current sample.
However, as noted in Sect. 5.2, there are no such stars in our sample. In this
respect, our observations appear inconsistent with an inhomogeneous ISM in
which the metal-poor stars in the Galactic halo were enriched only by a small
number of SNe, as indicated by the presence of HE $1424-0241$ at
[Si/Fe]$=-1.01$. The fact that the star found by Cohen et al. (2007) have such
a low [Si/Fe] and appears so detached from the rest of the halo stars, which
all have [Si/Fe]$\gtrsim 0$, may suggest that its Si abundance is not (only) a
result of enrichment by regular core collapse SNe (cf. (Cohen et al. 2007)).
If so, we should exclude it from the comparison between the observed and
simulated star-to-star scatter. This view is also supported by the findings
above that more such stars would likely have been detected in our sample if
this star was a “normal” outlier, enriched by a regular core collapse SN.
In what follows, we shall exclude HE $1424-0241$ in the discussion and only
consider the sample stars presented here (Table 3). Consequently, the observed
range in [Si/Fe] is significantly reduced, with a star-to-star scatter of
$\sigma=0.22$ below [Fe/H]$=-3$. As illustrated in the upper panel of Fig. 9,
the observed 1-$\sigma$ scatter is comparable to the theoretical dispersions
expected from the yield ratio of Si-to-Fe over the mass range of core collapse
SNe (the distributions in Fig. 9 are convolved with a gaussian
($\sigma=0.14$), to account for the random errors in the observations). The
yield calculations by Nomoto et al. (2006) infer a dispersion of $\sigma=0.33$
while the calculations by Heger & Woosley (2008) infer a dispersion of
$\sigma=0.23$, or $\sigma=0.27$, if the full mass range $10\leq
m/\mathcal{M_{\odot}}\leq 100$ is considered. Moreover, the observed range,
[Si/Fe]max – [Si/Fe]${}_{\mathrm{min}}=1.53,$ is larger than the expected,
theoretical range predicted by Heger & Woosley (2008, the observed range is
larger in $>99.9\%$ of the cases for $n=253$ stars, assuming SN progenitor
masses in the range $10\leq m/\mathcal{M_{\odot}}\leq 40$), while it is
comparable to the one predicted by Nomoto et al. (2006), larger in $38\%$ of
the cases).
Note that these are the maximum theoretical dispersions and ranges. In
reality, we expect the stars enriched by a single SNe to be distributed over a
range in [Fe/H]. In particular, a fraction of the stars below [Fe/H]$=-3$ are
expected to be enriched by more than one SN. These stars are closer to the
mean [Si/Fe] and the observed 1-$\sigma$ dispersion below [Fe/H]$=-3$ (see
Fig. 9, lower panel) is therefore expected to be lower than the dispersion of
the yield ratio depicted in Fig. 9, upper panel. The lower panel of Fig. 9
shows a simulation in which the turbulent mixing is turned off (i.e., minimal
mixing). The size (mixing volume) of the SN remnants is set to
$\sigma_{\mathrm{E}}=8.5\times 10^{-3}$, which corresponds to a mixing mass of
$1\times 10^{5}\leavevmode\nobreak\ \mathcal{M_{\odot}}$, for a particle
density of 1cm-3. The SN II yields are taken from Nomoto et al. (2006). Apart
from the overall trend, which is shallower in the simulation, the 1-$\sigma$
scatter in the metallicity three bins $[-4,-3]$, $[-3,-2]$, and $[-2,-1]$, are
found to be 0.23, 0.16, and 0.14, respectively, excluding the stars
predominantly enriched by electron capture SNe (see below). This is
significantly smaller than the scatter predicted by Argast et al. (2000) and
in close agreement with observations. Since we have turned off the turbulent
mixing in our simulations, the discrepancy between the two model results
should predominantly be due to differences in the adopted SN yields.
In conclusion, we cannot reject the possibility that the stars in our sample
were formed in a chemically inhomogeneous ISM, solely based on the
measurements of Si. Admittedly, our sample lack extremely Si-deficient stars,
but this may rather suggest that HE $1424-0241$ is very atypical, and should
not be included in the analysis. If this star was born with such a low Si
abundance, reflecting the nucleosynthesis of a rare SN, the early ISM must,
indeed, have been highly inhomogeneous. Note that gas that low in Si will
rapidly reach “chemical normality” as soon as SNe II enrich it. Low Si-stars
would therefore be relatively uncommon. Moreover, the observed scatter
increases faster with decreasing [Fe/H] than does the mean observational
uncertainty of the stars (see Fig. 6, lower panel). This suggests that the
scatter at the lowest metallicities has a small but non-negligible
contribution from real abundance inhomogeneities in the early star-forming
gas.
#### 5.3.3 Contribution from electron capture SNe
To find out the frequency of low-Si stars enriched by a rare type of SNe, we
included the contribution of electron capture SNe, which have masses in the
range $8-10\leavevmode\nobreak\ \mathcal{M_{\odot}}$. The electron capture SNe
are believed to constitute a fraction of $\sim 4-30\%$ of all SNe (Poelarends
et al. 2008; Wanajo et al. 2009, 2010). During the final stage of their
evolution, these objects develop a degenerate O-Ne-Mg core and their structure
and nucleosynthesis are distinctly different from the more massive Fe-core
collapse SNe, including a very low Si yield (Wanajo et al. 2009). Assuming
that all stars in the mass range $8-10\leavevmode\nobreak\
\mathcal{M_{\odot}}$ become electron capture SNe (i.e., $30\%$ of all SNe,
given a Salpeter IMF), the fraction of stars in the simulation with a
[Si/Fe]$<-0.5$ is $p_{\mathrm{xtrm}}\simeq 1.55\times 10^{-3}$. This gives a
probability of $32.4\%$ of finding a low [Si/Fe] star in our sample. It is a
relatively low probability but not extremely low, and the possibility to find
such a star in the combined sample of Galactic halo stars studied with
detailed spectroscopy is non-negligible. The lower panel of Fig. 9 is
truncated at [Si/Fe]$=-0.5$. Nevertheless, the few model stars below
[Si/Fe]$\sim 0$ do have a small contribution from electron capture SNe.
Figure 9: The expected star-to-star scatter in [Si/Fe] for core collapse SNe.
The top panel shows the expected maximum range in [Si/Fe] for stars enriched
by single Type II SNe. The solid curve denotes the probability density
function (PDF) assuming SN yields by Nomoto et al. (2006) while the dashed
curve (SN mass range $10\leq m/\mathcal{M_{\odot}}\leq 40$) and the dotted
curve ($10\leq m/\mathcal{M_{\odot}}\leq 100$) denote the PDFs assuming yields
by Heger & Woosley (2008). Each PDF is convolved with a gaussian
($\sigma=0.14$) to account for the observational uncertainty in [Si/Fe]. The
corresponding 1-$\sigma$ dispersions are shown as solid ($\sigma=0.33$),
dashed ($\sigma=0.23$), and dotted ($\sigma=0.27$) thin lines, centered at the
respective mean of each distribution. The gray thin line at [Si/Fe] = 0.53
denotes the observational star-to-star scatter ($\sigma=0.22$) below
[Fe/H]$=-3$. The bottom panel shows the full (convolved) distribution of model
stars (small black dots) in the [Fe/H] – [Si/Fe] plane. The observations are
shown as red dots (upper limits are shown as triangles), for comparison. The
star-to-star scatter below [Fe/H]$=-3$ in the simulation is $\sigma\simeq
0.23$. Note the small number of EMP stars below [Si/Fe]$\sim 0$. These stars
have partly been enriched by electron capture SNe in the mass range $8\leq
m/\mathcal{M_{\odot}}\leq 10$, which produce very small amounts of Si. The SN
II (Fe-core collapse) yields are taken from Nomoto et al. (2006) while the
electron capture SN yields are taken from Wanajo et al. (2009).
It should be noted that although the electron capture SNe indeed produce a low
Si yield and the fraction of low-Si stars enriched by this type of SN is
consistent with observations, the overall predicted abundance pattern (Wanajo
et al. 2009) provides quite a poor fit to that observed in HE $1424-0241$. The
situation improves if a few per mille of ejecta of an Fe-core collapse SN is
added to the gas. However, the fit to the light elements Na, Mg, and Al is
still poor. It is beyond the scope of this study to discuss the abundance
pattern and possible origin of HE $1424-0241$ in detail. The interested reader
is directed to Cohen et al. (2007).
#### 5.3.4 A note on trends and observed scatter
As discussed in 5.1 and the begining of 5.3, our results present not only a
slope in [Si/Fe] with metallicity but also a small cosmic scatter.
Trends, as well as scatters, are affected by the star formation and mixing
time scale of the ISM. Homogeneous chemical evolution models assume
instantaneous mixing. In these models, trends may, in the most metal-poor
regime, arise from the progenitor mass dependence of the SN yields. A given
abundance ratio, e.g., [Si/Fe], evolves with time, or metallicity, because the
most massive, short lived, SNe have a different [Si/Fe] yield ratio from those
of less massive, longer lived, SNe.
Mixing is, however, not instantaneous. In order to relax the assumption of
unphysically short mixing time scales, and still retain the small star-to-star
scatter observed in a number of abundance ratios, Arnone et al. (2005)
speculated that the cooling time scale of metal-poor gas may be long enough
for the ISM to mix before subsequent generations of stars are able to form.
However, since the star-forming gas, in this scenario, always has to be well
mixed, such a “global mixing” would have difficulties to explain any trends
with metallicity, like the one reported here, (see also, e.g., Cayrel et al.
2004), unless such trends are a result of a metallicity-dependency of the SN
yields. In the case of Si, the conclusion is ambiguous. Nomoto et al. (2006),
predict a trend in [Si/Fe] with metallicity which goes in the right direction,
although with a shallower slope than what is observed, while Chieffi & Limongi
(2004) predict almost no trend, however, with a very shallow slope in the
opposite direction.
An alternative explanation to the small observed scatter, without invoking an
unphysically short mixing time scale, is suggested by Bland-Hawthorn et al.
(2010). They present a new stochastic chemical evolution model in which stars
are formed in clusters, as is known to be the case in present-day star
formation. In this scenario, the mixing initially only occurs on a local
scale. However, as a result of stars being grouped together in clusters, the
ejecta of $\geq 1$ SNe are mixed together within each cluster, i.e, if the
clusters are massive enough to contain SNe. This may produce enough mixing to
explain the observations of, e.g., [Mg/Fe], while the large scatter observed
for a number of neutron-capture elements, e.g., [Ba/Fe] (Burris et al. 2000;
François et al. 2007), can still be accounted for. This will be discussed in a
forthcoming paper.
Karlsson & Gustafsson (2005) found trends with metallicity for certain
abundance ratios, while the scatter stayed small at all metallicities, and, in
particular cases, even decreased towards lower metallicity. These trends are
an effect of the local enrichment in which different regions are enriched by
SNe of different masses. A similar effect was noticed by Ryan et al. (1996).
If the metal-poor star-forming gas were not very well mixed, trends like these
are to be expected, depending on the SN yields. Note that the very same SN
mass dependence could, in homogeneous models, generate a trend with a
different, or even opposite slope to that in a stochastic, inhomogeneous
model. Finally, a change in the IMF, e.g. from a top-heavy to a Salpeter-like
IMF, may also, possibly, generate a trend with a non-zero slope. Clearly, in
order to fully unravel the origins of the observed trends at low
metallicities, a deeper understanding of the interplay between the mixing and
cooling processes in the ISM is necessary (Karlsson et al. 2011). This
knowledge must be incorporated in the modelling of chemical evolution.
###### Acknowledgements.
We thank Dr. J.R. Shi for useful suggestions and discussions on NLTE
corrections. This work is supported by the NSFC under grant 10821061, by the
National Basic Rsearch Program of China under grant 2007CB815103, and the
Global Networks program of the University of Heidelberg. T.K. is funded by ARC
FF grant 0776384 through the University of Sydney. T.K. is grateful to the
Beecroft Institute for Particle Astrophysics and Cosmology for their
hospitality. A.J.K. acknowledges support through grants by the Swedish
Research Council (VR) and the Swedish National Space Board (SNSB).P.S.B is
aRoyal Swedish Academy of Sciences Research Fellow supported by a grant from
the Knut and Alice Wallenberg Foundation. P.S.B also acknowledges additional
support from the Swedish Research Council. A number of comments and
suggestions by an anonymous referee helped improving the paper.
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3
Table 4: Abundance results of carbon and silicon. The last column is the average of [Si/Fe] from two Si I lines. If only upper limit can be got from one line, taking the value of the other line represents the average value. | | | | | $\log\epsilon(\rm Si)_{\rm NLTE}$ | [Si/H]NLTE | [Si/Fe]NLTE |
---|---|---|---|---|---|---|---|---
star | [Fe/H] | $\log\epsilon(\rm C)$ | [C/H] | [C/Fe] | 3905 | 4103 | 3905 | 4103 | 3905 | 4103 | $\overline{\rm[Si/Fe]_{NLTE}}$
CS22175-007 | –2.81 | 5.80 | –2.72$\pm$0.14 | 0.09$\pm$0.16 | 5.16 | $<$5.19 | –2.39$\pm$0.13 | $<$–2.36$\pm$0.15 | 0.42$\pm$0.14 | $<$0.45$\pm$0.16 | 0.42$\pm$0.14
CS22186-023 | –2.72 | 6.00 | –2.52$\pm$0.10 | 0.20$\pm$0.12 | 5.26 | 5.17 | –2.29$\pm$0.09 | –2.38$\pm$0.11 | 0.43$\pm$0.10 | 0.34$\pm$0.13 | 0.39$\pm$0.12
CS22186-025 | –2.87 | 5.35 | –3.17$\pm$0.15 | –0.30$\pm$0.17 | 5.22 | 5.28 | –2.33$\pm$0.14 | –2.27$\pm$0.16 | 0.54$\pm$0.15 | 0.60$\pm$0.17 | 0.57$\pm$0.17
CS22886-042 | –2.68 | 5.71 | –2.81$\pm$0.11 | –0.13$\pm$0.13 | 5.46 | 5.22 | –2.09$\pm$0.10 | –2.33$\pm$0.12 | 0.59$\pm$0.11 | 0.35$\pm$0.13 | 0.47$\pm$0.13
CS22892-052 | –2.95 | 6.35 | –2.17$\pm$0.11 | 0.78$\pm$0.13 | 5.31 | 5.13 | –2.24$\pm$0.10 | –2.42$\pm$0.12 | 0.71$\pm$0.11 | 0.53$\pm$0.13 | 0.62$\pm$0.13
CS22945-028 | –2.66 | 6.11 | –2.41$\pm$0.13 | 0.25$\pm$0.15 | 5.38 | 5.36 | –2.17$\pm$0.12 | –2.19$\pm$0.14 | 0.49$\pm$0.13 | 0.47$\pm$0.15 | 0.48$\pm$0.15
CS22957-013 | –2.64 | 5.90 | –2.62$\pm$0.12 | 0.02$\pm$0.14 | 5.34 | 5.34 | –2.21$\pm$0.11 | –2.21$\pm$0.13 | 0.43$\pm$0.12 | 0.43$\pm$0.14 | 0.43$\pm$0.14
CS22958-083 | –2.79 | 6.28 | –2.24$\pm$0.13 | 0.55$\pm$0.15 | 5.44 | 5.25 | –2.11$\pm$0.12 | –2.30$\pm$0.14 | 0.68$\pm$0.13 | 0.49$\pm$0.15 | 0.58$\pm$0.15
CS22960-010 | –2.65 | 6.57 | –1.95$\pm$0.11 | 0.70$\pm$0.13 | 5.61 | $<$5.57 | –1.94$\pm$0.10 | $<$–1.98$\pm$0.12 | 0.71$\pm$0.11 | $<$0.67$\pm$0.13 | 0.71$\pm$0.11
CS29491-069 | –2.81 | 5.93 | –2.59$\pm$0.10 | 0.22$\pm$0.12 | 5.23 | 5.05 | –2.32$\pm$0.09 | –2.50$\pm$0.11 | 0.49$\pm$0.10 | 0.31$\pm$0.13 | 0.40$\pm$0.12
CS29491-109 | –2.90 | 5.32 | –3.20$\pm$0.09 | –0.30$\pm$0.11 | 5.15 | 5.11 | –2.40$\pm$0.08 | –2.44$\pm$0.10 | 0.50$\pm$0.09 | 0.46$\pm$0.12 | 0.48$\pm$0.11
CS29497-004 | –2.81 | 5.84 | –2.68$\pm$0.10 | 0.13$\pm$0.12 | 5.11 | 5.09 | –2.44$\pm$0.09 | –2.46$\pm$0.11 | 0.37$\pm$0.10 | 0.35$\pm$0.13 | 0.36$\pm$0.12
CS29510-058 | –2.61 | 6.20 | –2.32$\pm$0.12 | 0.29$\pm$0.14 | 5.38 | 5.35 | –2.17$\pm$0.11 | –2.20$\pm$0.13 | 0.44$\pm$0.12 | 0.41$\pm$0.14 | 0.42$\pm$0.14
CS30308-035 | –3.35 | 5.10 | –3.42$\pm$0.15 | –0.07$\pm$0.17 | 4.66 | 4.57 | –2.89$\pm$0.14 | –2.98$\pm$0.16 | 0.46$\pm$0.15 | 0.37$\pm$0.17 | 0.42$\pm$0.17
CS30315-001 | –2.98 | 5.04 | –3.48$\pm$0.13 | –0.50$\pm$0.15 | 5.05 | 5.04 | –2.50$\pm$0.12 | –2.51$\pm$0.14 | 0.48$\pm$0.13 | 0.47$\pm$0.15 | 0.47$\pm$0.15
CS30315-029 | –3.33 | 4.64 | –3.88$\pm$0.12 | –0.55$\pm$0.14 | 4.77 | 4.80 | –2.78$\pm$0.11 | –2.75$\pm$0.13 | 0.55$\pm$0.12 | 0.58$\pm$0.14 | 0.56$\pm$0.14
CS30337-097 | –2.73 | 5.67 | –2.85$\pm$0.11 | –0.12$\pm$0.13 | 5.38 | 5.33 | –2.17$\pm$0.10 | –2.22$\pm$0.12 | 0.56$\pm$0.11 | 0.51$\pm$0.13 | 0.54$\pm$0.13
CS30339-041 | –2.20 | 6.22 | –2.30$\pm$0.12 | –0.10$\pm$0.14 | 5.75 | 5.70 | –1.80$\pm$0.11 | –1.85$\pm$0.13 | 0.40$\pm$0.12 | 0.35$\pm$0.14 | 0.38$\pm$0.14
CS30343-063 | –2.95 | 4.69 | –3.83$\pm$0.12 | –0.88$\pm$0.14 | 5.10 | 4.92 | –2.45$\pm$0.11 | –2.63$\pm$0.13 | 0.50$\pm$0.12 | 0.32$\pm$0.14 | 0.41$\pm$0.14
CS31060-047 | –2.72 | 5.45 | –3.07$\pm$0.17 | –0.35$\pm$0.18 | 5.35 | 5.39 | –2.20$\pm$0.17 | –2.16$\pm$0.18 | 0.52$\pm$0.18 | 0.56$\pm$0.19 | 0.54$\pm$0.19
CS31062-041 | –2.67 | 6.30 | –2.22$\pm$0.11 | 0.45$\pm$0.13 | 5.42 | 5.46 | –2.13$\pm$0.10 | –2.09$\pm$0.12 | 0.54$\pm$0.11 | 0.58$\pm$0.13 | 0.56$\pm$0.13
CS31072-118 | –3.06 | 4.90 | –3.62$\pm$0.11 | –0.56$\pm$0.13 | 5.14 | 5.18 | –2.41$\pm$0.10 | –2.37$\pm$0.12 | 0.65$\pm$0.11 | 0.69$\pm$0.13 | 0.67$\pm$0.13
CS31082-001 | –2.78 | 5.91 | –2.61$\pm$0.09 | 0.17$\pm$0.11 | 5.35 | 5.30 | –2.20$\pm$0.08 | –2.25$\pm$0.10 | 0.58$\pm$0.09 | 0.53$\pm$0.12 | 0.55$\pm$0.11
HD20 | –1.58 | 6.51 | –2.01$\pm$0.09 | –0.43$\pm$0.11 | 6.57 | 6.43 | –0.98$\pm$0.08 | –1.12$\pm$0.10 | 0.60$\pm$0.09 | 0.46$\pm$0.12 | 0.53$\pm$0.11
HD221170 | –2.14 | 5.81 | –2.71$\pm$0.10 | –0.57$\pm$0.12 | 5.54 | 5.56 | –2.01$\pm$0.09 | –1.99$\pm$0.11 | 0.13$\pm$0.10 | 0.15$\pm$0.13 | 0.14$\pm$0.12
HE0005-0002 | –3.09 | 5.54 | –2.98$\pm$0.11 | 0.11$\pm$0.13 | 5.17 | 4.82 | –2.38$\pm$0.10 | –2.73$\pm$0.12 | 0.42$\pm$0.11 | 0.36$\pm$0.13 | 0.39$\pm$0.13
HE0008-3842 | –3.35 | 4.20 | –4.32$\pm$0.11 | –0.97$\pm$0.13 | 4.81 | 4.59 | –2.74$\pm$0.10 | –2.96$\pm$0.12 | 0.61$\pm$0.11 | 0.39$\pm$0.13 | 0.50$\pm$0.13
HE0017-4838 | –3.23 | 5.39 | –3.13$\pm$0.16 | 0.10$\pm$0.17 | 4.79 | 4.67 | –2.76$\pm$0.15 | –2.88$\pm$0.17 | 0.47$\pm$0.16 | 0.35$\pm$0.18 | 0.41$\pm$0.18
HE0018-1349 | –2.26 | 6.48 | –2.04$\pm$0.11 | 0.22$\pm$0.13 | 5.37 | 5.31 | –2.18$\pm$0.10 | –2.24$\pm$0.12 | 0.08$\pm$0.11 | 0.02$\pm$0.13 | 0.05$\pm$0.13
HE0023-4825 | –2.06 | 6.76 | –1.76$\pm$0.11 | 0.30$\pm$0.13 | 5.90 | 5.81 | –1.65$\pm$0.10 | –1.74$\pm$0.12 | 0.41$\pm$0.11 | 0.32$\pm$0.13 | 0.36$\pm$0.13
HE0029-1839 | –2.50 | 6.31 | –2.21$\pm$0.10 | 0.29$\pm$0.12 | 5.33 | 5.25 | –2.22$\pm$0.09 | –2.30$\pm$0.11 | 0.28$\pm$0.10 | 0.20$\pm$0.13 | 0.24$\pm$0.12
HE0037-2657 | –3.22 | 5.49 | –3.03$\pm$0.11 | 0.19$\pm$0.13 | 5.01 | 4.99 | –2.54$\pm$0.10 | –2.56$\pm$0.12 | 0.68$\pm$0.11 | 0.66$\pm$0.13 | 0.67$\pm$0.13
HE0039-4154 | –3.38 | 5.07 | –3.45$\pm$0.11 | –0.07$\pm$0.13 | 4.50 | 4.56 | \- 3.05$\pm$0.10 | –2.99$\pm$0.12 | 0.33$\pm$0.11 | 0.39$\pm$0.13 | 0.36$\pm$0.13
HE0043-2845 | –2.91 | 5.85 | –2.67$\pm$0.10 | 0.24$\pm$0.12 | 5.13 | $<$5.15 | –2.42$\pm$0.09 | $<$–2.40$\pm$0.11 | 0.49$\pm$0.10 | $<$0.51$\pm$0.13 | 0.49$\pm$0.10
HE0044-2459 | –3.28 | 5.67 | –2.85$\pm$0.11 | 0.43$\pm$0.13 | 4.94 | $<$4.82 | –2.61$\pm$0.10 | $<$–2.73$\pm$0.12 | 0.67$\pm$0.11 | $<$0.55$\pm$0.13 | 0.67$\pm$0.11
HE0044-4023 | –2.56 | 6.24 | –2.28$\pm$0.15 | 0.28$\pm$0.17 | 5.23 | $<$5.02 | –2.32$\pm$0.14 | $<$–2.53$\pm$0.16 | 0.24$\pm$0.15 | $<$0.03$\pm$0.17 | 0.24$\pm$0.15
HE0045-2430 | –1.77 | 6.55 | –1.97$\pm$0.10 | –0.20$\pm$0.12 | 5.87 | 5.80 | –1.68$\pm$0.09 | –1.75$\pm$0.11 | 0.09$\pm$0.10 | 0.02$\pm$0.13 | 0.06$\pm$0.12
HE0049-5700 | –2.41 | 6.49 | –2.03$\pm$0.13 | 0.38$\pm$0.15 | 5.55 | $<$5.59 | –2.00$\pm$0.12 | $<$–1.96$\pm$0.14 | 0.41$\pm$0.13 | $<$0.45$\pm$0.15 | 0.41$\pm$0.13
HE0051-2304 | –2.41 | 5.49 | –3.03$\pm$0.10 | –0.62$\pm$0.12 | 5.49 | 5.70 | –2.06$\pm$0.09 | –1.85$\pm$0.11 | 0.35$\pm$0.10 | 0.56$\pm$0.13 | 0.46$\pm$0.12
HE0054-0657 | –2.00 | 6.77 | –1.75$\pm$0.13 | 0.25$\pm$0.15 | 5.80 | 5.96 | –1.75$\pm$0.12 | –1.59$\pm$0.14 | 0.25$\pm$0.13 | 0.41$\pm$0.15 | 0.33$\pm$0.15
HE0057-4541 | –2.32 | 6.37 | –2.15$\pm$0.10 | 0.17$\pm$0.12 | 5.58 | 5.41 | –1.97$\pm$0.09 | –2.14$\pm$0.11 | 0.35$\pm$0.10 | 0.18$\pm$0.13 | 0.27$\pm$0.12
HE0104-4007 | –3.30 | 5.72 | –2.80$\pm$0.13 | 0.50$\pm$0.15 | 5.03 | 4.98 | –2.52$\pm$0.12 | –2.57$\pm$0.14 | 0.78$\pm$0.13 | 0.73$\pm$0.15 | 0.76$\pm$0.15
HE0104-5300 | –3.42 | 5.22 | –3.30$\pm$0.13 | 0.12$\pm$0.15 | 4.98 | 4.81 | –2.57$\pm$0.12 | –2.74$\pm$0.14 | 0.85$\pm$0.13 | 0.68$\pm$0.15 | 0.77$\pm$0.15
HE0105-6141 | –2.55 | 6.12 | –2.40$\pm$0.10 | 0.15$\pm$0.12 | 5.41 | 5.34 | –2.14$\pm$0.09 | –2.21$\pm$0.11 | 0.41$\pm$0.10 | 0.34$\pm$0.13 | 0.38$\pm$0.12
HE0109-0742 | –2.53 | 5.97 | –2.55$\pm$0.12 | –0.02$\pm$0.14 | 5.49 | 5.38 | –2.06$\pm$0.11 | –2.17$\pm$0.13 | 0.47$\pm$0.12 | 0.36$\pm$0.14 | 0.41$\pm$0.14
HE0109-3711 | –1.91 | 6.63 | –1.89$\pm$0.18 | 0.02$\pm$0.19 | $<$6.05 | $<$6.00 | $<$–1.50$\pm$0.18 | $<$–1.55$\pm$0.19 | $<$0.41$\pm$0.19 | $<$0.36$\pm$0.20 | $<$0.39$\pm$0.20
HE0111-1454 | –2.99 | 5.19 | –3.33$\pm$0.10 | –0.34$\pm$0.12 | 5.21 | 5.02 | –2.34$\pm$0.09 | –2.53$\pm$0.11 | 0.65$\pm$0.10 | 0.46$\pm$0.13 | 0.56$\pm$0.12
HE0121-2826 | –2.97 | 6.03 | –2.49$\pm$0.11 | 0.48$\pm$0.13 | 5.26 | 5.16 | –2.29$\pm$0.10 | –2.39$\pm$0.12 | 0.68$\pm$0.11 | 0.58$\pm$0.13 | 0.63$\pm$0.13
HE0131-2740 | –3.08 | 5.62 | –2.90$\pm$0.16 | 0.18$\pm$0.17 | $<$5.02 | $<$4.98 | $<$–2.53$\pm$0.15 | $<$–2.57$\pm$0.17 | $<$0.55$\pm$0.16 | $<$0.51$\pm$0.18 | $<$0.53$\pm$0.18
HE0131-3953 | –2.71 | 8.29 | –0.23$\pm$0.11 | 2.48$\pm$0.13 | 5.85 | $<$5.76 | –1.70$\pm$0.10 | $<$–1.73$\pm$0.12 | 1.01$\pm$0.11 | $<$0.92$\pm$0.13 | 1.01$\pm$0.11
HE0143-1135 | –2.13 | 6.62 | –1.90$\pm$0.10 | 0.23$\pm$0.12 | 5.97 | 6.08 | –1.58$\pm$0.09 | –1.47$\pm$0.11 | 0.55$\pm$0.10 | 0.66$\pm$0.13 | 0.60$\pm$0.12
HE0143-4108 | –2.62 | 6.12 | –2.40$\pm$0.10 | 0.22$\pm$0.12 | 5.20 | 5.02 | –2.35$\pm$0.09 | –2.53$\pm$0.11 | 0.27$\pm$0.10 | 0.09$\pm$0.13 | 0.18$\pm$0.12
HE0143-4146 | –2.94 | 5.64 | –2.88$\pm$0.13 | 0.06$\pm$0.15 | 4.93 | 4.98 | –2.62$\pm$0.12 | –2.57$\pm$0.14 | 0.32$\pm$0.13 | 0.37$\pm$0.15 | 0.34$\pm$0.15
HE0157-3335 | –3.08 | 5.22 | –3.30$\pm$0.10 | –0.22$\pm$0.12 | 5.01 | 4.99 | –2.54$\pm$0.09 | –2.56$\pm$0.11 | 0.54$\pm$0.10 | 0.52$\pm$0.13 | 0.53$\pm$0.12
HE0200-0955 | –2.46 | 6.34 | –2.18$\pm$0.13 | 0.28$\pm$0.15 | 5.59 | 5.43 | –1.96$\pm$0.12 | –2.12$\pm$0.14 | 0.50$\pm$0.13 | 0.34$\pm$0.15 | 0.42$\pm$0.15
HE0202-2204 | –1.98 | 7.66 | –0.86$\pm$0.16 | 1.12$\pm$0.17 | 5.70 | 5.53 | –1.85$\pm$0.15 | –2.02$\pm$0.17 | 0.13$\pm$0.16 | –0.04$\pm$0.18 | 0.04$\pm$0.18
HE0231-4016 | –2.08 | 7.64 | –0.88$\pm$0.11 | 1.20$\pm$0.13 | 6.11 | 6.01 | –1.44$\pm$0.10 | –1.54$\pm$0.12 | 0.64$\pm$0.11 | 0.51$\pm$0.13 | 0.64$\pm$0.13
HE0240-0807 | –2.68 | 5.44 | –3.08$\pm$0.12 | –0.40$\pm$0.14 | 5.54 | 5.37 | –2.01$\pm$0.11 | –2.18$\pm$0.13 | 0.67$\pm$0.12 | 0.50$\pm$0.14 | 0.58$\pm$0.14
HE0240-6105 | –3.23 | 4.94 | –3.58$\pm$0.10 | –0.35$\pm$0.12 | 5.09 | 5.02 | –2.46$\pm$0.09 | –2.53$\pm$0.11 | 0.77$\pm$0.10 | 0.70$\pm$0.13 | 0.73$\pm$0.12
HE0243-0753 | –2.49 | 6.29 | –2.23$\pm$0.11 | 0.26$\pm$0.13 | 5.53 | 5.47 | –2.02$\pm$0.10 | –2.08$\pm$0.12 | 0.47$\pm$0.11 | 0.41$\pm$0.13 | 0.44$\pm$0.13
HE0243-5238 | –3.04 | 5.81 | –2.71$\pm$0.12 | 0.33$\pm$0.14 | 5.14 | 4.93 | –2.41$\pm$0.11 | –2.62$\pm$0.13 | 0.63$\pm$0.12 | 0.42$\pm$0.14 | 0.53$\pm$0.14
HE0244-4111 | –2.56 | 6.36 | –2.16$\pm$0.11 | 0.40$\pm$0.13 | 5.54 | 5.5 | –2.01$\pm$0.10 | –2.05$\pm$0.12 | 0.55$\pm$0.11 | 0.51$\pm$0.13 | 0.53$\pm$0.13
HE0248+0039 | –2.53 | 6.06 | –2.46$\pm$0.20 | 0.07$\pm$0.21 | 5.43 | 5.35 | –2.12$\pm$0.20 | –2.20$\pm$0.20 | 0.41$\pm$0.21 | 0.33$\pm$0.21 | 0.37$\pm$0.21
HE0256-1109 | –2.73 | 6.53 | –1.99$\pm$0.12 | 0.74$\pm$0.14 | $<$5.36 | $<$5.44 | $<$–2.19$\pm$0.11 | $<$–2.11$\pm$0.13 | $<$0.54$\pm$0.12 | $<$0.62$\pm$0.14 | $<$0.58$\pm$0.14
HE0300-0751 | –2.27 | 6.38 | –2.14$\pm$0.13 | 0.13$\pm$0.15 | 5.76 | 5.78 | –1.79$\pm$0.12 | –1.77$\pm$0.14 | 0.48$\pm$0.13 | 0.50$\pm$0.15 | 0.49$\pm$0.15
HE0305-4520 | –2.91 | 5.81 | –2.71$\pm$0.11 | 0.20$\pm$0.13 | 5.15 | 5.07 | –2.40$\pm$0.10 | –2.48$\pm$0.12 | 0.51$\pm$0.11 | 0.43$\pm$0.13 | 0.47$\pm$0.13
HE0308-1154 | –2.82 | 6.08 | –2.44$\pm$0.13 | 0.38$\pm$0.15 | 6.23 | 6.17 | –1.32$\pm$0.12 | –1.38$\pm$0.14 | 1.50$\pm$0.13 | 1.44$\pm$0.15 | 1.47$\pm$0.15
HE0315+0000 | –2.73 | 5.95 | –2.57$\pm$0.15 | 0.16$\pm$0.17 | 5.20 | 5.27 | –2.35$\pm$0.14 | –2.28$\pm$0.16 | 0.38$\pm$0.15 | 0.45$\pm$0.17 | 0.42$\pm$0.17
HE0316+0214 | –3.13 | 4.64 | –3.88$\pm$0.10 | –0.75$\pm$0.12 | 5.27 | 5.27 | –2.28$\pm$0.09 | –2.28$\pm$0.11 | 0.85$\pm$0.10 | 0.85$\pm$0.13 | 0.85$\pm$0.12
HE0317-4640 | –2.33 | 6.44 | –2.08$\pm$0.17 | 0.25$\pm$0.18 | 5.73 | 5.63 | –1.82$\pm$0.17 | –1.92$\pm$0.18 | 0.51$\pm$0.18 | 0.41$\pm$0.19 | 0.46$\pm$0.19
HE0323-4529 | –3.15 | 5.81 | –2.71$\pm$0.10 | 0.44$\pm$0.12 | 4.55 | $<$4.58 | –3.00$\pm$0.09 | $<$–2.97$\pm$0.11 | 0.15$\pm$0.10 | $<$0.18$\pm$0.13 | 0.15$\pm$0.10
HE0328-1047 | –2.25 | 6.38 | –2.14$\pm$0.12 | 0.11$\pm$0.14 | 5.63 | 5.65 | –1.92$\pm$0.11 | –1.90$\pm$0.13 | 0.33$\pm$0.12 | 0.35$\pm$0.14 | 0.34$\pm$0.14
HE0330-4004 | –2.20 | 6.40 | –2.12$\pm$0.11 | 0.08$\pm$0.13 | 5.70 | $<$5.50 | –1.85$\pm$0.10 | $<$–2.05$\pm$0.12 | 0.35$\pm$0.11 | $<$0.15$\pm$0.13 | 0.35$\pm$0.11
HE0330-4144 | –1.90 | 6.70 | –1.82$\pm$0.14 | 0.08$\pm$0.16 | 5.90 | 5.90 | –1.65$\pm$0.13 | –1.65$\pm$0.15 | 0.25$\pm$0.14 | 0.25$\pm$0.16 | 0.25$\pm$0.16
HE0331-4939 | –2.90 | 5.97 | –2.55$\pm$0.11 | 0.35$\pm$0.13 | 5.24 | 5.14 | –2.31$\pm$0.10 | –2.41$\pm$0.12 | 0.59$\pm$0.11 | 0.49$\pm$0.13 | 0.54$\pm$0.13
HE0333-4001 | –2.64 | 6.18 | –2.34$\pm$0.14 | 0.30$\pm$0.16 | 5.37 | $<$7.31 | –2.18$\pm$0.13 | $<$–2.24$\pm$0.15 | 0.46$\pm$0.14 | $<$0.40$\pm$0.16 | 0.46$\pm$0.14
HE0336-3829 | –2.75 | 6.15 | –2.37$\pm$0.11 | 0.38$\pm$0.13 | 5.14 | $<$5.19 | –2.41$\pm$0.10 | $<$–2.36$\pm$0.12 | 0.34$\pm$0.11 | $<$0.39$\pm$0.13 | 0.34$\pm$0.11
HE0337-5127 | –2.62 | 6.09 | –2.43$\pm$0.12 | 0.19$\pm$0.14 | 5.52 | 5.50 | –2.03$\pm$0.11 | –2.05$\pm$0.13 | 0.59$\pm$0.12 | 0.57$\pm$0.14 | 0.59$\pm$0.14
HE0338-3945 | –2.41 | 8.24 | –0.28$\pm$0.10 | 2.13$\pm$0.12 | 5.70 | $<$5.51 | –1.85$\pm$0.09 | $<$–2.04$\pm$0.11 | 0.56$\pm$0.10 | $<$0.37$\pm$0.13 | 0.47$\pm$0.10
HE0339-4027 | –1.81 | 6.87 | –1.65$\pm$0.11 | 0.16$\pm$0.13 | 6.03 | 6.09 | –1.52$\pm$0.10 | –1.46$\pm$0.12 | 0.29$\pm$0.11 | 0.35$\pm$0.13 | 0.32$\pm$0.13
HE0340-3430 | –1.95 | 6.79 | –1.73$\pm$0.12 | 0.22$\pm$0.14 | 6.13 | 6.19 | –1.42$\pm$0.11 | –1.36$\pm$0.13 | 0.53$\pm$0.12 | 0.59$\pm$0.14 | 0.56$\pm$0.14
HE0340-5355 | –2.89 | 5.41 | –3.11$\pm$0.10 | –0.22$\pm$0.12 | 4.91 | 4.85 | –2.64$\pm$0.09 | –2.70$\pm$0.11 | 0.25$\pm$0.10 | 0.19$\pm$0.13 | 0.22$\pm$0.12
HE0341-4024 | –1.82 | 6.84 | –1.68$\pm$0.11 | 0.14$\pm$0.13 | 6.12 | 6.06 | –1.43$\pm$0.10 | –1.49$\pm$0.12 | 0.39$\pm$0.11 | 0.33$\pm$0.13 | 0.36$\pm$0.13
HE0344+0139 | –1.81 | 7.10 | –1.42$\pm$0.10 | 0.39$\pm$0.12 | 6.31 | 6.14 | –1.24$\pm$0.09 | –1.41$\pm$0.11 | 0.56$\pm$0.10 | 0.40$\pm$0.13 | 0.48$\pm$0.12
HE0347-1819 | –2.78 | 5.78 | –2.74$\pm$0.12 | 0.04$\pm$0.14 | 5.21 | 5.19 | –2.34$\pm$0.11 | –2.36$\pm$0.13 | 0.44$\pm$0.12 | 0.42$\pm$0.14 | 0.43$\pm$0.14
HE0353-6024 | –3.17 | 5.64 | –2.88$\pm$0.11 | 0.29$\pm$0.13 | 4.97 | 4.91 | –2.58$\pm$0.10 | –2.64$\pm$0.12 | 0.59$\pm$0.11 | 0.53$\pm$0.13 | 0.56$\pm$0.13
HE0400-2917 | –2.88 | 5.72 | –2.80$\pm$0.13 | 0.08$\pm$0.15 | 4.83 | 4.60 | –2.72$\pm$0.12 | –2.95$\pm$0.14 | 0.16$\pm$0.13 | –0.07$\pm$0.15 | 0.05$\pm$0.15
HE0401-0138 | –3.34 | 5.38 | –3.14$\pm$0.10 | 0.20$\pm$0.12 | 4.81 | 4.76 | –2.74$\pm$0.09 | –2.79$\pm$0.11 | 0.60$\pm$0.10 | 0.55$\pm$0.13 | 0.57$\pm$0.12
HE0417-0821 | –2.33 | 6.58 | –1.94$\pm$0.13 | 0.39$\pm$0.15 | 5.69 | 5.58 | –1.86$\pm$0.12 | –1.97$\pm$0.14 | 0.47$\pm$0.13 | 0.36$\pm$0.15 | 0.41$\pm$0.15
HE0430-4404 | –2.07 | 7.58 | –0.94$\pm$0.11 | 1.13$\pm$0.13 | 5.90 | $<$5.85 | –1.65$\pm$0.10 | $<$–1.77$\pm$0.12 | 0.42$\pm$0.11 | $<$0.30$\pm$0.13 | 0.42$\pm$0.11
HE0430-4901 | –2.72 | 5.80 | –2.72$\pm$0.10 | 0.00$\pm$0.12 | 5.06 | 5.02 | –2.49$\pm$0.09 | –2.53$\pm$0.11 | 0.23$\pm$0.10 | 0.19$\pm$0.13 | 0.21$\pm$0.12
HE0432-0923 | –3.19 | 5.60 | –2.92$\pm$0.12 | 0.27$\pm$0.14 | 4.86 | 4.80 | –2.69$\pm$0.11 | –2.75$\pm$0.13 | 0.50$\pm$0.12 | 0.44$\pm$0.14 | 0.47$\pm$0.14
HE0436-4008 | –2.35 | 6.61 | –1.91$\pm$0.12 | 0.44$\pm$0.14 | 5.76 | 5.67 | –1.79$\pm$0.11 | –1.88$\pm$0.13 | 0.56$\pm$0.12 | 0.47$\pm$0.14 | 0.52$\pm$0.14
HE0441-4343 | –2.52 | 6.41 | –2.11$\pm$0.10 | 0.41$\pm$0.12 | 5.55 | 5.56 | –2.00$\pm$0.09 | –1.99$\pm$0.11 | 0.52$\pm$0.10 | 0.53$\pm$0.13 | 0.53$\pm$0.12
HE0442-1234 | –2.41 | 5.46 | –3.06$\pm$0.10 | –0.65$\pm$0.12 | 5.49 | 5.51 | –2.06$\pm$0.09 | –2.04$\pm$0.11 | 0.35$\pm$0.10 | 0.37$\pm$0.13 | 0.36$\pm$0.12
HE0447-4858 | –1.69 | 6.81 | –1.71$\pm$0.12 | –0.02$\pm$0.14 | $<$6.57 | 6.72 | $<$–0.98$\pm$0.11 | –0.83$\pm$0.13 | $<$0.71$\pm$0.12 | 0.86$\pm$0.14 | 0.71$\pm$0.14
HE0450-4705 | –3.10 | 6.36 | –2.16$\pm$0.10 | 0.94$\pm$0.12 | 4.86 | 4.86 | –2.69$\pm$0.09 | –2.69$\pm$0.11 | 0.41$\pm$0.10 | 0.41$\pm$0.13 | 0.41$\pm$0.12
HE0454-4758 | –3.10 | 5.87 | –2.65$\pm$0.18 | 0.45$\pm$0.19 | 4.90 | 4.81 | –2.65$\pm$0.18 | –2.74$\pm$0.19 | 0.45$\pm$0.19 | 0.36$\pm$0.20 | 0.41$\pm$0.20
HE0501-5139 | –2.38 | 6.48 | –2.04$\pm$0.12 | 0.34$\pm$0.14 | $<$6.12 | $<$6.61 | $<$–1.43$\pm$0.11 | $<$–1.68$\pm$0.13 | $<$0.95$\pm$0.12 | $<$0.70$\pm$0.14 | $<$0.95$\pm$0.14
HE0501-5644 | –2.41 | 6.33 | –2.19$\pm$0.12 | 0.22$\pm$0.14 | 5.60 | 5.51 | –1.95$\pm$0.11 | –2.04$\pm$0.13 | 0.46$\pm$0.12 | 0.37$\pm$0.14 | 0.42$\pm$0.14
HE0512-3835 | –2.40 | 5.82 | –2.70$\pm$0.26 | –0.30$\pm$0.27 | 5.64 | 5.57 | –1.91$\pm$0.26 | –1.98$\pm$0.26 | 0.49$\pm$0.26 | 0.42$\pm$0.27 | 0.45$\pm$0.27
HE0513-4557 | –2.79 | 5.84 | –2.68$\pm$0.11 | 0.11$\pm$0.13 | $<$5.30 | $<$5.39 | $<$–2.25$\pm$0.10 | $<$–2.16$\pm$0.12 | $<$0.54$\pm$0.11 | $<$0.63$\pm$0.13 | $<$0.54$\pm$0.13
HE0516-3820 | –2.33 | 6.56 | –1.96$\pm$0.11 | 0.37$\pm$0.13 | 5.71 | 5.72 | –1.84$\pm$0.10 | –1.83$\pm$0.12 | 0.49$\pm$0.11 | 0.50$\pm$0.13 | 0.50$\pm$0.13
HE0517-1952 | –2.61 | 5.46 | –3.06$\pm$0.13 | –0.45$\pm$0.15 | 5.22 | 5.21 | –2.33$\pm$0.12 | –2.34$\pm$0.14 | 0.28$\pm$0.13 | 0.27$\pm$0.15 | 0.28$\pm$0.15
HE0519-5525 | –2.52 | 6.28 | –2.24$\pm$0.10 | 0.28$\pm$0.12 | 5.65 | $<$5.41 | –1.90$\pm$0.09 | $<$–2.14$\pm$0.11 | 0.62$\pm$0.10 | $<$0.38$\pm$0.13 | 0.50$\pm$0.10
HE0520-1748 | –2.52 | 6.40 | –2.12$\pm$0.10 | 0.40$\pm$0.12 | 5.41 | 5.42 | –2.14$\pm$0.09 | –2.13$\pm$0.11 | 0.38$\pm$0.10 | 0.39$\pm$0.13 | 0.39$\pm$0.12
HE0524-2055 | –2.58 | 5.59 | –2.93$\pm$0.10 | –0.35$\pm$0.12 | 5.40 | 5.30 | –2.15$\pm$0.09 | –2.25$\pm$0.11 | 0.43$\pm$0.10 | 0.33$\pm$0.13 | 0.38$\pm$0.12
HE0534-4615 | –2.01 | 6.66 | –1.86$\pm$0.10 | 0.15$\pm$0.12 | 6.02 | 5.93 | –1.53$\pm$0.09 | –1.62$\pm$0.11 | 0.48$\pm$0.10 | 0.39$\pm$0.13 | 0.44$\pm$0.12
HE0538-4515 | –1.52 | 7.14 | –1.38$\pm$0.10 | 0.14$\pm$0.12 | 6.48 | 6.48 | –1.07$\pm$0.09 | –1.07$\pm$0.11 | 0.45$\pm$0.10 | 0.45$\pm$0.13 | 0.45$\pm$0.12
HE0547-4539 | –3.01 | 5.99 | –2.53$\pm$0.12 | 0.48$\pm$0.14 | 4.93 | 4.80 | –2.62$\pm$0.11 | –2.75$\pm$0.13 | 0.39$\pm$0.12 | 0.24$\pm$0.14 | 0.32$\pm$0.14
HE0858-0016 | –2.73 | 4.91 | –3.61$\pm$0.10 | –0.88$\pm$0.12 | 5.36 | 5.50 | –2.19$\pm$0.09 | –2.05$\pm$0.11 | 0.54$\pm$0.10 | 0.68$\pm$0.13 | 0.61$\pm$0.12
HE0926-0508 | –2.78 | 6.36 | –2.16$\pm$0.09 | 0.62$\pm$0.11 | 5.06 | $<$4.90 | –2.49$\pm$0.08 | $<$–2.65$\pm$0.10 | 0.29$\pm$0.09 | $<$0.13$\pm$0.12 | 0.29$\pm$0.09
HE0938+0114 | –2.51 | 6.53 | –1.99$\pm$0.10 | 0.52$\pm$0.12 | 5.60 | $<$5.57 | –1.95$\pm$0.09 | $<$–1.98$\pm$0.11 | 0.56$\pm$0.10 | $<$0.53$\pm$0.13 | 0.56$\pm$0.10
HE0951-1152 | –2.62 | 5.98 | –2.54$\pm$0.10 | 0.08$\pm$0.12 | 5.56 | 5.55 | –1.99$\pm$0.09 | –2.00$\pm$0.11 | 0.63$\pm$0.10 | 0.62$\pm$0.13 | 0.63$\pm$0.12
HE1006-2218 | –2.69 | 6.41 | –2.11$\pm$0.12 | 0.58$\pm$0.14 | 5.44 | $<$5.31 | –2.11$\pm$0.11 | $<$–2.24$\pm$0.13 | 0.58$\pm$0.12 | $<$0.45$\pm$0.14 | 0.58$\pm$0.12
HE1015-0027 | –2.66 | 6.53 | –1.99$\pm$0.11 | 0.67$\pm$0.13 | 5.66 | $<$5.29 | –1.89$\pm$0.10 | $<$–2.26$\pm$0.12 | 0.77$\pm$0.11 | $<$0.40$\pm$0.13 | 0.77$\pm$0.11
HE1044-2509 | –2.89 | 6.03 | –2.49$\pm$0.10 | 0.40$\pm$0.12 | 5.20 | 5.11 | –2.35$\pm$0.09 | –2.44$\pm$0.11 | 0.54$\pm$0.10 | 0.45$\pm$0.13 | 0.50$\pm$0.12
HE1052-2548 | –2.29 | 6.76 | –1.76$\pm$0.13 | 0.53$\pm$0.15 | 5.96 | $<$5.76 | –1.59$\pm$0.12 | $<$–1.79$\pm$0.14 | 0.70$\pm$0.13 | $<$0.50$\pm$0.15 | 0.70$\pm$0.13
HE1054-0059 | –3.34 | 4.48 | –4.04$\pm$0.10 | –0.70$\pm$0.12 | 4.73 | 4.66 | –2.82$\pm$0.09 | –2.89$\pm$0.11 | 0.52$\pm$0.10 | 0.45$\pm$0.13 | 0.48$\pm$0.12
HE1059-0118 | –2.81 | 5.98 | –2.54$\pm$0.12 | 0.27$\pm$0.14 | 5.38 | 5.32 | –2.17$\pm$0.11 | –2.23$\pm$0.13 | 0.64$\pm$0.12 | 0.58$\pm$0.14 | 0.61$\pm$0.14
HE1100-0137 | –2.92 | 6.16 | –2.36$\pm$0.14 | 0.56$\pm$0.16 | 5.12 | $<$5.23 | –2.43$\pm$0.13 | $<$–2.32$\pm$0.15 | 0.49$\pm$0.14 | $<$0.40$\pm$0.16 | 0.49$\pm$0.14
HE1105+0027 | –2.42 | 8.00 | –0.52$\pm$0.09 | 1.90$\pm$0.11 | 6.01 | 5.96 | –1.54$\pm$0.08 | –1.59$\pm$0.10 | 0.88$\pm$0.04 | 0.83$\pm$0.12 | 0.85$\pm$0.11
HE1120-0153 | –2.77 | 6.33 | –2.19$\pm$0.13 | 0.58$\pm$0.15 | 5.31 | $<$5.33 | –2.24$\pm$0.12 | $<$–2.22$\pm$0.14 | 0.53$\pm$0.13 | $<$0.55$\pm$0.15 | 0.53$\pm$0.13
HE1122-1429 | –2.65 | 6.29 | –2.23$\pm$0.11 | 0.42$\pm$0.13 | 5.55 | $<$5.41 | –2.00$\pm$0.10 | $<$–2.14$\pm$0.12 | 0.65$\pm$0.11 | $<$0.51$\pm$0.13 | 0.65$\pm$0.11
HE1124-2335 | –2.95 | 6.43 | –2.09$\pm$0.13 | 0.86$\pm$0.15 | 5.16 | 5.05 | –2.39$\pm$0.12 | –2.50$\pm$0.14 | 0.56$\pm$0.13 | 0.45$\pm$0.15 | 0.51$\pm$0.15
HE1126-1735 | –2.69 | 6.11 | –2.41$\pm$0.12 | 0.28$\pm$0.14 | 5.22 | $<$5.20 | –2.33$\pm$0.11 | $<$–2.35$\pm$0.13 | 0.36$\pm$0.12 | $<$0.34$\pm$0.14 | 0.35$\pm$0.12
HE1127-1143 | –2.73 | 6.25 | –2.27$\pm$0.11 | 0.46$\pm$0.13 | 5.25 | $<$5.09 | –2.30$\pm$0.10 | $<$–2.46$\pm$0.12 | 0.43$\pm$0.11 | $<$0.27$\pm$0.13 | 0.35$\pm$0.11
HE1128-0823 | –2.71 | 6.41 | –2.11$\pm$0.11 | 0.60$\pm$0.13 | 5.32 | 5.30 | –2.23$\pm$0.10 | –2.25$\pm$0.12 | 0.48$\pm$0.11 | 0.46$\pm$0.13 | 0.48$\pm$0.13
HE1131+0141 | –2.48 | 6.26 | –2.26$\pm$0.10 | 0.22$\pm$0.12 | 5.65 | 5.80 | –1.90$\pm$0.09 | –1.75$\pm$0.11 | 0.58$\pm$0.10 | 0.73$\pm$0.13 | 0.66$\pm$0.12
HE1132+0125 | –2.42 | 6.35 | –2.17$\pm$0.11 | 0.25$\pm$0.13 | 5.78 | 5.72 | –1.77$\pm$0.10 | –1.83$\pm$0.12 | 0.65$\pm$0.11 | 0.59$\pm$0.13 | 0.62$\pm$0.13
HE1132+0204 | –2.55 | 6.10 | –2.42$\pm$0.15 | 0.13$\pm$0.17 | 5.28 | 5.19 | –2.27$\pm$0.14 | –2.36$\pm$0.16 | 0.28$\pm$0.15 | 0.19$\pm$0.17 | 0.24$\pm$0.17
HE1135+0139 | –2.33 | 7.20 | –1.32$\pm$0.13 | 1.01$\pm$0.15 | 5.61 | $<$5.48 | –1.94$\pm$0.12 | $<$–2.07$\pm$0.14 | 0.39$\pm$0.13 | $<$0.26$\pm$0.15 | 0.39$\pm$0.13
HE1135-0344 | –2.63 | 6.79 | –1.73$\pm$0.10 | 0.90$\pm$0.12 | 5.28 | $<$5.24 | –2.27$\pm$0.09 | $<$–2.31$\pm$0.11 | 0.36$\pm$0.10 | $<$0.32$\pm$0.13 | 0.36$\pm$0.10
HE1148-0037 | –3.47 | 5.92 | –2.60$\pm$0.11 | 0.87$\pm$0.13 | 4.62 | $<$4.67 | –2.93$\pm$0.10 | $<$–2.88$\pm$0.12 | 0.54$\pm$0.11 | $<$0.59$\pm$0.13 | 0.54$\pm$0.11
HE1207-2031 | –2.82 | 6.53 | –1.99$\pm$0.13 | 0.83$\pm$0.15 | 5.36 | $<$5.43 | –2.19$\pm$0.12 | $<$–2.12$\pm$0.14 | 0.63$\pm$0.13 | $<$0.70$\pm$0.15 | 0.63$\pm$0.13
HE1210+0048 | –2.28 | 6.72 | –1.80$\pm$0.12 | 0.48$\pm$0.14 | 6.06 | $<$5.87 | –1.49$\pm$0.11 | $<$–1.68$\pm$0.13 | 0.79$\pm$0.12 | $<$0.60$\pm$0.14 | 0.79$\pm$0.12
HE1210-1956 | –2.57 | 6.10 | –2.42$\pm$0.11 | 0.15$\pm$0.13 | 5.49 | $<$5.33 | –2.06$\pm$0.10 | $<$–2.22$\pm$0.12 | 0.51$\pm$0.11 | $<$0.35$\pm$0.13 | 0.51$\pm$0.11
HE1212-0127 | –2.15 | 5.97 | –2.55$\pm$0.12 | –0.40$\pm$0.14 | 5.67 | 5.63 | –1.88$\pm$0.11 | –1.92$\pm$0.13 | 0.27$\pm$0.12 | 0.23$\pm$0.14 | 0.25$\pm$0.14
HE1214-1819 | –3.01 | 5.86 | –2.66$\pm$0.13 | 0.35$\pm$0.15 | 5.13 | 4.99 | –2.42$\pm$0.12 | –2.56$\pm$0.14 | 0.59$\pm$0.13 | 0.45$\pm$0.15 | 0.52$\pm$0.15
HE1215+0149 | –2.90 | 5.86 | –2.66$\pm$0.11 | 0.24$\pm$0.13 | 5.25 | 5.06 | –2.30$\pm$0.10 | –2.49$\pm$0.12 | 0.60$\pm$0.11 | 0.41$\pm$0.13 | 0.51$\pm$0.13
HE1217-0540 | –2.95 | 6.39 | –2.13$\pm$0.13 | 0.82$\pm$0.15 | 5.12 | 5.11 | –2.43$\pm$0.12 | –2.44$\pm$0.14 | 0.52$\pm$0.13 | 0.51$\pm$0.15 | 0.52$\pm$0.15
HE1219-0312 | –2.81 | 5.89 | –2.63$\pm$0.11 | 0.18$\pm$0.13 | 5.11 | 4.92 | –2.44$\pm$0.10 | –2.63$\pm$0.12 | 0.37$\pm$0.11 | 0.18$\pm$0.13 | 0.28$\pm$0.13
HE1221-0522 | –2.84 | 6.26 | –2.26$\pm$0.11 | 0.58$\pm$0.13 | 5.22 | $<$5.14 | –2.33$\pm$0.10 | $<$–2.41$\pm$0.12 | 0.51$\pm$0.11 | $<$0.43$\pm$0.13 | 0.51$\pm$0.11
HE1221-1948 | –3.36 | 6.46 | –2.06$\pm$0.12 | 1.30$\pm$0.14 | 5.11 | $<$4.89 | –2.44$\pm$0.11 | $<$–2.66$\pm$0.13 | 0.92$\pm$0.12 | $<$0.70$\pm$0.14 | 0.92$\pm$0.12
HE1222-0200 | –2.45 | 6.24 | –2.28$\pm$0.11 | 0.17$\pm$0.13 | 5.78 | 5.77 | –1.77$\pm$0.10 | –1.78$\pm$0.12 | 0.68$\pm$0.11 | 0.67$\pm$0.13 | 0.68$\pm$0.13
HE1222-0336 | –2.04 | 6.54 | –1.98$\pm$0.09 | 0.06$\pm$0.11 | 5.83 | 5.84 | –1.72$\pm$0.08 | –1.71$\pm$0.10 | 0.32$\pm$0.09 | 0.33$\pm$0.12 | 0.33$\pm$0.11
HE1225+0155 | –2.75 | 5.98 | –2.54$\pm$0.12 | 0.21$\pm$0.14 | 5.23 | 5.21 | –2.32$\pm$0.11 | –2.34$\pm$0.13 | 0.43$\pm$0.12 | 0.41$\pm$0.14 | 0.42$\pm$0.14
HE1225-0515 | –1.96 | 7.14 | –1.38$\pm$0.11 | 0.58$\pm$0.13 | 5.96 | 5.93 | –1.59$\pm$0.10 | –1.62$\pm$0.12 | 0.37$\pm$0.11 | 0.34$\pm$0.13 | 0.35$\pm$0.13
HE1230-1724 | –2.30 | 6.42 | –2.10$\pm$0.14 | 0.20$\pm$0.16 | 5.66 | $<$5.63 | –1.89$\pm$0.13 | $<$–1.92$\pm$0.15 | 0.41$\pm$0.14 | $<$0.38$\pm$0.16 | 0.41$\pm$0.14
HE1237-3103 | –2.91 | 5.51 | –3.01$\pm$0.12 | –0.10$\pm$0.14 | 4.84 | 4.84 | –2.71$\pm$0.11 | –2.71$\pm$0.13 | 0.20$\pm$0.12 | 0.20$\pm$0.14 | 0.20$\pm$0.14
HE1243-1425 | –2.67 | 6.25 | –2.27$\pm$0.11 | 0.40$\pm$0.13 | 5.10 | 5.21 | –2.45$\pm$0.10 | –2.34$\pm$0.12 | 0.22$\pm$0.11 | 0.33$\pm$0.13 | 0.28$\pm$0.13
HE1245-1616 | –2.98 | 6.71 | –1.81$\pm$0.12 | 1.17$\pm$0.14 | 5.31 | $<$5.17 | –2.24$\pm$0.11 | $<$–2.38$\pm$0.13 | 0.74$\pm$0.12 | $<$0.60$\pm$0.14 | 0.74$\pm$0.12
HE1246-1344 | –3.40 | 5.00 | –3.52$\pm$0.11 | –0.12$\pm$0.13 | 5.20 | 5.08 | –2.35$\pm$0.10 | –2.47$\pm$0.12 | 1.05$\pm$0.11 | 0.93$\pm$0.13 | 0.99$\pm$0.13
HE1247-2114 | –2.61 | 6.26 | –2.26$\pm$0.12 | 0.35$\pm$0.14 | 5.47 | 5.52 | –2.08$\pm$0.11 | –2.03$\pm$0.13 | 0.53$\pm$0.12 | 0.58$\pm$0.14 | 0.55$\pm$0.14
HE1248-1800 | –2.89 | 6.19 | –2.33$\pm$0.11 | 0.56$\pm$0.13 | 5.25 | 5.04 | –2.30$\pm$0.10 | –2.51$\pm$0.12 | 0.59$\pm$0.11 | 0.38$\pm$0.13 | 0.48$\pm$0.13
HE1249-2932 | –2.65 | 5.40 | –3.12$\pm$0.12 | –0.47$\pm$0.14 | 5.35 | 5.40 | –2.20$\pm$0.11 | –2.15$\pm$0.13 | 0.45$\pm$0.12 | 0.50$\pm$0.14 | 0.47$\pm$0.14
HE1249-3121 | –3.23 | 7.11 | –1.41$\pm$0.12 | 1.82$\pm$0.14 | 4.78 | $<$4.67 | –2.77$\pm$0.11 | $<$–2.88$\pm$0.13 | 0.46$\pm$0.12 | $<$0.35$\pm$0.14 | 0.46$\pm$0.12
HE1251-0104 | –2.73 | 5.98 | –2.54$\pm$0.13 | 0.19$\pm$0.15 | 5.22 | 5.07 | –2.33$\pm$0.12 | –2.48$\pm$0.14 | 0.40$\pm$0.13 | 0.25$\pm$0.15 | 0.33$\pm$0.15
HE1252+0044 | –3.28 | 5.81 | –2.71$\pm$0.13 | 0.57$\pm$0.15 | 4.98 | 4.87 | –2.57$\pm$0.12 | –2.58$\pm$0.14 | 0.71$\pm$0.13 | 0.70$\pm$0.15 | 0.71$\pm$0.15
HE1252-0117 | –2.89 | 5.45 | –3.07$\pm$0.12 | –0.18$\pm$0.14 | 4.93 | $<$4.95 | –2.62$\pm$0.11 | $<$–2.60$\pm$0.13 | 0.27$\pm$0.12 | $<$0.29$\pm$0.14 | 0.28$\pm$0.14
HE1254+0009 | –2.94 | 5.43 | –3.09$\pm$0.10 | –0.15$\pm$0.12 | 5.26 | 5.24 | –2.29$\pm$0.09 | –2.31$\pm$0.11 | 0.65$\pm$0.10 | 0.63$\pm$0.13 | 0.64$\pm$0.12
HE1256-0228 | –2.07 | 6.33 | –2.19$\pm$0.12 | –0.12$\pm$0.14 | 5.55 | 5.31 | –2.00$\pm$0.11 | –2.24$\pm$0.13 | 0.07$\pm$0.12 | –0.17$\pm$0.14 | –0.05$\pm$0.14
HE1256-0651 | –2.36 | 6.69 | –1.83$\pm$0.12 | 0.53$\pm$0.14 | 5.42 | $<$5.49 | –2.13$\pm$0.11 | $<$–2.06$\pm$0.13 | 0.23$\pm$0.12 | $<$0.30$\pm$0.14 | 0.23$\pm$0.12
HE1259-0621 | –2.64 | 6.35 | –2.17$\pm$0.12 | 0.47$\pm$0.14 | 5.36 | 5.32 | –2.19$\pm$0.11 | –2.23$\pm$0.13 | 0.45$\pm$0.12 | 0.41$\pm$0.14 | 0.43$\pm$0.14
HE1300+0157 | –3.76 | 5.82 | –2.70$\pm$0.14 | 1.06$\pm$0.16 | 4.55 | 4.34 | –3.00$\pm$0.13 | –3.21$\pm$0.15 | 0.76$\pm$0.14 | 0.55$\pm$0.16 | 0.66$\pm$0.16
HE1300-0641 | –3.14 | 6.53 | –1.99$\pm$0.14 | 1.15$\pm$0.16 | 4.38 | $<$4.51 | –3.17$\pm$0.13 | $<$–3.04$\pm$0.15 | –0.03$\pm$0.14 | $<$0.10$\pm$0.16 | –0.03$\pm$0.14
HE1300-0642 | –3.03 | 5.90 | –2.62$\pm$0.11 | 0.41$\pm$0.13 | 5.07 | 5.15 | –2.48$\pm$0.10 | –2.40$\pm$0.12 | 0.55$\pm$0.11 | 0.63$\pm$0.13 | 0.59$\pm$0.13
HE1300-2201 | –2.61 | 7.10 | –1.42$\pm$0.13 | 1.19$\pm$0.15 | 5.45 | 5.26 | –2.10$\pm$0.12 | –2.29$\pm$0.14 | 0.51$\pm$0.13 | 0.32$\pm$0.15 | 0.42$\pm$0.15
HE1300-2431 | –3.25 | 5.17 | –3.35$\pm$0.11 | –0.10$\pm$0.13 | 4.71 | 4.59 | –2.84$\pm$0.10 | –2.96$\pm$0.12 | 0.41$\pm$0.11 | 0.29$\pm$0.13 | 0.35$\pm$0.13
HE1305-0331 | –3.26 | 6.53 | –1.99$\pm$0.11 | 1.27$\pm$0.13 | 4.64 | $<$4.59 | –2.91$\pm$0.10 | $<$–2.96$\pm$0.12 | 0.35$\pm$0.11 | $<$0.30$\pm$0.13 | 0.35$\pm$0.11
HE1311-1412 | –2.91 | 5.41 | –3.11$\pm$0.10 | –0.20$\pm$0.12 | 5.11 | 4.97 | –2.44$\pm$0.09 | –2.58$\pm$0.11 | 0.47$\pm$0.10 | 0.33$\pm$0.13 | 0.40$\pm$0.12
HE1314-3036 | –2.99 | 5.30 | –3.22$\pm$0.09 | –0.23$\pm$0.11 | 5.15 | 5.05 | –2.40$\pm$0.08 | –2.50$\pm$0.10 | 0.59$\pm$0.09 | 0.49$\pm$0.12 | 0.54$\pm$0.11
HE1320-1339 | –2.78 | 5.15 | –3.37$\pm$0.12 | –0.59$\pm$0.14 | 5.24 | 5.13 | –2.31$\pm$0.11 | –2.42$\pm$0.13 | 0.47$\pm$0.12 | 0.36$\pm$0.14 | 0.41$\pm$0.14
HE1330-0354 | –2.29 | 7.01 | –1.51$\pm$0.12 | 0.78$\pm$0.14 | 5.82 | 5.67 | –1.73$\pm$0.11 | –1.88$\pm$0.13 | 0.56$\pm$0.12 | 0.41$\pm$0.14 | 0.48$\pm$0.14
HE1330-0607 | –2.33 | 6.37 | –2.15$\pm$0.13 | 0.18$\pm$0.15 | 5.59 | 5.67 | –1.96$\pm$0.12 | –1.88$\pm$0.14 | 0.37$\pm$0.13 | 0.45$\pm$0.15 | 0.41$\pm$0.15
HE1332-0309 | –2.46 | 6.19 | –2.33$\pm$0.12 | 0.13$\pm$0.14 | 5.58 | 5.44 | –1.97$\pm$0.11 | –2.11$\pm$0.13 | 0.49$\pm$0.12 | 0.35$\pm$0.14 | 0.42$\pm$0.14
HE1333-0340 | –2.64 | 6.26 | –2.26$\pm$0.10 | 0.38$\pm$0.12 | 5.23 | $<$5.21 | –2.32$\pm$0.09 | $<$–2.34$\pm$0.11 | 0.32$\pm$0.10 | $<$0.30$\pm$0.13 | 0.32$\pm$0.10
HE1335+0135 | –2.47 | 6.15 | –2.37$\pm$0.09 | 0.10$\pm$0.11 | 5.53 | 5.50 | –2.02$\pm$0.08 | –2.05$\pm$0.10 | 0.45$\pm$0.09 | 0.42$\pm$0.12 | 0.44$\pm$0.11
HE1337+0012 | –3.44 | 5.96 | –2.56$\pm$0.11 | 0.88$\pm$0.13 | 4.53 | $<$4.82 | –3.02$\pm$0.10 | $<$–2.73$\pm$0.12 | 0.42$\pm$0.11 | $<$0.35$\pm$0.13 | 0.42$\pm$0.11
HE1337-0453 | –2.34 | 6.51 | –2.01$\pm$0.11 | 0.33$\pm$0.13 | 5.63 | $<$5.57 | –1.92$\pm$0.10 | $<$–1.98$\pm$0.12 | 0.42$\pm$0.11 | $<$0.36$\pm$0.13 | 0.42$\pm$0.11
HE1343-0640 | –1.90 | 7.33 | –1.19$\pm$0.15 | 0.71$\pm$0.17 | 6.02 | 6.08 | –1.53$\pm$0.14 | –1.47$\pm$0.16 | 0.37$\pm$0.15 | 0.43$\pm$0.17 | 0.40$\pm$0.17
HE1345-0206 | –2.82 | 6.05 | –2.47$\pm$0.12 | 0.35$\pm$0.14 | 5.07 | 5.01 | –2.48$\pm$0.11 | –2.54$\pm$0.13 | 0.34$\pm$0.12 | 0.28$\pm$0.14 | 0.31$\pm$0.14
HE1351-1049 | –3.46 | 6.70 | –1.82$\pm$0.11 | 1.64$\pm$0.13 | 4.61 | $<$4.54 | –2.94$\pm$0.10 | $<$–3.01$\pm$0.12 | 0.52$\pm$0.11 | $<$0.45$\pm$0.13 | 0.52$\pm$0.11
HE1413-1954 | –3.22 | 6.88 | –1.64$\pm$0.12 | 1.58$\pm$0.14 | 5.09 | $<$4.93 | –2.46$\pm$0.11 | $<$–2.62$\pm$0.13 | 0.76$\pm$0.12 | $<$0.60$\pm$0.14 | 0.76$\pm$0.12
HE1419-1759 | –3.18 | 5.04 | –3.48$\pm$0.11 | –0.30$\pm$0.13 | 5.22 | 5.02 | –2.33$\pm$0.10 | –2.53$\pm$0.12 | 0.85$\pm$0.11 | 0.65$\pm$0.13 | 0.75$\pm$0.13
HE1421-2006 | –2.65 | 6.20 | –2.32$\pm$0.10 | 0.33$\pm$0.12 | 5.46 | 5.22 | –2.09$\pm$0.09 | –2.33$\pm$0.11 | 0.56$\pm$0.10 | 0.33$\pm$0.13 | 0.45$\pm$0.12
HE1430+0053 | –3.03 | 5.78 | –2.74$\pm$0.11 | 0.29$\pm$0.13 | 5.05 | 4.94 | –2.50$\pm$0.10 | –2.61$\pm$0.12 | 0.53$\pm$0.11 | 0.42$\pm$0.13 | 0.47$\pm$0.13
HE1430-0026 | –2.79 | 6.21 | –2.31$\pm$0.12 | 0.48$\pm$0.14 | 5.44 | 5.46 | –2.11$\pm$0.11 | –2.09$\pm$0.13 | 0.68$\pm$0.12 | 0.70$\pm$0.14 | 0.69$\pm$0.14
HE1430-1123 | –2.71 | 7.56 | –0.96$\pm$0.12 | 1.75$\pm$0.14 | 5.87 | $<$5.74 | –1.68$\pm$0.11 | $<$–1.81$\pm$0.13 | 1.03$\pm$0.12 | $<$0.90$\pm$0.14 | 1.03$\pm$0.12
HE1431-2142 | –2.60 | 6.40 | –2.12$\pm$0.10 | 0.48$\pm$0.12 | 5.44 | $<$5.35 | –2.11$\pm$0.09 | $<$–2.20$\pm$0.11 | 0.49$\pm$0.10 | $<$0.40$\pm$0.13 | 0.49$\pm$0.10
HE1500-1628 | –2.31 | 6.30 | –2.22$\pm$0.11 | 0.09$\pm$0.13 | 5.38 | 5.32 | –2.17$\pm$0.10 | –2.23$\pm$0.12 | 0.14$\pm$0.11 | 0.08$\pm$0.13 | 0.11$\pm$0.13
HE2133-1432 | –2.02 | 6.63 | –1.89$\pm$0.11 | 0.13$\pm$0.13 | 6.18 | 5.94 | –1.37$\pm$0.10 | –1.61$\pm$0.12 | 0.65$\pm$0.11 | 0.41$\pm$0.13 | 0.53$\pm$0.13
HE2134+0001 | –2.22 | 6.51 | –2.01$\pm$0.12 | 0.21$\pm$0.14 | 5.85 | 5.73 | –1.70$\pm$0.11 | –1.82$\pm$0.13 | 0.52$\pm$0.12 | 0.40$\pm$0.14 | 0.46$\pm$0.14
HE2139-1851 | –3.25 | 5.73 | –2.79$\pm$0.21 | 0.46$\pm$0.22 | 4.78 | 4.65 | –2.77$\pm$0.21 | –2.90$\pm$0.21 | 0.48$\pm$0.22 | 0.35$\pm$0.22 | 0.41$\pm$0.22
HE2143+0030 | –2.43 | 5.73 | –2.79$\pm$0.12 | –0.36$\pm$0.14 | 5.38 | 5.62 | –2.17$\pm$0.11 | –1.93$\pm$0.13 | 0.26$\pm$0.12 | 0.50$\pm$0.14 | 0.38$\pm$0.14
HE2145-3025 | –2.69 | 5.63 | –2.89$\pm$0.09 | –0.20$\pm$0.11 | 4.88 | 4.94 | –2.67$\pm$0.08 | –2.61$\pm$0.10 | 0.02$\pm$0.09 | 0.08$\pm$0.12 | 0.05$\pm$0.11
HE2150-0825 | –1.98 | 7.93 | –0.59$\pm$0.10 | 1.39$\pm$0.12 | 6.14 | 6.06 | –1.41$\pm$0.09 | –1.49$\pm$0.11 | 0.57$\pm$0.10 | 0.49$\pm$0.13 | 0.53$\pm$0.12
HE2151-2858 | –2.38 | 6.36 | –2.16$\pm$0.10 | 0.22$\pm$0.12 | 5.72 | 5.74 | –1.83$\pm$0.09 | –1.81$\pm$0.11 | 0.55$\pm$0.10 | 0.57$\pm$0.13 | 0.56$\pm$0.12
HE2153-2719 | –2.49 | 6.07 | –2.45$\pm$0.10 | 0.04$\pm$0.12 | 5.57 | 5.68 | –1.98$\pm$0.09 | –1.87$\pm$0.11 | 0.51$\pm$0.10 | 0.62$\pm$0.13 | 0.56$\pm$0.12
HE2154-2838 | –1.85 | 6.63 | –1.89$\pm$0.11 | –0.04$\pm$0.13 | 6.06 | 6.26 | –1.49$\pm$0.10 | –1.19$\pm$0.12 | 0.36$\pm$0.11 | 0.56$\pm$0.13 | 0.46$\pm$0.13
HE2155+0136 | –2.07 | 6.38 | –2.14$\pm$0.10 | –0.07$\pm$0.12 | 5.76 | 5.65 | –1.79$\pm$0.09 | –1.90$\pm$0.11 | 0.28$\pm$0.10 | 0.17$\pm$0.13 | 0.23$\pm$0.12
HE2156-3130 | –3.13 | 5.98 | –2.54$\pm$0.13 | 0.59$\pm$0.15 | 5.16 | 4.91 | –2.39$\pm$0.12 | –2.64$\pm$0.14 | 0.74$\pm$0.13 | 0.51$\pm$0.15 | 0.63$\pm$0.15
HE2158-3112 | –2.75 | 5.65 | –2.87$\pm$0.13 | –0.12$\pm$0.15 | 5.51 | 5.60 | –2.04$\pm$0.12 | –1.95$\pm$0.14 | 0.71$\pm$0.13 | 0.80$\pm$0.15 | 0.76$\pm$0.15
HE2200-2030 | –2.00 | 6.73 | –1.79$\pm$0.14 | 0.21$\pm$0.16 | 6.09 | $<$5.95 | –1.46$\pm$0.13 | $<$–1.60$\pm$0.15 | 0.54$\pm$0.14 | $<$0.40$\pm$0.16 | 0.54$\pm$0.14
HE2201-0637 | –2.61 | 6.04 | –2.48$\pm$0.11 | 0.13$\pm$0.13 | 5.26 | 5.40 | –2.29$\pm$0.10 | –2.15$\pm$0.12 | 0.32$\pm$0.11 | 0.46$\pm$0.13 | 0.39$\pm$0.13
HE2204-1703 | –2.79 | 5.88 | –2.64$\pm$0.16 | 0.15$\pm$0.17 | 5.37 | 5.20 | –2.18$\pm$0.15 | –2.35$\pm$0.17 | 0.61$\pm$0.16 | 0.44$\pm$0.18 | 0.53$\pm$0.18
HE2206-2245 | –2.73 | 5.99 | –2.53$\pm$0.10 | 0.20$\pm$0.12 | 5.32 | 5.18 | –2.23$\pm$0.09 | –2.37$\pm$0.11 | 0.50$\pm$0.10 | 0.36$\pm$0.13 | 0.43$\pm$0.12
HE2216-0621 | –3.23 | 4.72 | –3.80$\pm$0.11 | –0.57$\pm$0.13 | 4.89 | 4.70 | –2.66$\pm$0.10 | –2.85$\pm$0.12 | 0.57$\pm$0.11 | 0.38$\pm$0.13 | 0.47$\pm$0.13
HE2216-1548 | –1.70 | 6.34 | –2.18$\pm$0.12 | –0.48$\pm$0.14 | 5.94 | 5.90 | –1.61$\pm$0.11 | –1.65$\pm$0.13 | 0.09$\pm$0.12 | 0.05$\pm$0.14 | 0.07$\pm$0.14
HE2217-0706 | –2.56 | 5.33 | –3.19$\pm$0.11 | –0.63$\pm$0.13 | 5.52 | 5.39 | –2.03$\pm$0.10 | –2.16$\pm$0.12 | 0.53$\pm$0.11 | 0.40$\pm$0.13 | 0.47$\pm$0.13
HE2217-1523 | –2.62 | 5.87 | –2.65$\pm$0.10 | –0.03$\pm$0.12 | 5.39 | 5.32 | –2.16$\pm$0.09 | –2.23$\pm$0.11 | 0.46$\pm$0.10 | 0.39$\pm$0.13 | 0.43$\pm$0.12
HE2219-0713 | –2.91 | 5.37 | –3.15$\pm$0.11 | –0.24$\pm$0.13 | 5.04 | 4.82 | –2.51$\pm$0.10 | –2.73$\pm$0.12 | 0.40$\pm$0.11 | 0.18$\pm$0.13 | 0.29$\pm$0.13
HE2221-4150 | –2.03 | 6.68 | –1.84$\pm$0.10 | 0.19$\pm$0.12 | 5.84 | 5.84 | –1.71$\pm$0.09 | –1.71$\pm$0.11 | 0.32$\pm$0.10 | 0.32$\pm$0.13 | 0.32$\pm$0.12
HE2222-4156 | –2.73 | 6.04 | –2.48$\pm$0.09 | 0.25$\pm$0.11 | 5.40 | 5.16 | –2.15$\pm$0.08 | –2.39$\pm$0.10 | 0.58$\pm$0.09 | 0.34$\pm$0.12 | 0.46$\pm$0.11
HE2224+0143 | –2.58 | 6.21 | –2.31$\pm$0.12 | 0.27$\pm$0.14 | 5.54 | 5.52 | –2.01$\pm$0.11 | –2.03$\pm$0.13 | 0.57$\pm$0.12 | 0.55$\pm$0.14 | 0.56$\pm$0.14
HE2224-4103 | –2.64 | 6.08 | –2.44$\pm$0.10 | 0.20$\pm$0.12 | 5.46 | 5.45 | –2.09$\pm$0.09 | –2.10$\pm$0.11 | 0.55$\pm$0.10 | 0.54$\pm$0.13 | 0.55$\pm$0.12
HE2226-4102 | –2.87 | 6.07 | –2.45$\pm$0.11 | 0.42$\pm$0.13 | 5.27 | 5.11 | –2.28$\pm$0.10 | –2.44$\pm$0.12 | 0.59$\pm$0.11 | 0.43$\pm$0.13 | 0.51$\pm$0.13
HE2227-4044 | –2.32 | 7.80 | –0.72$\pm$0.10 | 1.60$\pm$0.12 | 5.78 | 5.68 | –1.77$\pm$0.09 | –1.87$\pm$0.11 | 0.55$\pm$0.10 | 0.45$\pm$0.13 | 0.50$\pm$0.12
HE2228-3806 | –3.07 | 5.79 | –2.73$\pm$0.15 | 0.34$\pm$0.17 | 5.04 | 4.97 | –2.51$\pm$0.14 | –2.58$\pm$0.16 | 0.56$\pm$0.15 | 0.49$\pm$0.17 | 0.53$\pm$0.17
HE2229-4153 | –2.62 | 6.28 | –2.24$\pm$0.12 | 0.38$\pm$0.14 | 5.44 | 5.46 | –2.11$\pm$0.11 | –2.09$\pm$0.13 | 0.51$\pm$0.12 | 0.53$\pm$0.14 | 0.52$\pm$0.14
HE2231-0622 | –2.12 | 6.40 | –2.12$\pm$0.10 | 0.00$\pm$0.12 | 5.76 | 5.77 | –1.79$\pm$0.09 | –1.78$\pm$0.11 | 0.33$\pm$0.10 | 0.34$\pm$0.13 | 0.34$\pm$0.12
HE2234-0521 | –2.78 | 6.16 | –2.36$\pm$0.11 | 0.42$\pm$0.13 | 5.51 | 5.49 | –2.04$\pm$0.10 | –2.06$\pm$0.12 | 0.74$\pm$0.11 | 0.72$\pm$0.13 | 0.73$\pm$0.13
HE2238-2152 | –2.40 | 6.27 | –2.25$\pm$0.11 | 0.15$\pm$0.13 | 5.57 | 5.49 | –1.98$\pm$0.10 | –2.06$\pm$0.12 | 0.42$\pm$0.11 | 0.34$\pm$0.13 | 0.38$\pm$0.13
HE2240-0412 | –2.20 | 7.69 | –0.83$\pm$0.11 | 1.37$\pm$0.13 | 5.76 | 5.68 | –1.79$\pm$0.10 | –1.87$\pm$0.12 | 0.41$\pm$0.11 | 0.33$\pm$0.13 | 0.37$\pm$0.13
HE2242-1930 | –2.21 | 6.37 | –2.15$\pm$0.13 | 0.06$\pm$0.15 | 5.69 | 5.69 | –1.86$\pm$0.12 | –1.86$\pm$0.14 | 0.35$\pm$0.13 | 0.35$\pm$0.15 | 0.35$\pm$0.15
HE2243-0151 | –1.61 | 7.09 | –1.43$\pm$0.11 | 0.18$\pm$0.13 | 6.25 | 6.24 | –1.30$\pm$0.10 | –1.31$\pm$0.12 | 0.31$\pm$0.11 | 0.30$\pm$0.13 | 0.30$\pm$0.13
HE2244-1503 | –2.88 | 5.76 | –2.76$\pm$0.13 | 0.12$\pm$0.15 | 5.21 | 5.12 | –2.34$\pm$0.12 | –2.43$\pm$0.14 | 0.54$\pm$0.13 | 0.45$\pm$0.15 | 0.50$\pm$0.15
HE2247-3705 | –2.27 | 6.63 | –1.89$\pm$0.10 | 0.38$\pm$0.12 | 5.69 | 5.66 | –1.86$\pm$0.09 | –1.89$\pm$0.11 | 0.41$\pm$0.10 | 0.38$\pm$0.13 | 0.40$\pm$0.12
HE2248-3345 | –2.74 | 5.95 | –2.57$\pm$0.09 | 0.17$\pm$0.11 | 4.91 | 4.90 | –2.64$\pm$0.08 | –2.65$\pm$0.10 | 0.10$\pm$0.09 | 0.09$\pm$0.12 | 0.10$\pm$0.11
HE2250-2132 | –2.22 | 6.61 | –1.91$\pm$0.11 | 0.31$\pm$0.13 | 5.73 | 5.87 | –1.82$\pm$0.10 | –1.68$\pm$0.12 | 0.40$\pm$0.11 | 0.54$\pm$0.13 | 0.47$\pm$0.13
HE2252-4157 | –1.93 | 6.45 | –2.07$\pm$0.13 | –0.14$\pm$0.15 | 5.84 | 5.67 | –1.71$\pm$0.12 | –1.88$\pm$0.14 | 0.22$\pm$0.13 | 0.05$\pm$0.15 | 0.14$\pm$0.15
HE2252-4225 | –2.83 | 5.24 | –3.28$\pm$0.11 | –0.45$\pm$0.13 | 5.03 | 5.00 | –2.52$\pm$0.10 | –2.55$\pm$0.12 | 0.31$\pm$0.11 | 0.28$\pm$0.13 | 0.30$\pm$0.13
HE2258-3456 | –2.97 | 5.36 | –3.16$\pm$0.10 | –0.19$\pm$0.12 | 5.18 | 4.95 | –2.37$\pm$0.09 | –2.60$\pm$0.11 | 0.60$\pm$0.10 | 0.37$\pm$0.13 | 0.48$\pm$0.12
HE2259-3407 | –2.29 | 6.85 | –1.67$\pm$0.12 | 0.62$\pm$0.14 | 5.72 | 5.78 | –1.83$\pm$0.11 | –1.77$\pm$0.13 | 0.46$\pm$0.12 | 0.52$\pm$0.14 | 0.49$\pm$0.14
HE2301-4024 | –2.11 | 6.66 | –1.86$\pm$0.12 | 0.25$\pm$0.14 | 5.89 | 5.87 | –1.66$\pm$0.11 | –1.68$\pm$0.13 | 0.45$\pm$0.12 | 0.43$\pm$0.14 | 0.44$\pm$0.14
HE2301-4126 | –2.37 | 6.51 | –2.01$\pm$0.11 | 0.36$\pm$0.13 | 5.46 | 5.59 | –2.09$\pm$0.10 | –1.96$\pm$0.12 | 0.28$\pm$0.11 | 0.41$\pm$0.13 | 0.34$\pm$0.13
HE2304-4153 | –3.02 | 4.86 | –3.66$\pm$0.13 | –0.64$\pm$0.15 | 4.83 | 4.88 | –2.72$\pm$0.12 | –2.67$\pm$0.14 | 0.30$\pm$0.13 | 0.35$\pm$0.15 | 0.32$\pm$0.15
HE2311+0129 | –2.78 | 6.02 | –2.50$\pm$0.16 | 0.28$\pm$0.17 | 5.40 | 5.24 | –2.15$\pm$0.15 | –2.31$\pm$0.17 | 0.63$\pm$0.16 | 0.47$\pm$0.18 | 0.55$\pm$0.18
HE2314-1554 | –3.27 | 5.80 | –2.72$\pm$0.12 | 0.55$\pm$0.14 | 5.24 | 5.43 | –2.31$\pm$0.11 | –2.12$\pm$0.13 | 0.96$\pm$0.12 | 1.15$\pm$0.14 | 1.10$\pm$0.14
HE2319-0852 | –3.38 | 4.74 | –3.78$\pm$0.11 | –0.40$\pm$0.13 | 4.73 | 4.72 | –2.82$\pm$0.10 | –2.83$\pm$0.12 | 0.56$\pm$0.11 | 0.55$\pm$0.13 | 0.56$\pm$0.13
HE2325-0755 | –2.85 | 5.99 | –2.53$\pm$0.11 | 0.32$\pm$0.13 | 5.14 | $<$5.03 | –2.41$\pm$0.10 | $<$–2.52$\pm$0.12 | 0.44$\pm$0.11 | $<$0.33$\pm$0.13 | 0.44$\pm$0.11
HE2326+0038 | –2.77 | 6.01 | –2.51$\pm$0.13 | 0.26$\pm$0.15 | 5.21 | 5.23 | –2.34$\pm$0.12 | –2.32$\pm$0.14 | 0.43$\pm$0.13 | 0.45$\pm$0.15 | 0.44$\pm$0.15
HE2327-5642 | –2.95 | 5.94 | –2.58$\pm$0.17 | 0.37$\pm$0.18 | 4.96 | 4.67 | –2.59$\pm$0.17 | –2.88$\pm$0.18 | 0.36$\pm$0.18 | 0.24$\pm$0.19 | 0.30$\pm$0.19
HE2329-3702 | –2.16 | 6.64 | –1.88$\pm$0.12 | 0.28$\pm$0.14 | 5.76 | 5.74 | –1.79$\pm$0.11 | –1.81$\pm$0.13 | 0.37$\pm$0.12 | 0.35$\pm$0.14 | 0.36$\pm$0.14
HE2333-1358 | –3.34 | 5.64 | –2.88$\pm$0.16 | 0.46$\pm$0.17 | 4.66 | 4.49 | –2.89$\pm$0.15 | –3.06$\pm$0.17 | 0.45$\pm$0.16 | 0.28$\pm$0.18 | 0.36$\pm$0.18
HE2334-0604 | –3.41 | 4.14 | –4.38$\pm$0.17 | –0.97$\pm$0.18 | 4.09 | 4.07 | –3.46$\pm$0.17 | –3.48$\pm$0.18 | –0.05$\pm$0.18 | –0.07$\pm$0.19 | –0.06$\pm$0.19
HE2335-5958B | –2.33 | 6.52 | –2.00$\pm$0.11 | 0.33$\pm$0.13 | 5.46 | 5.52 | –2.09$\pm$0.10 | –2.03$\pm$0.12 | 0.24$\pm$0.11 | 0.30$\pm$0.13 | 0.27$\pm$0.13
HE2338-1311 | –2.86 | 6.11 | –2.41$\pm$0.11 | 0.45$\pm$0.13 | 5.24 | 5.13 | –2.31$\pm$0.10 | –2.42$\pm$0.12 | 0.55$\pm$0.11 | 0.44$\pm$0.13 | 0.50$\pm$0.13
HE2338-1618 | –2.65 | 6.31 | –2.21$\pm$0.10 | 0.44$\pm$0.12 | 5.41 | 5.25 | –2.14$\pm$0.09 | –2.30$\pm$0.11 | 0.51$\pm$0.10 | 0.35$\pm$0.13 | 0.43$\pm$0.12
HE2345-1919 | –2.46 | 6.40 | –2.12$\pm$0.10 | 0.34$\pm$0.12 | 5.58 | 5.60 | –1.97$\pm$0.09 | –1.95$\pm$0.11 | 0.49$\pm$0.10 | 0.51$\pm$0.13 | 0.50$\pm$0.12
HE2347-1254 | –1.83 | 7.02 | –1.50$\pm$0.14 | 0.33$\pm$0.16 | 6.07 | 6.11 | –1.48$\pm$0.13 | –1.44$\pm$0.15 | 0.35$\pm$0.14 | 0.39$\pm$0.16 | 0.37$\pm$0.16
HE2347-1334 | –2.55 | 5.20 | –3.32$\pm$0.13 | –0.77$\pm$0.15 | 5.36 | 5.26 | –2.19$\pm$0.12 | –2.29$\pm$0.14 | 0.36$\pm$0.13 | 0.26$\pm$0.15 | 0.31$\pm$0.15
HE2347-1448 | –2.31 | 6.84 | –1.68$\pm$0.11 | 0.63$\pm$0.13 | 5.79 | $<$5.74 | –1.76$\pm$0.10 | $<$–1.81$\pm$0.12 | 0.55$\pm$0.11 | $<$0.50$\pm$0.13 | 0.55$\pm$0.11
Table 4: continued.
|
arxiv-papers
| 2010-06-18T02:29:10 |
2024-09-04T02:49:11.013342
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L. Zhang, T. Karlsson, N. Christlieb, A. J. Korn, P. S. Barklem, and\n G. Zhao",
"submitter": "Lan Zhang",
"url": "https://arxiv.org/abs/1006.3594"
}
|
1006.3700
|
# Universal Enveloping Algebras of Braided m-Lie Algebras
Shouchuan Zhang, Jieqiong He
Department of Mathematics, Hunan University
Changsha 410082, P.R. China
###### Abstract
Universal enveloping algebras of braided m-Lie algebras and PBW theorem are
obtained by means of combinatorics on words.
2000 Mathematics Subject Classification: 16W30, 16G10
keywords: Braided Lie algebra, Universal enveloping algebras.
## 0 Introduction
The theory of Lie superalgebras has been developed systematically, which
includes the representation theory and classifications of simple Lie
superalgebras and their varieties [8] [3]. In many physical applications or in
pure mathematical interest, one has to consider not only ${\bf Z}_{2}$\- or
${\bf Z}$\- grading but also $G$-grading of Lie algebras, where $G$ is an
abelian group equipped with a skew symmetric bilinear form given by a
2-cocycle. Lie algebras in symmetric and more general categories were
discussed in [6] and [5]. A sophisticated multilinear version of the Lie
bracket was considered in [9] [15]. Various generalized Lie algebras have
already appeared under different names, e.g. Lie color algebras, $\epsilon$
Lie algebras [14], quantum and braided Lie algebras, generalized Lie algebras
[2] and $H$-Lie algebras [1].
In [12], Majid introduced braided Lie algebras from geometrical point of view,
which have attracted attention in mathematics and mathematical physics (see
e.g. [13] and references therein).
In paper [17], braided m-Lie algebras was introduced, which generalize Lie
algebras, Lie color algebras and quantum Lie algebras. Two classes of braided
m-Lie algebras are given, which are generalized matrix braided m-Lie algebras
and braided m-Lie subalgebras of $End_{F}M$, where $M$ is a Yetter-Drinfeld
module over $B$ with dim $B<\infty$ . In particular, generalized classical
braided m-Lie algebras $sl_{q,f}(GM_{G}(A),F)$ and $osp_{q,t}(GM_{G}(A),M,F)$
of generalized matrix algebra $GM_{G}(A)$ are constructed and their connection
with special generalized matrix Lie superalgebra $sl_{s,f}(GM_{{\bf
Z}_{2}}(A^{s}),F)$ and orthosymplectic generalized matrix Lie super algebra
$osp_{s,t}(GM_{{\bf Z}_{2}}(A^{s}),M^{s},F)$ are established. The relationship
between representations of braided m-Lie algebras and their associated
algebras are established. In this paper we follow paper [17] and obtain
universal enveloping algebras of braided m-Lie algebras and PBW theorem by
means of combinatorics on words (see [10]).
Throughout, $F$ is a field,
## 1 Braided m-Lie Algebras
We recalled two concepts.
###### Definition 1.1.
(See [17]) Let $(L,[\ \ ])$ be an object in the braided tensor category
$({\cal C},C)$ with morphism $[\ \ ]:L\otimes L\rightarrow L$. If there exists
an algebra $(A,m)$ in $({\cal C},C)$ and monomorphism $\phi:L\rightarrow A$
such that $\phi[\ \ ]=m(\phi\otimes\phi)-m(\phi\otimes\phi)C_{L,L},$ then
$(L,[\ \ ])$ is called a braided m-Lie algebra in $({\cal C},C)$ induced by
multiplication of $A$ through $\phi$. Algebra $(A,m)$ is called an algebra
associated to $(L,[\ \ ])$.
A Lie algebra is a braided m-Lie algebra in the category of ordinary vector
spaces, a Lie color algebra is a braided m-Lie algebra in symmetric braided
tensor category $({\cal M}^{FG},C^{r})$ since the canonical map
$\sigma:L\rightarrow U(L)$ is injective (see [14, Proposition 4.1]), a quantum
Lie algebra is a braided m-Lie algebra in the Yetter-Drinfeld category
$(^{B}_{B}{\cal YD},C)$ by [4, Definition 2.1 and Lemma 2.2]), and a “good”
braided Lie algebra is a braided m-Lie algebra in the Yetter-Drinfeld category
$(^{B}_{B}{\cal YD},C)$ by [4, Definition 3.6 and Lemma 3.7]). For a
cotriangular Hopf algebra $(H,r)$, the $(H,r)$-Lie algebra defined in [1, 4.1]
is a braided m-Lie algebra in the braided tensor category $({}^{H}{\cal
M},C^{r})$. Therefore, the braided m-Lie algebras generalize most known
generalized Lie algebras.
For an algebra $(A,m)$ in $({\cal C},C)$, obviously $L=A$ is a braided m-Lie
algebra under operation $[\ \ ]=m-mC_{L,L}$, which is induced by $A$ through
$id_{A}$. This braided m-Lie algebra is written as $A^{-}$.
###### Definition 1.2.
(see [19]) Let H be a Hopf algebra, $(V,\alpha)$ and $(V,\delta)$ be a left
$H$-module and a left $H$-comodule, respectively. If
$\displaystyle\delta(\alpha(h\otimes v))=\delta(h\cdot v)=\sum
h_{(1)}v_{(-1)}S(h_{(3)})\otimes h_{(2)}.v_{0}$ (1.1)
$\forall v\in V$, $h\in G$, then $(V,\alpha,\delta)$ is called a Yetter-
Drinfeld module over $H$, or a $H$\- YD module in short. All of $H$\- YD
module construct a braided tensor category, called the Yetter-Drinfeld module
category, denoted as $(^{H}_{H}{\cal YD},C)$, where $C$ is the braiding.
If $H=FG$ is a group algebra and $(V,\alpha,\delta)$ is an $FG$\- YD module,
then $V$ becomes a $G$-graded space $V=\oplus_{g\in G}V_{g}$ and the condition
(1.1) becomes
$\displaystyle\delta(\alpha(h\otimes v))=\delta(h\cdot v)=\sum hgh^{-1}\otimes
h\cdot v$ (1.2)
for any $h,g\in G,$ $v\in V_{g}.$
Let $G$ be a group and $\chi$ a bicharacter of $G$, i.e. $\chi$ is a map from
$G\times G$ to $F$ satisfying $\chi(ab,c)=\chi(a,c)\chi(b,c)$,
$\chi(a,bc)=\chi(a,b)\chi(a,c)$ and $\chi(a,e)=1=\chi(e,a)$ for any $a,b,c\in
G$, where $e$ is the unit element of $G$. The braiding of $FG$-YD module
$(V,\alpha,\delta)$ is determined by bicharacter $\chi$ if $h\cdot
x=\chi(h,g)x$ for any $h,g\in G,$ $x\in V_{g}.$ In this case, $C(x\otimes
y)=\chi(g,h)y\otimes x$ for any homogeneous elements $x\in V_{g},$ $y\in
V_{h}$. Obviously, if the braiding of $FG$-YD module $(V,\alpha,\delta)$ is
determined by bicharacter $\chi$, then the braiding of $(V,\alpha,\delta)$ is
diagonal. Conversely, if the braiding of a braided vector space $V$ is
diagonal, then $V$ can becomes an $F\mathbb{Z}[I]$-YD module, which braiding
is determined by a bicharacter (see [7]). If $G$ is a finite abelian group and
$V$ is a $kG$-YD module, then the braiding of $V$ is diagonal (see [18]). In
this paper we only consider the braiding determined by bicharacter $\chi.$
## 2 Jacobi identity
###### Lemma 2.1.
(See [9]) If $L$ is braided m-Lie algebra, then Jacobi identity holds:
$\displaystyle[[ab]c]-[a[bc]]+\chi(a,b)b[ac]-\chi(b,c)[ac]b=0$ (2.1)
for any homogeneous elements $a,b,c\in L$, where $\chi(a,b)$ denotes
$\chi(g,h)$ for $a\in V_{g}$, $b\in V_{h}.$
Proof.
the left side $\displaystyle=$ $\displaystyle
abc-\chi(a,b)bac-\chi(ab,c)cab+\chi(ab,c)\chi(a,b)cba$ $\displaystyle-
abc+\chi(b,c)acb+\chi(a,bc)bca-\chi(a,bc)\chi(b,c)cba$
$\displaystyle+\chi(a,b)bac-\chi(a,b)\chi(a,c)bca-\chi(b,c)acb+\chi(b,c)\chi(a,c)cab$
$\displaystyle=$ $\displaystyle 0.\Box$
## 3 Universal enveloping algebras of braided m-Lie algebras and PBW theorem
Let $E$ be a homogeneous basis of braided m-Lie algebra $L$ and $B$ a set. Let
$B^{*}$ denote the set of all words (see [10]) on $B$ and $\varphi$ a
bijective map from $E$ to $B$. Define $[bc]=\varphi([ef])$ for any
$b=\varphi(e)$, $c=\varphi(f)$, $e,f\in E$. Let $\prec$ be an order of $B$ and
$P=:\\{b_{1}b_{2}\cdots b_{n}\ |\ b_{i}\in B,b_{n}\prec
b_{n-1}\prec\cdots\prec b_{1},n\in\mathbb{N}\\}$.
For any $w\in B^{*}$, let $\nu(w)$ denote the number of elements in set
$\\{(r,s,t)\ |\ w=rasbt;a,b\in B,r,s,t\in B^{*},a\prec b,\\}$ and $\nu(w)$ is
called the index of $w$. Obviously, we have
$v(ubav)=v(uabv)-1$
for any $a,b\in B,u,v\in B^{*},a\prec b$. We also have that $\nu(w)=0$ if and
only if $w\in F$.
For any a set $X$, let $FX$ denote the vector space spanned by $X$ with basis
$X.$ It is clear that $FB^{*}$ is the free algebra on $B$. Meantime $FB^{*}$
also is the tensor algebra $T(FB)$ over $FB$.
###### Lemma 3.1.
There exists $\lambda:FB^{*}\rightarrow FP$ such that
(i) $\lambda(f)=f,$ $f\in P$;
(ii) $\lambda(ubcv)=\chi(b,c)\lambda(ucbv)+\lambda(u[bc]v),u,v\in
B^{\ast},b,c\in B$;
(iii) $\lambda(uv)=\lambda(\lambda(u)v)=\lambda(u\lambda(v)),u,v\in FB^{*}$.
Proof. For $w\in B^{*}$, we define $\lambda(w)$ using an induction first on
the length and second on the index. If $w\in B$, define $\lambda(w)=w$. Let
the length of $w$ be larger than 1 and define
$\lambda(w)=:\chi(b,c)\lambda(ucbv)+\lambda(u[bc]v)$ for $w=ubcv$ with $b,c\in
B$, $u,v\in B^{*}$. Now we show that the definition is well-defined. For
$w=ubcv=u^{\prime}b^{\prime}c^{\prime}v^{\prime}$ with
$b,c,b^{\prime},c^{\prime}\in B$, $u,v,u^{\prime},v^{\prime}\in B^{*}$, we
only need show that
$\displaystyle\chi(b,c)\lambda(ucbv)+\lambda(u[bc]v)=\chi(b^{\prime},c^{\prime})\lambda(u^{\prime}c^{\prime}b^{\prime}v^{\prime})+\lambda(u^{\prime}[b^{\prime}c^{\prime}]v^{\prime}).$
(3.1)
We show this by following two steps.
($1^{\circ}$) If $|u|\leq|u^{\prime}|-2$, then
$u^{\prime}=ubct,v=tb^{\prime}c^{\prime}v^{\prime},t\in B^{*}$. By induction
hypothesis we have
the left side $\displaystyle=$
$\displaystyle\chi(b,c)\chi(b^{\prime},c^{\prime})\lambda(ucbtc^{\prime}b^{\prime}v^{\prime})+\chi(b,c)\lambda(ucbt[b^{\prime}c^{\prime}]v^{\prime})$
$\displaystyle+\chi(b^{\prime},c^{\prime})\lambda(u[bc]tc^{\prime}b^{\prime}v^{\prime})+\lambda(u[bc]t[b^{\prime}c^{\prime}]v^{\prime})$
and
the right side $\displaystyle=$
$\displaystyle\chi(b,c)\chi(b^{\prime},c^{\prime})\lambda(ucbtc^{\prime}b^{\prime}v^{\prime})+\chi(b,c)\lambda(ucbt[b^{\prime}c^{\prime}]v^{\prime})$
$\displaystyle+\chi(b^{\prime},c^{\prime})\lambda(u[bc]tc^{\prime}b^{\prime}v^{\prime})+\lambda(u[bc]t[b^{\prime}c^{\prime}]v^{\prime}).$
Thus (3.1) holds.
($2^{\circ}$) If $|u|=|u^{\prime}|-1$ then
$u^{\prime}=ub,c=b^{\prime},v=c^{\prime}v^{\prime}$. We only need show
$\chi(a,b)\lambda(rbacs)+\lambda(r[ab]cs)=\chi(b,c)\lambda(racbs)+\lambda(ra[bc]s)$.
By induction hypothesis we have
the left side $\displaystyle=$
$\displaystyle\chi(a,b)\\{\chi(a,c)\lambda(rbcas)+\lambda(rb[ac]s)\\}+\lambda(r[ab]cs)$
$\displaystyle=$
$\displaystyle\chi(a,b)\chi(a,c)\lambda(rbcas)+\chi(a,b)\lambda(rb[ac]s)+\lambda(r[ab]cs)$
$\displaystyle=$
$\displaystyle\chi(a,b)\chi(a,c)\\{chi(b,c)\lambda(rcbas)+\lambda(r[bc]as)\\}+\chi(a,b)\lambda(rb[ac]s)+\lambda(r[ab]cs)$
$\displaystyle=$
$\displaystyle\chi(a,b)\chi(a,c)chi(b,c)\lambda(rcbas)+\chi(a,b)\chi(a,c)\lambda(r[bc]as)+\chi(a,b)\lambda(rb[ac]s)$
$\displaystyle+\lambda(r[ab]cs)$
and
the right side $\displaystyle=$
$\displaystyle\chi(b,c)\\{\chi(a,c)\lambda(rcabs)+\lambda(r[ac]bs)\\}+\lambda(ra[bc]s)$
$\displaystyle=$
$\displaystyle\chi(b,c)\chi(a,c)\lambda(rcabs)+\chi(b,c)\lambda(r[ac]bs)+\lambda(ra[bc]s)$
$\displaystyle=$
$\displaystyle\chi(b,c)\chi(a,c)\\{\chi(a,b)\lambda(rcbas)+\lambda(rc[ab]s)\\}+\chi(b,c)\lambda(r[ac]bs)+\lambda(ra[bc]s)$
$\displaystyle=$
$\displaystyle\chi(b,c)\chi(a,c)\chi(a,b)\lambda(rcbas)+\chi(b,c)\chi(a,c)\lambda(rc[ab]s)+\chi(b,c)\lambda(r[ac]bs)$
$\displaystyle+\lambda(ra[bc]s).$
Thus
$\displaystyle\mbox{the left side }-\mbox{the right side }$ $\displaystyle=$
$\displaystyle\\{\chi(a,b)\chi(a,c)\lambda(r[bc]as)-\lambda(ra[bc]s)\\}+\\{\chi(a,b)\lambda(rb[ac]s)-\chi(b,c)\lambda(r[ac]bs)\\}$
$\displaystyle+\\{\lambda(r[ab]cs)-\chi(b,c)\chi(a,c)\lambda(rc[ab]s)\\}$
$\displaystyle=$
$\displaystyle-\lambda(r[a[bc]]s)+\lambda(\chi(a,b)rb[ac]s-\chi(b,c)r[ac]bs)+\lambda(r[[ab]c]s)$
$\displaystyle=$ $\displaystyle 0\ \ \ \ {\mbox{(by Jacobi identity)}}.$
For (iii), we use an induction first on the length and second on the index.
Assume $|w_{1}|\neq|w|$ and $w=w_{1}w_{2}$. If $w_{1}=ubct$, $b,c\in B$,
$u,t\in B^{*}$, then
$\displaystyle\lambda(w)$
$\displaystyle=\chi(b,c)\lambda(ucbtw_{2})+\lambda(u[bc]tw_{2})$
$\displaystyle=\chi(b,c)\lambda(\lambda(ucbt)w_{2})+\lambda(\lambda(u[bc]t)w_{2})$
$\displaystyle=\lambda(\lambda(w_{1})w_{2})$
If $w_{1}=b$, $w_{2}=cv$, $b,c\in B$, $v\in B^{*}$, then
$\lambda(w)=\lambda(\lambda(w_{1})w_{2}).$ $\Box$
###### Definition 3.2.
Suppose that ${L}$ is a braided m-Lie algebra in $({\cal C},C)$ and $U$ is a
algebra with Lie algebra homomorphism $i:L\rightarrow U^{-}$. $(U,i)$ is
called the universal algebra of braided m -Lie algebra $L$, if the following
condition holds: If for any an algebra $W$ in $({\cal C},C)$ with a Lie
algebra homomorphism $\psi:L\rightarrow W^{-}$ in $({\cal C},C)$, there exists
the unique algebra homomorphism $\bar{\psi}:U\rightarrow W$ in $({\cal C},C)$
such that the following is commutative:
$\begin{array}[]{lcccr}{}&\varphi&{}\hfil\\\ L&\longrightarrow&U\\\
&\psi\searrow&\downarrow\bar{\psi}\\\ &&W&.\end{array}.$
Obviously, $\varphi$ in section above is a Lie algebra monomorphism from $L$
to $FP$ in ${}^{FG}_{FG}{\mathcal{Y}D}$.
Let $U(L)=:FP$. Define the multiplication of $U(L)$ as follows:
$u*v=\lambda(uv)$ for any $u,v\in P.$ By Lemma 3.1 (iii), $U(L)$ is an
associative algebra: $u\ast(v\ast
w)=\lambda(u\lambda(vw))=\lambda(uvw)=\lambda(\lambda(uv)w)=(u\ast v)\ast w$
for any $u,v,w\in P.$ Obviously, $\lambda$ is an algebra homomorphism.
###### Lemma 3.3.
If $(V,\alpha,\delta)$ is an $FG$-YD module, then tensor algebra $T(V)$ over
$V$ is an $FG$-YD module.
Proof. By the universal property of tensor algebra, we can construct the
module operation $\alpha^{(T(V))}$ and comodule operation $\delta^{(T(V))}$of
$T(V)$ as follows:
i)
$\begin{array}[]{lcccr}{}&\delta^{(T(V))}&{}\hfil\\\
T(V)&\longrightarrow&FG\otimes T(V)\\\ i\uparrow&\nearrow({\rm id}\otimes
i)\delta^{(V)}&\uparrow{\rm id}\otimes i\\\ V&\longrightarrow&FG\otimes V\\\
&\delta^{(V)}&\ \ \ \ \ \ \ \ \ .\end{array}$
ii)
$\begin{array}[]{lcccr}{}&\alpha_{g}^{(V)}&{}\hfil\\\ V&\longrightarrow&V\\\
i\downarrow&\searrow i\alpha_{g}^{(V)}&i\downarrow\\\
T(V)&\longrightarrow&T(V)\\\ &\alpha_{g}^{(T(V))}&\ \ \ \ \ \ \ \ \
,\end{array}$
where $\alpha^{(V)}_{g}(v)=:\alpha(g\otimes v)=g\cdot v$ for any $v\in V,$
$g\in G.$
iii) For $\forall g\in G,$ $x_{j}\in V_{g_{j}}$, $1\leq j\leq r$, See that
$\displaystyle\delta(g\cdot(x_{1}\cdot\cdot\cdot x_{r}))$ $\displaystyle=$
$\displaystyle\delta(\alpha_{g}(x_{1}\cdot\cdot\cdot x_{r}))$ $\displaystyle=$
$\displaystyle\delta((g\cdot x_{1})\cdot\cdot\cdot(g\cdot
x_{r}))=\delta(g\cdot x_{1})\cdot\cdot\cdot\delta(g\cdot x_{r})$
$\displaystyle=$ $\displaystyle(gg_{1}g^{-1}\otimes(g\cdot
x_{1}))\cdot\cdot\cdot(gg_{r}g^{-1}\otimes(g\cdot x_{r}))$ $\displaystyle=$
$\displaystyle(gg_{1}g^{-1})\cdot\cdot\cdot(gg_{r}g^{-1})\otimes
x_{1}\cdot\cdot\cdot x_{r}$ $\displaystyle=$ $\displaystyle
g(g_{1}\cdot\cdot\cdot g_{r})g^{-1}\otimes x_{1}\cdot\cdot\cdot x_{r}.$
Thus $(T(V),\alpha,\delta)$ is an $FG$-YD module. Furthermore, considering (i)
and (ii), we have that $T(V)$ is an algebra in ${}^{FG}_{FG}{\mathcal{Y}D}$.
$\Box$
###### Lemma 3.4.
(i) $FB^{*}$ is an $FG$-YD module.
(ii) $FP$ is an $FG$-YD sub-module of $FB^{*}.$
(iii) $FP$ is an algebra in ${}^{FG}_{FG}{\cal YD}$.
Proof. (i) It follows from Lemma 3.3.
(ii) and (iii) are clear. $\Box$
###### Theorem 3.5.
(PBW). $(U(L),\varphi$) is the universal enveloping algebra of braided m-Lie
algebra $L$.
Proof. For any an algebra $W$ in ${}^{FG}_{FG}{\mathcal{Y}D}$ with a Lie
algebra homomorphism $\psi:L\rightarrow W^{-}$ in
${}^{FG}_{FG}{\mathcal{Y}D}$, define $\bar{\psi}:FB^{*}\rightarrow FP$ such
that $\bar{\psi}\varphi=\psi$ and $\theta=:\bar{\psi}\mid_{FP}$, the
restriction of $\bar{\psi}$ on $FP.$ It is clear that the following is
commutative.
$\begin{array}[]{lcccr}{}&\varphi&{}\hfil&\lambda&{}\hfil\\\
L&\longrightarrow&FB^{*}&\longrightarrow&FP\\\
&\psi\searrow&\bar{\psi}\downarrow&\swarrow\theta\\\ &&W&\ \ \ \ \ \
.\end{array}$
Now we show that $\theta$ is an algebra homomorphism, i.e.
$\displaystyle\theta(r\ast s)$ $\displaystyle=$
$\displaystyle\theta(r)\theta(s)$
for any $r,s\in B^{*}$. We show this using induction by following several
steps.
$(1^{\circ})$ If $rs\in P,$ then $\theta(r\ast
s)=\theta(\lambda(rs))=\theta(rs)=\theta(r)\theta(s)$.
$(2^{\circ})$ $r,s\in B$ and $r\prec s$. See that
$\displaystyle\theta(r\ast s)$ $\displaystyle=$
$\displaystyle\theta(\lambda(rs))$ $\displaystyle=$
$\displaystyle\theta(\lambda(sr\chi(r,s)+[rs]))$ $\displaystyle=$
$\displaystyle\theta(\lambda(sr))\chi(r,s)+\theta(\lambda([rs]))$
$\displaystyle=$ $\displaystyle\theta(\lambda(sr))\chi(r,s)+\theta([rs])\ \ \
(\mbox{ since the length of }[rs]<2\mbox{ and }\nu(sr)<\nu(rs))$
$\displaystyle=$ $\displaystyle\theta(\lambda(sr))\chi(r,s)+\theta([rs])$
$\displaystyle=$ $\displaystyle\theta(sr)\chi(r,s)+\theta([rs])\ \ \ \ (\mbox{
since }sr\in P)$ $\displaystyle=$
$\displaystyle\theta(rs)=\theta(r)\theta(s).$
$(3^{\circ})$ If $r=ub,$ $s=cv,u,v\in B^{\ast}$, $b,c\in B,b\prec c$,
$uv\not=1$, then
$\displaystyle\theta(r\ast s)$
$\displaystyle=\theta(\lambda(rs))=\chi(b,c)\theta(\lambda(ucbv))+\theta(\lambda(u[bc]v))$
$\displaystyle=\chi(b,c)\theta((uc)*(bv))+\theta((u[bc])*v)$
$\displaystyle=\chi(b,c)\theta(uc)\theta(bv)+\theta(u[bc])\theta(v)\ \ \
{\mbox{(by induction hypothesis)}}$
$\displaystyle=\chi(b,c)\theta(u)\theta(cb)\theta(v)+\theta(u)\theta([bc])\theta(v)$
$\displaystyle=\theta(u)\theta(bc)\theta(v)$
$\displaystyle=\theta(u)\theta(b)\theta(c)\theta(v)$
$\displaystyle=\theta(r)\theta(s).\ \ \Box$
## References
* [1] Y. Bahturin, D. Fischman and S. Montgomery, Bicharacter, twistings and Scheunert’s theorem for Hopf algebra, J. Alg. 236 (2001), 246-276.
* [2] Y. Bahturin, D. Fischman and S. Montgomery. On the generalized Lie structure of associative algebras. Israel J. of Math., 96(1996) , 27–48.
* [3] Y. Bahturin, D. Mikhalev, M. Zaicev and V. Petrogradsky, Infinite dimensional Lie superalgebras, Walter de Gruyter Publ. Berlin, New York, 1992.
* [4] X. Gomez and S. Majid, Braided Lie algebras and bicovariant differential calculi over coquasitriangular Hopf algebras, J. Alg. 261(2003), 334–388.
* [5] D. Gurevich, A. Radul and V. Rubtsov, Noncommutative differential geometry related to the Yang-Baxter equation, Zap. Nauchn. Sem. S.-Peterburg Otdel. Mat. Inst. Steklov. (POMI) 199 (1992); translation in J. Math. Sci. 77 (1995), 3051–3062.
* [6] D. I. Gurevich, The Yang-Baxter equation and the generalization of formal Lie theory, Dokl. Akad. Nauk SSSR, 288 (1986), 797–801.
* [7] I. Heckenberger, Classification of arithmetic root systems, preprint, arXiv:math.QA/0605795.
* [8] V. G. Kac. Lie superalgebras. Adv. in Math., 26(1977) , 8–96.
* [9] V. K. Kharchenko, An existence condition for multilinear quantum operations, J. Alg. 217 (1999), 188–228.
* [10] M. Lothaire, Combinatorics on words. London:Cambridge University Press, 1983.
* [11] S. Majid, Free braided differential calculus, braided binomial theorem, and the braided exponential map. J. Math. Phys., 34, 1993, 4843–4856.
* [12] S. Majid, Quantum and braided Lie algebras, J. Geom. Phys. 13 (1994), 307–356.
* [13] S. Majid, Foundations of Quantum Group Theory, Cambradge University Press, 1995.
* [14] M. Scheunert. Generalized Lie algebras. J. Math. Phys., 20 (1979), 712–720.
* [15] B. Pareigis, On Lie algebras in the category of Yetter-Drinfeld modules. Appl. Categ. Structures, 6 (1998), 151–175.
* [16] S. L. Woronowicz, Differential calculus on compact matrix pseudogroups(quantum groups). Commun. Math. Phys, 122(1989)1, 125-170.
* [17] S. C. Zhang, Y. Z. Zhang, Braided m-Lie algebras. Letters in Mathematical Physics, 70 (2004), 155-167. Also in math.RA/0308095.
* [18] Shouchuan Zhang, Y-Z Zhang, H.X. Chen, Classification of PM Quiver Hopf Algebras, Journal of Algebra and Its Applications, 6(2007)4, 1-32. Also in arXiv, math.QA/0410150.
* [19] Shouchuan Zhang, Braided Hopf Algebras, Hunan Normal University Press, 1999. Also in math.RA/0511251.
|
arxiv-papers
| 2010-06-18T14:24:01 |
2024-09-04T02:49:11.039687
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lingwei Guo, Shouchuan Zhang, Jieqiong He",
"submitter": "Shouchuan Zhang",
"url": "https://arxiv.org/abs/1006.3700"
}
|
1006.3923
|
###### Abstract
In this review we establish various connections between complex networks and
symmetry. While special types of symmetries (e.g., automorphisms) are studied
in detail within discrete mathematics for particular classes of deterministic
graphs, the analysis of more general symmetries in real complex networks is
far less developed. We argue that real networks, as any entity characterized
by imperfections or errors, necessarily require a stochastic notion of
invariance. We therefore propose a definition of stochastic symmetry based on
graph ensembles and use it to review the main results of network theory from
an unusual perspective. The results discussed here and in a companion paper
show that stochastic symmetry highlights the most informative topological
properties of real networks, even in noisy situations unaccessible to exact
techniques.
###### keywords:
complexity; networks; symmetry
10.3390/—— xx Received: xx / Accepted: xx / Published: xx Complex Networks and
Symmetry I: A Review Diego Garlaschelli 1, Franco Ruzzenenti 2,⋆ and Riccardo
Basosi 3 E-Mail: ruzzenenti@unisi.it; Tel.: +39-0577-234240; Fax:
+39-0577-234239.
## 1 Introduction
In this review and in a companion paper symmetry2 , we study several
connections between symmetry and network theory. Most complex systems
encountered in a diverse range of domains, from biology through sociology to
technology, consist of networks of elements (_vertices_) connected together
(by _links_ , or _edges_) in an intricate way guidosbook ; largescalestructure
; dynamicalprocessesoncomplexnetworks ; ecologicalnetworks ; internet ;
networksincellbiology ; adaptivenetworks . While graph theory started dealing
with the mathematical description of network properties long ago harary , only
recently massive datasets about large real-world complex networks have become
available. This allowed an unprecedented activity of data analysis, which
resulted in the establishment of some key ‘stylized facts’ about the structure
of real networks, and motivated an intense theoretical activity aimed at
explaining them.
Surprisingly (at least at the time when this was first observed), the
empirically observed structure of real networks is strikingly different from
what is obtained assuming simple homogeneous mechanisms of network formation,
such as the traditional Erdős-Rényi random graph model guidosbook ;
largescalestructure . In the latter, which will be an important reference
throughout this review, every pair of vertices has the same probability $p$ to
be connected. This generates homogeneous topological features, such as a
constant link density across the network, and a narrow (binomial) distribution
of the degree $k$ (number of edges reaching a vertex). By contrast, virtually
any real network is found to display a modular structure, with vertices
organized in communities tightly connected internally and loosely connected to
each other, and a broad degree distribution, typically featuring a power-law
tail of the form $P(k)\propto k^{-\gamma}$. Networks characterized by the
latter property are called _scale-free_.
Besides the purely topological level, networks are also characterized by
heterogeneous link weights. That is, the intensity of the connections is again
broadly distributed, and non-trivially correlated to the topology. Capturing
the richness of the information encoded in the weighted structure of networks
is a hard task, and the definition of proper weighted structural properties an
open problem vespy_weighted ; myensemble ; kertesz_clustering . At both the
topological and the weighted level, many real networks are also characterized
by an intrinsic directionality of their connections, which again implies that
proper quantities must be introduced and measured in order to fully understand
directed structural patterns myreciprocity ; mymultispecies ;
giorgioclustering .
As an additional level of complexity, dynamical processes generally take place
on networks dynamicalprocessesoncomplexnetworks . Remarkably, the
heterogeneous structure of real networks has been found to determine major
deviations from the behavior expected in the homogeneous case, which is the
traditional assumption used to obtain predictions about the dynamics. As a
consequence, most of these predictions have been shown to be incorrect when
applied to real-world networks. A prototypic example of this discrepancy is
found in models of epidemic disease spreading. When these models are defined
on regular graphs, one finds that the transmission rate must overcome a finite
_epidemic threshold_ in order to guarantee the persistence of an infection. By
contrast, on scale-free networks the value of the epidemic threshold vanishes,
implying that a large class of diseases can escape extinction no matter their
transmission rates, even if extremely low dynamicalprocessesoncomplexnetworks
.
Finally, in some networks a feedback is present between the topology and the
dynamics taking place on it. This is the case of adaptive networks, whose
structure changes in response to their dynamical behavior, which is in turn
affected by the structure itself adaptivenetworks . Generally, adaptive
networks cannot be properly understood by studying their topology and their
dynamics separately, as simple models show myselforganized .
It may appear that, due to the various levels of complexity encountered in the
description of real networks, performing a symmetry analysis of these highly
heterogeneous systems is likely to lead to a dead end. This is probably the
reason why, although network theory developed very rapidly in recent years
guidosbook ; largescalestructure ; dynamicalprocessesoncomplexnetworks and
established tight connections with many other disciplines ecologicalnetworks ;
internet ; networksincellbiology ; adaptivenetworks , its many relations to
symmetry concepts have not been made explicit yet, apart from isolated
examples symmetry ; quotient ; redundancy ; symmetry_wtw . On the other hand,
one expects the formation of real networks to be guided by some organising
principle, maybe non-obvious but surely not completely random, and possibly
the result of evolutionary or optimisation mechanisms. This implies that
network structure should encode some degree of order and symmetry, even if
more general and challenging than the type found in geometrical objects. It is
therefore important to introduce proper definitions of symmetries capturing
the possible forms of organisation of real networks, and enabling a simplified
understanding of the latter.
In this review we explore the connections between real networks and symmetry
in more detail. We show that many of the approaches that have been proposed to
characterize both real and model-generated networks can be rephrased more
firmly in terms of symmetry concepts. To this end, in Section 2 we first
clarify the peculiar notions of symmetry pertinent to real networks, which
(unlike formal graphs studied in discrete mathematics) are always
characterized by errors or imperfections. Then, in Section 3 we shall
establish several connections between network theory and symmetry. Symmetry
will be investigated over a wide range of invariances related to topological
variables. The empirical result that in real networks some topological
properties tend to distribute in structurally different ways from random
networks, thus emphasizing a complex structure, will be rephrased in terms of
symmetry concepts. Interestingly, Section 3 can be regarded as a brief review
of network theory from the unusual perspective of the symmetry properties of
real networks. Finally, in Section 4 we summarize our survey of network
symmetries. In the companion paper symmetry2 , we exploit the concepts
developed here to study stochastic symmetry in great detail in a particular
case, and to address the problem of symmetry breaking in networks.
## 2 Types of Graph Symmetries
Before proceeding with a review of the empirical symmetries of real networks,
we first distinguish between different notions of invariance we will be
interested in. The mathematical definition of symmetry of an object is the set
of transformations that leave the properties of the object unchanged. For
instance, a straight line of infinite length is unchanged after displacing it
along its own direction, and a circle is unchanged after rotating it around
its center. Conversely, the transformations involved in the symmetries of an
object can be exploited to define and construct the object itself: a straight
line can be drawn by displacing a point along a chosen direction, and a circle
can be drawn by rotating a point around a chosen center. In this case, one
needs a _unit_ (in both examples, a point) to iterate through the
transformation. More complicated units lead to more complicated objects (for
instance, rotating a whole disk rather than a single point leads to a torus).
### 2.1 Discreteness, Permutations, and More General Symmetries
In the case of graph symmetries, various considerations are in order. Firstly,
a graph is a discrete object, and therefore the relevant transformations are
discrete and not continuous as in the previous examples. An example of
discrete transformation is the rotation of a square by an angle of $\pi/2$
radians (or a multiple) around its center: the square is symmetric under this
discrete rotation but not under one with different angle, or under a
continuous one. Similarly, a lattice (see Figure 1) is symmetric under a
discrete displacement by a multiple of the lattice spacing (which maps all
vertices to their nearest neighbours in a specified direction), but not under
one with different length, or under a continuous one. We will discuss the
discrete translational symmetry of networks in Section 3.1.
Figure 1: A two-dimensional lattice (in principle of infinite size)
constructed by assigning equally spaced planar coordinates to vertices, and
connecting each vertex to its nearest neighbours. (a) The lattice is
visualized by drawing vertices according to their coordinates in the embedding
space. (b) The same graph is drawn by arbitrarily positioning vertices,
irrespective of their coordinates in the embedding space. Mathematically, the
two graphs are indistinguishable and are therefore characterized by the same
automorphisms (permutations of vertices leading to the same topology). The
knowledge of the vertices’ positions [evident in (a)] directly indicates which
are the permutations corresponding to the symmetries of the graph: only those
that map all vertices to their nearest neighbours in a specified direction.
Even if it is natural to regard such transformations as translations or
displacements (with respect to the embedding space), topologically they are
mere permutations of vertices.
Secondly, graphs are topological objects, not geometrical entities: their
properties are independent on the positions of vertices in some metric space,
even if the graph itself may be the result of some position-dependent
construction rule. Changing the positions (and sizes) of vertices, as well as
changing the lengths (and widths) of links, only leads to a different
visualisation of the same graph (see Figure 1), and has no effect on the
topology of the latter (provided each link remains attached to the original
vertices). Therefore the properties in terms of which one can check the
symmetries of a graph are purely topological, and the set of transformations
involved in such symmetries are purely relational. Whereas a geometric
transformation (such as a translation or rotation) maps each point in a circle
to a different point determined by its coordinates in the plane, a topological
transformation maps each vertex in a graph to a vertex determined not by its
coordinates but simply by its identity (i.e. its label): this transformation
is merely a permutation of vertices. Permutations of vertices leading to the
same topology are the _automorphisms_ of a graph, and we will discuss them in
Section 3.3. Nonetheless, if vertices are assigned coordinates in some
embedding space, and the network construction depends on those coordinates (as
for lattices), then the automorphisms are permutations induced by proper
coordinate transformations (e.g. a translation). This is illustrated in Figure
1. Similar considerations apply if the graph construction depends on other
properties, rather than positions, assigned to vertices (we will consider this
case explicitly in Section 3.5). However, in many real-world cases one only
knows the topology of the graph, and not the properties of vertices. In
general, one does not even know whether vertices are actually assigned
properties on which the structure of the network depends. In this case, all
the automorphisms of the graph must be looked for by enumeration. The
possibility that the complexity of real networks might be traced back to some
simpler description involving _hidden_ variables attached to vertices, whose
transformations may induce symmetries that are not evident _a priori_ , is an
important aspect of network research, that we will discuss in Section 3.5.
Therefore the general problem of graph automorphisms (Section 3.3) can take
different forms depending on the nature of the (possible) properties inducing
the symmetries of a particular network, as the two examples of translational
symmetry (Section 3.1) and permutation of vertex properties (Section 3.5)
show.
Thirdly, graphs may (or may not) exhibit symmetries under transformations that
are not necessarily vertex permutations. An example is _scale invariance_
(Section 3.2), which also applies to self-similar geometric objects, or
_fractals_. In this case, the transformation is a change of scale in the
description of the system. We will also encounter transformations that
drastically change the topology of a graph, and only preserve some specified
property such as the total number of links or the degrees of all vertices
(Section 3.6). Finally, the transformations we will consider in Sections 3.7
and 3.8 are vertex partitions and edge (rather than vertex) permutations
respectively.
### 2.2 Stochastic Symmetry
As a final important remark we note that, when considering real networks
rather than abstract graphs, one must take into account that the observed
symmetry is in general only approximate. To illustrate this concept, let us
consider the example of a real object of circular shape. While a perfect
circle, as a mathematical entity, displays an exact rotational symmetry around
its center, a real circle is unavoidably an approximate object, characterized
by small imperfections. If we look for exact rotational symmetry in real
circles, we have to conclude that no real object is circular, as perfect
circles do not exist in reality. The paradox can only be solved by introducing
an approximate notion of rotational symmetry, i.e. one where we allow rotated
points to fall _nearby_ existing points of the circle. Ultimately, this
changes the picture substantially, since while a perfect circle can only be
drawn by a perfectly rotating point (i.e. there is a unique trajectory
defining the circle), an imperfect circle can be drawn following (infinitely)
many trajectories. While there is a single perfect circle of given radius,
there are infinite imperfect circles of given radius (and the definitions of
circle and radius themselves also acquire an approximate meaning).
Thus, while we started investigating the symmetry of a single object, we
naturally end up with a _family_ of objects (containing the original one), all
different from each other but nonetheless characterized by the same
approximate symmetry. Remarkably, to define the symmetry of the single object,
we need the entire family of its variants: while a rotation maps a perfect
circle to itself, it maps an imperfect circle to a different imperfect circle.
In particular, if we assume that a probability is associated with each
approximate object (for instance, if we draw a circle by adding a small noise
term in the radial direction), we end up with what is known as a _statistical
ensemble_ of objects. Objects ‘closer’ to the perfectly symmetric one are
assigned larger probability, and objects deviating from the perfectly
symmetric one by the same amount are equiprobable. A given real object is
symmetric under the transformation considered if it is a _typical_ (i.e. not
unlikely) member of the ensemble defined by the transformation itself. This
also means that, in order to detect deviations from symmetry in real objects,
one needs an ensemble of imperfectly symmetric objects as a reference or _null
model_. For instance, suppose one is investigating the properties of a real
circle and a real square under rotational symmetry. If a perfect circle is
assumed as the reference for a rotationally symmetric object, than both the
real square and the real circle will be classified as non-symmetric. By
contrast, if an ensemble of imperfect circles is considered as the null model,
then the real square will still be classified as non-symmetric (since it is a
very unlikely outcome of a circular null model) but the real circle will now
be correctly classified as symmetric.
When applied to networks, the above considerations naturally lead to the
notion of _statistical ensembles of graphs_ , i.e. families of networks where
each graph $G$ is assigned a probability $P(G)$. We will encounter graph
ensembles when considering either approximate equivalences or null models of
real networks. As in the example above, a graph will be classified as _exactly
symmetric_ under a given transformation if it is mapped onto itself by the
transformation (graph automorphism are an example of exact vertex permutation
symmetry). By contrast, a graph will be classified as _stochastically
symmetric_ under a given transformation if it is a typical member of (i.e.
well reproduced by) a graph ensemble which is stochastically symmetric under
the same transformation. In the last definition, we consider a graph ensemble
as stochastically symmetric under a transformation if the latter maps a graph
$G_{1}$ in the ensemble into an equiprobable graph $G_{2}$ with
$P(G_{2})=P(G_{1})$. Graph ensembles as null models of real networks will be
introduced and discussed in Section 3.6, where we will also illustrate in more
detail the idea of stochastic symmetry. We will also show that stochastic
symmetry and entropy are intimately related in graph ensembles.
## 3 Symmetries in Real Networks
Thus there are various possible notions of symmetry one can look for in
networks. In what follows, rather than discussing them in the order presented
above, we follow a more pedagogical ordering, which allows us to trace the
main results of network theory from the unusual perspective of symmetry. As we
will try to elucidate, some symmetries are generally present in real networks,
others are generally absent, and others are strongly network-dependent and
variably observed. In some cases, even when a symmetry is present, it only
holds within a limited range. All these situations are equally important, as
they suggest what is relevant and what is not to plausible formation
mechanisms involving a particular network. Our discussion provides a somewhat
unconventional overview of this problem, and list a few examples (among
possibly many more) of symmetries relevant to networks. Readers interested in
a more comprehensive account of the results of network theory are referred to
the relevant literature guidosbook ; largescalestructure ;
dynamicalprocessesoncomplexnetworks ; adaptivenetworks and to the
publications cited in the following text.
### 3.1 Translational Symmetry
As we mentioned, some graphs may be embedded in a metric space where vertices
are assigned positions. In this case, the symmetries (automorphisms) of a
graph are induced by the transformations of coordinates in the embedding
space, even if topologically their are simply permutations of vertices. This
means that the topological properties of the graph, which are independent of
the embedding space, will nonetheless reflect the properties of the latter.
For instance, lattices are naturally formed by connecting vertices to their
nearest neighbours in some embedding space (see Figure 1). A simple type of
discrete symmetry encountered in (either infinite or periodic) regular
lattices is translational symmetry. That is, the fact that the topology of a
lattice embedded in some $D$-dimensional space does not change after a
displacement by an integer multiple of the lattice spacing. Lattices are a
particular type of _regular graphs_ , i.e. graphs where every vertex has the
same number of neighbours. In Figure 2 we show three examples of regular
graphs embedded in different dimensions ($D=1$, $D=2$ and $D=\infty$) and with
differently ranged connections (nearest neighbours, nearest and second-nearest
neighbours, infinite neighbours).
Figure 2: Examples of regular graphs. (a) A periodic one-dimensional ($D=1$)
lattice (i.e. a ring) where each vertex is connected to its nearest and
second-nearest neighbours. (b) A two-dimensional ($D=2$) lattice (in principle
of infinite size) where each vertex is connected only to its nearest
neighbours. (c) A complete graph where every vertex is connected to all other
vertices. A complete graph can be regarded either as a lattice embedded in
some space of finite dimension $D<\infty$ (as in the two previous examples)
with infinite-ranged connections, or as a lattice embedded in infinite
dimension $D=\infty$ with finite-ranged (as in the two previous examples)
connections.
If the labeling of vertices reflects their position in space, then
translational symmetry is reflected in some regularities of the adjacency
matrix $A$ of the network (for undirected graphs, where no orientation is
defined on the edges, the adjacency matrix $A$ is a binary matrix whose
entries equal $a_{ij}=1$ if a link between vertex $i$ and vertex $j$ is
present, and $a_{ij}=0$ otherwise; here $i=1,\dots,N$ where $N$ is the total
number of vertices, i.e. the size of the network). For instance, if the
vertices are numbered cyclically along the ring, the adjacency matrices
$A_{a}$ and $A_{c}$ of the graphs shown in Figure 2a and c read
$A_{a}=\left(\begin{array}[]{cccccccc}0&1&1&0&0&0&1&1\\\ 1&0&1&1&0&0&0&1\\\
1&1&0&1&1&0&0&0\\\ 0&1&1&0&1&1&0&1\\\ 0&0&1&1&0&1&1&0\\\ 0&0&0&1&1&0&1&1\\\
1&0&0&0&1&1&0&1\\\ 1&1&0&1&0&1&1&0\end{array}\right)\qquad
A_{c}=\left(\begin{array}[]{cccccccc}0&1&1&1&1&1&1&1\\\ 1&0&1&1&1&1&1&1\\\
1&1&0&1&1&1&1&1\\\ 1&1&1&0&1&1&1&1\\\ 1&1&1&1&0&1&1&1\\\ 1&1&1&1&1&0&1&1\\\
1&1&1&1&1&1&0&1\\\ 1&1&1&1&1&1&1&0\end{array}\right)\vspace{0.3cm}$ (1)
respectively. Translational symmetry is one of the traditional assumptions
used in the theoretical study of discrete (or discretized) dynamical systems,
and most of the available analytical results about dynamical processes are
only valid under the assumption of the existence of this symmetry.
However, as one moves beyond the simple case of atoms regularly embedded in
crystal lattices, virtually all real-world networks strongly violate
translational symmetry. An important deviation from lattice-like topology in
real networks is signaled by a surprisingly small value of the average _inter-
vertex distance_ , i.e. the average number of links one needs to traverse
along the shortest path connecting two vertices. In most real networks, this
quantity increases at most logarithmically with the number $N$ of vertices, a
phenomenon known as the _small-world_ effect guidosbook . This behavior is
also encountered in the random graph model mentioned in Section 1 but not in
lattices, where the average distance (if infinite-ranged connections are not
allowed, e.g. for the graphs in Figure 2a and b but not for that in Figure 2c)
grows as $N^{1/D}$, thus much faster. The breakdown of translational symmetry
implies that the wealth of knowledge accumulated in the literature about the
outcome of dynamical processes on lattices cannot be applied to the same
processes when they take place on real networks
dynamicalprocessesoncomplexnetworks . We already mentioned epidemic spreading
processes as an example of the surprising deviation between dynamics on
lattices and on more complicated networks. Nonetheless, real networks bear an
interesting similarity with regular graphs, namely a large average value of
the _clustering coefficient_ , defined as the number of triangles (loops of
length three) starting at a vertex, divided by its maximum possible value.
The simultaneous presence of a small average distance and of a large
clustering coefficient (which is sometimes taken as a stronger definition of
the _small-world_ effect) has motivated the introduction of an important and
popular network model which is somehow ‘intermediate’ between regular lattices
and random graphs. In the model proposed by Watts and Strogatz smallworld ,
one starts with a regular lattice and then, with fixed probability $p$, goes
through every edge and rewires one of its two end-point connections to a new,
randomly chosen vertex. Clearly, when $p=0$ one has the original lattice
(large clustering and large distance), while when $p=1$ one has a completely
random graph (small clustering and small distance). Thus the parameter $p$ can
be viewed as a measure of the deviation from complete translational symmetry
in the model. Interestingly, in a broad intermediate range of values one
simultaneously obtains a large clustering and a small distance, thus
recovering the empirically observed effect. This suggests that real networks
may be partially, but surely not completely, affected by translational
symmetry (due for instance to the existence of a natural spatial embedding).
As we shall discuss in Section 3.5, translational symmetry, and in general the
dependence of structural properties on the vertices’ positions in some
embedding space, is an example of a more general situation where vertices are
characterised by some non-topological quantity that may determine or condition
their connectivity patterns.
### 3.2 Scale Invariance
As we mentioned, one of the most striking and ubiquitous features of real
networks is the power-law form $P(k)\propto k^{-\gamma}$ of the degree
distribution. This property means that vertices are extremely heterogeneous in
terms of their number of connections: many vertices have a few links, and a
few vertices (the _hubs_) have incredibly many links. An example of a small
network with highly heterogeneous degree distribution is shown in Figure 3.
Importantly, most of the empirically observed values of the exponent $\gamma$
are found to be in the range $2<\gamma<3$, where the variance of the
distribution diverges. This implies that there is no typical scale for the
degree $k$ in the system, and motivates the expression _scale-free network_
guidosbook .
Figure 3: Example of a network with $N=9$ vertices and highly heterogeneous
degree distribution. Vertex $4$ is a highly connected hub with degree
$k_{4}=7$ (the maximum possible value is $N-1=8$), whereas all other vertices
have only $k=1$ or $k=2$ connections.
The above property is an example of a remarkable type of symmetry, precisely
scale invariance. It is found across different domains powerlaws , and in
particular in fractal objects. In fractals, scale invariance is manifest in
the fact that iterated magnifications of an object all have the same shape,
i.e. the system ‘looks the same’ at all scales. Similarly, in networks one
finds that if the scale of the observation is changed (e.g. one switches from
degree $k$ to degree $ak$, with $a$ positive), the number of vertices with
given degree only changes by a (magnification) factor, from $P(k)$ to
$P(ak)=a^{-\gamma}P(k)$. This is very different from exponential
distributions, characterized by a strong variation in the number of counts as
the scale is changed. In networks, power laws have also been found to describe
the distribution of link weights, of the sum of link weights (the so-called
strength) of vertices, and of many more quantities guidosbook . They also
appear to hold across various coarse-grained levels of description of the same
network, if groups of vertices are iteratively merged into ‘supervertices’ and
the original connections collapsed into links among these supervertices shlomo
. The symmetry group associated to scale invariance, i.e. the _renormalization
group_ renormalization , has therefore been used many times to theoretically
understand power-law distributed network properties.
The presence of a scale-free topology across several real-world networks,
which is not reproduced by the Erdős-Rényi model and by the Watts-Strogatz
one, has led to the introduction of new theoretical mechanisms that could
possibly explain the onset of this widespread phenomenon. The earliest (even
if analogous mechanisms were already known in different contexts powerlaws )
and most popular scale-free network model is the one proposed by Barabasi and
Albert BA . It is based on two key ideas: firstly, networks can grow in time,
therefore one can assume that new vertices are continuously added to a
preexisting network; secondly, already popular (highly connected) vertices are
likely to become more and more popular (‘rich get richer’). The latter idea,
known as _preferential attachment_ , is modeled as a multiplicative process in
degree space: the probability that newly introduced vertices establish a
connection to a preexisting vertex $i$ is proportional to the degree $k_{i}$
of that vertex. The iteration of this elementary process of growth and
preferential attachment eventually generates a power-law degree distribution
of the form $P(k)\propto k^{-3}$. In degree space, preferential attachment is
a symmetry-breaking mechanism: vertices are not equally likely to receive new
connections as the network grows. Even if all vertices are identical _a
priori_ , preferential attachment determines and amplifies heterogeneities in
the degree, and eventually vertices with different degrees become subject to
different probabilistic rules. Since in the model there is a tight
relationship between the degree of a vertex and the time the same vertex
entered the network, one could also say that different injection times imply
different expected topological properties. On the other hand, with respect to
scale invariance, preferential attachment is symmetry-preserving and gives
rise to a stationary process. Indeed, as the network grows infinitely in size
over time, its scale-free degree distribution remains unchanged. This
highlights how the same network properties may bear different meanings in
relation to different symmetries. There are now many alternative models that
reproduce scale-free networks with any value of the power-law exponent
$\gamma$, not only $\gamma=3$ guidosbook ; largescalestructure ;
adaptivenetworks . In all of them, there is some mechanism that eventually
sets on and drives the network to converge to an extremely heterogeneous
topology. We shall describe one of these models fitness in Section 3.5.
Before doing that, in the following Sections 3.3 and 3.4 we shall make a more
general discussion about symmetry breaking due to differences in topological
properties in a model-free and real-world framework.
### 3.3 Graph Automorphisms and Structural Equivalence
Various types of vertex permutation symmetry can be defined for graphs. Some
of these symmetries are trivial, while others can be very interesting and
informative. A trivial example is the symmetry under any overall permutation
of vertex labels: if all vertices are relabelled differently, and a new
adjacency matrix is defined accordingly, the resulting graph will have exactly
the same topology of the original one (i.e. the two graphs are _isomorphic_ to
each other harary ). Since one is always free to assign any labelling to
vertices, permutation symmetry trivially holds in any network (in mathematical
words, an unlabelled graph is invariant under vertex relabelling). In this
sense, a graph with $N$ vertices is trivially invariant under the possible
$N!$ permutations of vertex labels, if all edges are relabelled accordingly.
However, a far less trivial problem is whether, after a given labelling has
been chosen (and the graph has therefore become a labelled one), the network
still remains invariant under further vertex permutations. As we mentioned in
Section 2, this is the _graph automorphism_ problem, i.e. the analysis of the
isomorphisms of a graph with itself harary . Suppose the identity of every
vertex has been fixed by assigning a unique label to each of them (as we
mentioned, this labelling is arbitrary and every choice leads to an equivalent
description of the same network). Once a labelling is chosen, one may still
find that a particular graph is unchanged after permutations of some vertices
(without exchanging the identity of the latter). Graph automorphisms are
studied in detail by discrete mathematics. Technically, the set of vertex
permutations defining the automorphisms of a graph forms a symmetry group,
denoted as the _automorphism group_ of the graph. Given a particular graph,
the analysis of its automorphism groups provides a characterization of its
properties, and in particular its symmetries. Traditionally, automorphism
groups are studied for specific classes of graphs generated according to
deterministic rules, which represent standard examples in graph theory harary
. The analysis of the automorphism groups of real-world networks is instead
very recent symmetry ; quotient ; redundancy ; symmetry_wtw . One of the
reasons why it is interesting to look for automorphisms in real networks is
their relation to the following important problem. If two vertices $i$ and $j$
have exactly the same set of neighbors (irrespective of whether they are
neighbors of each other), then a permutation exchanging $i$ and $j$, and
leaving all other vertices unchanged, leads to exactly the same graph. In
social science, when this occurs the vertices $i$ and $j$ are said to be
_structurally equivalent_ wasserman . In food web ecology (where also the
direction of each link to the common neighbours must be the same), they are
said to belong to the same _trophic species_ ecologicalnetworks ; myfoodwebs .
An illustration of structural equivalence is shown in Figure 4. The adjacency
matrix of a graph where $i$ and $j$ are structurally equivalent is unchanged
after exchanging its $i$th and $j$th row, and its $i$th and $j$th column. In
doing so, we are not interchanging the identity of $i$ and $j$, which still
represent the original vertices (for instance, two particular persons in a
social network).
Figure 4: In the example shown, vertices $1$ and $4$ are structurally
equivalent because they have the same set of neighbours (vertices $2$ and
$3$). Similarly, vertices $5$ and $7$ are structurally equivalent because they
are both connected only to vertices $3$ and $6$.
Structural equivalence, which may or may not be present in a given real
network, is very important for many disciplines. It is directly related to the
problem of network robustness: if a vertex is removed from the network, the
presence of at least one structurally equivalent vertex warrants that there
are no secondary effects (other vertices becoming disconnected) or major
topological changes. By contrast, the effects can be dramatic if the removed
vertex is a special one with no structurally equivalent peers (for instance, a
highly connected hub). The analysis of the automorphism groups of real
networks has revealed that, unlike random graphs, real networks are highly
symmetric and contain a significant amount of structural redundancy symmetry ;
quotient ; redundancy ; symmetry_wtw . This property may naturally arise from
growth processes involved in the formation of many networks, and affects local
topological properties such as network motifs (subgraphs of three or four
vertices recurring in real networks much more often than in random graphs
motifs ). Graph automorphisms have also been used to simplify the topology of
real networks by collapsing redundant information and obtaining _network
quotients_ quotient , i.e. coarse grained graphs without structural
repetition. Despite quotients of real networks are substantially smaller than
the original graphs, they are found to preserve various structural properties
(degree heterogeneity, small distance, etc.), effectively capturing a sort of
skeleton of the entire empirical networks quotient ; symmetry_wtw .
### 3.4 Statistical Equivalence
Structural equivalence is a very strict definition of similarity between two
vertices. A more relaxed condition that is usually of interest in sufficiently
large networks is whether two vertices are _statistically equivalent_ , i.e.
whether their topological properties are the same in an average or weak sense.
For instance, one could ask whether two vertices $i$ and $j$ have simply the
same degree (irrespective of the identity of their neighbors), and/or the same
number of second neighbours, or whether they participate in the same number of
triangles and/or longer loops. Similarly, one could be interested in finding
two vertices whose neighbours have the same average degree, irrespective of
the numbers of neighbours of each vertex, and of the individual values of the
degrees of these neighbours (this is explained in more detail below). In all
these examples, one focuses on a subset (or some average value) of the
possible topological properties involving $i$ and $j$, and defines an
equivalence with respect to it only. According to this relaxed condition, a
number of statistically equivalent vertices are found in real networks. The
structure of the resulting equivalence classes determines the symmetry of a
particular network. While permutations of structurally equivalent vertices are
exact symmetries of the graph (i.e. automorphisms), permutations of
statistically equivalent vertices are stochastic symmetries in the sense
introduced in Section 2. Such transformations do not map a network to itself,
but to another member of the family of networks with the same statistical
properties. Importantly, while even small errors such as a missing link in the
data have a dramatic effect on structural equivalence, statistical equivalence
is more robust to fluctuations in network structure. Moreover, introducing
this stochastic type of symmetry gives rise to identify more general patterns
than those accessible to the analysis of structural equivalence. We discuss
this concept by making some examples of the main scientific questions related
to statistical equivalence in networks.
_Do all vertices in a network have the same degree?_ As already discussed in
Section 3.2, this type of symmetry is strongly violated in real networks. A
weaker question would be: are the degrees of all vertices _nearly_ the same?
In this case, one could speak of a typical degree of vertices, and interpret
the deviations from the average value as finite fluctuations due either to
external noise or some intrinsic stochasticity. However, as we mentioned, the
majority of real networks are scale-free, with degrees being broadly
distributed and wildly fluctuating. There are many vertices with small degree,
among which one can in principle find vertices with exactly the same number of
neighbors, but also a few vertices with extremely large degree, which strongly
break the symmetry.
_Is the average degree of the neighbors of all vertices (nearly) the same?_
After recognizing that some vertices attract many more links than others, one
can move one step forward and wonder what is the average degree of the
neighbors of a given vertex (the so-called _average nearest neighbor degree_ ,
or ANND guidosbook ). This quantity encodes some information about the
matching patterns in the network: if the degree plays no role in deciding
whether two vertices are connected, then one expects that the ANND is
independent of the degree itself (as we discuss below, this is not completely
true). By contrast, one finds the presence of strong correlations between the
degrees of neighboring vertices. These correlations can be either positive or
negative, and have opposite effects on the ANND. In networks where large-
degree vertices are more likely to be connected to each other than to low-
degree ones, one observes an increasing trend of the ANND as a function of the
degree. This property is known as _assortativity_ newman_assortative . In
networks where the opposite is true, the ANND decreases with the degree, a
situation denoted _disassortativity_. Importantly, degree-degree correlations
have profound effects on the outcomes of dynamical processes taking place on
networks dynamicalprocessesoncomplexnetworks .
_Do all vertices have (nearly) the same clustering coefficient?_ Again, this
symmetry is generally not observed, as vertices with different degree also
have different values of the clustering coefficient. The latter usually
displays a decreasing trend with the degree $k$. This behavior has been
interpreted as the signature of a hierarchically organised topology, where a
simple wiring pattern is repeated at different scales in a bottom-up fashion:
first creating modules of vertices, then modules of modules, etc. hierarchy .
Since both the clustering coefficient and the ANND strongly depend on the
degree, and since the latter is broadly distributed, it appears that real
networks are characterised by a high level of complexity, with no
characteristic scale associated to any of the simplest topological properties
one can define.
However, the last observation also leads to a reverse, possibly simplifying,
approach to the problem. Interestingly, it has been shown that some of the
correlations mentioned above are partly an unavoidable, ‘spurious’ outcome of
enforcing some topological constraints in the network maslov ; newman_origin .
That is, exactly because many properties ultimately depend on the degree, a
number of structural patterns are automatically generated once the degrees of
all vertices are fixed to specified values. For instance, in networks with
power-law degree distribution the ANND and the clustering coefficient both
decrease with the degree. These patterns do not signal ‘true’ higher-order
correlations, as they are natural outcomes due to the presence of simpler
constraints. If an explanation from the latter exists, it also automatically
explains the former. This highlights the importance of separating low-order
effects from more fundamental higher-order structural patterns. This problem
leads to the definition of suitable _null models_ of networks, a point that we
shall discuss in Section 3.6.
### 3.5 Invariance under Permutation of External Properties
An important type of permutation symmetry can be defined when some external,
non-topological property is attached to vertices (or to edges, or to other
subgraphs; but we will consider the case of vertices for simplicity). This
situation is particularly relevant when one is interested in studying the
relation between the topology and some other property characterising the
vertices of a network, and is tightly related (even if in a nontrivial way) to
structural and statistical equivalence, as the example in Figure 5 shows. Note
that translational symmetry (described in Section 3.1) can be viewed as a
particular case of this problem, if vertices are assigned positions in some
metric space. Translational symmetry is in principle an exact symmetry (the
graph is mapped onto itself) since it is the effect of a deterministic graph
formation rule. However, symmetries due to external properties are in general
stochastic in the sense discussed in Section 2, since real networks are always
best understood as a result of non-deterministic rules. We therefore expect
that stochastic symmetry is more powerful in detecting patterns in real
networks than exact symmetry, and the following discussion confirms this
expectation.
The impact of external factors is an extremely important problem, related to
key questions about network formation, for many research areas. Typical
examples include: _is a social network partly determined by factors such as
race, gender, age, etc.?_ _Is wealth or income relevant to the formation of
economic networks?_ In order to answer the above questions, one needs a way to
assess the structural impact of properties which are in some sense external to
the network.
There have been many attempts in this direction. Social network analysis has a
long tradition in dealing with this problem, firmly based on statistical
theory. The role of vertex properties is generally inspected through the
values of regression parameters used in suitable graph models that are fitted
to the real network wasserman . More recently, in the physics community
different approaches have been proposed. Techniques have been introduced
newman_assortative ; pin_jackson in order to capture whether the connections
observed in a particular network occur mainly between vertices with similar
properties (this is a generalised notion of _assortativity_ , not necessarily
related to vertices’ degrees, also known as _homophily_ in social science) or
between vertices with different properties (_disassortativity_) . More
generally, there have been attempts in understanding whether a specification
of vertex properties effectively reduces the available configuration space for
a real network pin_ginestra and can thus be interpreted as a structurally
important factor. All these different approaches to the same problem could be
restated in more general terms as follows: _is the network (stochastically)
symmetric under a permutation of the properties attached to vertices?_ If this
is the case, the properties under consideration have no statistically
significant impact on network structure. Otherwise, vertex-specific features
are symmetry-breaking, as vertices with different properties are no longer
equivalent under a somewhat generalised notion of the statistical equivalence
described in Section 3.4. In particular, the overall permutation symmetry of
vertex properties is broken and the network is only symmetric under a
restricted set of permutations exchanging vertices within the same equivalence
classes (sets of vertices with the same external properties). It is therefore
clear that the behaviour of a network under the permutations associated to
this type of permutation symmetry is determined by, and carries information
about, the effects that external quantities have on the topology.
In general, the behaviour of a real network under permutation of external
properties can be very complicated and lead to a variety of different symmetry
properties. However, it is possible to understand the problem clearly in
simplified models. Indeed, the idea that vertex properties may be crucial to
network formation has led to the definition of an important class of network
models known as _fitness_ or _hidden variable_ models fitness . Unlike the
Barabasi-Albert model mentioned in Section 3.2, fitness models are static and
do not require the hypothesis of network growth. In these models, one assumes
that the probability $p_{ij}$ that a link is present between vertex $i$ and
vertex $j$ is a function $p(x_{i},x_{j})$ of some property $x$, or _fitness_ ,
attached to these vertices (see Figure 5). Therefore the model requires the
specification of a list of fitness values $\\{x_{i}\\}$, usually assumed to be
drawn independently from some probability distribution $\rho(x)$, and of the
connection function $p(x_{i},x_{j})$. All the expected topological properties
crucially depend on $\\{x_{i}\\}$. For instance, the expected degree of two
vertices $i$ and $j$ with different fitness values ($x_{i}\neq x_{j}$) is in
general different. On the other hand, two vertices with $x_{i}=x_{j}$ are
statistically equivalent. However, due to the probabilistic nature of the
model, in a particular realization of the network the statistical equivalence
of vertices with equal fitness values does not necessarily reflects in their
structural equivalence (see example in Figure 5). This model specification
successfully reproduces the situation mentioned above, as the permutation
symmetry of vertex properties is broken down to disjoint equivalence classes
represented by sets of vertices with identical hidden values. Moreover, the
flexibility in the choice of the fitness values and connection probability
allows to reproduce various topological properties of real-world networks. For
instance, a power-law distribution of fitness values (mimicking some
heterogeneously distributed real-world feature such as individual wealth,
country population, etc.) and a connection probability that linearly depends
on the fitness naturally lead to a scale-free network topology fitness .
Besides providing a valid route to network modelling, hidden variable models
can also be fitted to real networks and shed light on the presence of external
factors case by case mylikelihood ; ramasco . In particular, inverse methods
have been devised in order to extract, only from the topology of a real
network, the values of the hidden variables $\\{x_{i}\\}$ potentially related
to network formation. These values can then be compared with the values of
candidate external properties relevant to that particular network, a strategy
that has been shown to successfully identify key factors related to structure
in real-world cases mylikelihood .
Figure 5: The topological properties of a network may depend on some external
property $x$ attached to vertices. (a) For instance, in the _fitness_ model
fitness one starts with an empty network where each vertex $i$ is assigned a
fitness value $x_{i}$ drawn from some specified distribution $\rho(x)$. (b)
Then, a link between vertices $i$ and $j$ is drawn with probability
$p(x_{i},x_{j})$. Vertices with identical values of $x$ are statistically
equivalent: all their topological properties have the same expected values.
However, the probabilistic nature of the model implies that, in a particular
realization of the network, two vertices $i$ and $j$ with $x_{i}=x_{j}$ are
not necessarily structurally equivalent, and conversely two structurally
equivalent vertices (for instance, $x_{2}$ and $x_{3}$ in the example shown)
do not necessarily have identical fitness values (as we may have $x_{2}\neq
x_{3}$; indeed, this is typically the case if $x$ is drawn from a continuous
probability density). This highlights the difference between structural
equivalence and statistical equivalence.
### 3.6 Ensemble Equiprobability
As we anticipated in Section 2, there are important symmetries associated not
to a single graph, but to a _statistical ensemble_ of graphs (we will define a
graph ensemble rigorously below). If the ensemble is a good model of a real
network, these symmetries can then be naturally related to the real network
itself. This possibility allows us to illustrate in more detail our idea of
stochastically symmetric ensemble, and the definition of stochastically
symmetric graph as a network which is well reproduced by a stochastically
symmetric ensemble (see Section 2). Null models automatically come into play
when one is interested in understanding whether, in a given network,
complicated high-order topological properties can be traced back to simpler
low-level constraints. We already mentioned this problem in Section 3.4. In
order to answer this question, it is necessary to consider a null model by
generating a collection of graphs having some property in common with the real
network (these properties act therefore as constraints), and being completely
random otherwise. This amounts to generate an ensemble of graphs that
maximizes an _entropy_ , that we shall define in a moment, under the enforced
constraints. Then, one can compare the properties of the real network with the
corresponding averages over the randomised ensemble. If there is no
statistically significant difference, one can conclude that the constraints
considered are indeed enough in order to generate all the other properties of
the real network. If differences are significant, then there are other factors
shaping the observed topology. We now rephrase this idea more formally, and
show how it highlights an intimate and instructive connection between
symmetry, entropy and complexity in networks.
A statistical ensemble of graphs newman_statistical is a collection of $M$
graphs $\\{G_{1},G_{2},\dots,G_{M}\\}$, each with an associated occurrence
probability $P(G)$ satisfying
$\sum_{G}P(G)\equiv\sum_{m=1}^{M}P(G_{m})=1$ (2)
We already mentioned examples of graph ensembles, without explicitly noticing
it: Erdős-Rényi model (Sections 1 and 3.1), the Watts-Strogatz model (Section
3.1), the Barabasi-Albert model (Section 3.2) and the fitness model (3.5) are
all examples of collections of possible graphs generated by probabilistic
rules. The Barabasi-Albert model is a non-equilibrium ensemble, as it
generates networks growing indefinitely in time; all the other examples
mentioned above are instead equilibrium ensembles. In what follows, we
restrict ourselves to the equilibrium case. Each graph $G$ is uniquely
specified by its adjacency (or weight) matrix, so we can think of $G$ as of a
matrix. For instance, if one is interested in the ensemble of binary
undirected graphs with $N$ vertices and no self-loops (edges starting and
ending at the same vertex), then $G$ will be a symmetric Boolean matrix with
zeroes along the diagonal, and there will be $M=2^{N(N-1)/2}$ possible such
matrices in the ensemble. In order to generate a maximally random ensemble of
graphs with given constraints newman_statistical ; ginestra_entropy ;
mybosefermi , one needs to find the form of the probability $P(G)$ that
maximises the Shannon-Gibbs entropy
$S\equiv-\sum_{G}P(G)\ln P(G)$ (3)
(a standard measure of disorder or uncertainty) under the enforced
constraints. The latter are a collection $\\{c_{1},\dots,c_{K}\\}$ of $K$
topological properties, forming a $K$-dimensional vector $\vec{c}$. Each
property $c_{a}$ ($a=1,\dots,K$) evaluates to $c_{a}(G)$ when measured on the
particular graph $G$. If the ensemble is meant as a null model of an empirical
network $G^{*}$, the constraints will be chosen as the properties
$\vec{c}(G^{*})$ evaluated on the particular graph $G^{*}$.
There are various possible choices to solve the entropy maximisation problem,
and different ensembles that one can define accordingly. If one is interested
in matching the constraints _exactly_ , i.e. in picking out only the graphs
that have exactly the same properties as a given network $G^{*}$, then the
solution is given by the probability
$P(G)=\left\\{\begin{array}[]{ll}1/\mathcal{N}[\vec{c}(G^{*})]&\textrm{if
}\vec{c}(G)=\vec{c}(G^{*})\\\ 0&\textrm{otherwise}\end{array}\right.$ (4)
where $\mathcal{N}[\vec{c}(G^{*})]$ is the number of graphs matching the
constraints $\vec{c}(G^{*})$. The above probability is uniform over the set of
configurations matching the constraints exactly, and the resulting ensemble is
known in statistical physics as the _microcanonical_ ensemble. With the above
choice, the entropy defined in Equation (3) takes the form
$S=-\mathcal{N}[\vec{c}(G^{*})]\frac{1}{\mathcal{N}[\vec{c}(G^{*})]}\ln\frac{1}{\mathcal{N}[\vec{c}(G^{*})]}=\ln\mathcal{N}[\vec{c}(G^{*})]$
(5)
which is known as the _microcanonical entropy_ and is simply the logarithm of
the number of configurations exactly matching the constraints.
A second alternative consists in requiring that the constraints $\vec{c}$ are
matched _on average_ , i.e. allowing any graph to occur with non-zero
probability, provided that the expected value
$\langle\vec{c}\rangle=\sum_{G}P(G)\vec{c}(G)$ of the constraints matches the
required value $\vec{c}(G^{*})$. This problem can be solved introducing
Lagrange multipliers $\\{\theta_{1},\dots,\theta_{K}\\}$, each associated to
one of the constraints. The solution is the probability distribution
$P(G)=\frac{e^{-H(G)}}{Z}$ (6)
where $H(G)$ (the _graph Hamiltonian_) is a linear combination of the
constraints
$H(G)\equiv\sum_{a=1}^{K}\theta_{a}c_{a}(G)$ (7)
and $Z$ is the _partition function_ that properly normalizes the probability:
$Z\equiv\sum_{G}e^{-H(G)}$ (8)
Thus both $Z$ and $P(G)$ depend on the $K$ parameters
$\\{\theta_{1},\dots,\theta_{K}\\}$. The ensemble generated by the above
probability is known in physics as the _canonical_ ensemble. For a given
choice of the parameters $\\{\theta_{1},\dots,\theta_{K}\\}$, the expected
value of a topological property $X$ across the ensemble is
$\langle X(\theta_{1},\dots,\theta_{K})\rangle\equiv\sum_{G}P(G)X(G)$ (9)
(throughout this review, the angular brackets $\langle\cdot\rangle$ will
denote ensemble averages). In order to match the constraints $\vec{c}(G^{*})$
on average, the $K$ parameters $\\{\theta_{1},\dots,\theta_{K}\\}$ must be set
to the particular values $\\{\theta^{*}_{1},\dots,\theta^{*}_{K}\\}$ such that
$\langle c_{a}(\theta^{*}_{1},\dots,\theta^{*}_{K})\rangle=c_{a}(G^{*})\qquad
a=1,\dots,K$ (10)
Importantly, the above parameter choice corresponds with what the _maximum
likelihood principle_ would indicate mylikelihood , i.e. with the values
maximising the probability $P(G^{*})$ to obtain the real network $G^{*}$ under
the model considered. We will indicate the maximum-likelihood parameter choice
explicitly in the examples considered later on. It has been shown
newman_statistical that the canonical ensemble of networks coincides with the
_exponential random graph models_ that have been first introduced in social
science wasserman . The Hamiltonian $H(G)$ represents the _energy_ , or _cost_
, associated with a given configuration, and contains all the information
required in order to formally obtain $P(G)$. This means that any two graphs
$G_{1}$ and $G_{2}$ for which
$H(G_{1})=H(G_{2})$ (11)
have the same ensemble probability $P(G_{1})=P(G_{2})$. Thus, the symmetries
of $H(G)$ are transformations connecting equiprobable graphs in the ensemble.
Such transformations map a graph $G_{1}$ into a graph $G_{2}$ which has a
different topology but exactly the same values of the enforced constraints.
According to our definition in Section 2, a canonical graph ensemble is
stochastically symmetric under such transformations. If a canonical graph
ensemble is a good model of a real network $G^{*}$, the latter is also
stochastically symmetric. Maximally random graphs with constraints therefore
represent ideal candidates to illustrate the concept of stochastic symmetry.
The symmetries of the Hamiltonian, together with the parameter values
enforcing the constraints, determine the entropy $S$ of the ensemble. This
entropy is a measure of the residual uncertainty about the detailed topology
of a network, once the constraints are fixed.
In statistical physics, there is also a third class of ensembles, i.e.
_grandcanonical_ ensembles. In the latter, the number of particles of the
system is also allowed to vary, and it is treated as one of the properties to
be matched on average. In the case of networks, the role of particles is
played by links newman_statistical , whose number is allowed to vary already
in the canonical ensemble, as the examples considered below illustrate.
Therefore there is no fundamental difference between the canonical and
grandcanonical ensembles of graphs, unless one is interested in networks with
different types of links mymultispecies . For large systems, the
microcanonical, canonical and grandcanonical ensembles give very similar
results. The canonical and grandcanonical ensembles have the enormous
advantage to be analytically treatable, as a consequence of the relaxed
requirement on the constraints. For this reason, in what follows we shall
consider (grand)canonical ensembles of graphs.
We now discuss some examples. If we consider again the ensemble of all
possible undirected graphs with $N$ vertices, the completely symmetric case is
the one where each graph $G$ has the same energy
$H(G)=H_{0}$ (12)
where $H_{0}$ is a constant. In other words, in this case there are no
constraints. Clearly, each of the $M$ possible graphs has the same probability
$P(G)=2^{-N(N-1)/2}$ (13)
and therefore the graph probability is uniformly distributed across the
ensemble (in this particular case, the microcanonical and canonical ensembles
coincide). Transformations changing a graph $G$ into any other graph in the
ensemble are symmetries of the Hamiltonian, and lead to the same ensemble
probability. Thus this ensemble is stochastically symmetric under any
transformation. The entropy is the maximum possible, and its value is
$S=\frac{N(N-1)}{2}\ln 2$ (14)
A different case is when there is a constraint on the total number of links
$L=\sum_{i<j}a_{ij}$. Then
$H(G)=\theta L(G)$ (15)
and it can be easily shown that
$P(G)=p^{L(G)}(1-p)^{N(N-1)/2-L(G)}$ (16)
where $p\equiv e^{-\theta}/(1+e^{-\theta})$. This shows that, as expected, two
graphs $G_{1}$ and $G_{2}$ with the same number of links $L(G_{1})=L(G_{2})$
are equiprobable. Graph transformations preserving this number are symmetries
of the Hamiltonian, and the ensemble is stochastically symmetric under such
transformations. Equation (16) indicates that, for each of the $N(N-1)/2$
pairs of vertices, the probability of an undirected link being there is $p$.
The probability of exactly $L(G)$ realised edges is $p^{L(G)}$ multiplied by
the probability $(1-p)^{N(N-1)/2-L(G)}$ of the complementary number
$N(N-1)/2-L(G)$ of missing edges. This case is therefore equivalent the Erdős-
Rényi random graph model that we already mentioned in Section 1, in which each
edge is drawn, independently of each other, with probability $p$. The entropy
of the ensemble now depends on $p$, and one can easily see that if $p=1/2$,
Equation (14) is recovered. Indeed, this is the case where each edge is
equally likely to be present and absent, which is another way to say that no
constraint has been enforced and the entropy is maximum. By contrast, in the
two cases $p=0$ and $p=1$ the entropy is $S=0$ as there is no uncertainty
about the resulting structure of the network. Indeed, in these cases the
ensemble completely shrinks to the only possible network, i.e., the empty
graph and the complete graph respectively. If one wants to use the random
graph model as a null model of a real network $G^{*}$, the maximum likelihood
principle applied to Equation (16) indicates mylikelihood that the parameter
$p$ must be set to the value
$p^{*}=\frac{2L(G^{*})}{N(N-1)}$ (17)
which ensures that the expected number of links $\langle L\rangle$, as defined
by Equation (9), reproduces the number of links $L(G^{*})$ of that particular
network:
$\langle L\rangle=p^{*}\frac{N(N-1)}{2}=L(G^{*})$ (18)
In the random graph model, the expected degree distribution is binomial (in
the large network limit with fixed average degree, Poissonian) with mean
$p^{*}(N-1)=2L(G^{*})/N$. The failure of the random graph model in reproducing
the properties of real networks, according to our discussion in Section 1, can
then be restated as the inefficacy of specifying the number of links as the
only property of a network. This also means that real networks are generally
not stochastically symmetric under transformations preserving the total number
of links. A less trivial choice is the so-called _configuration model_ maslov
; configuration . Assuming we are still interested in undirected binary
networks, the configuration model is a maximally random graph ensemble where
the degrees of all vertices, i.e. the _degree sequence_ $\\{k_{i}\\}$, are
specified. Note that, in terms of the adjacency matrix $A$ of the graph, the
degree of vertex $i$ is $k_{i}=\sum_{j}a_{ij}$, and the total number of links
is twice the sum of the degrees of all vertices:
$L=\sum_{i<j}a_{ij}=\sum_{i}k_{i}/2$. Therefore specifying the degree sequence
automatically fixes also the total number of links, which confirms that this
model is more constraining than the random graph one. The configuration model
naturally comes into play in the problem we described in Section 3.4, when we
stressed the importance of comparing a real network to a null model in order
to separate genuine higher-order correlations from mere effects of low-level
constraints. The degree sequence is an important constraint to consider,
because the widespread occurrence of scale-free architectures implies that
major topological differences across real networks must be looked for in other
properties beyond the degree distribution. Note that specifying the degree
sequence $\\{k_{i}\\}$ is different from specifying the degree distribution
$P(k)$. A given degree sequence generates a unique degree distribution, but
there are many degree sequences ($N!$ permutations) generating the same degree
distribution. Therefore fixing the degree distribution is less informative
than specifying the entire degree sequence, and we do not consider it here.
For directed graphs, the configuration model is naturally extended by
simultaneously considering as constraints the number of incoming links (_in-
degree_) and the number of outgoing links (_out-degree_) of all vertices.
Similarly, for weighted networks the constraints become the _strength_ (total
edge weight) of all vertices (the _strength sequence_), or the corresponding
directed quantities when applicable.
In the binary undirected case, the Hamiltonian of the configuration model
contains the degrees of all vertices:
$H(G)=\sum_{i=1}^{N}\theta_{i}k_{i}(G)$ (19)
and it can be shown newman_origin that the form of $P(G)$ determined by the
above choice is
$P(G)=\prod_{i<j}p_{ij}^{a_{ij}(G)}(1-p_{ij})^{1-a_{ij}(G)}=\frac{\prod_{i}x_{i}^{k_{i}(G)}}{\prod_{i<j}(1+x_{i}x_{j})}$
(20)
where
$p_{ij}=\frac{x_{i}x_{j}}{1+x_{i}x_{j}}$ (21)
and $x_{i}\equiv e^{-\theta_{i}}$ is another way to write the Lagrange
multiplier associated to $k_{i}$. In this model, edges are still independent,
but have different probabilities.
The probability $P(G)$ of a graph $G$ only depends on its degree sequence, as
evident from Equation (20). Thus any two graphs $G_{1}$ and $G_{2}$ with the
same degree sequence $\\{k_{i}(G_{1})\\}=\\{k_{i}(G_{2})\\}$ are equiprobable
in the ensemble specified by Equation (19). A consequence of this property is
illustrated in Figure 6, where we show two graphs $G_{1}$ and $G_{2}$ that
have exactly the same topology, except for the two edges shown. Graph $G_{2}$
can be obtained from $G_{1}$ by replacing the two edges $(A-B)$ and $(C-D)$
with the two edges $(A-C)$ and $(B-D)$. Since this transformation preserves
the degree sequence, it is a symmetry of the Hamiltonian defined in Equation
(19) and connects equiprobable graphs. According to our definition in Section
2, the ensemble is stochastically symmetric under such transformation. The
equivalence classes of this symmetry are sets of graphs with the same degree
sequence.
Figure 6: The two undirected graphs $G_{1}$ and $G_{2}$ are identical, except
for the two pairs of edges shown. In the configuration model, $G_{1}$ and
$G_{2}$ occur with the same probability since their degree sequences are the
same. Reference maslov exploits this property as a recipe to iteratively
randomize a real network while preserving its degree sequence: in an
elementary step, a graph like $G_{1}$ is transformed into the graph $G_{2}$
(_local rewiring algorithm_).
This property has been used to constructively define an algorithm that
randomises a real network $G^{*}$ by iteratively selecting a pair of edges and
swapping the end-point vertices exactly as in Figure 6 maslov . This
procedure, known as the _local rewiring algorithm_ , ergodically explores the
equivalence class where the real network $G^{*}$ belongs. Any topological
property of interest can be averaged across the set of graphs produced by the
algorithm and compared with the value of the same property in the original
graph $G^{*}$. This allows to check the effects of the degree sequence alone
on the other topological properties. As we mentioned, this null model is
restricted to only one equivalence class of the symmetry (it is a
_microcanonical ensemble_), and requires that averages are numerically
performed over the graphs sampled by the local rewiring algorithm. By
contrast, the null model defined by Equation (19) explores the entire set of
$2^{N(N-1)/2}$ undirected graphs (it is a _(grand)canonical ensemble_), and
allows to obtain the expectation values analytically through Equation (9).
This requires that the parameters $\\{x_{1},\dots,x_{N}\\}$ are set to the
values $\\{x^{*}_{1},\dots,x^{*}_{N}\\}$ that maximise the likelihood to
obtain the real network $G^{*}$ mylikelihood ; myrandomization . These values
are found by solving the following $N$ coupled equations
$\langle k_{i}\rangle=\sum_{j\neq
i}\frac{x^{*}_{i}x^{*}_{j}}{1+x^{*}_{i}x^{*}_{j}}=k_{i}(G^{*})\qquad\forall i$
(22)
ensuring that the expected degree sequence coincides with the observed one,
and thus generalising Equation (18). As we already anticipated in Section 3.4,
an important conclusion drawn from the analysis of the configuration model is
that, if real-world scale-free degree distributions are specified, higher-
order patterns are automatically generated. In particular, the average nearest
neighbour degree and the clustering coefficient of a vertex with degree $k$
are both found to decrease with $k$ maslov ; newman_origin ; myrandomization .
These patterns should not be interpreted necessarily as the result of
additional mechanisms, beyond those required to explain the form of the degree
distribution. Note that if a real network is found to be well reproduced by
the configuration model, then it is stochastically symmetric under
transformations preserving the degree sequence. Also note that any two
vertices $i$ and $j$ with the same degree $k_{i}(G^{*})=k_{j}(G^{*})$ in the
real network are statistically equivalent in the sense specified in Section
3.4. This is because Equation (22) implies that those vertices would be
assigned the same parameter value $x_{i}^{*}=x^{*}_{j}$, and would therefore
have the same expected topological properties as discussed for the fitness
model in Section 3.5. Whereas permutations of structurally equivalent vertices
lead to exactly the same topology and are therefore automorphisms (exact
symmetries) of the network, permutations of statistically equivalent vertices
(here, vertices with the same degree) are stochastic symmetries of the
network, if the latter is in accordance with the configuration model. This is
an interesting and important relation between ensemble equiprobability,
symmetry under permutation of vertex properties, and statistical equivalence.
If the ensemble is not a good model of the real network, which signals the
presence of mechanisms that break the postulated equiprobability symmetry,
then the real network is not stochastically symmetric under transformations
preserving the degree sequence, and vertices displaying the same values of the
enforced constraints are no longer statistical equivalent.
Note that Equation (20) generalises Equation (16), and also that Equation (21)
can be viewed as a particular case of the connection probability
$p(x_{i},x_{j})$ introduced in the fitness model we described in Section 3.5.
Indeed, the configuration model and the fitness model both reduce to the
random graph case if $x_{i}=x_{0}$ $\forall i$, i.e. if all vertices have the
same properties. In this case, the entropy associated with Equation (21)
coincides with the one associated with Equation (16). By contrast, if the
$x_{i}$’s are heterogeneously distributed, the entropy is significantly
decreased. In particular, the values of the $x_{i}$’s required in order to
enforce a scale-free degree distribution as observed in real networks are
approximately power-law distributed, a result implying a strong reduction of
the entropy of the ensemble associated with the degree sequence of real
networks. In particular, it was shown that networks with degree distribution
$P(k)\propto k^{-2}$ have remarkably small entropy ginestra_entropy and can
be generated deterministically fitness like regular graphs. We therefore see
that network complexity, as signalled in this example by a scale-free degree
distribution, can lead to a decrease in the stochastic symmetry associated
with ensemble equiprobability, and to a substantial decrease in the
corresponding entropy. From the perspective of the amount of information
required in order to reproduce them, real networks (and possibly many real
complex systems) turn out to achieve an unsuspected degree of order by
following a nontrivial path, which is completely different from that taken by
regular structures.
### 3.7 Symmetry under Network Partitioning: Modularity and Communities
As we briefly mentioned in Section 1, real networks are found to display
inhomogeneous link density, and to be partitioned into _communities_ of
vertices santo_communities . Several different definitions of a community have
been introduced. Generally, these definitions try to capture different aspects
of the same simple idea: that communities are more densely connected
internally than with other communities, so that intra-community links are
typically denser than inter-community ones. An example is shown in Figure 7 to
illustrate this concept. This simple idea can however give rise to technical
difficulties when implemented into community detection algorithms and applied
to large networks, and as a result different methods have been developed, each
dealing with a different aspect of the problem. For instance, some methods try
to identify the _optimal partition_ of vertices into non-overlapping subsets
representing communities; others recognise that the optimality of a partition
depends on the resolution adopted, and give a _multi-resolution_ output where
communities are hierarchically nested into each other; others are devised to
identify _overlapping_ communities, etc. Presenting the subtleties and
diversity of the community detection problem is beyond the scope of the
present review, and the interested reader is referred to the relevant
literature santo_communities . We simply note here that the community
structure of a network is connected to a particular type of symmetry: the
invariance under network partitioning. To illustrate this idea, we consider as
an example a widely used quantity that measures the goodness of a partition of
a real undirected network into non-overlapping communities, i.e. the
_modularity_
$Q\equiv\frac{1}{L}\sum_{i<j}(a_{ij}-p_{ij})c_{ij}$ (23)
In the above definition, $a_{ij}$ is the entry of the adjacency matrix $A$ of
the real network, $L=\sum_{i<j}a_{ij}$ is the observed number of links,
$p_{ij}$ is the probability that vertices $i$ and $j$ are connected under a
null model chosen as a reference, and $c_{ij}$ indicates if in the partition
under consideration vertices $i$ and $j$ are placed in the same community
($c_{ij}=1$) or in different communities ($c_{ij}=0$). Typically, the null
model considered is the configuration model (see Section 3.6). Since different
partitions of the same network correspond to different sets of values
$\\{c_{ij}\\}$, $Q$ can be used to assess the performance of a partition in
correctly placing in the same community ($c_{ij}=1$) pairs of vertices that
are connected ($a_{ij}=1$) despite the null model predicts a low connection
probability ($p_{ij}\approx 0$), and in placing in different communities
($c_{ij}=0$) pairs of vertices that are not connected ($a_{ij}=0$) despite the
null model predicts a high connection probability ($p_{ij}\approx 1$). Larger
values of $Q$ represent better partitions. If the network is well reproduced
by the null model, then one expects a value of $Q$ close to zero,
independently of the partition. To see this, imagine that the network has
indeed been generated by the null model. If several realisations of the
network are generated, then the expected value pf $a_{ij}$ is $p_{ij}$ and the
expected modularity is
$\langle Q\rangle=0$ (24)
independently of $c_{ij}$. This means that a network with no community
structure is stochastically invariant (in the sense specified in Section 2)
under vertex partitioning, as all reassignments of vertices to different
communities preserve on average the modularity. The modular structure of real
networks can be therefore seen as a symmetry-breaking property. In some
networks, the maximisation of the modularity can be very complicated
numerically, as there are many competing partitions with similar values of $Q$
(computationally, finding the partition corresponding to the global maximum of
$Q$ is a NP-hard problem). This indicates that in real networks the overall
invariance under partitioning is often broken down to equivalence classes
containing partitions with approximately equal modularity.
Figure 7: Example of an undirected network with $N=9$ vertices, that can be
clearly grouped into $2$ non-overlapping communities: vertices $1$ to $4$ form
one community, and vertices $5$ to $9$ form a second community. Intra-
community links are denser than inter-community ones.
### 3.8 Edge Weight Permutation Invariance
As the last example of symmetries in networks, we consider an invariance that
naturally comes into play in the analysis of weighted networks. Weighted
networks are described by a non-negative matrix $W$ rather than by a binary
adjacency matrix $A$. The entry $w_{ij}$ of the matrix $W$ represents the
weight of the edge from vertex $i$ to vertex $j$ (if $w_{ij}=0$ no edge is
there). In the analysis of weighted networks, a crucial point is assessing
whether the knowledge of edge weights indeed conveys additional information
with respect to the knowledge of the binary topology. This problem has been
tackled by introducing suitable definitions of structural properties that make
explicit use of the empirical edge weights and that distinguish between the
real network and suitably randomised counterparts vespy_weighted ; myensemble
; kertesz_clustering .
The randomised case can be either a weighted generalisation of the maximally
random networks described in Section 3.6 mybosefermi , or a different null
model providing a reference where weights and topology are uncorrelated, so
that weighted properties reduce to simpler binary properties vespy_weighted .
The latter null model consists in taking the real network, keeping its
topology fixed, and randomly reshuffling the values of the weights across the
edges (see Figure 8).
Figure 8: Construction of a null model, alternative to the weighted
generalization of the local rewiring algorithm defined in Figure 6, against
which the properties of a real weighted network can be compared. (a) A real
network is considered, where each link $(i-j)$ has an observed weight
$w_{ij}$. (b) The empirical weights $\\{w_{ij}\\}$ are randomly shuffled
across the links of the network, which are kept in the original positions (the
topology is unchanged). Iterating this procedure generates an ensemble of
randomized weighted networks. In such a way, the correlations between weights
and topology are removed, and one has a family of uncorrelated benchmarks for
the empirical network.
Iterating this procedure generates an ensemble of randomised networks where
any correlation existing between weights and topology is destroyed. This
provides a reference for the analysis of the original real network. A
prototypical example of the deviation of real networks from the uncorrelated
case is the generally observed power-law relation between the degree
$k_{i}=\sum_{j\neq i}a_{ij}$ and the strength $s_{i}=\sum_{j\neq i}w_{ij}$ of
vertices:
$s_{i}\propto k_{i}^{\beta}$ (25)
where usually $\beta>1$. By contrast, in the uncorrelated case provided by the
null model, the strength is simply proportional to the degree, which is its
unweighted counterpart. This yields $\beta=1$. Similar results are found for
other quantities. In general, if suitable weighted structural properties are
defined and averaged across the uncorrelated ensemble, the output is in a
trivial relation with the purely binary counterparts of these properties
vespy_weighted .
We note that the above problem can be rephrased as a generalisation of the
symmetry we introduced in Section 3.5. Indeed, weights can be considered as
non-topological properties attached to edges (rather than to vertices).
Nontrivial correlations between weights and topology correspond to a lack of
invariance of the real network under permutations of weights across the edges.
Whereas uncorrelated weighted networks are stochastically symmetric under such
permutations, real networks are found to display strong correlations.
Therefore, we find again that network complexity, now at the level of weights,
can manifest itself in terms of symmetry-breaking correlations restricting
possible network invariances to smaller equivalence classes.
## 4 Conclusions
In this review we have discussed various types of symmetries encountered in
the analysis of real networks. Symmetry concepts turn out to offer an
insightful review of network theory from an unusual perspective. In
particular, we have shown that many empirical properties of complex networks
can be rephrased in terms of (the lack of) exact or stochastic symmetries.
Exact symmetries of a network are transformations that map the network onto
itself. If such transformations are permutations of vertices, they are the
automorphisms of the graph. Special cases include symmetries induced by
structural equivalence (Section 3.3) or by an embedding of vertices in some
space, such as translational symmetry (Section 3.1). Stochastic symmetries of
a network are transformations that map the network onto a different one in the
same statistical ensemble, and are therefore associated with a family of
graphs with similar properties, rather than with a single graph. We have
discussed stochastic vertex permutation symmetries in the context of
statistical equivalence (Section 3.4) and invariance under permutation of
vertex properties (Section 3.5). We have also discussed transformations not
associated with permutations of vertices, such as scale invariance (Section
3.2), ensemble equiprobability (Section 3.6), invariance under vertex
partitions (Section 3.7), and edge weight permutations (Section 3.8). We have
shown that various correlation patterns observed in real networks imply that
the above symmetries only hold within disjoint equivalence classes, specified
for instance by some property of vertices. This often indicates which are the
most informative topological properties of real networks: those that partition
vertices (or other parts of the graph) into the equivalence classes of some
(stochastic) symmetry. Therefore we believe that the study of symmetry in
networks is a promising field of research, which deserves more attention in
future investigations. While automorphism groups are well studied within
discrete mathematics for particular classes of graphs generated according to
deterministic rules, the analysis of symmetry in real heterogeneous networks
is far less developed. We suggested that real networks—as any real entity
characterized by imperfections or errors—necessarily require a stochastic
notion of symmetry. Our preliminary investigation shows that such an expanded
scenario may lead to very informative results, as it can detect ordered
patterns in intrinsically noisy contexts, where exact techniques fail. In the
companion paper symmetry2 , we apply our ideas in more detail and show the
full power of stochastic symmetry in a particular case.
## Acknowledgements
D.G. acknowledges financial support from the European Commission 6th FP
(Contract CIT3-CT-2005-513396), Project: DIME - Dynamics of Institutions and
Markets in Europe.
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|
arxiv-papers
| 2010-06-20T09:49:33 |
2024-09-04T02:49:11.051472
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Diego Garlaschelli, Franco Ruzzenenti, Riccardo Basosi",
"submitter": "Diego Garlaschelli",
"url": "https://arxiv.org/abs/1006.3923"
}
|
1006.4024
|
# Monte-Carlo Simulations of Thermal Comptonization Process in a Two Component
Accretion Flow Around a Black Hole in presence of an Outflow
Himadri Ghosh S.N. Bose National Centre for Basic Sciences,
JD-Block, Sector III, Salt Lake, Kolkata 700098, India.
himadri@bose.res.in Sudip K. Garain S.N. Bose National Centre for Basic
Sciences,
JD-Block, Sector III, Salt Lake, Kolkata 700098, India.
sudip@bose.res.in Sandip K. Chakrabarti111Also at Indian Centre for Space
Physics, Chalantika 43, Garia Station Rd., Kolkata 700084 S.N. Bose National
Centre for Basic Sciences,
JD-Block, Sector III, Salt Lake, Kolkata 700098, India.
chakraba@bose.res.in Philippe Laurent IRFU, Service d’Astrophysique, Bat.
709 Orme des Merisiers, CEA Saclay, 91191
Gif-sur-Yvette Cedex, France, philippe.laurent@cea.fr
(Day Month Year; Day Month Year)
###### Abstract
A black hole accretion may have both the Keplerian and the sub-Keplerian
components. The Keplerian component supplies low-energy (soft) photons while
the sub-Keplerian component supplies hot electrons which exchange their energy
with the soft photons through Comptonization or inverse Comptonization
processes. In the sub-Keplerian flow, a shock is generally produced due to the
centrifugal force. The post-shock region is known as the CENtrifugal pressure
supported BOundary Layer or CENBOL. We compute the effects of the thermal and
the bulk motion Comptonization on the soft photons emitted from a Keplerian
disk by the CENBOL, the pre-shock sub-Keplerian disk and the outflowing jet.
We study the emerging spectrum when both the converging inflow and the
diverging outflow (generated from the CENBOL) are simultaneously present. From
the strength of the shock, we calculate the percentage of matter being carried
away by the outflow and determined how the emerging spectrum depends on the
the outflow rate. The pre-shock sub-Keplerian flow was also found to
Comptonize the soft photons significantly. The interplay among the up-
scattering and down-scattering effects determines the effective shape of the
emerging spectrum. By simulating several cases with various inflow parameters,
we conclude that whether the pre-shock flow, or the post-shock CENBOL or the
emerging jet is dominant in shaping the emerging spectrum, strongly depends on
the geometry of the flow and the strength of the shock in the sub-Keplerian
flow.
###### keywords:
accretion disk, black hole physics, shock waves, radiative processes, Monte-
Carlo simulations
Managing Editor
## 1 Introduction
It is well known (Chakrabarti 1990, hereafter C90) that the flow velocity is
the same as the velocity of light $c$ as the matter enters through the event
horizon. However, the sound speed is never so high. Thus the incoming flow on
a black hole is always supersonic and thus these solutions are likely to be
most relevant in the study of the physical processes around black holes. As
the flow begins its journey sub-sonically very far away, and becomes
supersonic on the horizon, the flow is also known as a transonic flow. In the
context of the spherical flows, Bondi (1952) solution of accretion and Parker
(1959) solution of winds are clear examples of transonic flows. But they have
only one sonic points. In presence of angular momenta, the flow may have two
saddle type sonic points with a shock in between (C90, Chakrabarti, 1996). The
solutions with shocks have been extensively studied in both the accretion and
the winds even when rotation, heating, cooling etc. are included (Chakrabarti,
1990, 1996). The study demonstrates that the accretion and the winds are
inter-related – the outflows are generated from the post-shock region.
Subsequently, in Chakrabarti (1999, hereafter C99), Das & Chakrabarti (1999)
and Das et al. (2001), the mass outflow rate was computed as a function of the
shock strength and other flow parameters. Meanwhile, in the so-called two
component advective flow (TCAF) model of Chakrabarti & Titarchuk (1995) and
Chakrabarti (1997), the spectral states were shown to depend on the location
and strength of the shock. Thus, C99 for the first time, brought out the
relationship between the jets and outflows with the presence or absence of
shocks, and therefore with the spectral states of a black hole candidates.
This paves the way to study the relative importance between the Compton cloud
and the outflow as far as emerging spectrum is concerned.
Computation of the spectral characteristics have so far concentrated only on
the advective accretion flows (Chakrabarti & Titarchuk, 1995; Chakrabarti &
Mandal, 2006) and the outflow or the base of the jet was not included. In the
Monte-Carlo simulations of Laurent & Titarchuk (2007) outflows in isolation
were used, but not in conjunction with inflows. In Ghosh, Chakrabarti &
Laurent (2009, hereafter Paper I), the results of Monte-Carlo simulations in a
setup similar to that of Chakrabarti & Titarchuk (1995) was presented. In the
present paper, we improve this and obtain the outgoing spectrum in presence of
both inflows and outflows. We also include a Keplerian disk inside an
advective flow which is the source of soft photons. We show how the spectrum
depends on the flow parameters of the inflow, such as the accretion rates of
the two components and the shock strength. The post-shock region being denser
and hotter, it behaves like the so-called ’Compton cloud’ in the classical
model of Sunyaev and Titarchuk (1980). This region is known as the CENtrifugal
pressure supported BOundary Layer or CENBOL. Since the shock location and its
strength depends on the inflow parameters, the variation of the size of the
Compton cloud, and then the basic Comptonized component of the spectrum is
thus a function of the basic parameters of the flow, such as the specific
energy, the accretion rate and the specific angular momentum. Since the
intensity of soft photons determines the Compton cloud temperature, the result
depends on the accretion rate of the Keplerian component also. In our result,
we see the effects of the bulk motion Comptonization (Chakrabarti & Titarchuk,
1995) because of which even a cooler CENBOL produces a harder spectrum. At the
same time, the effect of down-scattering due the outflowing electrons is also
seen, because of which even a hotter CENBOL causes the disk-jet system to emit
lesser energetic photons. Thus, the net spectrum is a combination of all these
effects.
In the next section, we discuss the geometry of the soft photon source and the
Compton cloud in our Monte-Carlo simulations. In §3, we present the variation
of the thermodynamic quantities and other vital parameters inside the
Keplerian disk and the Compton cloud which are required for the Monte-Carlo
simulations. In §4, we describe the simulation procedure and in §5, we present
the results of our simulations. Finally in §6, we make concluding remarks.
## 2 Geometry of the electron cloud and the soft photon source
The problem at hand is very complex and thus we need to simplify the geometry
of the inflow-outflow configuration without sacrificing the salient features.
In Fig. 1, we present a cartoon diagram of our simulation set up. The
components of the hot electron clouds, namely, the CENBOL, the outflow and the
sub-Keplerian flow, intercept the soft photons emerging out of the Keplerian
disk and reprocess them via inverse Compton scattering. An injected photon may
undergo a single, multiple or no scattering at all with the hot electrons in
between its emergence from the Keplerian disk and its detection by the
telescope at a large distance. The photons which enter the black holes are
absorbed. The CENBOL, though toroidal in nature, is chosen to be of spherical
shape for simplicity. The sub-Keplerian inflow in the pre-shock region is
assumed to be of wedge shape of a constant angle $\Psi$. The outflow, which
emerges from the CENBOL in this picture is also assumed to be of constant
conical angle $\Phi$. In reality, inflow and outflow both could have somewhat
different shapes, depending on the balance of the force components. However,
the final result is not expected to be sensitive to such assumptions.
Figure 1: A cartoon diagram of the geometry of our Monte-Carlo simulations
presented in this paper. The spherical inflowing post-shock region (CENBOL)
surrounds the black hole and it is surrounded by the Keplerian disk on the
equatorial plane and a sub-Keplerian halo above and below. A diverging conical
outflow is formed from the CENBOL. Typical path of a photon is shown by zig-
zag paths.
### 2.1 Distribution of temperature and density inside the Compton cloud
We assume the black hole to be non-rotating and we use the pseudo-Newtonian
potential (Paczyński & Wiita, 1980) to describe the geometry around a black
hole. This potential is $-\frac{1}{2(r-1)}$ (Here, $r$ is in the unit of
Schwarzschild radius $r_{g}=2GM/c^{2}$). Velocities and angular momenta are
measured are in units of $c$, the velocity of light and $r_{g}c$ respectively.
For simplicity, we chose the Bondi accretion solution in pseudo-Newtonian
geometry to describe both the accretion and winds. The equation of motion of
the sub-Keplerian matter around the black hole in the steady state is assumed
to be given by,
$u\frac{du}{dr}+\frac{1}{\rho}\frac{dP}{dr}+\frac{1}{2(r-1)^{2}}=0.$
Integrating this equation, we get the expression of the conserved specific
energy as,
$\epsilon=\frac{u^{2}}{2}+na^{2}-\frac{1}{2(r-1)}.$ (1)
Here $P$ is the thermal pressure and $a$ is the adiabatic sound speed, given
by $a=\sqrt{\gamma P/\rho}$, $\gamma$ being the adiabatic index and is equal
to $\frac{4}{3}$ in our case. The conserved mass flux equation, as obtained
from the continuity equation, is given by
$\dot{M}=\Omega\rho ur^{2},$ (2)
where, $\rho$ is the density of the matter and $\Omega$ is the solid angle
subtended by the flow. For an inflowing matter, $\Omega$ is given by,
$\Omega_{in}=4\pi Sin\Psi,$
where, $\Psi$ is the half-angle of the conical inflow. For the outgoing
matter, the solid angle is given by,
$\Omega_{out}=4\pi(1-cos\Phi),$
where $\Phi$ is the half-angle of the conical outflow. From Eqn. 2, we get
$\dot{\mu}=a^{2n}ur^{2}.$ (3)
The quantity $\dot{\mu}=\frac{\dot{m}\gamma^{n}K^{n}}{\Omega}$ is the
Chakrabarti rate (Chakrabarti, 1989, C90, 1996) which includes the entropy,
$K$ being the constant measuring the entropy of the flow, and
$n=\frac{1}{\gamma-1}$ is called the polytropic index. We take derivative of
equations (1) and (3) with respect to $r$ and eliminating $\frac{da}{dr}$ from
both the equations, we get the gradient of the velocity as,
$\frac{du}{dr}=\frac{\frac{1}{2(r-1)^{2}}-\frac{2a^{2}}{r}}{\frac{a^{2}}{u}-u}.$
(4)
From this, we obtain the Bondi accretion and wind solutions in the usual
manner (C90). Solving these equations we obtain, $u$, $a$ and finally the
temperature profile of the electron cloud ($T_{e}$) using $T_{e}=\frac{\mu
a^{2}m_{p}}{\gamma k_{B}}$, where $\mu=0.5$ is the mean molecular weight,
$m_{p}$ is the proton mass and $k_{B}$ is the Boltzmann constant. Using Eq.
(2), we calculate the mass density $\rho$, and hence, the number density
variation of electrons inside the Compton cloud. We ignore the electron-
positron pair formation inside the cloud.
The flow is supersonic in the pre-shock region and sub-sonic in the post-shock
(CENBOL) region. We chose this surface at a location where the pre-shock Mach
number $M=2$. This location depends on the specific energy $\epsilon$ (C90).
In our simulation, we have chosen $\epsilon=0.015$ so that we get $R_{s}=10$.
We simulated a total of six cases. For Cases 1(a-c), we chose $\dot{m_{h}}=1$,
$\dot{m_{d}}=0.01$ and for Cases 2(a-c), the values are listed in Table 2. The
velocity variation of the sub-Keplerian flow is the inflowing Bondi solution
(pre-sonic point). The density and the temperature of this flow have been
calculated according to the above mentioned formulas. Inside the CENBOL, both
the Keplerian and the sub-Keplerian components are mixed together. The
velocity variation of the matter inside the CENBOL is assumed to be the same
as the Bondi accretion flow solution reduced by the compression ratio due to
the shock. The compression ratio (i.e., the ratio between the post-shock and
pre-shock densities) $R$ is also used to compute the density and the
temperature profile has been calculated accordingly. When the outflow is
adiabatic, the ratio of the outflow to the inflow rate is (Das et al. 2001)
given by,
$R_{\dot{m}}=\frac{\Omega_{out}}{\Omega_{in}}\left(\frac{f_{0}}{4\gamma}\right)^{3}\frac{R}{2}\left[\frac{4}{3}\left(\frac{8(R-1)}{R^{2}}-1\right)\right]^{3/2}$
(5)
here, we have used $n=3$ for a relativistic flow. From this, and the velocity
variation obtained from the outflow branch of Bondi solution, we compute the
density variation inside the jet. In our simulation, we have used
$\Phi=58^{\circ}$ and $\Psi=32^{\circ}$. Fig. 2 shows the variation of the
percentage of matter in the outflow for these particular parameters.
Figure 2: Ratio of the outflow and the inflow rates as a function of the
compression ratio $R$ of the inflow when the outflow is adiabatic. In our
simulations, we have used the jet angle to be 58∘.
### 2.2 Keplerian disk
The soft photons are produced from a Keplerian disk whose inner edge coincides
with CENBOL surface, while the outer edge is located at $500r_{g}$. The source
of soft photons have a multi-color blackbody spectrum coming from a standard
(Shakura & Sunyaev, 1973, hereafter SS73) disk. We assume the disk to be
optically thick and the opacity due to free-free absorption is more important
than the opacity due to scattering. The emission is black body type with the
local surface temperature (SS73):
$\displaystyle T(r)\approx 5\times
10^{7}(M_{bh})^{-1/2}(\dot{M_{d}}_{17})^{1/4}(2r)^{-3/4}\left[1-\sqrt{\frac{3}{r}}\right]^{1/4}K,$
(6)
The total number of photons emitted from the disk surface is obtained by
integrating over all frequencies ($\nu$) and is given by,
$\displaystyle
n_{\gamma}(r)=\left[16\pi\left(\frac{k_{b}}{hc}\right)^{3}\times
1.202057\right]\left(T(r)\right)^{3}$ (7)
The disk between radius $r$ to $r+\delta r$ injects $dN(r)$ number of soft
photons.
$\displaystyle dN(r)=2\pi r\delta rH(r)n_{\gamma}(r),$ (8)
where, $H(r)$ is the half height of the disk given by:
$\displaystyle
H(r)=10^{5}\dot{M_{d}}_{17}\left[1-\sqrt{\frac{3}{r}}\right]{\rm cm}.$ (9)
The soft photons are generated isotropically between the inner and outer edge
of the Keplerian disk but their positions are randomized using the above
distribution function (Eq. 8) of black body temperature $T(r)$. All the
results of the simulations presented here have used the number of injected
photons to be $6.4\times 10^{8}$. In the above equations, the mass of the
black hole $M_{bh}$ is measured in units of the mass of the Sun ($M_{\odot}$),
the disk accretion rate $\dot{M_{d}}_{17}$ is in units of $10^{17}$ gm/s. We
chose $M_{bh}=10$ and $\delta r=0.5r_{g}$.
### 2.3 Simulation Procedure
In a given simulation, we assume a given Keplerian rate and a given sub-
Keplerian halo rate. The specific energy of the halo provides hydrodynamic
properties (such as number density of the electrons and the velocity
variation) and the thermal properties of matter. Since we chose the Paczynski-
Wiita (1980) potential, the radial velocity is not exactly unity at $r=1$, the
horizon, but it becomes unity just outside. In order not to over estimate the
effects of bulk motion Comptonization which is due to the momentum transfer of
the moving electrons to the horizon, we shift the horizon just outsize $r=1$
where the velocity is unity. The shock location of the CENBOL is chosen where
the Mach number $M=2$ for simplicity and the compression ratio at the shock is
assumed to be a free parameter. These simplifying assumptions are not expected
to affect our conclusions. Photons are generated from the Keplerian disk
according to the prescription in SS73 as mentioned before and are injected
into the sub-Keplerian halo, the CENBOL and the outflowing jet.
In a simulation, we randomly generated a soft photon out of the Keplerian
disk. The energy of the soft photon at radiation temperature $T(r)$ are
calculated using the Planck’s distribution formula, where the number density
of the photons ($n_{\gamma}(E)$) having an energy $E$ is expressed by
$\displaystyle n_{\gamma}(E)=\frac{1}{2\zeta(3)}b^{3}E^{2}(e^{bE}-1)^{-1},$
(10)
where $b=1/kT(r)$; $\zeta(3)=\sum^{\infty}_{1}{l}^{-3}=1.202$ is the Riemann
zeta function.
Using another set of random numbers we obtained the direction of the injected
photons and with yet another random number we obtained a target optical depth
$\tau_{c}$ at which the scattering takes place. The photon was followed within
the CENBOL till the optical depth ($\tau$) reached $\tau_{c}$. The increase in
optical depth ($d\tau$) during its traveling of a path of length $dl$ inside
the electron cloud is given by: $d\tau=\rho_{n}\sigma dl$, where $\rho_{n}$ is
the electron number density.
The total scattering cross section $\sigma$ is given by Klein-Nishina formula:
$\sigma=\frac{2\pi r_{e}^{2}}{x}\\\
\left[\left(1-\frac{4}{x}-\frac{8}{x^{2}}\right)ln\left(1+x\right)+\frac{1}{2}+\frac{8}{x}-\frac{1}{2\left(1+x\right)^{2}}\right],$
(11)
where, $x$ is given by,
$x=\frac{2E}{mc^{2}}\gamma\left(1-\mu\frac{v}{c}\right),$ (12)
$r_{e}=e^{2}/mc^{2}$ is the classical electron radius and $m$ is the mass of
the electron.
We have assumed here that a photon of energy $E$ and momentum
$\frac{E}{c}\bf{\widehat{\Omega}}$ is scattered by an electron of energy
$\gamma mc^{2}$ and momentum $\overrightarrow{\bf{p}}=\gamma
m\overrightarrow{\bf{v}}$, with
$\gamma=\left(1-\frac{v^{2}}{c^{2}}\right)^{-1/2}$ and
$\mu=\bf{\widehat{\Omega}}.\widehat{\bf{v}}$. At this point a scattering is
allowed to take place. The photon selects an electron and the energy exchange
is computed through Compton or inverse Compton scattering formula. The
electrons are assumed to obey relativistic Maxwell distribution inside the
CENBOL. The number $dN(p)$ of Maxwellian electrons having momentum between
$\vec{p}$ to $\vec{p}+d\vec{p}$ is expressed by,
$\displaystyle
dN(\vec{p})=exp[-(p^{2}c^{2}+m^{2}c^{4})^{1/2}/kT_{e}]d\vec{p}.$ (13)
Generally, the same procedure as in Paper I was used, except that we are now
focusing on those photons also photons which were scattered at least once by
the outflow. We are especially choosing the cases when the jet could play a
major role in shaping the spectrum.
## 3 Results and Discussions
In Fig. 3(a-c) we present the velocity, electron number density and
temperature variations as a function of the radial distance from the black
hole for specific energy $\epsilon=0.015$. $\dot{m_{d}}=0.01$ and
$\dot{m_{h}}=1$ were chosen. Three cases were run by varying the compression
ratio $R$. These are given in Col. 2 of Table 1. The corresponding percentage
of matter going in the outflow is also given in Col. 2. In the left panel, the
bulk velocity variation is shown. The solid, dotted and dashed curves are the
same for $R=2$ (Case 1a), $4$ (Case 1b) and $6$ (Case 1c) respectively. The
same line style is used in other panels. The velocity variation within the jet
does not change with $R$, but the density (in the unit of $cm^{-3}$) does
(middle panel). The doubledot-dashed line gives the velocity variation of the
matter within the jet for all the above cases. The arrows show the direction
of the bulk velocity (radial direction in accretion, vertical direction in
jets). The last panel gives the temperature (in keV) of the electron cloud in
the CENBOL, jet, sub-Keplerian and Keplerian disk. Big dash-dotted line gives
the temperature profile inside the Keplerian disk.
Figure 3: (a-c): Velocity (left), density (middle) and the temperature (right)
profiles of Cases 1(a-c) as described in Table 1 are shown with solid ($R=2$),
dotted ($R=4$) and dashed ($R=6$) curves. $\dot{m_{d}}=0.01$ and
$\dot{m_{h}}=1$ were used. Figure 4: (a-c): Velocity (left), density (middle)
and the temperature (right) profiles of Cases 2(a-c) as described in Table 2
are shown with solid ($\dot{m_{h}}=0.5$), dotted ($1$) and dashed ($1.5$)
curves. $\dot{m_{d}}=1.5$ was used throughout. Velocities are the same for all
the disk accretion rates.
In Figs. 4(a-c), we show the velocity (left), number density of electrons
(middle) and temperature (right) profiles of Cases 2(a-c) as described in
Table 2. Here we have fixed $\dot{m_{d}}=1.5$ and $\dot{m_{h}}$ is varied:
${\dot{m_{h}}}=\ 0.5$ (solid), $1$ (dotted) and $1.5$ (dashed). No jet is
present in this case ($R=1$). To study the effects of bulk motion
Comptonization, the temperature of the electron cloud has been kept low for
these cases. The temperature profile in the different cases has been chosen
according to the Fig. 3b of CT95. The temperature profile of the Keplerian
disk for the above cases has been marked as ‘Disk’ .
Table 1
---
Case | R, $P_{m}$ | $N_{int}$ | $N_{cs}$ | $N_{cenbol}$ | $N_{jet}$ | $N_{subkep}$ | $N_{cap}$ | $p$ | $\alpha$
1a | 2, 58 | 2.7E+08 | 4.03E+08 | 1.35E+07 | 7.48E+07 | 8.39E+08 | 3.35E+05 | 63 | 0.43
1b | 4, 97 | 2.7E+08 | 4.14E+08 | 2.39E+06 | 1.28E+08 | 8.58E+08 | 3.27E+05 | 65 | 1.05
1c | 6, 37 | 2.7E+08 | 3.98E+08 | 5.35E+07 | 4.75E+07 | 8.26E+08 | 3.07E+05 | 62 | -0.4
In Table 1, we summarize the details of all the Cases results of which were
depicted in Fig. 3(a-c). In Col. 1, various Cases are marked. In Col. 2, the
compression ratio ($R$) and percentage $P_{m}$ of the total matter that is
going out as outflow (see, Fig. 2) are listed. In Col. 3, we show the total
number of photons (out of the total injection of $6.4\times 10^{8}$)
intercepted by the CENBOL and jet ($N_{int}$) combined. Column 4 gives the
number of photons ($N_{cs}$) that have suffered Compton scattering inside the
flow. Columns 5, 6 and 7 show the number of scatterings which took place in
the CENBOL ($N_{cenbol}$), in the jet ($N_{jet}$) and in the pre-shock sub-
Keplerian halo ($N_{subkep}$) respectively. A comparison of them will give the
relative importance of these three sub-components of the sub-Keplerian disk.
The number of photons captured ($N_{cap}$) by the black hole is given in Col.
8. In Col. 9, we give the percentage $p$ of the total injected photons that
have suffered scattering through CENBOL and the jet. In Col. 10, we present
the energy spectral index $\alpha$ ($I(E)\sim E^{-\alpha}$) obtained from our
simulations.
Table 2
---
Case | $\dot{m_{h}}$, $\dot{m_{d}}$ | $N_{int}$ | $N_{cs}$ | $N_{ms}$ | $N_{subkep}$ | $N_{cap}$ | $p$ | $\alpha_{1},\alpha_{2}$
2a | 0.5, 1.5 | 1.08E+06 | 2.13E+08 | 7.41E+05 | 3.13E+08 | 1.66E+05 | 33.34 | -0.09, 0.4
2b | 1.0, 1.5 | 1.22E+06 | 3.37E+08 | 1.01E+06 | 6.82E+08 | 2.03E+05 | 52.72 | -0.13, 0.75
2c | 1.5, 1.5 | 1.34E+06 | 4.15E+08 | 1.26E+06 | 1.11E+09 | 2.29E+05 | 64.87 | -0.13, 1.3
In Table 2, we summarize the results of simulations where we have varied
$\dot{m_{d}}$, for a fixed value of $\dot{m_{h}}$. In all of these cases no
jet comes out of the CENBOL (i.e., $R=1$). In the last column, we listed two
spectral slopes $\alpha_{1}$ (from $10$ to $100$keV) and $\alpha_{2}$ (due to
the bulk motion Comptonization). Here, $N_{ms}$ represents the photons that
have suffered scattering between $r_{g}=3$ and the horizon of the black hole.
In Fig. 5, we show the variation of the spectrum in the three simulations
presented in Fig. 3(a-c). The dashed, dash-dotted and doubledot-dashed lines
are for $R=2$ (Case 1a), $R=4$ (Case 1b) and $R=6$ (Case 1c) respectively. The
solid curve gives the spectrum of the injected photons. Since the density,
velocity and temperature profiles of the pre-shock, sub-Keplerian region and
the Keplerian flow are the same in all these cases, we find that the
difference in the spectrum is mainly due to the CENBOL and the jet. In the
case of the strongest shock (compression ratio $R=6$), only $37\%$ of the
total injected matter goes out as the jet. At the same time, due to the shock,
the density of the post-shock region increases by a factor of $6$. Out of the
three cases, the effective density of the matter inside CENBOL is the highest
and that inside the jet is the lowest in this case. Again, due to the shock,
the temperature increases inside the CENBOL and hence the spectrum is the
hardest. Similar effects are seen for moderate shock ($R=4$) and to a lesser
extent, the low strength shock ($R=2$) also. When $R=4$, the density of the
post-shock region increases by the factor of $4$ while almost $97\%$ of total
injected matter (Fig. 2) goes out as the jet reducing the matter density of
the CENBOL significantly. From Table 1 we find that the $N_{cenbol}$ is the
lowest and $N_{jet}$ is the highest in this case (Case 1b). This decreases the
up-scattering and increases the down-scattering of the photons. This explains
why the spectrum is the softest in this case. In the case of low strength
shock ($R=2$), $57\%$ of the inflowing matter goes out as jet, but due to the
shock the density increases by factor of $2$ in the post-shock region. This
makes the density similar to a non-shock case as far as the density is
concerned, but with a little higher temperature of the CENBOL due to the
shock. So the spectrum with the shock would be harder than when the shock is
not present. The disk and the halo accretion rates used for these cases are
$\dot{m_{d}}=0.01$ and $\dot{m_{h}}=1$.
Figure 5: Variation of the emerging spectrum for different compression ratios.
The solid curve is the injected spectrum from the Keplerian disk. The dashed,
dash-dotted and doubledot-dashed lines are for $R=2$ (Case 1a), $R=4$ (Case
1b) and $R=6$ (Case 1c) respectively. The disk and halo accretion rates used
for these cases are $\dot{m_{d}}=0.01$ and $\dot{m_{h}}=1$. See, text for
details.
In Fig. 6, we show the components of the emerging spectrum for all the three
cases presented in Fig. 5. The solid curve is the intensity of all the photons
which suffered at least one scattering. The dashed curve corresponds to the
photons emerging from the CENBOL region and the dash-dotted curve is for the
photons coming out of the jet region. We find that the spectrum from the jet
region is softer than the spectrum from the CENBOL. As $N_{jet}$ increases and
$N_{cenbol}$ decreases, the spectrum from the jet becomes softer because of
two reasons. First, the temperature of the jet is lesser than that of the
CENBOL, so the photons get lesser amount of energy from thermal Comptonization
making the spectrum softer. Second, the photons are down-scattered by the
outflowing jet which eventually make the spectrum softer. We note that a
larger number of photons are present in the spectrum from the jet than the
spectrum from the CENBOL, which shows the photons have actually been down-
scattered. The effect of down-scattering is larger when $R=4$. For $R=2$ also
there is significant amount of down scattered photons. But this number is very
small for the case $R=6$ as $N_{cenbol}$ is much larger than $N_{jet}$ so most
of the photons get up-scattered. The difference between total (solid) and the
sum of the other two regions gives an idea of the contribution from the sub-
Keplerian halo located in the pre-shock region. In our choice of geometry
(half angles of the disk and the jet), the contribution of the pre-shock flow
is significant. In general it could be much less. This is especially true when
the CENBOL is farther out.
Figure 6: (a-c): Variation of the components of the emerging spectrum with the
shock strength (R). The dashed curves correspond to the photons emerging from
the CENBOL region and the dash-dotted curves are for the photons coming out of
the jet region. The solid curve is the spectrum for all the photons that have
suffered scatterings. See, the text for details.
In Fig. 7, the emerging spectra due to the bulk motion Comptonization when the
halo rate is varied. The solid curve is the injected spectrum (modified black
body). The dotted, dashed, and dash-dotted curves are for $\dot{m_{h}}=0.5,\
1$ and $1.5$ respectively. $\dot{m_{d}}=1.5$ for all the cases. The Keplerian
disk extends up to $3r_{g}$. Table 2 summarizes the parameter used and the
results of the simulation. As the halo rate increases, the density of the
CENBOL also increases causing a larger number of scattering. From Fig. 4a, we
noticed that the bulk velocity variation of the electron cloud is the same for
all the four cases. Hence, the case where the density is maximum, the photons
got energized to a very high value due to repeated scatterings with that high
velocity cold matter. As a result, there is a hump in the spectrum around 100
keV energy for all the cases. We find the signature of two power-law regions
in the higher energy part of the spectrum. The spectral indices are given in
Table 2. It is to be noted that $\alpha_{2}$ increases with $\dot{m_{h}}$ and
becomes softer for high $\dot{m_{h}}$. Our geometry here at the inner edge is
conical which is more realistic, unlike a sphere (perhaps nonphysically so) in
Laurent & Titarchuk (2001). This may be the reason why our slope is not the
same as in Laurent & Titarchuk (2001) where $\alpha_{2}=1.9$. In Fig. 8, we
present the components of the emerging spectra. As in Fig. 6, solid curves are
the spectra of all the photons that have suffered scattering. The dashed and
dash-dotted curves are the spectra of photons emitted from inside and outside
of the marginally stable orbit ($3r_{g}$) respectively. The photons from
inside the marginally stable radius are Comptonized by the bulk motion of the
converging infalling matter and produces the power-law tail whose spectral
index is given by $\alpha_{2}$ (Table 2).
Figure 7: Bulk motion Comptonization spectrum. Solid (Injected), dotted
($\dot{M_{h}}=0.5$), dashed ($\dot{M_{h}}=1$), dash-dotted
($\dot{M_{h}}=1.5$). $\dot{M_{d}}=1.5$ for all the cases. Keplerian disk
extends up to $3.1r_{g}$. Table 2 summarizes the parameters used and the
simulation results for these cases. Figure 8: Components of the emerging
spectrum for the Cases 2(a-c). Solid curves are the spectra of all the photons
that have suffered scattering. The dashed and dash-dotted curves are the
spectra of photons which are emitted from inside and outside of the marginally
stable orbit ($3r_{g}$) respectively. The photons from inside the marginally
stable radius are Comptonized by the bulk motion of the infalling matter. Here
the jet is absent.
## 4 Concluding remarks
In this paper, we extended the results of our previous work on Monte-Carlo
simulations (Paper I). We included the outflow in conjunction with the inflow.
The outflow rate was self-consistently computed from the inflow rate using
well-known considerations present in the literature (Das et al. 2001 and
references therein). We compute the effects of the thermal and the bulk motion
Comptonization on the soft photons emitted from a Keplerian disk around a
black hole by the post-shock region of a sub-Keplerian flow which surrounds
the Keplerian disk. A shock in the inflow increases the CENBOL temperature,
increases the electron number density and reduces the bulk velocity. Thermal
Comptonization and bulk motion Comptonization inside the CENBOL increases
photon energy. However, the CENBOL also generates the outflow of matter which
down-scatters the photons to lower energy. We show that the thermal
Comptonization and the bulk motion Comptonization were possible by both the
accretion and the outflows. While the converging flow up-scatters the
radiation, the outflow down-scatters. However, the net effect is not simple.
The outflow parameters are strongly coupled to the inflow parameters and thus
for a given inflow and outflow geometry, the strength of the shock can also
determine whether the net scattering by the jets would be significant or not.
Sometimes the spectrum may become very complex with two power-law indices, one
from thermal and the other from the bulk motion Comptonization. Since the
volume of the jet may be larger than that of the CENBOL, sometimes the number
of scatterings suffered by softer photons from the electrons in the jet may be
high. However, whether the CENBOL or the jet emerging from it will dominate in
shaping the spectrum strongly depends on the geometry of the flow and the
strength of the shock. We also found that the halo can Comptonize and harden
the spectrum even without the CENBOL.
## Acknowledgments
The work of HG is supported by a RESPOND project.
## References
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|
arxiv-papers
| 2010-06-21T10:24:13 |
2024-09-04T02:49:11.068839
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Himadri Ghosh, Sudip K. Garain, Sandip K. Chakrabarti, Philippe\n Laurent",
"submitter": "Himadri Ghosh Mr.",
"url": "https://arxiv.org/abs/1006.4024"
}
|
1006.4071
|
A Generalization of
Seifert-Van Kampen Theorem for Fundamental Groups
Linfan MAO
(Chinese Academy of Mathematics and System Science, Beijing 100080, P.R.China)
E-mail: maolinfan@163.com
Abstract: As we known, the Seifert-Van Kampen theorem handles fundamental
groups of those topological spaces $X=U\cup V$ for open subsets $U,\ V\subset
X$ such that $U\cap V$ is arcwise connected. In this paper, this theorem is
generalized to such a case of maybe not arcwise-connected, i.e., there are
$C_{1}$, $C_{2}$,$\cdots,\ C_{m}$ arcwise-connected components in $U\cap V$
for an integer $m\geq 1$, which enables one to find fundamental groups of
combinatorial spaces by that of spaces with theirs underlying topological
graphs, particularly, that of compact manifolds by their underlying graphs of
charts.
Key Words: Fundamental group, Seifert-Van Kampen theorem, topological space,
combinatorial manifold, topological graph.
AMS(2010): 51H20.
§$1.$ Introduction
All spaces $X$ considered in this paper are arcwise-connected, graphs are
connected topological graph, maybe with loops or multiple edges and interior
of an arc $a:(0,1)\rightarrow X$ is opened. For terminologies and notations
not defined here, we follow the reference [1]-[3] for topology and [4]-[5] for
topological graphs.
Let $X$ be a topological space. A fundamental group $\pi_{1}(X,x_{0})$ of $X$
based at a point $x_{0}\in X$ is formed by homotopy arc classes in $X$ based
at $x_{0}\in X$. For an arcwise-connected space $X$, it is known that
$\pi_{1}(X,x_{0})$ is independent on the base point $x_{0}$, that is, for
$\forall x_{0},y_{0}\in X$,
$\pi_{1}(X,x_{0})\cong\pi_{1}(X,y_{0}).$
Find the fundamental group of a space $X$ is a difficult task in general.
Until today, the basic tool is still the Seifert-Van Kampen theorem following.
Theorem $1.1$(Seifert and Van-Kampen) Let $X=U\cup V$ with $U,\ V$ open
subsets and let $X,\ U,\ V$, $U\cap V$ be non-empty arcwise-connected with
$x_{0}\in U\cap V$ and $H$ a group. If there are homomorphisms
$\phi_{1}:\pi_{1}(U,x_{0})\rightarrow H\ \ {and}\ \
\phi_{2}:\pi_{1}(V,x_{0})\rightarrow H$
and
with $\phi_{1}\cdot i_{1}=\phi_{2}\cdot i_{2}$, where $i_{1}:\pi_{1}(U\cap
V,x_{0})\rightarrow\pi_{1}(U,x_{0})$, $i_{2}:\pi_{1}(U\cap
V,x_{0})\rightarrow\pi_{1}(V,x_{0})$,
$j_{1}:\pi_{1}(U,x_{0})\rightarrow\pi_{1}(X,x_{0})$ and
$j_{2}:\pi_{1}(V,x_{0})\rightarrow\pi_{1}(X,x_{0})$ are homomorphisms induced
by inclusion mappings, then there exists a unique homomorphism $\Phi:\
\pi_{1}(X,x_{0})\rightarrow H$ such that $\Phi\cdot j_{1}=\phi_{1}$ and
$\Phi\cdot j_{2}=\phi_{2}$.
Applying Theorem $1.1$, it is easily to determine the fundamental group of
such spaces $X=U\cup V$ with $U\cap V$ an arcwise-connected following.
Theorem $1.2$(Seifert and Van-Kampen theorem, classical version) Let spaces
$X,U,V$ and $x_{0}$ be in Theorem $1.1$. If
$j:\pi_{1}(U,x_{0})*\pi_{1}(V,x_{0})\rightarrow\pi_{1}(X,x_{0})$
is an extension homomorphism of $j_{1}$ and $j_{2}$, then $j$ is an
epimorphism with kernel Ker$j$ generated by $i_{1}^{-1}(g)i_{2}(g),\
g\in\pi_{1}(U\cap V,x_{0})$, i.e.,
$\pi_{1}(X,x_{0})\cong\frac{\pi_{1}(U,x_{0})*\pi_{1}(V,x_{0})}{\left[i_{1}^{-1}(g)\cdot
i_{2}(g)|\ g\in\pi_{1}(U\cap V,x_{0})\right]},$
where $\left[A\right]$ denotes the minimal normal subgroup of a group
$\mathscr{G}$ included $A\subset\mathscr{G}$.
Now we use the following convention.
Convention $1.3$ Assume that
($1$) $X$ is an arcwise-connected spaces, $x_{0}\in X$;
($2$) $\\{U_{\lambda}:\lambda\in\Lambda\\}$ is a covering of $X$ by arcwise-
connected open sets such that $x_{0}\in U_{\lambda}$ for
$\forall\lambda\in\Lambda$;
($3$) For any two indices $\lambda_{1},\lambda_{2}\in\Lambda$ there exists an
index $\lambda\in\Lambda$ such that $U_{\lambda_{1}}\cap
U_{\lambda_{2}}=U_{\lambda}$
If $U_{\lambda}\subset U_{\mu}\subset X$, then the notation
$\phi_{\lambda\mu}:\pi_{1}(U_{\lambda},x_{0})\rightarrow\pi_{1}(U_{\mu},x_{0})\
\ {\rm and}\ \
\phi_{\lambda}:\pi_{1}(U_{\lambda},x_{0})\rightarrow\pi_{1}(X,x_{0})$
denote homomorphisms induced by the inclusion mapping $U_{\lambda}\rightarrow
U_{\mu}$ and $U_{\lambda}\rightarrow X$, respectively. It should be noted that
the Seifert-Van Kampen theorem has been generalized in [2] under Convention
$1.3$ by any number of open subsets instead of just by two open subsets
following.
Theorem $1.4$([2]) Let $X,U_{\lambda},\ \lambda\in\Lambda$ be arcwise-
connected space with Convention $1.3$ satisfies the following universal
mappping condition: Let $H$ be a group and let
$\rho_{\lambda}:\pi_{1}(U_{\lambda},x_{0})\rightarrow H$ be any collection of
homomorphisms defined for all $\lambda\in\Lambda$ such that the following
diagram is commutative for $U_{\lambda}\subset U_{\mu}$:
Then there exists a unique homomorphism $\Phi:\pi_{1}(X,x_{0})\rightarrow H$
such that for any $\lambda\in\Lambda$ the following diagram is commutative:
Moreover, this universal mapping condition characterizes $\pi_{1}(X,x_{0})$ up
to a unique isomorphism.
Theorem $1.4$ is useful for determining the fundamental groups of CW-
complexes, particularly, the adjunction of $n$-dimensional cells to a space
for $n\geq 2$. Notice that the essence in Theorems $1.2$ and $1.4$ is that
$\cap_{\lambda\in\Lambda}U_{\lambda}$ is arcwise-connected, which limits the
application of such kind of results. The main object of this paper is to
generalize the Seifert-Van Kampen theorem to such an intersection maybe non-
arcwise connected, i.e., there are $C_{1}$, $C_{2}$,$\cdots,\ C_{m}$ arcwise-
connected components in $U\cap V$ for an integer $m\geq 1$. This enables one
to determine the fundamental group of topological spaces, particularly,
combinatorial manifolds introduced in [6]-[8] following which is a special
case of Smarandache multi-space ([9]-[10]).
Definition $1.4$ A combinatorial Euclidean space
$\mathscr{E}_{G}(n_{\nu};\nu\in\Lambda)$ underlying a connected graph $G$ is a
topological spaces consisting of ${\bf R}^{n_{\nu}}$, $\nu\in\Lambda$ for an
index set $\Lambda$ such that
$V(G)=\\{{\bf R}^{n_{\nu}}|\nu\in\Lambda\\}$;
$E(G)=\\{\ ({\bf R}^{n_{\mu}},{\bf R}^{n_{\nu}})|\ {\bf R}^{n_{\mu}}\cap{\bf
R}^{n_{\nu}}\not=\emptyset,\mu,\nu\in\Lambda\\}$.
A combinatorial fan-space $\widetilde{\bf R}(n_{\nu};\nu\in\Lambda)$ is a
combinatorial Euclidean space
$\mathscr{E}_{K_{|\Lambda|}}(n_{\nu};\nu\in\Lambda)$ of ${\bf R}^{n_{\nu}},\
\nu\in\Lambda$ such that for any integers $\mu,\nu\in\Lambda,\ \mu\not=\nu$,
${\bf R}^{n_{\mu}}\bigcap{\bf
R}^{n_{\nu}}=\bigcap\limits_{\lambda\in\Lambda}{\bf R}^{n_{\lambda}},$
which enables us to generalize the conception of manifold to combinatorial
manifold, also a locally combinatorial Euclidean space following.
Definition $1.5$ For a given integer sequence $0<n_{1}<n_{2}<\cdots<n_{m}$,
$m\geq 1$, a topological combinatorial manifold $\widetilde{M}$ is a Hausdorff
space such that for any point $p\in\widetilde{M}$, there is a local chart
$(U_{p},\varphi_{p})$ of $p$, i.e., an open neighborhood $U_{p}$ of $p$ in
$\widetilde{M}$ and a homoeomorphism
$\varphi_{p}:U_{p}\rightarrow\widetilde{\bf
R}(n_{1}(p),n_{2}(p),\cdots,n_{s(p)}(p))=\bigcup\limits_{i=1}^{s(p)}{\bf
R}^{n_{i}(p)}$ with
$\\{n_{1}(p),n_{2}(p),\cdots,n_{s(p)}(p)\\}\subseteq\\{n_{1},n_{2},\cdots,n_{m}\\}$
and
$\bigcup\limits_{p\in\widetilde{M}}\\{n_{1}(p),n_{2}(p),\cdots,n_{s(p)}(p)\\}=\\{n_{1},n_{2},\cdots,n_{m}\\}$,
denoted by $\widetilde{M}(n_{1},n_{2},\cdots,n_{m})$ or $\widetilde{M}$ on the
context and
$\widetilde{{\mathcal{A}}}=\\{(U_{p},\varphi_{p})|p\in\widetilde{M}(n_{1},n_{2},\cdots,n_{m}))\\}$
an atlas on $\widetilde{M}(n_{1},n_{2},\cdots,n_{m})$.
A topological combinatorial manifold $\widetilde{M}(n_{1},n_{2},\cdots,n_{m})$
is finite if it is just combined by finite manifolds without one manifold
contained in the union of others.
If these manifolds $M_{i},\ 1\leq i\leq m$ in
$\widetilde{M}(n_{1},n_{2},\cdots,n_{m})$ are Euclidean spaces ${\bf
R}^{n_{i}},\ 1\leq i\leq m$, then $\widetilde{M}(n_{1},n_{2},\cdots,n_{m})$ is
nothing but a combinatorial Euclidean space
$\mathscr{E}_{G}(n_{\nu};\nu\in\Lambda)$ with $\Lambda=\\{1,2,\cdots,m\\}$.
Furthermore, If $m=1$ and $n_{1}=n$, or $n_{\nu}=n$ for $\nu\in\Lambda$, then
$\widetilde{M}(n_{1},n_{2},\cdots,n_{m})$ or
$\mathscr{E}_{G}(n_{\nu};\nu\in\Lambda)$ is exactly a manifold $M^{n}$ by
definition.
§$2.$ Topological Space Attached Graphs
A topological graph $G$ is itself a topological space formally defined as
follows.
Definition $2.1$ A topological graph $G$ is a pair $(S,S^{0})$ of a Hausdorff
space $S$ with its a subset $S^{0}$ such that
($1$) $S^{0}$ is discrete, closed subspaces of $S$;
($2$) $S-S^{0}$ is a disjoint union of open subsets
$e_{1},e_{2},\cdots,e_{m}$, each of which is homeomorphic to an open interval
$(0,1)$;
($3$) the boundary $\overline{e}_{i}-e_{i}$ of $e_{i}$ consists of one or two
points. If $\overline{e}_{i}-e_{i}$ consists of two points, then
$(\overline{e}_{i},e_{i})$ is homeomorphic to the pair $([0,1],(0,1))$; if
$\overline{e}_{i}-e_{i}$ consists of one point, then
$(\overline{e}_{i},e_{i})$ is homeomorphic to the pair
$(S^{1},S^{1}-\\{1\\})$;
($4$) a subset $A\subset G$ is open if and only if $A\cap\overline{e}_{i}$ is
open for $1\leq i\leq m$.
Fig.$2.1$
Notice that a topological graph maybe with semi-edges, i.e., those edges
$e^{+}\in E(G)$ with $e^{+}:[0,1)\ {\rm or}\ (0,1]\rightarrow S$. A
topological space $X$ attached with a graph $G$ is such a space $X\odot G$
such that
$X\bigcap G\not=\emptyset,\ \ G\not\subset X$
and there are semi-edges $e^{+}\in(X\bigcap G)\setminus G$, $e^{+}\in
G\setminus X$. An example for $X\odot G$ can be found in Fig.$2.1$. In this
section, we characterize the fundamental groups of such topological spaces
attached with graphs.
Theorem $2.2$ Let $X$ be arc-connected space, $G$ a graph and $H$ the subgraph
$X\cap G$ in $X\odot G$. Then for $x_{0}\in X\cap G$,
$\pi_{1}(X\odot
G,x_{0})\cong\frac{\pi_{1}(X,x_{0})*\pi_{1}(G,x_{0})}{\left[i_{1}^{-1}(\alpha_{e_{\lambda}})i_{2}(\alpha_{e_{\lambda}})|\
e_{\lambda}\in E(H)\setminus T_{span})\right]},$
where $i_{1}:\pi_{1}(H,x_{0})\rightarrow X$,
$i_{2}:\pi_{1}(H,x_{0})\rightarrow G$ are homomorphisms induced by inclusion
mappings, $T_{span}$ is a spanning tree in $H$,
$\alpha_{\lambda}=A_{\lambda}e_{\lambda}B_{\lambda}$ is a loop associated with
an edge $e_{\lambda}=a_{\lambda}b_{\lambda}\in H\setminus T_{span}$, $x_{0}\in
G$ and $A_{\lambda}$, $B_{\lambda}$ are unique paths from $x_{0}$ to
$a_{\lambda}$ or from $b_{\lambda}$ to $x_{0}$ in $T_{span}$.
Proof This result is an immediately conclusion of Seifert-Van Kampen theorem.
Let $U=X$ and $V=G$. Then $X\odot G=X\cup G$ and $X\cap G=H$. By definition,
there are both semi-edges in $G$ and $H$. Whence, they are opened. Applying
the Seifert-Van Kampen theorem, we get that
$\pi_{1}(X\odot
G,x_{0})\cong\frac{\pi_{1}(X,x_{0})*\pi_{1}(G,x_{0})}{\left[i_{1}^{-1}(g)i_{2}(g)|\
g\in\pi_{1}(X\cap G,x_{0})\right]},$
Notice that the fundamental group of a graph $H$ is completely determined by
those of its cycles ([2]), i.e.,
$\pi_{1}(H,x_{0})=\left<\alpha_{\lambda}|e_{\lambda}\in E(H)\setminus
T_{span}\right>,$
where $T_{span}$ is a spanning tree in $H$,
$\alpha_{\lambda}=A_{\lambda}e_{\lambda}B_{\lambda}$ is a loop associated with
an edge $e_{\lambda}=a_{\lambda}b_{\lambda}\in H\setminus T_{span}$, $x_{0}\in
G$ and $A_{\lambda}$, $B_{\lambda}$ are unique paths from $x_{0}$ to
$a_{\lambda}$ or from $b_{\lambda}$ to $x_{0}$ in $T_{span}$. We finally get
the following conclusion,
$\hskip 56.9055pt\pi_{1}(X\odot
G,x_{0})\cong\frac{\pi_{1}(X,x_{0})*\pi_{1}(G,x_{0})}{\left[i_{1}^{-1}(\alpha_{e_{\lambda}})i_{2}(\alpha_{e_{\lambda}})|\
e_{\lambda}\in E(H)\setminus T_{span})\right]}\hskip 56.9055pt\Box$
Corollary $2.3$ Let $X$ be arc-connected space, $G$ a graph. If $X\cap G$ in
$X\odot G$ is a tree, then
$\pi_{1}(X\odot G,x_{0})\cong\pi_{1}(X,x_{0})*\pi_{1}(G,x_{0}).$
Particularly, if $G$ is graphs shown in Fig.$2.2$ following
Fig.$2.2$
and $X\cap G=K_{1,m}$, Then
$\pi_{1}(X\odot B_{m}^{T},x_{0})\cong\pi_{1}(X,x_{0})*\left<L_{i}|1\leq i\leq
m\right>,$
where $L_{i}$ is the loop of parallel edges $(x_{0},x_{i})$ in $B_{m}^{T}$ for
$1\leq i\leq m-1$ and
$\pi_{1}(X\odot S_{m}^{T},x_{0})\cong\pi_{1}(X,x_{0}).$
Theorem $2.4$ Let $\mathscr{X}_{m}\odot G$ be a topological space consisting
of $m$ arcwise-connected spaces $X_{1},X_{2},\cdots,X_{m}$, $X_{i}\cap
X_{j}=\emptyset$ for $1\leq i,j\leq m$ attached with a graph $G$,
$V(G)=\\{x_{0},x_{1},\cdots,x_{l-1}\\}$, $m\leq l$ such that $X_{i}\cap
G=\\{x_{i}\\}$ for $0\leq i\leq l-1$. Then
$\displaystyle\pi_{1}(\mathscr{X}_{m}\odot G,x_{0})$ $\displaystyle\cong$
$\displaystyle\left(\prod\limits_{i=1}^{m}\pi_{1}(X_{i}^{*},x_{0})\right)*\pi_{1}(G,x_{0})$
$\displaystyle\cong$
$\displaystyle\left(\prod\limits_{i=1}^{m}\pi_{1}(X_{i},x_{i})\right)*\pi_{1}(G,x_{0}),$
where $X_{i}^{*}=X_{i}\bigcup(x_{0},x_{i})$ with
$X_{i}\cap(x_{0},x_{i})=\\{x_{i}\\}$ for $(x_{0},x_{i})\in E(G)$, integers
$1\leq i\leq m$.
Proof The proof is by induction on $m$. If $m=1$, the result is hold by
Corollary $2.3$.
Now assume the result on $\mathscr{X}_{m}\odot G$ is hold for $m\leq k<l-1$.
Consider $m=k+1\leq l$. Let $U=\mathscr{X}_{k}\odot G$ and $V=X_{k+1}$. Then
we know that $\mathscr{X}_{k+1}\odot G=U\cup V$ and $U\cap V=\\{x_{k+1}\\}$.
Applying the Seifert-Van Kampen theorem, we find that
$\displaystyle\pi_{1}(\mathscr{X}_{k+1}\odot G,x_{k+1})$ $\displaystyle\cong$
$\displaystyle\frac{\pi_{1}(U,x_{k+1})*\pi_{1}(V,x_{k+1})}{\left[i_{1}^{-1}(g)i_{2}(g)|\
g\in\pi_{1}(U\cap V,x_{k+1})\right]}$ $\displaystyle\cong$
$\displaystyle\frac{\pi_{1}(\mathscr{X}_{k}\odot
G,x_{0})*\pi_{1}(X_{k+1},x_{k+1})}{\left[i_{1}^{-1}(g)i_{2}(g)|\ g\in\\{{\bf
e}_{x_{k+1}}\\}\right]}$ $\displaystyle\cong$
$\displaystyle\left(\left(\prod\limits_{i=1}^{k}\pi_{1}(X_{i}^{*},x_{0})\right)*\pi_{1}(G,x_{0})\right)*\pi_{1}(X_{k+1},x_{k+1})$
$\displaystyle\cong$
$\displaystyle\left(\prod\limits_{i=1}^{k+1}\pi_{1}(X_{i}^{*},x_{0})\right)*\pi_{1}(G,x_{0})$
$\displaystyle\cong$
$\displaystyle\left(\prod\limits_{i=1}^{m}\pi_{1}(X_{i},x_{i})\right)*\pi_{1}(G,x_{0}),$
by the induction assumption. $\Box$
Particularly, for the graph $B_{m}^{T}$ or star $S_{m}^{T}$ in Fig.$2.2$, we
get the following conclusion.
Corollary $2.5$ Let $G$ be the graph $B_{m}^{T}$ or star $S_{m}^{T}$. Then
$\displaystyle\pi_{1}(\mathscr{X}_{m}\odot B_{m}^{T},x_{0})$
$\displaystyle\cong$
$\displaystyle\left(\prod\limits_{i=1}^{m}\pi_{1}(X_{i}^{*},x_{0})\right)*\pi_{1}(B_{m}^{T},x_{0})$
$\displaystyle\cong$
$\displaystyle\left(\prod\limits_{i=1}^{m}\pi_{1}(X_{i},x_{i-1})\right)*\left<L_{i}|1\leq
i\leq m\right>,$
where $L_{i}$ is the loop of parallel edges $(x_{0},x_{i})$ in $B_{m}^{T}$ for
integers $1\leq i\leq m-1$ and
$\pi_{1}(\mathscr{X}_{m}\odot
S_{m}^{T},x_{0})\cong\prod\limits_{i=1}^{m}\pi_{1}(X_{i}^{*},x_{0})\cong\prod\limits_{i=1}^{m}\pi_{1}(X_{i},x_{i-1}).$
Corollary $2.6$ Let $X=\mathscr{X}_{m}\odot G$ be a topological space with
simply-connected spaces $X_{i}$ for integers $1\leq i\leq m$ and $x_{0}\in
X\cap G$. Then we know that
$\pi_{1}(X,x_{0})\cong\pi_{1}(G,x_{0}).$
§$3.$ A Generalization of Seifert-Van Kampen Theorem
These results and graph $B_{m}^{T}$ shown in Section $2$ enables one to
generalize the Seifert-Van Kampen theorem to the case of $U\cap V$ maybe not
arcwise-connected.
Theorem $3.1$ Let $X=U\cup V$, $U,V\subset X$ be open subsets, $X,\ U,\ V$
arcwise-connected and let $C_{1},C_{2},\cdots,C_{m}$ be arcwise-connected
components in $U\cap V$ for an integer $m\geq 1$, $x_{i-1}\in C_{i}$,
$b(x_{0},x_{i-1})\subset V$ an arc $:I\rightarrow X$ with
$b(0)=x_{0},b(1)=x_{i-1}$ and $b(x_{0},x_{i-1})\cap U=\\{x_{0},x_{i-1}\\}$,
$C_{i}^{E}=C_{i}\bigcup b(x_{0},x_{i-1})$ for any integer $i,\ 1\leq i\leq m$,
$H$ a group and there are homomorphisms
$\phi_{1}^{i}:\pi_{1}(U\bigcup b(x_{0},x_{i-1}),x_{0})\rightarrow H,\ \
\phi_{2}^{i}:\pi_{1}(V,x_{0})\rightarrow H$
such that
with $\phi_{1}^{i}\cdot i_{i1}=\phi_{2}^{i}\cdot i_{i2}$, where
$i_{i1}:\pi_{1}(C_{i}^{E},x_{0})\rightarrow\pi_{1}(U\cup
b(x_{0},x_{i-1}),x_{0})$,
$i_{i2}:\pi_{1}(C_{i}^{E},x_{0})\rightarrow\pi_{1}(V,x_{0})$ and
$j_{i1}:\pi_{1}(U\cup b(x_{0},x_{i-1},x_{0}))\rightarrow\pi_{1}(X,x_{0})$,
$j_{i2}:\pi_{1}(V,x_{0}))\rightarrow\pi_{1}(X,x_{0})$ are homomorphisms
induced by inclusion mappings, then there exists a unique homomorphism $\Phi:\
\pi_{1}(X,x_{0})\rightarrow H$ such that $\Phi\cdot j_{i1}=\phi_{1}^{i}$ and
$\Phi\cdot j_{i2}=\phi_{2}^{i}$ for integers $1\leq i\leq m$.
Proof Define $U^{E}=U\bigcup\\{\ b(x_{0},x_{i})\ |\ 1\leq i\leq m-1\\}$. Then
we get that $X=U^{E}\cup V$, $U^{E},V\subset X$ are still opened with an
arcwise-connected intersection $U^{E}\cap V=\mathscr{X}_{m}\odot S_{m}^{T}$,
where $S_{m}^{T}$ is a graph formed by arcs $b(x_{0},x_{i-1})$, $1\leq i\leq
m$.
Notice that $\mathscr{X}_{m}\odot Sm^{T}=\bigcup\limits_{i=1}^{m}C_{i}^{E}$
and $C_{i}^{E}\bigcap C_{j}^{E}=\\{x_{0}\\}$ for $1\leq i,j\leq m,\ i\not=j$.
Therefore, we get that
$\pi_{1}(\mathscr{X}_{m}\odot
S_{m}^{T},x_{0})=\bigotimes\limits_{i=1}^{m}\pi_{1}(C_{i}^{E},x_{0}).$
This fact enables us knowing that there is a unique $m$-tuple
$(h_{1},h_{2},\cdots,h_{m})$, $h_{i}\in\pi_{1}(C_{i}^{E},x_{i-1}),\ 1\leq
i\leq m$ such that
$\mathscr{I}=\prod\limits_{i=1}^{m}h_{i}$
for $\forall\mathscr{I}\in\pi_{1}(\mathscr{X}_{m}\odot S_{m}^{T},x_{0})$.
By definition,
$i_{i1}:\pi_{1}(C_{i}^{E},x_{0})\rightarrow\pi_{1}(U\cap
b(x_{0},x_{i-1}),x_{0}),$
$i_{i2}:\pi_{1}(C_{i}^{E},x_{0})\rightarrow\pi_{1}(V,x_{0})$
are homomorphisms induced by inclusion mappings. We know that there are
homomorphisms
$i_{1}^{E}:\pi_{1}(\mathscr{X}_{m}\odot
S_{m}^{T},x_{0})\rightarrow\pi_{1}(U^{E},x_{0}),$
$i_{2}^{E}:\pi_{1}(\mathscr{X}_{m}\odot
S_{m}^{T},x_{0})\rightarrow\pi_{1}(V,x_{0})$
with $i_{1}^{E}|_{\pi_{1}(C_{i}^{E},x_{0})}=i_{i1}$,
$i_{2}^{E}|_{\pi_{1}(C_{i}^{E},x_{0})}=i_{i2}$ for integers $1\leq i\leq m$.
Similarly, because of
$\pi_{1}(U^{E},x_{0})=\bigcup\limits_{i=1}^{m}\pi_{1}(U\cup
b(x_{0},x_{i-1},x_{0}))$
and
$j_{i1}:\pi_{1}(U\cup b(x_{0},x_{i-1},x_{0}))\rightarrow\pi_{1}(X,x_{0}),$
$j_{i2}:\pi_{1}(V\rightarrow\pi_{1}(X,x_{0})$
being homomorphisms induced by inclusion mappings, there are homomorphisms
$j_{1}^{E}:\pi_{1}(U^{E},x_{0})\rightarrow\pi_{1}(X,x_{0}),\ \
j_{2}^{E}:\pi_{1}(V,x_{0})\rightarrow\pi_{1}(X,x_{0})$
induced by inclusion mappings with $j_{1}^{E}|_{\pi_{1}(U\cup
b(x_{0},x_{i-1},x_{0}))}=j_{i1}$, $j_{2}^{E}|_{\pi_{1}(V,x_{0})}=j_{i2}$ for
integers $1\leq i\leq m$ also.
Define $\phi_{1}^{E}$ and $\phi_{2}^{E}$ by
$\phi_{1}^{E}(\mathscr{I})=\prod\limits_{i=1}^{m}\phi_{1}^{i}(i_{i1}(h_{i})),\
\
\phi_{2}^{E}(\mathscr{I})=\prod\limits_{i=1}^{m}\phi_{2}^{i}(i_{i2}(h_{i})).$
Then they are naturally homomorphic extensions of homomorphisms
$\phi_{1}^{i},\ \phi_{2}^{i}$ for integers $1\leq i\leq m$. Notice that
$\phi_{1}^{i}\cdot i_{i1}=\phi_{2}^{i}\cdot i_{i2}$ for integers $1\leq i\leq
m$, we get that
$\displaystyle\phi_{1}^{E}\cdot i_{1}^{E}(\mathscr{I})$ $\displaystyle=$
$\displaystyle\phi_{1}^{E}\cdot
i_{1}^{E}\left(\prod\limits_{i=1}^{m}h_{i}\right)$ $\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{m}\left(\phi_{1}^{i}\cdot
i_{i1}(h_{i})\right)=\prod\limits_{i=1}^{m}\left(\phi_{2}^{i}\cdot
i_{i2}(h_{i})\right)$ $\displaystyle=$ $\displaystyle\phi_{2}^{E}\cdot
i_{2}^{E}\left(\prod\limits_{i=1}^{m}h_{i}\right)=\phi_{2}^{E}\cdot
i_{2}^{E}(\mathscr{I}),$
i.e., the following diagram
is commutative with $\phi_{1}^{E}\cdot i_{1}^{E}=\phi_{2}^{E}\cdot i_{2}^{E}$.
Applying Theorem $1.1$, we know that there exists a unique homomorphism
$\Phi:\ \pi_{1}(X,x_{0})\rightarrow H$ such that $\Phi\cdot
j_{1}^{E}=\phi_{1}^{E}$ and $\Phi\cdot j_{2}^{E}=\phi_{2}^{E}$. Whence,
$\Phi\cdot j_{i1}=\phi_{1}^{i}$ and $\Phi\cdot j_{i2}=\phi_{2}^{i}$ for
integers $1\leq i\leq m$. $\Box$
The following result is a generalization of the classical Seifert-Van Kampen
theorem to the case of maybe non-arcwise connected.
Theorem $3.2$ Let $X$, $U$, $V$, $C_{i}^{E}$, $b(x_{0},x_{i-1})$ be arcwise-
connected spaces for any integer $i,\ 1\leq i\leq m$ as in Theorem $3.1$,
$U^{E}=U\bigcup\\{\ b(x_{0},x_{i})\ |\ 1\leq i\leq m-1\\}$ and $B_{m}^{T}$ a
graph formed by arcs $a(x_{0},x_{i-1})$, $b(x_{0},x_{i-1})$, $1\leq i\leq m$,
where $a(x_{0},x_{i-1})\subset U$ is an arc $:I\rightarrow X$ with
$a(0)=x_{0},a(1)=x_{i-1}$ and $a(x_{0},x_{i-1})\cap V=\\{x_{0},x_{i-1}\\}$.
Then
$\pi_{1}(X,x_{0})\cong\frac{\pi_{1}(U,x_{0})*\pi_{1}(V,x_{0})*\pi_{1}(B_{m}^{T},x_{0})}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}(g)|\ g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i}^{E},x_{0})\ \right]},$
where $i_{1}^{E}:\pi_{1}(U^{E}\cap V,x_{0})\rightarrow\pi_{1}(U^{E},x_{0})$
and $i_{2}^{E}:\pi_{1}(U^{E}\cap V,x_{0})\rightarrow\pi_{1}(V,x_{0})$ are
homomorphisms induced by inclusion mappings.
Proof Similarly, $X=U^{E}\cup V$, $U^{E},V\subset X$ are opened with
$U^{E}\cap V=\mathscr{X}_{m}\odot S_{m}^{T}$. By the proof of Theorem $3.1$ we
have known that there are homomorphisms $\phi_{1}^{E}$ and $\phi_{2}^{E}$ such
that $\phi_{1}^{E}\cdot i_{1}^{E}=\phi_{2}^{E}\cdot i_{2}^{E}$. Applying
Theorem $1.2$, we get that
$\pi_{1}(X,x_{0})\cong\frac{\pi_{1}(U^{E},x_{0})*\pi_{1}(V,x_{0})}{\left[(i_{1}^{E})^{-1}(\mathscr{I})\cdot
i_{2}^{E}(\mathscr{I})|\mathscr{I}\in\pi_{1}(U^{E}\cap V,x_{0})\right]}.$
Notice that $U^{E}\cap V^{E}=\mathscr{X}_{m}\odot S_{m}^{T}$. We have known
that
$\pi_{1}(U^{E},x_{0})\cong\pi_{1}(U,x_{0})*\pi_{1}(B_{m}^{T},x_{0})$
by Corollary $2.3$. As we have shown in the proof of Theorem $3.1$, an element
$\mathscr{I}$ in $\pi_{1}(\mathscr{X}_{m}\odot S_{m}^{T},x_{0})$ can be
uniquely represented by
$\mathscr{I}=\prod\limits_{i=1}^{m}h_{i},$
where $h_{i}\in\pi_{1}(C_{i}^{E},x_{0}),\ 1\leq i\leq m$. We finally get that
$\hskip
56.9055pt\pi_{1}(X,x_{0})\cong\frac{\pi_{1}(U,x_{0})*\pi_{1}(V,x_{0})*\pi_{1}(B_{m}^{T},x_{0})}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i}^{E},x_{0})\
\right]}.\hskip 56.9055pt\Box$
The form of elements in $\pi_{1}(\mathscr{X}_{m}\odot S_{m}^{T},x_{0})$
appeared in Corollary $2.5$ enables one to obtain another generalization of
classical Seifert-Van Kampen theorem following.
Theorem $3.3$ Let $X$, $U$, $V$, $C_{1},C_{2},\cdots,C_{m}$ be arcwise-
connected spaces, $b(x_{0},x_{i-1})$ arcs for any integer $i,\ 1\leq i\leq m$
as in Theorem $3.1$, $U^{E}=U\bigcup\\{\ b(x_{0},x_{i-1})\ |\ 1\leq i\leq
m\\}$ and $B_{m}^{T}$ a graph formed by arcs $a(x_{0},x_{i-1})$,
$b(x_{0},x_{i-1})$, $1\leq i\leq m$. Then
$\pi_{1}(X,x_{0})\cong\frac{\pi_{1}(U,x_{0})*\pi_{1}(V,x_{0})*\pi_{1}(B_{m}^{T},x_{0})}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i},x_{i-1})\right]},$
where $i_{1}^{E}:\pi_{1}(U^{E}\cap V,x_{0})\rightarrow\pi_{1}(U^{E},x_{0})$
and $i_{2}^{E}:\pi_{1}(U^{E}\cap V,x_{0})\rightarrow\pi_{1}(V,x_{0})$ are
homomorphisms induced by inclusion mappings.
Proof Notice that $U^{E}\cap V=\mathscr{X}_{m}\odot S^{T}_{m}$. Applying
Corollary $2.5$, replacing
$\pi_{1}(\mathscr{X}_{m}\odot S_{m}^{T},x_{0})=\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i}^{E},x_{0})\right]$
by
$\pi_{1}(\mathscr{X}_{m}\odot S_{m}^{T},x_{0})=\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i},x_{i-1})\right]$
in the proof of Theorem $3.2$. We get this conclusion. $\Box$
Particularly, we get corollaries following by Theorems $3.1$, $3.2$ and $3.3$.
Corollary $3.4$ Let $X=U\cup V$, $U,V\subset X$ be open subsets and $X,\ U,\
V$ and $U\cap V$ arcwise-connected. Then
$\pi_{1}(X,x_{0})\cong\frac{\pi_{1}(U,x_{0})*\pi_{1}(V,x_{0})}{\left[i_{1}^{-1}(g)\cdot
i_{2}(g)|\ g\in\pi_{1}(U\cap V,x_{0})\right]},$
where $i_{1}:\pi_{1}(U\cap V,x_{0})\rightarrow\pi_{1}(U,x_{0})$ and
$i_{2}:\pi_{1}(U\cap V,x_{0})\rightarrow\pi_{1}(V,x_{0})$ are homomorphisms
induced by inclusion mappings.
Corollary $3.5$ Let $X$, $U$, $V$, $C_{i}$, $a(x_{0},x_{i})$, $b(x_{0},x_{i})$
for integers $i,\ 1\leq i\leq m$ be as in Theorem $3.1$. If each $C_{i}$ is
simply-connected, then
$\pi_{1}(X,x_{0})\cong\pi_{1}(U,x_{0})*\pi_{1}(V,x_{0})*\pi_{1}(B_{m}^{T},x_{0}).$
Proof Notice that $C_{1}^{E},C_{2}^{E},\cdots,C_{m}^{E}$ are all simply-
connected by assumption. Applying Theorem $3.3$, we easily get this
conclusion. $\Box$
Corollary $3.6$ Let $X$, $U$, $V$, $C_{i}$, $a(x_{0},x_{i})$, $b(x_{0},x_{i})$
for integers $i,\ 1\leq i\leq m$ be as in Theorem $3.1$. If $V$ is simply-
connected, then
$\pi_{1}(X,x_{0})\cong\frac{\pi_{1}(U,x_{0})*\pi_{1}(B_{m}^{T},x_{0})}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i}^{E},x_{0})\ \right]},$
where $i_{1}^{E}:\pi_{1}(U^{E}\cap V,x_{0})\rightarrow\pi_{1}(U^{E},x_{0})$
and $i_{2}^{E}:\pi_{1}(U^{E}\cap V,x_{0})\rightarrow\pi_{1}(V,x_{0})$ are
homomorphisms induced by inclusion mappings.
§$4.$ Fundamental Groups of Combinatorial Spaces
4.1 Fundamental groups of combinatorial manifolds
By definition, a combinatorial manifold $\widetilde{M}$ is arcwise-connected.
So we can apply Theorems $3.2$ and $3.3$ to find its fundamental group
$\pi_{1}(\widetilde{M})$ up to isomorphism in this section.
Definition $4.1$ Let $\widetilde{M}$ be a combinatorial manifold underlying a
graph $G[\widetilde{M}]$. An edge-induced graph $G^{\theta}[\widetilde{M}]$ is
defined by
$V(G^{\theta}[\widetilde{M}])=\\{x_{M},x_{M^{\prime}},x_{1},x_{2},\cdots,x_{\mu(M,M^{\prime})}|\
for\ \forall(M,M^{\prime})\in E(G[\widetilde{M}])\\},$
$E(G^{\theta}[\widetilde{M}])=\\{(x_{M},x_{M^{\prime}}),(x_{M},x_{i}),(x_{M^{\prime}},x_{i})|\
1\leq i\leq\mu(M,M^{\prime})\\},$
where $\mu(M,M^{\prime})$ is called the edge-index of $(M,M^{\prime})$ with
$\mu(M,M^{\prime})+1$ equal to the number of arcwise-connected components in
$M\cap M^{\prime}$.
By the definition of edge-induced graph, we finally get
$G^{\theta}[\widetilde{M}]$ of a combinatorial manifold $\widetilde{M}$ if we
replace each edge $(M,M^{\prime})$ in $G[\widetilde{M}]$ by a subgraph
$TB_{\mu(M,M^{\prime})}^{T}$ shown in Fig.$4.1$ with $x_{M}=M$ and
$x_{M^{\prime}}=M^{\prime}$.
Fig.$4.1$
Then we have the following result.
Theorem $4.2$ Let $\widetilde{M}$ be a finitely combinatorial manifold. Then
$\pi_{1}(\widetilde{M})\cong\frac{\left(\prod\limits_{M\in
V(G[\widetilde{M}])}\pi_{1}(M)\right)*\pi_{1}(G^{\theta}[\widetilde{M}])}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ g\in\prod\limits_{(M_{1},M_{2})\in
E(G[\widetilde{M}])}\pi_{1}(M_{1}\bigcap M_{2})\right]},$
where $i_{1}^{E}$ and $i_{2}^{E}$ are homomorphisms induced by inclusion
mappings $i_{M}:\pi_{1}(M\cap M^{\prime})\rightarrow\pi_{1}(M)$,
$i_{M^{\prime}}:\pi_{1}(M\cap M^{\prime})\rightarrow\pi_{1}(M^{\prime})$ such
as those shown in the following diagram:
for $\forall(M,M^{\prime})\in E(G[\widetilde{M}])$.
Proof This result is obvious for $|G[\widetilde{M}]|=1$. Notice that
$G^{\theta}[\widetilde{M}]=B_{\mu(M,M^{\prime})+1}^{T}$ if
$V(G[\widetilde{M}])=\\{M,\ M^{\prime}\\}$. Whence, it is an immediately
conclusion of Theorem $3.2$ for $|G[\widetilde{M}]|=2$.
Now let $k\geq 3$ be an integer. If this result is true for
$|G[\widetilde{M}]|\leq k$, we prove it hold for $|G[\widetilde{M}]|=k$. It
should be noted that for an arcwise-connected graph $H$ we can always find a
vertex $v\in V(H)$ such that $H-v$ is also arcwise-connected. Otherwise, each
vertex $v$ of $H$ is a cut vertex. There must be $|H|=1$, a contradiction.
Applying this fact to $G[\widetilde{M}]$, we choose a manifold $M\in
V(G[\widetilde{M}])$ such that $\widetilde{M}-M$ is arcwise-connected, which
is also a finitely combinatorial manifold.
Let $U=\widetilde{M}\setminus(M\setminus\widetilde{M})$ and $V=M$. By
definition, they are both opened. Applying Theorem $3.2$, we get that
$\pi_{1}(\widetilde{M})\cong\frac{\pi_{1}(\widetilde{M}-M)*\pi_{1}(M)*\pi_{1}(B_{m}^{T})}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i})\ \right]},$
where $C_{i}$ is an arcwise-connected component in $M\cap(\widetilde{M}-M)$
and
$m=\sum\limits_{(M,M^{\prime})\in E(G[\widetilde{M}])}\mu(M,M^{\prime}).$
Notice that
$\pi_{1}(B_{m}^{T})\cong\prod\limits_{(M,M^{\prime})\in
E(G[\widetilde{M}]}\pi_{1}(TB_{\mu(M,M^{\prime})}).$
By the induction assumption, we know that
$\pi_{1}(\widetilde{M}-M)\cong\frac{\left(\displaystyle\prod\limits_{M\in
V(G[\widetilde{M}-M])}\pi_{1}(M)\right)*\pi_{1}(G^{\theta}[\widetilde{M}-M])}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ \displaystyle g\in\prod\limits_{(M_{1},M_{2})\in
E(G[\widetilde{M}-M])}\pi_{1}(M_{1}\cap M_{2})\right]},$
where $i_{1}^{E}$ and $i_{2}^{E}$ are homomorphisms induced by inclusion
mappings $i_{M_{1}}:\pi_{1}(M_{1}\cap M_{2})\rightarrow\pi_{1}(M_{1})$,
$i_{M_{2}}:\pi_{1}(M_{1}\cap M_{2})\rightarrow\pi_{1}(M_{2})$ for
$\forall(M_{1},M_{2})\in E(G[\widetilde{M}-M])$. Therefore, we finally get
that
$\displaystyle\pi_{1}(\widetilde{M})$ $\displaystyle\cong$
$\displaystyle\frac{\pi_{1}(\widetilde{M}-M)*\pi_{1}(M)*\pi_{1}(B_{m}^{T})}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ \displaystyle g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i})\
\right]}$ $\displaystyle\cong$
$\displaystyle\frac{\frac{\left(\displaystyle\prod\limits_{M\in
V(G[\widetilde{M}-M])}\pi_{1}(M)\right)\displaystyle*\pi_{1}(G^{\theta}[\widetilde{M}-M])}{\left[\displaystyle(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ g\in\prod\limits_{(M_{1},M_{2})\in
E(G[\widetilde{M}-M])}\pi_{1}(M_{1}\cap
M_{2})\right]}}{\left[\displaystyle(i_{1}^{E})^{-1}(g)\cdot i_{2}(g)|\
g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i})\ \right]}$ $\displaystyle*$
$\displaystyle\frac{\pi_{1}(M)*\displaystyle\prod\limits_{(M,M^{\prime})\in
E(G[\widetilde{M}]}\pi_{1}(TB_{\mu(M,M^{\prime})})}{\left[\displaystyle(i_{1}^{E})^{-1}(g)\cdot
i_{2}(g)|\ g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i})\ \right]}$
$\displaystyle\cong$ $\displaystyle\frac{\left(\displaystyle\prod\limits_{M\in
V(G[\widetilde{M}])}\pi_{1}(M)\right)*\pi_{1}(G^{\theta}[\widetilde{M}])}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ \displaystyle g\in\prod\limits_{(M_{1},M_{2})\in
E(G[\widetilde{M}])}\pi_{1}(M_{1}\bigcap M_{2})\right]}$
by facts
$\left(\mathscr{G}/\mathscr{H}\right)*H\cong\mathscr{G}*H/\mathscr{H}$
for groups $\mathscr{G,\ H}$, $G$ and
$G^{\theta}[\widetilde{M}]=G^{\theta}[\widetilde{M}-M]\bigcup\limits_{(M,M^{\prime})\in
E(G[\widetilde{M}]}TB_{\mu(M,M^{\prime})},$
$\pi_{1}(G^{\theta}[\widetilde{M}])=\pi_{1}(G^{\theta}[\widetilde{M}-M])*\prod\limits_{(M,M^{\prime})\in
E(G[\widetilde{M}]}\pi_{1}(TB_{\mu(M,M^{\prime})}),$ $\prod\limits_{M\in
V(G[\widetilde{M}])}\pi_{1}(M)=\left(\prod\limits_{M\in
V(G[\widetilde{M}-M])}\pi_{1}(M)\right)*\pi_{1}(M),$
where $i_{1}^{E}$ and $i_{2}^{E}$ are homomorphisms induced by inclusion
mappings $i_{M}:\pi_{1}(M\cap M^{\prime})\rightarrow\pi_{1}(M)$,
$i_{M^{\prime}}:\pi_{1}(M\cap M^{\prime})\rightarrow\pi_{1}(M^{\prime})$ for
$\forall(M,M^{\prime})\in E(G[\widetilde{M}])$. This completes the proof.
$\Box$
Applying Corollary $3.5$, we get a result known in [8] by noted that
$G^{\theta}[\widetilde{M}]=G[\widetilde{M}]$ if $\forall(M_{1},M_{2})\in
E(G^{L}[\widetilde{M}])$, $M_{1}\cap M_{2}$ is simply connected.
Corollary $4.3$([8]) Let $\widetilde{M}$ be a finitely combinatorial manifold.
If for $\forall(M_{1},M_{2})\in E(G^{L}[\widetilde{M}])$, $M_{1}\cap M_{2}$ is
simply connected, then
$\pi_{1}(\widetilde{M})\cong\left(\bigoplus\limits_{M\in
V(G[\widetilde{M}])}\pi_{1}(M)\right)\bigoplus\pi_{1}(G[\widetilde{M}]).$
4.2 Fundamental groups of manifolds
Notice that $\pi_{1}({\bf R}^{n})=identity$ for any integer $n\geq 1$. If we
choose $M\in V(G[\widetilde{M}])$ to be a chart
$(U_{\lambda},\varphi_{\lambda})$ with
$\varphi_{\lambda}:U_{\lambda}\rightarrow{\bf R}^{n}$ for $\lambda\in\Lambda$
in Theorem $4.2$, i.e., an $n$-manifold, we get the fundamental group of
$n$-manifold following.
Theorem $4.4$ Let $M$ be a compact $n$-manifold with charts
$\\{(U_{\lambda},\varphi_{\lambda})|\
\varphi_{\lambda}:U_{\lambda}\rightarrow{\bf R}^{n},\lambda\in\Lambda)\\}$.
Then
$\pi_{1}(M)\cong\frac{\pi_{1}(G^{\theta}[M])}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ g\in\prod\limits_{(U_{\mu},U_{\nu})\in
E(G[M])}\pi_{1}(U_{\mu}\cap U_{\nu})\right]},$
where $i_{1}^{E}$ and $i_{2}^{E}$ are homomorphisms induced by inclusion
mappings $i_{U_{\mu}}:\pi_{1}(U_{\mu}\cap
U_{\nu})\rightarrow\pi_{1}(U_{\mu})$, $i_{U_{\nu}}:\pi_{1}(U_{\mu}\cap
U_{\nu})\rightarrow\pi_{1}(U_{\nu})$, $\mu,\nu\in\Lambda$.
Corollary $4.5$ Let $M$ be a simply connected manifold with charts
$\\{(U_{\lambda},\varphi_{\lambda})|\
\varphi_{\lambda}:U_{\lambda}\rightarrow{\bf R}^{n},\lambda\in\Lambda)\\}$,
where $|\Lambda|<+\infty$. Then $G^{\theta}[M]=G[M]$ is a tree.
Particularly, if $U_{\mu}\cap U_{\nu}$ is simply connected for
$\forall\mu,\nu\in\Lambda$, then we obtain an interesting result following.
Corollary $4.6$ Let $M$ be a compact $n$-manifold with charts
$\\{(U_{\lambda},\varphi_{\lambda})|\
\varphi_{\lambda}:U_{\lambda}\rightarrow{\bf R}^{n},\lambda\in\Lambda)\\}$. If
$U_{\mu}\cap U_{\nu}$ is simply connected for $\forall\mu,\nu\in\Lambda$, then
$\pi_{1}(M)\cong\pi_{1}(G[M]).$
Therefore, by Theorem $4.4$ we know that the fundamental group of a manifold
$M$ is a subgroup of that of its edge-induced graph $G^{\theta}[M]$.
Particularly, if $G^{\theta}[M]=G[\widetilde{M}]$, i.e., $U_{\mu}\cap U_{\nu}$
is simply connected for $\forall\mu,\nu\in\Lambda$, then it is nothing but the
fundamental group of $G[\widetilde{M}]$.
References
[1] Munkres J.R., Topology (2nd edition), Prentice Hall, Inc, 2000.
[2] W.S.Massey, Algebraic Topology: An Introduction, Springer-Verlag, New
York, etc.(1977).
[3] John M.Lee, Introduction to Topological Manifolds, Springer-Verlag New
York, Inc., 2000.
[4] J.L.Gross and T.W.Tucker, Topological Graph Theory, John Wiley & Sons,
1987.
[5] L.F.Mao, Automorphism Groups of Maps, Surfaces and Smarandache Geometries,
American Research Press, 2005.
[6] L.F.Mao, Geometrical theory on combinatorial manifolds, JP J.Geometry and
Topology, Vol.7, No.1(2007),65-114.
[7] L.F.Mao, Combinatorial fields - an introduction, International J.Math.
Combin. Vol.3 (2009), 01-22.
[8] L.F.Mao, Combinatorial Geometry with Applications to Field Theory,
InfoQuest, USA, 2009.
[9] L.F.Mao, Smarandache Multi-Space Theory, Hexis, Phoenix, USA 2006.
[10] Smarandache F. Mixed noneuclidean geometries. arXiv: math/0010119,
10/2000.
|
arxiv-papers
| 2010-06-18T05:13:48 |
2024-09-04T02:49:11.079270
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Linfan Mao",
"submitter": "Linfan Mao l.f.m",
"url": "https://arxiv.org/abs/1006.4071"
}
|
1006.4118
|
# Ab initio theory of coherent phonon generation by laser excitation
Y. Shinohara Graduate School of Science and Technology, University of
Tsukuba, Tsukuba 305-8571, Japan K. Yabana Graduate School of Science and
Technology, University of Tsukuba, Tsukuba 305-8571, Japan Center for
Computational Sciences, University of Tsukuba, Tsukuba 305-8571, Japan Y.
Kawashita Graduate School of Science and Technology, University of Tsukuba,
Tsukuba 305-8571, Japan J.-I. Iwata Center for Computational Sciences,
University of Tsukuba, Tsukuba 305-8571, Japan T. Obote Advanced Photon
Research Center, Japan Atomic Energy Agency, Kizugawa, Kyoto 619-0215, Japan
G.F. Bertsch Institute for Nuclear Theory and Dept. of Physics, University of
Washington, Seattle, Washington
###### Abstract
We show that time-dependent density functional theory (TDDFT) is applicable to
coherent optical phonon generation by intense laser pulses in solids. The two
mechanisms invoked in phenomenological theories, namely impulsively stimulated
Raman scattering and displacive excitation, are present in the TDDFT. Taking
the example of crystalline Si, we find that the theory reproduces the
phenomena observed experimentally: dependence on polarization, strong growth
at the direct band gap, and the change of phase from below to above the band
gap. We conclude that the TDDFT offers a predictive ab initio framework to
treat coherent optical phonon generation.
There has been much experimental progress in the study of intense
electromagnetic fields interacting with condensed matter using pump-probe
techniques on femtosecond time scales ro02 . These interactions are a
challenging subject for theory, in view of the need to go beyond perturbative
methods in dealing with strong fields. One promising theoretical approach
useful to describe electron dynamics on femtosecond time scales is time-
dependent density functional theory (TDDFT) rg84 . In this Letter we apply the
TDDFT to the generation of coherent phonons by strong laser pulses. Our goals
are both to test the utility of the TDDFT in this domain and to assess the
validity of phenomenological models that are in current use. In the past, two
mechanisms have been invoked to explain the generation of coherent phonon me97
; st02 . The impulsively stimulated Raman scattering was proposed for the
coherent phonon generation in dielectrics with a laser pulse whose frequency
is lower than the direct band gap. In this mechanism, electrons are virtually
excited following adiabatically the laser electric field. The crucial quantity
is the Raman tensor, the derivative of dielectric function with respect to the
phonon coordinate. The other mechanism, called displacive excitation, requires
higher frequencies to generate real electron-hole excitations in the final
state ze92 ; sc93 ; ku94 . These excitations then shift the equilibrium
position of the phonon coordinates. In this work we consider a bulk Si under
irradiation of laser pulses of frequencies below and above the direct band
gap, and show that the TDDFT is computationally feasible, includes two above-
mentioned mechanisms, and produces results that are in qualitative agreement
with experiments ha03 ; ri07 . The TDDFT calculations prove to be also useful
to evaluate phenomenological and macroscopic models for the phonon generation
process.
Our computational framework is based on equations of motion derived from a
Lagrangian for a periodic crystalline system under a time-dependent, spatially
uniform electric field biry . The Lagrangian is
$\displaystyle L$ $\displaystyle=$
$\displaystyle\sum_{i}\int_{\Omega}d\vec{r}\left\\{\psi_{i}^{*}i\frac{\partial}{\partial
t}\psi_{i}-\frac{1}{2m}\left|\left(-i\vec{\nabla}+\frac{e}{c}\vec{A}\right)\psi_{i}\right|^{2}\right\\}$
(1) $\displaystyle-\int_{\Omega}d\vec{r}\left\\{(en_{ion}-en_{e})\phi-
E_{xc}[n_{e}]\right\\}$
$\displaystyle+\frac{1}{8\pi}\int_{\Omega}d\vec{r}(\vec{\nabla}\phi)^{2}+\frac{\Omega}{8\pi
c^{2}}\left(\frac{d\vec{A}}{dt}\right)^{2}$
$\displaystyle+\frac{1}{2}\sum_{\alpha}M_{\alpha}\left(\frac{d\vec{R}_{\alpha}}{dt}\right)^{2}+\frac{1}{c}\sum_{\alpha}Z_{\alpha}e\frac{d\vec{R}_{\alpha}}{dt}\vec{A}\,.$
Here $\psi_{i}$ is the time-dependent electron orbitals, taken as Bloch
orbitals in a unit cell of volume $\Omega$.
$n_{e}(\vec{r},t)=\sum_{i}|\psi_{i}(\vec{r},t)|^{2}$ represents the electron
density distribution. $\vec{R}_{\alpha}$ are atomic positions. The
electromagnetic field terms are split into a long-range spatially uniform part
$\vec{A}(t)$ and a periodic part given by a Coulomb potential $\phi$.
Variations with respect to the orbitals $\psi_{i}$, potential $\phi$, and
atomic coordinates $\vec{R}_{\alpha}$ result in the time-dependent Kohn-Sham
equation for $\psi_{i}$, the Poisson equation for $\phi$, and the Newton
equation for $\vec{R}_{\alpha}$, respectively. All the equations except those
for $\vec{R}_{\alpha}$ are the same as those employed in biry and ot08 .
To introduce the external laser field, we express the vector potential
$\vec{A}(t)$ as a sum of an external field $\vec{A}_{\rm ext}(t)$ and the
induced field $\vec{A}_{\rm ind}(t)$, with $\vec{A}(t)=\vec{A}_{\rm
ext}(t)+\vec{A}_{\rm ind}(t)$ and treat $\vec{A}_{\rm ind}(t)$ as dynamic. The
variation with respect to $\vec{A}_{\rm ind}(t)$ yields the following equation
of motion,
$\frac{\Omega}{4\pi
c^{2}}\frac{d^{2}\vec{A}_{ind}(t)}{dt^{2}}=\frac{e}{c}\int_{\Omega}d\vec{r}\left\\{\vec{j}_{ion}-\vec{j}_{e}\right\\}-\frac{e^{2}}{mc^{2}}N_{e}\vec{A}(t)$
(2)
To simulate the time-dependent electric field of the laser pulse, we take
$\vec{A}_{\rm ext}(t)$ to have the form
$\vec{A}_{\rm ext}=\int^{t}dt^{\prime}{\cal E}_{0}\sin^{2}\left({\pi
t^{\prime}\over T_{p}}\right)\,\sin\omega t^{\prime}$ (3)
for $0<t<T_{p}$ and zero otherwise, with $T_{p}=16$ fs and ${\cal E}_{0}$
corresponding to peak intensity $I=10^{12}$ W/cm2.
Figure 1: Geometry of the electric field and the optical phonon displacement
in the 8-atom unit cell. The $[011]\times[100]$ plane and atoms on the plane
are drawn with small arrows which show the direction of the optical phonon
coordinate.
The laser pulse is directed on a $[100]$ Si surface at normal incidence with a
linear polarization oriented along the $[011]$ axis. We show in Fig. 1 the
atomic positions of Si atoms in the plane defined by the $[011]$ and $[100]$
axes. The 4 atoms lying on the plane are shown. The optical phonon coordinate
which couples to the laser field is shown by vertical blue arrows.
Figure 2: Electric fields are shown as a function of time. The red solid line
shows the applied laser pulse, Eq. (3), characterized by the peak intensity,
$I=10^{12}$W/cm2, frequency $\hbar\omega=2.5$ eV, and the pulse duration,
$T_{p}=16$ fs. The green dashed line shows the summed electric field of
applied and induced ones, multiplied by a factor 15.
Figure 3: Top panel shows the ground-state electron density in the plane
shown in Fig. 1. The middle and bottom panels show the change of the electron
density from that in the ground state by the laser pulse described in Fig. 2.
The middle panel corresponds to the time $A$ and the bottom panel to the time
$B$ in Fig. 2, respectively. In the middle and bottom panels, the red color
indicates the increase of the electron density, while blue color indicates the
decrease.
Our calculations are based on the LDA density functional pz81 , treating the
four valence electrons of Silicon explicitly and using the Troullier-Martins
pseudopotential TM . We employ the real-time and real-space scheme which was
developed by us yb96 . The geometry is taken to be a simple cubic unit cell
containing 8 Si atoms, with lattice constant $a=10.26$ au. We have carefully
examined the convergence of the results with respect to numerical parameters.
We find that a spatial division of $16^{3}$, $k$-space grid of $24^{3}$, and
the time step of $\Delta t=0.08$ au is adequate for our purposes, and these
numerical parameters are adopted for the results reported below. To make the
present calculation feasible, parallel computation distributing $k$-points
into processors is indispensable. We note the calculated direct band gap of Si
is 2.4 eV, smaller than the measured value of 3.3 eV.
We first show the electron dynamics induced by a laser pulse. Figure 2 shows
the time dependence of the electric fields. The red solid curve shows the
electric field of applied laser pulse $E_{\rm ext}(t)=-(1/c)dA_{\rm ext}/dt$.
We choose the laser frequency $\hbar\omega=2.5$ eV, close to the value of the
direct band gap. The green dashed curve shows the sum of the applied and
induced electric fields, $E_{\rm tot}(t)=E_{\rm ext}(t)+E_{\rm ind}(t)$. The
difference of the magnitudes of the two fields comes from a dielectric
screening.
Figure 3 shows the electron density in the plane of Fig. 1. The top panel
shows the ground-state electron density, and the middle and bottom panels show
the change of electron density from that in the ground state at two times,
marked $A$ and $B$ in Fig. 2, respectively. In the middle and bottom panels,
red and blue indicate an increase or decrease of electron density,
respectively. At time $A$, the electric field is maximum and there is a strong
virtual excitation of the electrons. In the middle panel of Fig. 3, a movement
of electrons is seen in the bond connecting two Si atoms. At the time $B$, the
external electric field ended. Since the ultrashort laser pulse includes
frequency components above the direct band gap, there appear real electron-
hole excitations. In the bottom panel of Fig. 3, one can see that the
excitation results in a decreased density in the bond region and an increase
near the Si atoms but away from the bond. One should note that the coloring of
the middle and bottom figures are different by a factor of 40 to improve the
visibility of the density change at time $B$.
Figure 4: Electron excitation of the crystal during and after the pulse for
several laser frequencies across the direct band gap. The top panel (a) shows
the energy in the unit cell including electron-hole excitation energy and the
electric field energy. The middle panel (b) shows the the number of electron-
hole pairs in the unit cell. The bottom panel (c) shows the force on the
optical phonon coordinate.
We next examine how the character of the electronic excitation changes as the
laser frequency increases from below to above the direct band gap.
Characteristics of the excitation as a function of time are shown for
frequencies $\hbar\omega=2.25$ eV, 2.5 eV, and 2.75 eV in Fig. 4. The top
panel shows the total increase in energy in the unit cell, including both
electronic excitation energy and the electromagnetic field energy. The red
solid curve shows the results for a frequency below the band gap. Here the
energy drops almost to zero after the pulse is over, as to be expected. The
green dashed curve, corresponding to a frequency at the band gap, shows that
some excitation energy remains after the end of the pulse, comparable in
magnitude to the total energy at the peak. Finally, the blue dotted curve
shows that above the gap the laser-electron interaction is highly dissipative,
leaving a large excitation energy in the final state. The middle panel in the
figure shows the number of excited electrons as a function of time. This is
calculated by taking the overlaps of the time-dependent occupied orbitals with
the initial state static orbitals as in Ref. ot08 . The results are
qualitatively very similar to what we found for the energy. Below the direct
band gap, the excited electron shows a peak during the pulse and then drops
off to a very small value in the final state. At higher frequencies, the
excitations remain in the final state and it is not possible to distinguish
the real excitation from the virtual one during the pulse. In summary, one
sees an adiabatic response below the gap switching rather abruptly to a
strongly dissipative response above the gap.
Finally, in the bottom figure, we show the calculated induced force for the
three frequencies. Note that the ion positions are fixed in these
calculations; the accelerations are small and the resulting displacements
would be inconsequential. The lowest frequency, shown by the red solid curve,
gives a force envelope that follows the shape of the pulse intensity. This is
just what one would expect from the adiabatic formula (me97, , Eq. (2)). One
also sees high frequency oscillations superimposed on the envelope of the
curve. The frequency of these oscillations are twice the laser frequency,
again as expected from the adiabatic formula. The green dashed curve shows the
force for a laser frequency of $\hbar\omega=2.5$, nearly at the direct band
gap. One still sees a large peak at 10 fs associated with instantaneous high
field intensity. However, there is a residual force after the end of the pulse
which is rather constant with time. This is just what one expects for
displacive mechanism. At this point, we have shown that TDDFT reproduces at a
qualitative level the role of the two mechanisms. Beyond that, the relative
sign associated with them can be extracted from the graph. The last case
shown, $\hbar\omega=2.75$, is $0.35$ eV above the direct gap. Here the
displacive mechanism is completely dominant, although one can still see an
enhancement of the force during the pulse.
We now integrate the time-dependent force to get the lattice distortion
associated with the phonon coordinate. In principle, the restoring potential
for the lattice vibration is included in the evolution equations, but the
amplitude of the lattice displacement is too small numerically to include it
in the direct integration. So for this part of the analysis we simply assume a
harmonic restoring potential consistent with the observed optical phonon
frequency, $f_{phonon}=15.3$ THz.
To analyze the characteristics of the coherent phonon, we fit the oscillation
of the displacement to a cosine function as in conventional parametrization of
the experimental reflectivity measurements ha03 ,
$q(t)=-q_{0}\cos(\omega_{ph}t+\phi)+\bar{q}\,.$ (4)
Figure 5: The amplitude (a) and the phase (b) of the phonon oscillation Eq.
(4) as a function of laser frequency $\omega$.
Fig. 5 shows the amplitude and phase as a function of laser frequency, fitted
in the time interval 40-90 fs. Below the direct gap energy the phase is close
to $\pi/2$ as expected for the Raman mechanism. The amplitude remains almost
constant in this frequency region, also consistent with the Raman mechanism.
One sees a quite sharp drop from that value to $\phi=0$ as the direct gap is
crossed, showing the transition to the displacive behavior. The amplitude also
shows a sudden increase across the direct gap. Several experimental
measurements are also shown on the figure for the phase. Two of them ha03 ;
ka09 are in the Raman regime. The theory supports the results of Ref. ha03 ,
which reports a value close to $\pi/2$. The other measurement does not appear
consistent with our theory or indeed with the other experiment. The phase has
also been measured in the gap region ri07 , shown by the square on Fig. 5(b).
This point should be compared with the theory at the corresponding calculated
gap energy, 2.4 eV. In both theory and experiment the phase has decreased from
the Raman value, but decrease seems larger for the experimental measurement.
Both results are in a range where the mechanism is changing rapidly. All in
all, we find the agreement quite satisfactory on a qualitative level,
particularly since the phase could have come out with an opposite sign
($\phi\approx\pi$).
At higher frequencies, the theoretical phase goes to zero as expected for the
displacive mechanism, but then it rises again beyond 4.5 eV, approaching $\pi$
at $\hbar\omega=5$ eV. There is a corresponding dip and growth in the
amplitudes associated with a change in the sign of the displacive force.
Different electron orbitals are excited at the high frequency, and apparently
those orbitals have an opposite sign contribution to the displacive shift.
We also examined the dependence of amplitude and phase of the coherent phonon
on the intensity of the laser pulse. At all frequencies we examined, the
amplitude of the phonon is proportional to the laser intensity. In the
impulsive Raman mechanism which is applicable below the band gap, this
dependence is expected from the adiabatic formula (me97, , Eq. (2)). In the
displacive mechanism, it is also expected if the medium is excited by a one-
photon absorption process. The phase of the coherent phonon is found to be
sensitive only on the frequency but not on the intensity until the multiphoton
absorption processes become significant.
In summary, we have derived and carried out a computational method to apply
time-dependent density functional theory to laser-lattice interactions, taking
as an example the excitation of coherent optical phonon by femtosecond-scale
laser pulse in silicon. The qualitative agreement between theory and measured
phase of the coherent phonon confirms the utility of the TDDFT to describe
electron dynamics resulting from intense laser pulse in solids.
The numerical calculation were performed on the massively parallel cluster
T2K-Tsukuba, University of Tsukuba, and the supercomputer at the Institute of
Solid State Physics, University of Tokyo. TO acknowledges support by the
Grant-in-Aid for Scientific Research No. 21740303. GFB acknowledges support by
the National Science Foundation under Grant PHY-0835543 and by the DOE under
grant DE-FG02-00ER41132.
## References
* (1) F. Rossi and T. Kuhn, Rev. Mod. Phys. 74 895 (2002).
* (2) E. Runge and E.K.U. Gross, Phys. Rev. Lett. 52, 997 (1984).
* (3) R. Merlin, Solid State Comm. 102 207 (1997).
* (4) T.E. Stevens, J. Kuhl and R. Merlin, Phys. Rev. B65 144304 (2002).
* (5) H.J. Zeiger, et al., Phys. Rev. B45 768 (1992).
* (6) R. Scholz, T. Pfeifer and H. Kurz, Phys. Rev. B47 16229 (1993).
* (7) A.V. Kuznetsov and C.J. Stanton, Phys. Rev. Lett. 73 3243 (1994).
* (8) M. Hase, M. Kitajima, A. Constantinescu, and H. Petek, Nature 426 51 (2003).
* (9) D.M. Riffe and A.J. Sabbah, Phys. Rev. B76 085207 (2007).
* (10) G.F. Bertsch, J.I. Iwata, A. Rubio, and K. Yabana, Phys. Rev. B62 7998 (2000).
* (11) T. Otobe, et al., Phys. Rev. B 77 165104 (2008).
* (12) J.P. Perdew, A. Zunger, Phys. Rev. B 23, 5048 (1981).
* (13) N. Troullier and J. Martins, Phys. Rev B 43 1993 (1991).
* (14) K. Yabana, G.F. Bertsch, Phys. Rev. B 54, 4484 (1996).
* (15) K. Kato, A. Ishizawa, K. Oguri, K. Tateno, T. Tawara, H. Gotoh, M. Kitajima, H. Nakano, Japanese J. Appl. Phys. 48 100205 (2009).
|
arxiv-papers
| 2010-06-21T16:56:22 |
2024-09-04T02:49:11.087808
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Y. Shinohara, Y. Kawashita, K. Yabana, J.-I. Iwata, T. Obote, G.F.\n Bertsch",
"submitter": "George F. Bertsch",
"url": "https://arxiv.org/abs/1006.4118"
}
|
1006.4128
|
Physica A 390 (2011) 1009–1025
# Kinetic Path Summation,
Multi–Sheeted Extension of Master Equation,
and Evaluation of Ergodicity Coefficient
A. N. Gorban ag153@le.ac.uk University of Leicester, UK Corresponding author:
University of Leicester, LE1 7RH, UK
###### Abstract
We study the Master equation with time–dependent coefficients, a linear
kinetic equation for the Markov chains or for the monomolecular chemical
kinetics. For the solution of this equation a path summation formula is
proved. This formula represents the solution as a sum of solutions for simple
kinetic schemes (kinetic paths), which are available in explicit analytical
form. The relaxation rate is studied and a family of estimates for the
relaxation time and the ergodicity coefficient is developed. To calculate the
estimates we introduce the multi–sheeted extensions of the initial kinetics.
This approach allows us to exploit the internal (“micro”)structure of the
extended kinetics without perturbation of the base kinetics.
###### keywords:
Path summation , Master Equation , ergodicity coefficient , transition graph ,
reaction network , kinetics , relaxation time , replica
## 1 Introduction
### 1.1 The problem
First-order kinetics form the simplest and well-studied class of kinetic
systems. It includes the continuous-time Markov chains [1, 2] (the Master
Equation [3]), kinetics of monomolecular and pseudomonomolecular reactions
[4], provides a natural language for description of fluxes in networks and has
many other applications, from physics and chemistry to biology, engineering,
sociology, and even political science.
At the same time, the first-order kinetics are very fundamental and provide
the background for kinetic description of most of nonlinear systems: we almost
always start from the Master Equation (it may be very high-dimensional) and
then reduce the description to a lower level but with nonlinear kinetics.
For the description of the first order kinetics we select the
species–concentration language of chemical kinetics, which is completely
equivalent to the states–probabilities language of the Markov chains theory
and is a bit more flexible in the normalization choice: the sum of
concentration could be any positive number, while for the Markov chains we
have to introduce special “incomplete states”.
The first-order kinetic system is weakly ergodic if it allows the only
conservation law: the sum of concentration. Such a system forgets its initial
condition: the distance between any two trajectories with the same value of
the conservation law tends to zero when time goes to infinity. Among all
possible distances, the $l_{1}$ distance ($\|x\|_{l_{1}}=\sum_{i}|x_{i}|$)
plays a special role: it decreases monotonically in time for any first order
kinetic system. Further in this paper, we use the $l_{1}$ norm on the space of
concentrations.
Straightforward analysis of the relaxation rate for a linear system includes
computation of the spectrum of the operator of the shift in time. For an
autonomous system, we have to find the “slowest” nonzero eigenvalue of the
kinetic (generator) matrix. For a system with time–dependent coefficients, we
have to solve the linear differential equations for the fundamental operator
(the shift in time). After that, we have to analyze the spectrum of this
operator. Beyond the simplest particular cases there exist no analytical
formulas for such calculations.
Nevertheless, there exists the method for evaluation of the contraction rate
for the first order kinetics, based on the analysis of transition graph. For
this evaluation, we need to solve kinetic equations for some irreversible
acyclic subsystems which we call the kinetic paths (10). These kinetic paths
are combined from simple fragments of the initial kinetic systems. For such
systems, it is trivial to solve the kinetic equations in quadratures even if
the coefficients are time–dependent. The explicit recurrent formulas for these
solutions are given (12).
We construct the explicit formula for the solution of the kinetic equation for
an arbitrary system with time–dependent coefficients by the summation of
solutions of an infinite number of kinetic paths (15).
On the basis of this summation formula we produce a representation of the
$l_{1}$ contraction rate for weakly ergodic systems (23). Because of
monotonicity, any partial sum of this formula gives an estimate for this
contraction.
To calculate the estimates we introduce the multi–sheeted extensions of the
initial kinetics. Such a multi–sheeted extension is a larger Markov chain
together with a projection of its (the larger) state space on the initial
state space and the following property: the projection of the extended random
walk is a random walk for the initial chain (Section 4.2).
This approach allows us to exploit the internal (“micro”)structure of the
extended kinetics without perturbation of the base kinetics.
It is difficult to find, who invented the kinetic path approach. We have used
it in 1980s [5], but consider this idea as a scientific “folklore”.
In this paper we study the backgrounds of the kinetic path methods. This
return to backgrounds is inspired, in particular, by the series of work [6,
7], where the kinetic path summation formula was introduced (independently, on
another material and with different argumentation) and applied to analysis of
large stochastic systems. The method was compared to the kinetic Gillespie
algorithm [8] and on model systems it was demonstrated [7] that for ensembles
of rare trajectories far from equilibrium, the path sampling method performs
better.
For the linear chains of reversible semi-Markovian processes with nearest
neighbors hopping, the path summation formula was developed with counting all
possible trajectories in Laplace space [9]. Higher order propagators and the
first passage time were also evaluated. This problem statement was inspired,
in particular, by the evolving field of single molecules (for more detail see
[10]).
The idea of kinetic path with selection of the dominant paths gives an
effective generalization of the limiting step approximation in chemical
kinetics [11, 12].
## 2 Basic Notions
Let us recall the basic facts about the first-order kinetics. We consider a
general network of linear reactions. This network is represented as a directed
graph (digraph) ([13, 14]): vertices correspond to components $A_{i}$
($i=1,2,\ldots,n$, edges correspond to reactions $A_{i}\to A_{j}$ ($i\neq j$).
For the set of vertices we use notation $\mathcal{A}$, and for the set of
edges notation $\mathcal{E}$. For each vertex, $A_{i}\in\mathcal{A}$, a
positive real variable $c_{i}$ (concentration) is defined. Each reaction
$A_{i}\to A_{j}$ is represented by a pair of numbers $(i,j)$, $i\neq j$. For
each reaction $A_{i}\to A_{j}$ a nonnegative continuous bounded function, the
reaction rate coefficient (the variable “rate constant”) $k_{ji}(t)\geq 0$ is
given. To follow the standard notation of the matrix multiplication, the order
of indexes in $k_{ji}$ is always inverse with respect to reaction: it is
$k_{j\leftarrow i}$, where the arrow shows the direction of the reaction. The
kinetic equations have the form
$\frac{{\mathrm{d}}c_{i}}{{\mathrm{d}}t}=\sum_{j,\ j\neq
i}(k_{ij}(t)c_{j}-k_{ji}(t)c_{i}),$ (1)
or in the vector form: $\dot{c}=K(t)c$. The quantities $c_{i}$ are
concentrations of $A_{i}$ and $c$ is a vector of concentrations. We don’t
assume any special relation between constants, and consider them as
independent quantities.
For each $t$, the matrix of kinetic coefficients $K$ has the following
properties:
* •
non-diagonal elements of $K$ are non-negative;
* •
diagonal elements of $K$ are non-positive;
* •
elements in each column of $K$ have zero sum.
This family of matrices coincides with the family of generators of finite
Markov chains in continuous time ([1, 2]).
A linear conservation law is a linear function defined on the concentrations
$b(c)=\sum_{i}b_{i}c_{i}$, whose value is preserved by the dynamics (1).
Equation (1) always has a linear conservation law:
$b^{0}(c)=\sum_{i}c_{i}={\rm const}$.
Another important and simple property of this equation is the preservation of
positivity for the solution of (1) $c(t)$: if $c_{i}(t_{0})\geq 0$ for all $i$
then $c_{i}(t_{1})\geq 0$ for $t_{1}>t_{0}$.
For many technical reasons it is useful to discuss not only positive solutions
to (1) and further we do not automatically assume that $c_{i}\geq 0$.
The time shift operator which transforms $c(t_{0})$ into $c(t)$ is
$U(t,t_{0})$. This is a column-stochastic matrix:
$u_{ij}(t,t_{0})\geq 0\ ,\ \ \sum_{i}u_{ij}(t,t_{0})=1\ \ (t\geq t_{0})\ .$
This matrix satisfies the equation:
$\frac{{\mathrm{d}}U(t,t_{0})}{{\mathrm{d}}t}=KU(t,t_{0})\ \mbox{ or }\
\frac{{\mathrm{d}}u_{il}}{{\mathrm{d}}t}=\sum_{j}(k_{ij}(t)u_{jl}-k_{ji}(t)u_{il})$
(2)
with initial conditions $U(t_{0},t_{0})=\mathbf{1}$, where $\mathbf{1}$ is the
unit operator ($u_{ij}(t_{0},t_{0})=\delta_{ij}$).
Every stochastic in column operator $U$ is a contraction in the $l_{1}$ norm
on the invariant hyperplanes $\sum_{i}c_{i}=const$. It is sufficient to study
the restriction of $U$ on the invariant subspace $\\{x\ |\
\sum_{i}x_{i}=0\\}$:
$\|Ux\|\leq\delta\|x\|\;{\rm if}\;\sum_{i}x_{i}=0$
for some $\delta\leq 1$. The minimum of such $\delta$ is $\delta_{U}$, the
norm of the operator $U$ restricted to its invariant subspace $\\{x\ |\
\sum_{i}x_{i}=0\\}$. One of the definitions of weak ergodicity is $\delta<1$
[15]. The unit ball of the $l_{1}$ norm restricted to the subspace $\\{x\ |\
\sum_{i}x_{i}=0\\}$ is a polyhedron with vertices
$g^{ij}=\frac{1}{2}(e^{i}-e^{j}),\;\;i\neq j\ ,$ (3)
where $e^{i}$ are the standard basis vectors in $\mathbb{R}^{n}$:
$e^{i}_{k}=\delta_{ik}$, $\delta_{ik}$ is the Kronecker delta. For a norm with
the polyhedral unit ball, the norm of the operator $U$ is
$\max_{v\in V}\|U(v)\|\ ,$
where $V$ is the set of vertices of the unit ball. Therefore, for a ball with
vertices (3)
$\delta_{U}=\|U\|=\frac{1}{2}\max_{i,j}\sum_{k}|u_{ki}-u_{kj}|\leq 1\ .$ (4)
This is a half of the maximum of the $l_{1}$ distances between columns of $U$.
The ergodicity coefficient, $\varepsilon_{U}=1-\delta_{U}$, is zero for a
matrix with unit norm $\delta_{U}=1$ and one if $U$ transforms any two vectors
with the same sum of coordinates in one vector ($\delta_{U}=0$).
The contraction coefficient $\delta_{U}$ (4) is a norm of operator and
therefore has a “submultiplicative” property: for two stochastic in column
operators $U,W$ the coefficient $\delta_{UW}$ could be estimated through a
product of the coefficients
$\delta_{UW}\leq\delta_{U}\delta_{W}\ .$ (5)
We will systematically use this property in such a way. In many estimates we
find an upper border $1\geq\delta(\tau)\geq\delta_{U(t_{1}+\tau,t_{1})}$,
$t_{2}\geq t_{1}$. In such a case, $\delta_{U(t_{1}+\tau,t_{1})}\to 0$
exponentially with $\tau\to\infty$. Nevertheless, the estimate $\delta(\tau)$
may originally have a positive limit $\delta(\tau)\to\delta_{\infty}>0$ when
$\tau\to\infty$. In this situation we can use $\delta(\tau)$ for bounded
$\tau<\tau_{1}$ and for $\tau>\tau_{1}$ exploit the multiplicative estimate
(5). The moment $\tau_{1}$ may be defined, for example, by maximization of the
negative Lyapunov exponent:
$\tau_{1}={\rm
arg}\max_{\tau>0}\left\\{-\frac{\ln(\delta(\tau))}{\tau}\right\\}\ .$ (6)
For a system with external fluxes $\Pi_{i}(t)$ the kinetic equation has the
form
$\frac{{\mathrm{d}}c_{i}}{{\mathrm{d}}t}=\sum_{j}(k_{ij}(t)c_{j}-k_{ji}(t)c_{i})+\Pi_{i}(t)\
.$ (7)
The Duhamel integral gives for this system with initial condition $c(t_{0})$:
$c(t)=U(t,t_{0})c(t_{0})+\int_{t_{0}}^{t}U(t,\tau)\Pi(\tau)\ {\mathrm{d}}\tau\
,$
where $\Pi(\tau)$ is the vector of fluxes with components $\Pi_{i}(\tau)$.
In particular, for stochastic in column operators $U(t,t_{0})$ this formula
gives: an identity for the linear conservation law
$\sum_{i}c_{i}(t)=\sum_{i}c_{i}(t_{0})+\int_{t_{0}}^{t}\sum_{i}\Pi_{i}(\tau)\
{\mathrm{d}}\tau\ ,$ (8)
and an inequality for the $l_{1}$ norm
$\|c(t)\|\leq\|U(t,t_{0})c(t_{0})\|+\int_{t_{0}}^{t}\sum_{i}\|\Pi(\tau)\|\
{\mathrm{d}}\tau\leq\|c(t_{0})\|+\int_{t_{0}}^{t}\sum_{i}\|\Pi(\tau)\|\
{\mathrm{d}}\tau\ .$ (9)
We need the last formula for the estimation of contraction coefficients when
the vector $c(t)$ is not positive.
## 3 Kinetic Paths
Two vertices are called adjacent if they share a common edge. A directed path
is a sequence of adjacent edges where each step goes in direction of an edge.
A vertex $A$ is reachable from a vertex $B$, if there exists a directed path
from $B$ to $A$.
Formally, a path in a reaction graph is any finite sequence of indexes (a
multiindex) $I=\\{i_{1},i_{2},\ldots i_{q}\\}$ ($q\geq 1$, $1\leq i_{j}\leq
n$) such that $(i_{k},i_{k+1})\in\mathcal{E}$ for all $k=1,\ldots,q-1$ (i.e.
there exists a reaction $A_{i_{k}}\to A_{i_{k+1}}$). The number of the
vertices $|I|$ in the path $I$ may be any natural number (including 1), and
any vertex $A_{i}$ can be included in the path $I$ several times. If $q=1$
then we call the one-vertex path $I$ degenerated. There is a natural order on
the set of paths: $J>I$ if $J$ is continuation of $I$, i.e.
$I=\\{i_{1},i_{2},\ldots i_{q}\\}$ and $J=\\{i_{1},i_{2},\ldots
i_{q},\ldots\\}$. In this order, the antecedent element (or the parent) for
each $I$ is $I^{-}$, the path which we produce from $I$ by deletion of the
last step. With this definition of parents $I^{-}$, the set of the paths with
a given start point is a rooted tree.
###### Definition 1
For each path $I=\\{i_{1},i_{2},\ldots i_{q}\\}$ we define an auxiliary set of
reaction, the kinetic path $P_{I}$:
$\begin{CD}B^{I}_{1(i_{1})}@>{k_{i_{2}i_{1}}}>{}>B^{2}_{2(i_{2})}@>{k_{i_{3}i_{2}}}>{}>\ldots
@>{k_{i_{q}i_{q-1}}}>{}>B^{I}_{q(i_{q})}\\\
@V{}V{\kappa_{i_{1}\overline{i_{2}}}}V@V{}V{\kappa_{i_{2}\overline{i_{3}}}}V@V{}V{\kappa_{i_{q}}}V\\\
\end{CD}$ (10)
The vertices $B^{I}_{l(i_{l})}$ of the kinetic path (10) are auxiliary
components. Each of them is determined by the path multiindex $I$ and the
position in the path $l$. There is a correspondence between the auxiliary
component $B^{I}_{l(i_{l})}$ and the component $A_{i_{l}}$ of the original
network. The coefficient $\kappa_{i}$ is a sum of the reaction rate
coefficients for all outgoing reactions from the vertex $A_{i}$ of the
original network, and the coefficient $\kappa_{i\overline{j}}$ is this sum
without the term which corresponds to the reaction $A_{i}\to A_{j}$:
$\kappa_{i}=\sum_{l,\ l\neq i}k_{li},\;\;\kappa_{i\overline{j}}=\sum_{l,\
l\neq i,j}k_{li}\ .$
A quantity, the concentration $b^{I}_{l(i_{l})}$, corresponds to any vertex of
the kinetic path $B^{I}_{l(i_{l})}$ and a kinetic equation of the standard
form can be written for this path. The end vertex, $B^{I}_{q(i_{q})}$, plays a
special role in the further consideration and we use the special notations:
$i_{I}=i_{q}$, $A_{I}=A_{i_{q}}$, $\varsigma_{I}=b^{I}_{q(i_{q})}$,
$\kappa_{I}$ is the reaction rate coefficient of the last outgoing reactions
in (10) (the last vertical arrow) and $k_{I}$ is the reaction rate coefficient
of the last incoming reaction in (10) (the last horizontal arrow).
We use $P_{I}^{+}$ for the incoming flux for the terminal vertex of the
kinetic path (10) and $P_{I}^{-}$ for the outgoing flux for this vertex.
Let us consider the set $\mathcal{I}_{1}$ of all paths with the same start
point $i_{1}$ and the solutions of all the correspondent kinetic equations
with initial conditions:
$b^{I}_{1(i_{1})}=1,\;b^{I}_{l(i_{l})}=0\;{\rm for}\;l>1\ .$
For the concentrations of the terminal vertices this self-consistent set of
initial conditions gives the infinite chain (or, to be more precise, the tree)
of simple kinetic equations for the set of variables $\varsigma_{I}$,
$I\in\mathcal{I}_{1}$:
$\dot{\varsigma}_{1}=-\kappa_{1}(t)\varsigma_{1},\;\dot{\varsigma}_{I}=-\kappa_{I}(t)\varsigma_{I}+k_{I}(t)\varsigma_{I^{-}}\
,$ (11)
where index 1 corresponds to the degenerated path which consists of one vertex
(the start point only) and corresponds to $A_{i_{1}}$.
This simple chain of equations with initial conditions,
$\varsigma_{1}(t_{0})=1$ and $\varsigma_{I}(t_{0})=0$ for $|I|>1$, has a
recurrent representation of solution:
$\begin{split}&\varsigma_{1}(t)=\exp\left(-\int_{t_{0}}^{t}\kappa_{1}(\tau)\,{\mathrm{d}}\tau\right),\;\;\\\
&\varsigma_{I}(t)=\int_{t_{0}}^{t}\exp\left(-\int_{\theta}^{t}\kappa_{I}(\tau)\,{\mathrm{d}}\tau\right)k_{I}(\theta)\varsigma_{I^{-}}(\theta)\,{\mathrm{d}}\theta\
.\end{split}$ (12)
The analogues of the Kirchhoff rules from the theory of electric or hydraulic
circuits are useful for outgoing flux of a path $J\in\mathcal{I}_{1}$ and for
incoming fluxes of the paths which $I$ are the one-step continuations of this
path (i.e. $I^{-}=J$):
$\kappa_{J}\varsigma_{J}=\sum_{I,\ I^{-}=J}k_{I}\varsigma_{I^{-}}\ .$ (13)
Let us rewrite this formula as a relation between the outgoing flux
$P_{J}^{-}$ from the last vertex of $J$ and incoming fluxes $P_{I}^{+}$ for
the last vertices of paths $I$ ($I^{-}=J$):
$P_{J}^{-}=\sum_{I,\ I^{-}=J}P_{I}^{+}\ .$ (14)
The Kirchhoff rule (14) together with the kinetic equation for given initial
conditions immediately implies the following summation formula.
###### Theorem 1
Let us consider the solution to the initial kinetic equations (1) with the
initial conditions $c_{j}(t_{0})=\delta_{ji_{1}}$. Then
$c_{j}(t)=\sum_{I\in\mathcal{I}_{1},\ i_{I}=j}\varsigma_{I}(t)$ (15)
Proof. To prove this formula let us prove that the sum from the right hand
side (i) exists (ii) satisfies the initial kinetic equations (1) and (iii)
satisfies the selected initial conditions.
Convergence of the series with positive terms follows from the boundedness of
the set of the partial sums, which follows from the Kirchhoff rules. According
to them,
$\sum_{I\in\mathcal{I}_{1}}\varsigma_{I}(t)\equiv 1$
because $\mathcal{I}_{1}$ consists of the paths with the selected initial
point $i_{1}$ only.
The sum
$C_{j}=\sum_{I\in\mathcal{I}_{1},\ i_{I}=j}\varsigma_{I}$
satisfies the kinetic equation (1). Indeed, let
$\mathcal{I}_{1j}=\\{I\in\mathcal{I}_{1}\ |\ i_{I}=j\\}$ be the set of all
paths from $i_{1}$ to $j$. Let us find the set of all paths of the form
$\\{I^{-}\ |\ I\in\mathcal{I}_{1j}\\}$. This set (we call it
$\mathcal{I}_{1j}^{-}$) consists of all paths to all points which are
connected to $A_{j}$ by a reaction:
$\mathcal{I}_{1j}^{-}=\bigcup_{(l,j)\in\mathcal{E}}\mathcal{I}_{1l}\ .$
From this identity and the chain of the kinetic equations (11) we get
immediately that
$\frac{{\mathrm{d}}C_{i}}{{\mathrm{d}}t}=\sum_{j,\ j\neq
i}(k_{ij}(t)C_{j}-k_{ji}(t)C_{i}),$ (16)
The coincidence of the initial conditions for $c_{i}$ and $C_{i}$ is obvious.
Hence, because of the uniqueness theorem for equations (1) we proved that
$c_{i}\equiv C_{i}$. $\square$
It is convenient to reformulate Theorem 1 in the terms of the fundamental
operator $U(t,t_{0})$. The $i$th column of $U(t,t_{0})$ is a solution of (1)
$c_{j}(t)=u_{ji}(t,t_{0})$ $(j=1,\ldots,n)$ with initial conditions
$c_{j}(t_{0})=\delta_{ij}$. Therefore, we have proved the following theorem.
Let $\mathcal{I}_{ij}$ be the set of all paths with the initial vertex $A_{i}$
and the end vertex $A_{j}$ and $\varsigma_{I}(t)$ be the solutions of the
chain (11) for $i_{1}=i$ with initial conditions: $\varsigma_{1}(t_{0})=1$ and
$\varsigma_{I}(t_{0})=0$ for $|I|>1$.
###### Theorem 2
$u_{ji}(t,t_{0})=\sum_{I\in\mathcal{I}_{ij}}\varsigma_{I}(t)\ .\ \ \ \ \
\square$ (17)
Remark 1. It is important that all the terms in the sum (17) are non-negative,
and any partial sum gives the approximation to $u_{ji}(t,t_{0})$ from below.
Remark 2. If the kinetic coefficients are constant then the Laplace transform
gives a very simple representation for solution to the chain (11) (see also
computations in [9, 6]). The kinetic path $I$ (10) is a sequence of elementary
links
$\begin{CD}\ldots
@>{k_{i_{r}i_{r-1}}}>{}>B^{r}_{r(i_{r})}@>{k_{i_{r+1}i_{r}}}>{}>\ldots\\\
@V{}V{\kappa_{i_{r}\overline{i_{r+1}}}}V\\\ \end{CD}$ (18)
The transfer function $W_{i_{r}}(p)$ for the link (18) is the ratio of the
output Laplace Transform to the input Laplace Transform for the equation. Let
the input be a function $X_{i_{r}}(t)$ and the output be
$Y_{i_{r}}(t)=b_{i_{r}}(t)$, where $b_{i_{r}}(t)$ is the solution to equation
$\dot{b}_{i_{1}}=-\kappa_{i_{1}}{b}_{i_{r}}+X_{i_{1}}(t)\,;\;\dot{b}_{i_{r}}=-\kappa_{i_{r}}{b}_{i_{r}}+k_{i_{r}i_{r-1}}X_{i_{r}}(t)\;(r>1)$
with zero initial conditions. The Laplace transform gives
$W_{i_{1}}=\frac{1}{p+\kappa_{i_{1}}}\,,\;\;W_{i_{r}}=\frac{k_{i_{r}i_{r-1}}}{p+\kappa_{i_{r}}}\;(r>1)$
for a link (18) and for the whole path (10) we get
$W_{I}=\frac{1}{p+\kappa_{i_{1}}}\prod_{r=2}^{q}\frac{k_{i_{r}i_{r-1}}}{p+\kappa_{i_{r}}}\,.$
(19)
(compare, for example, to formula (9) in [6]). It is worth to mention
commutativity of this product: it does not change after a permutation of
internal links. For the infinite chain (11) with initial conditions,
$\varsigma_{1}(0)=1$ and $\varsigma_{I}(0)=0$ for $|I|>1$, the Laplace
transformation of solutions is
$\mathcal{L}\varsigma_{I}=W_{I}$ (20)
## 4 Evaluation of Ergodicity Coefficient
### 4.1 Preliminaries: Weak Ergodicity and Annihilation Formula
#### 4.1.1 Geometric Criterion of Weak Ergodicity
In this Subsection, let us consider a reaction kinetic system (1) with
constant coefficients $k_{ji}>0$ for $(i,j)\in\mathcal{E}$.
A set $E$ is positively invariant with respect to the kinetic equations (1),
if any solution $c(t)$ that starts in $E$ at time $t_{0}$ ($c(t_{0})\in E$)
belongs to $E$ for $t>t_{0}$ ($c(t)\in E$ if $t>t_{0}$). It is straightforward
to check that the standard simplex $\Sigma=\\{c\,|\,c_{i}\geq
0,\,\sum_{i}c_{i}=1\\}$ is a positively invariant set for kinetic equation
(1): just check that if $c_{i}=0$ for some $i$, and all $c_{j}\geq 0$ then
$\dot{c}_{i}\geq 0$. This simple fact immediately implies the following
properties of ${K}$:
* •
All eigenvalues $\lambda$ of ${K}$ have non-positive real parts,
$Re\lambda\leq 0$, because solutions cannot leave $\Sigma$ in positive time;
* •
If $Re\lambda=0$ then $\lambda=0$, because the intersection of $\Sigma$ with
any plane is a polygon, and a polygon cannot be invariant with respect to
rotations to sufficiently small angles;
* •
The Jordan cell of ${K}$ that corresponds to the zero eigenvalue is diagonal –
because all solutions should be bounded in $\Sigma$ for positive time.
* •
The shift in time operator $\exp({K}t)$ is a contraction in the $l_{1}$ norm
for $t>0$: there exists such a monotonically decreasing (non-increasing)
function $\delta(t)$ ($t>0$, $0<\delta(t)\leq 1$, that for any two solutions
of (1) $c(t),c^{\prime}(t)\in\Sigma$
$\sum_{i}|c_{i}(t)-c^{\prime}_{i}(t)|\leq\delta(t)\sum_{i}|c_{i}(0)-c^{\prime}_{i}(0)|.$
(21)
Moreover, if for $c(t),c^{\prime}(t)\in\Sigma$ the values of all linear
conservation laws coincide then $\sum_{i}|c_{i}(t)-c^{\prime}_{i}(t)|\to 0$
monotonically when $t\to\infty$.
The first-order kinetic system is weakly ergodic if it allows only the
conservation law: the sum of concentration. Such a system forgets its initial
condition: distance between any two trajectories with the same value of the
conservation law tends to zero when time goes to infinity.
The difference between weakly ergodic and ergodic systems is in obligatory
existence of a strictly positive stationary distribution: for an ergodic
system, in addition, a strictly positive steady state exists: $Kc=0$ and all
$c_{i}>0$. Examples of weakly ergodic but not ergodic systems: a chain of
reactions $A_{1}\to A_{2}\to\ldots\to A_{n}$ and symmetric random walk on an
infinite lattice.
The weak ergodicity of the network follows from its topological properties.
###### Theorem 3
The following properties are equivalent (and each one of them can be used as
an alternative definition of weak ergodicity):
1. 1.
There exists a unique independent linear conservation law for kinetic
equations (this is $b^{0}(c)=\sum_{i}c_{i}={\rm const}$).
2. 2.
For any normalized initial state $c(0)$ ($b^{0}(c)=1$) there exists a limit
state
$c^{*}=\lim_{t\rightarrow\infty}\exp(Kt)\,c(0)$
that is the same for all normalized initial conditions: For all $c$,
$\lim_{t\rightarrow\infty}\exp(Kt)\,c=b^{0}(c)c^{*}.$
3. 3.
For each two vertices $A_{i},\>A_{j}\>(i\neq j)$ we can find such a vertex
$A_{k}$ that is reachable both from $A_{i}$ and from $A_{j}$. This means that
the following structure exists:
$A_{i}\to\ldots\to A_{k}\leftarrow\ldots\leftarrow A_{j}\ .$ (22)
One of the paths can be degenerated: it may be $i=k$ or $j=k$.
4. 4.
For $t>0$ operator $\exp(Kt)$ is a strong contraction in the invariant
subspace $\sum_{i}c_{i}=0$ in the $l_{1}$ norm:
$\|\exp(Kt)x\|\leq\delta(t)<1$, the function $\delta(t)>0$ is strictly
monotonic and $\delta(t)\to 0$ when $t\to\infty$ $\square$.
The proof of this theorem could be extracted from detailed books about Markov
chains and networks ([1, 17]). In its present form it was published in [5]
with explicit estimations of the ergodicity coefficients.
Let us demonstrate how to prove the geometric criterion of weak ergodicity,
the equivalence $1\Leftrightarrow 3$.
Let us assume that there are several linearly independent conservation laws,
linear functionals $b^{0}(c),b^{1}(c),\ldots,b^{m}(c)$, $m\geq 1$. The linear
transform $c\mapsto(b^{1}(c),\ldots,b^{m}(c))$ maps the standard simplex
$\Sigma_{n}$ in $\mathbb{R}^{n}$ ($c_{i}\geq 0$, $\sum_{i}c_{i}=1$) onto a
polyhedron $D\subset\mathbb{R}^{m}$. Because of linear independence of the
system $b^{0}(c),b^{1}(c),\ldots,b^{m}(c)$, $m\geq 1$, this $D$ has nonempty
interior. Hence, it has no less than $m+1$ vertices $w_{1},\ldots,w_{q}$,
$q>m$.
The preimage of every point $x\in D$ in $\Sigma_{n}$ is a positively invariant
subset with respect to kinetic equations because the standard simplex is
positively invariant and the functionals $b^{i}(c)$ are the conservation laws.
In particular, preimage of every vertex $w_{q}$ is a positively invariant face
of $\Sigma_{n}$, $F_{q}$; $F_{q}\cap F_{r}=\emptyset$ if $q\neq r$.
Each vertex $v_{i}$ of the standard simplex corresponds to a component
$A_{i}$: at this vertex $c_{i}=1$ and other $c_{j}=0$ there. Let the vertices
from $F_{q}$ correspond to the components which form a set $S_{q}$; $S_{q}\cap
S_{r}=\emptyset$ if $q\neq r$.
For any $A_{i}\in S_{q}$ and every reaction $A_{i}\to A_{j}$ the component
$A_{j}$ also belongs to $S_{q}$ because $F_{q}$ is positively invariant and a
solution to kinetic equations cannot leave this face. Therefore, if $q\neq r$,
$A_{i}\in S_{q}$ and $A_{j}\in S_{r}$ then there is no such vertex $A_{k}$
that is reachable both from $A_{i}$ and from $A_{j}$. We proved the
implication $3\Rightarrow 1$.
Now, let us assume that the statement 3 is wrong and there exist two such
components $A_{i}$ and $A_{j}$ that no components are reachable both from
$A_{i}$ and $A_{j}$. Let $S_{i}$ and $S_{j}$ be the sets of components
reachable from $A_{i}$ and $A_{j}$ (including themselves), respectively;
$S_{i}\cap S_{j}=\emptyset$.
For every concentration vector $c\in\mathbb{R}^{n}$ a limit exists
$c^{*}(c)=\lim_{t\to\infty}\exp(Kt)\ c$ (because all eigenvalues of $K$ have
non-positive real part and the Jordan cell of ${K}$ that corresponds to the
zero eigenvalue is diagonal). The operator $c\mapsto c^{*}(c)$ is linear
operator in $\mathbb{R}^{n}$. Let us define two linear conservation laws:
$b^{i}(c)=\sum_{A_{r}\in S_{i}}c_{r}^{*}(c),\ \ b^{j}(c)=\sum_{A_{r}\in
S_{j}}c_{r}^{*}(c)\ .$
These functionals are linearly independent because for a vector $c$ with
coordinates $c_{r}=\delta_{ri}$ we get $b^{i}(c)=1$, $b^{j}(c)=0$ and for a
vector $c$ with coordinates $c_{r}=\delta_{rj}$ we get $b^{i}(c)=0$,
$b^{j}(c)=1$. Hence, the system has at least two linearly independent linear
conservation laws. Therefore, $1\Rightarrow 3$.
#### 4.1.2 Annihilation Formula
Let us return to general time–dependent kinetic equations (1).
In this Section, we find an exact expression for the contraction coefficients
$\delta(t,t_{0})$ for the time evolution operator $U(t,t_{0})$ in $l_{1}$ norm
on the invariant subspace $\\{x\ |\ \sum_{i}x_{i}=0\\}$. The unit $l_{1}$-ball
in this subspace is a polyhedron with vertices
$g^{ij}=\frac{1}{2}(e^{i}-e^{j})$, where $e_{i}$ are the standard basic
vectors in $\mathbb{R}^{n}$ (3). The contraction coefficient of an operator
$U$ is its norm on that subspace (4), this is half of the maximum of the
$l_{1}$ distances between columns of $U$.
The kinetic path summation formula (17) estimates the matrix elements of
$U(t,t_{0})$ from below, but this does not give the possibility to evaluate
the difference between these elements. To use the summation formula
efficiently, we need another expression for the contraction coefficient.
The $i$th column of $U(t,t_{0})$ is a solution of the kinetic equations (1)
$c_{j}(t)=u_{ji}(t,t_{0})$ $(j=1,\ldots,n)$ with initial conditions
$c_{j}(t_{0})=\delta_{ij}$. For each $j$ let us introduce the incoming flux
for the vertex $A_{j}$ in this solution:
$\Pi_{j}^{i}(t)=\sum_{q}k_{jq}(t)c_{q}(t)$
(the upper index indicates the number of column in $U(t,t_{0})$, the lower
index corresponds to the number of vertex $A_{j}$).
Formula (4) for the contraction coefficient gives
$\delta(t,t_{0})=\frac{1}{2}\max_{i,j}\|U(t,t_{0})(e^{i}-e^{j})\|\ .$
$U(t,t_{0})(e^{i}-e^{j})$ is a solution to the kinetic equation (1) with
initial conditions $c_{i}(t_{0})=1$, $c_{j}(t_{0})=-1$ and $c_{q}(t_{0})=0$
for $q\neq i,j$. This is the difference between two solutions,
$c^{+}_{q}(t)=u_{qi}(t,t_{0})$ and $c^{-}_{q}(t)=u_{qj}(t,t_{0})$. Let us use
the notation
$G^{ij}(t)=\frac{1}{2}U(t,t_{0})(e^{i}-e^{j})\ .$
For each $q$ we define
$\Pi^{+}_{q}=\sum_{l,c^{+}_{l}>c^{-}_{l}}k_{ql}(c^{+}_{l}-c^{-}_{l}),\;\;\Pi^{-}_{q}=\sum_{l,c^{+}_{l}<c^{-}_{l}}k_{ql}(c^{-}_{l}-c^{+}_{l}),\;\;\Pi^{\pm}_{q}\geq
0\ .$
The decrease in the $l_{1}$ norm of $c^{+}(t)-c^{-}(t)$ can be represented as
an annihilation of a flux $\Pi^{\pm}_{q}(t)$ with an equal amount of
concentration $c^{+}(t)-c^{-}(t)$ from the vertex $A_{q}$ by the following
rules:
1. 1.
If $c_{q}=c^{+}_{q}(t)-c^{-}_{q}(t)>0$ then the flux $\Pi^{-}_{q}$ annihilates
with an equal amount of positive concentration stored at vertex $A_{q}$ (Fig.
1a);
2. 2.
If $c_{q}=c^{+}_{q}(t)-c^{-}_{q}(t)<0$ then the flux $\Pi^{+}_{q}$ annihilates
with an equal amount of negative concentration stored at vertex $A_{q}$ (Fig.
1b);
3. 3.
If $c_{q}=c^{+}_{q}(t)-c^{-}_{q}(t)=0$ then the flux
$\min\\{\Pi^{+}_{q},\Pi^{-}_{q}\\}$ annihilates with the equal amount from the
opposite flux (Fig. 1c).
Let us summarize these rules in one formula:
(a) $c>0$, the negative flux annihilates
(b) $c<0$, the positive flux annihilates
(c) $c=0$, the minimal flux annihilates
Figure 1: Annihilation of fluxes.
###### Proposition 1
$\begin{split}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\|G^{ij}(t)\|_{l_{1}}=&-\sum_{q,\
c^{+}_{q}>c^{-}_{q}}\Pi^{-}_{q}(t)-\sum_{q,\
c^{+}_{q}<c^{-}_{q}}\Pi^{+}_{q}(t)\\\ &-\sum_{q,\
c^{+}_{q}=c^{-}_{q}}\min\\{\Pi^{+}_{q}(t),\Pi^{-}_{q}(t)\\}\
.\;\;\;\square\end{split}$ (23)
Immediately from (23) we obtain the following integral formula
$\begin{split}1-\|G^{ij}(t)\|_{l_{1}}=&\int_{t_{0}}^{t}\left(\sum_{q,\
c^{+}_{q}>c^{-}_{q}}\Pi^{-}_{q}(\tau)+\sum_{q,\
c^{+}_{q}<c^{-}_{q}}\Pi^{+}_{q}(\tau)\right.\\\ &+\left.\sum_{q,\
c^{+}_{q}=c^{-}_{q}}\min\\{\Pi^{+}_{q}(\tau),\Pi^{-}_{q}(\tau)\\})\right)\
{\mathrm{d}}\tau\ .\end{split}$ (24)
The annihilation formula gives us a better understanding of the nature of
contraction but is not fully constructive because it uses fluxes from
solutions to the initial kinetic equation (1).
### 4.2 Multi–Sheeted Extensions of Kinetic System
Let us introduce a multi–sheeted extension of a kinetic system.
###### Definition 2
The vertices of a multi–sheeted extension of the system (1) are
$\mathcal{A}\times K$ where $K$ is a finite or countable set. An individual
vertex is $(A_{i},l)$ ($l\in K$). The corresponding concentration is
$c_{(i,l)}$. The reaction rate constant for $(A_{i},l)\to(A_{j},r)$ is
$k_{(j,r)(i,l)}\geq 0$. This system is a multi–sheeted extension of the
initial system if an identity holds:
$\sum_{r}k_{(j,r)(i,l)}=k_{ji}\ \mbox{ for all }\ l\ .$ (25)
This means that the flux from each vertex is distributed between sheets, but
the sum through sheets is the same as for the initial system. We call the
kinetic behavior of the sum $c_{i}=\sum_{l}c_{(i,l)}$ the base kinetics.
A simple proposition is important for further consideration.
Figure 2: Redirection of a reaction from one sheet to another with
preservation of the base kinetics. The redirected reaction is highlighted by
bold.
###### Proposition 2
If $c_{(i,l)}(t)$ is a solution to the extended multi–sheeted system then
$c_{i}(t)=\sum_{l}c_{(i,l)}(t)$ (26)
is a solution to the initial system and
$\sum_{il}|c_{(i,l)}(t)|\geq\sum_{i}|c_{i}(t)|\ .$ (27)
(Here we do not assume positivity of all $c_{i}$). $\square$
Formula (25) allows us to redirect reactions from one sheet to another (Fig.
2) without any change of the base kinetics. In the next section we show how to
use this possibility for effective calculations.
Formula (26) means that kinetics of the extended system in projection on the
initial space is the base kinetics: the components $(A_{i},l)$ are projected
in $A_{i}$ the projected vector of concentrations is $c_{i}=\sum_{l}c_{(i,l)}$
and the projected kinetics is given by the initial Master equation with the
projected coefficients $k_{ji}=\sum_{r}k_{(j,r)(i,l)}$. “Recharging” is any
change of the non-negative extended coefficients $k_{(j,r)(i,l)}$ which does
not change the projected coefficients.
The key role in the further estimates plays formula (27). We will apply this
formula to the solutions with the zero sums of coordinates, they are
differences between the normalized positive solutions.
### 4.3 Fluxes and Mixers
In this Subsection, we present the system of estimates for the contraction
coefficient. The main idea is based on the following property which can be
used as an alternative definition of weak ergodicity (Theorem 3): For each two
vertices $A_{i},\>A_{j}\>(i\neq j)$ we can find a vertex $A_{q}$ that is
reachable both from $A_{i}$ and from $A_{j}$. This means that the following
structure exists:
$A_{i}\to\ldots\to A_{q}\leftarrow\ldots\leftarrow A_{j}.$
One of the paths can be degenerated: it may be $i=q$ or $j=q$. The positive
flux from $A_{i}$ meets the negative flux from $A_{j}$ at point $A_{q}$ and
one of them annihilates with the equal amount of the concentration of opposite
sign.
Let us generalize this construction. Let us fix three different vertices:
$A_{i}$ (the “positive source”), $A_{j}$ (the “negative source”) and $A_{q}$
(the “mixing point”). The degenerated case $q=i$ or $q=j$ we discuss
separately. Let $S^{+}$ be such a system of vertices that $A_{i}\in S^{+}$,
$A_{q}\notin S^{+}$ and there exists an oriented path in
$S^{+}\cup\\{A_{q}\\}$ from $A_{i}$ to $A_{q}$. Analogously, let $S^{-}$ be
such a system of vertices that $A_{j}\in S^{-}$, $A_{q}\notin S^{-}$ and there
exists an oriented path in $S^{-}\cup\\{A_{q}\\}$ from $A_{j}$ to $A_{q}$. We
assume that $S^{+}\cap S^{-}=\emptyset$.
With each subset of vertices $S$ we associate a kinetic system (subsystem):
for $A_{r}\in S$
$\dot{c}_{r}=\sum_{l,\ A_{l}\in S,\ r\neq
l}k_{rl}c_{l}-\sum_{p=1}^{n}k_{pr}c_{r}\ .$ (28)
In this subsystem, we retain all the outgoing reaction for $A_{r}\in S$ and
delete the reactions which lead to vertices in $S$ from “abroad”.
The flux $\Pi_{S}^{+}$ from $S^{+}$ to $A_{q}$ is
$\Pi_{S}^{+}=\sum_{r,\ A_{r}\in S^{+}}k_{qr}c_{r}(t)\ ,$
where $c_{r}(t)$ is a component of the solution of (28) for $S=S^{+}$ with
initial conditions $c_{r}(t_{0})=\delta_{ri}$. Analogously, we define the flux
$\Pi_{S}^{-}=\sum_{r,\ A_{r}\in S^{-}}k_{qr}c_{r}(t)\ ,$
where $c_{r}(t)$ is a component of the solution of (28) for $S=S^{-}$ with
initial conditions $c_{r}(t_{0})=\delta_{rj}$. Decrease of the norm
$\|G^{ij}(t)\|$ is estimated by the following theorem.
The system $S^{+},S^{-},A_{q}$ we call a mixer, that is a device for mixing.
An elementary mixer consists of two kinetic paths $A_{i}\to\ldots\to
A_{q}\leftarrow\ldots\leftarrow A_{j}$ (22) with the corespondent outgoing
reactions:
$\setcounter{MaxMatrixCols}{11}\begin{CD}A_{i_{1}}@>{k_{i_{2}i_{1}}}>{}>\ldots
@>{k_{i_{r}i_{r-1}}}>{}>A_{i_{r}}@<{k_{i_{r}i_{r+1}}}<{}<\ldots
@<{k_{i_{r+l-1}i_{r+l}}}<{}<A_{i_{r+l}}\\\
@V{}V{\kappa_{i_{1}\overline{i_{2}}}}V@V{}V{\kappa_{i_{r}}}V@V{\kappa_{i_{r+l}{\overline{i_{r+l-1}}}}}V{}V\\\
\end{CD}$ (29)
where $i_{1}=i$, $i_{r}=q$, $i_{r+l}=j$.
The degenerated elementary mixer consists of one kinetic path:
$\begin{CD}A_{i_{1}}@>{k_{i_{2}i_{1}}}>{}>A_{i_{2}}@>{k_{i_{3}i_{2}}}>{}>\ldots
@>{k_{i_{r}i_{r-1}}}>{}>A_{i_{r}}\\\
@V{}V{\kappa_{i_{1}\overline{i_{2}}}}V@V{}V{\kappa_{i_{2}\overline{i_{3}}}}V@V{}V{\kappa_{i_{r}}}V\\\
\end{CD}$ (30)
where $i_{1}=i$, $i_{r}=j$.
###### Theorem 4
$\|G^{ij}(t)\|\leq 1-\int_{t_{0}}^{t}\min\\{\Pi_{S}^{+},\Pi_{S}^{-}\\}\
{\mathrm{d}}t\ .$ (31)
Figure 3: A mixer: two subsystems, $S^{+}$ (includes $A_{i}$) and $S^{-}$
(includes $A_{j}$). There may be outgoing reactions from $S^{\pm}$ but all
incoming reactions to $S^{\pm}$ from outside are deleted. A mixing point
$A_{q}$ and two fluxes, positive from $S^{+}$ (marked by dark color) and
negative from $S^{-}$, meet at the mixing point.
Proof. To prove this theorem let us organize a 4–sheeted extension of the
initial kinetic system as it is demonstrated in Fig. 3. Subsystems $S^{\pm}$
including the positive source (initial concentration $+1$) and the negative
source (initial concentration $-1$) belong to level 0. Reactions from
$S^{\pm}$ to $A_{q}$ are redirected to the sheet $f$, reactions from $S^{+}$
to other vertices, which do not belong to $S^{+}$, go to sheet $+1$, reactions
from $S^{-}$ to other vertices, which do not belong to $S^{-}$, go to sheet
$-1$. The incoming flux to the sheet $f$ is $\Pi_{S}^{+}-\Pi_{S}^{-}$.
Let us introduce the following notations:
$C_{S}^{+}=\sum_{A_{p}\in S^{+}}c_{(p,0)}+\sum_{q=1}^{n}c_{(q,1)}\ ;$
$C_{S}^{-}=-\sum_{A_{p}\in S^{-}}c_{(p,0)}-\sum_{q=1}^{n}c_{(q,-1)}\ ;$
$C_{f}=\sum_{r=1}^{n}|c_{(r,f)}|\ .$
We consider solution to the kinetic equations for the multi–sheeted system
with initial conditions: $c_{(i,0)}(t_{0})=1$, $c_{(j,0)}(t_{0})=-1$ and all
other concentrations are equal to zero at time $t_{0}$. In this case, some of
the signs of concentrations are known for $t\geq t_{0}$ due to the
organization of the redirection of reactions (Fig. 3):
$\begin{split}&c_{(p,0)}\geq 0\ \ \mbox{for}\ \ A_{p}\in S^{+}\ ,\ \
c_{(p,0)}\leq 0\ \ \mbox{for}\ \ A_{p}\in S^{-}\ ,\\\ &c_{(p,0)}=0\ \
\mbox{for}\ \ A_{p}\notin S^{+}\cup S^{-}\ ,\\\ &c_{(q,1)}\geq 0,\ \
c_{(q,-1)}\leq 0\ .\end{split}$ (32)
Let us use (8) for $S^{+}$ with the sheet $+1$ and for $S^{-}$ with the sheet
$-1$. We get immediately
$\frac{{\mathrm{d}}C_{S}^{+}}{{\mathrm{d}}t}=\Pi_{S}^{+}\ ,\ \
\frac{{\mathrm{d}}C_{S}^{-}}{{\mathrm{d}}t}=\Pi_{S}^{-}$ (33)
Analogously, we can use (9) for the sheet $f$ and get
$\frac{{\mathrm{d}}C_{f}}{{\mathrm{d}}t}\leq|\Pi_{S}^{+}-\Pi_{S}^{-}|\ .$ (34)
For the norm of the base vector of concentration $c$ the inequality holds
(Proposition 2):
$\|c\|\leq C^{+}_{S}+C^{-}_{S}+C_{f}\ .$
Finally, we combine this inequality with (33), (34) and get
$\|c(t)\|\leq
2-2\int_{t_{0}}^{t}\min\\{\Pi_{S}^{+}(\tau),\Pi_{S}^{-}(\tau)\\}\
{\mathrm{d}}\tau\ \ \ \ \square$
For the degenerate case the path from $A_{i}$ goes directly to $A_{j}$ (or
inverse). let us assume that there is a subsystem $S^{+}$, $A_{i}\in S^{+}$,
the mixing point $A_{q}$ coincides with $A_{j}$ and the flux $\Pi^{+}_{S}$ is
$\Pi_{S}^{+}=\sum_{r,\ A_{r}\in S^{+}}k_{jr}c_{r}(t)\ ,$
where $c_{r}(t)$ is a component of the solution of (28) for $S=S^{+}$ with
initial conditions $c_{r}(t_{0})=\delta_{ri}$.
###### Theorem 5
$\|G^{ij}(t)\|\leq 1-\int_{t_{0}}^{\min\\{t,t_{1}\\}}{\Pi_{S}^{+}(\tau)}\
{\mathrm{d}}\tau,$ (35)
where $\kappa_{j}=\sum_{p}k_{pj}$ and $t_{1}$ is a solution to equation
$\int_{t_{0}}^{t}{\Pi_{S}^{+}}(\tau)\exp(-\kappa_{j}(t-\tau))\
{\mathrm{d}}\tau=\exp(-\kappa_{j}t)\ .$ (36)
Proof. This theorem is also proved by the construction of the appropriate
multi–sheeted extension of the kinetic system. For the degenerated case we
need only two additional sheets: subsystem $S^{+}$ including the positive
source $A_{i}$ (initial concentration $+1$) and the negative source $A_{j}$
(initial concentration $-1$) belong to level 0. Reactions from $S^{+}$ to
other vertices, which do not coincide with $A_{j}$, go to sheet $+1$,
reactions from $A_{j}$ to other vertices go to sheet $-1$. The concentration
of $A_{(j,0)}$ is
$c_{(j,0)}(t)=\int_{t_{0}}^{t}{\Pi_{S}^{+}}(\tau)\exp(-\kappa_{j}(t-\tau))\
{\mathrm{d}}\tau-\exp(-\kappa_{j}t)\ .$
Let us introduce the following notation:
$C_{S}^{+}=\sum_{A_{p}\in S^{+}}c_{(p,0)}+\sum_{q=1}^{n}c_{(q,1)}\ ;$
$C^{-}=-c_{(j,0)}-\sum_{q=1}^{n}c_{(q,-1)}\ .$
For $t\leq t_{1}$ concentrations $c_{(j,0)}(t)$ and all $c_{(q,-1)}$ are
negative, hence
$\frac{{\mathrm{d}}C_{S}^{+}}{{\mathrm{d}}t}=\frac{{\mathrm{d}}C^{-}}{{\mathrm{d}}t}=-\Pi_{S}^{+}(t)$
(37)
and for the norm of the correspondent solution for the base system we get the
inequality
$\|c(t)\|\leq 2-2\int_{t_{0}}^{\min\\{t,t_{1}\\}}{\Pi_{S}^{+}(\tau)}\
{\mathrm{d}}\tau\ \ \ \ \ \square$ (38)
The kinetic path summation formula gives us a family of estimates of
$\Pi_{S}^{\pm}$ from below. For each pair $i,j$ we can find the best of
available estimates of $\|G^{ij}(t)\|$ (the smallest one for various choices
of $A_{q}$ and subsets $S^{\pm}$) and then among all pairs of $i,j$ we should
choose the “most pessimistic” evaluation of $\|G^{ij}(t)\|$ (the biggest one).
It will give the evaluation of the contraction coefficient from above.
## 5 Simple example: Irreversible Cycle
Let us demonstrate all results for a simple kinetic system, a simple
irreversible cycle:
$A_{1}\xrightarrow{k_{1}}A_{2}\xrightarrow{k_{2}}\ldots\xrightarrow{k_{n-1}}A_{n}\xrightarrow{k_{n}}A_{1}$
(39)
All $k_{i}>0$ and are constant in time. For enumeration of $A_{i}$ we use the
standard cyclic order (mod$n$): $A_{n+j}\equiv A_{j}$.
The kinetic equations for this system are: $\dot{c}=Kc$ or
$\frac{{\mathrm{d}}}{{\mathrm{d}}t}\left[\begin{array}[]{l}c_{1}\\\ c_{2}\\\
\vdots\\\
c_{n}\end{array}\right]=\left[\begin{array}[]{llll}-k_{1}&0&\ldots&k_{n}\\\
k_{1}&-k_{2}&\ldots&0\\\ \vdots&\vdots&\vdots&\vdots\\\
0&0&0&-k_{n}\end{array}\right]\,\left[\begin{array}[]{l}c_{1}\\\ c_{2}\\\
\vdots\\\ c_{n}\end{array}\right]$ (40)
The characteristic equation for this system is
$\prod_{i=1}^{n}(k_{i}+\lambda)=\prod_{i=1}^{n}k_{i}\,.$
One eigenvalue for matrix $K$ is, obviously, $\lambda=0$, the correspondent
left eigenvector is the linear conservation law $l_{1}=(1,1,\ldots,1)$. The
right eigenvector for this $\lambda$ is the steady state
$r_{1}=\frac{1}{\sum_{i}\frac{1}{k_{i}}}(\frac{1}{k_{1}},\frac{1}{k_{2}},\ldots,\frac{1}{k_{n}})^{\rm
T}$ (normalized for $l_{1}r_{1}=1$). Other $n-1$ roots of the characteristic
equations have strictly negative real parts, $Re\lambda_{i}<0$ ($i>1$) and, in
general, cannot be found explicitly. For a given eigenvalue $\lambda$, the
eigenvectors have a simple structure:
$l_{{\lambda}\,i+1}=l_{{\lambda}\,i}\frac{\lambda+k_{i}}{k_{i}}\,\;\;r_{{\lambda}\,i}=\frac{\psi_{{\lambda}\,i}}{k_{i}}\,,\,\,\psi_{{\lambda}\,i-1}=\psi_{{\lambda}\,i}\frac{\lambda+k_{i}}{k_{i}}\,.$
(41)
With the normalization condition: for eigenvalues $\lambda$,
$\lambda^{\prime}$:
$l_{\lambda}r_{\lambda^{\prime}}=\delta_{\lambda\lambda^{\prime}}$, that is 1
for $\lambda=\lambda^{\prime}$ and 0 for $\lambda\neq\lambda^{\prime}$.
Two limit cases allow explicit analysis of eigenvalues and eigenvectors of
$K$:
1. 1.
Systems with limiting steps: one constant is much smaller than others, let it
be $k_{n}$, $k_{n}\ll k_{i}$, ($i=1,\ldots,n-1$);
2. 2.
Fully symmetric systems, $k_{1}=k_{2}=\ldots=k_{n}$.
For systems with limiting steps ($k_{n}\ll k_{i}$, ($i=1,\ldots,n-1$)) the
eigenvalues are close to $-k_{1},\ldots,-k_{n-1}$ and the relaxation is
limited by the second constant, the next to the minimal one (detailed analysis
is provided in [11, 12]).
For a symmetric system ($k_{1}=k_{2}=\ldots=k_{n}=k$), the eigenvalues are:
$\lambda_{q}=k\exp\left(\frac{2\pi iq}{n}\right)-1$ for $q=1,\ldots,n$. There
are $n$ distinct eigenvalues, one of them, $\lambda_{n}=0$, the other have
negative real part: $Re\lambda_{q}=k\left[\cos\left(\frac{2\pi
iq}{n}\right)-1\right]$. Let us further take $k=1$ for this system (include
$k$ into dimensionless time). For the left and right eigenvectors (41) we have
two waves moving in opposite directions,
$l_{q\,j+1}=l_{qj}\exp\left(\frac{2\pi iq}{n}\right)$,
$r_{q\,j-1}=r_{q\,j}\exp\left(\frac{2\pi iq}{n}\right)$. We can take with
respect to the normalization condition, $l_{q}r_{p}=\delta_{qp}$:
$\begin{split}&l_{q}=\left(1,\exp\left(\frac{2\pi
iq}{n}\right),\exp\left(2\frac{2\pi
iq}{n}\right),\ldots,\exp\left((n-1)\frac{2\pi iq}{n}\right)\right)\,,\\\
&r_{q}=\frac{1}{n}\left(1,\exp\left(-\frac{2\pi
iq}{n}\right),\exp\left(-2\frac{2\pi
iq}{n}\right),\ldots,\exp\left(-(n-1)\frac{2\pi iq}{n}\right)\right)^{\rm
T}\,.\end{split}$ (42)
For constant coefficients, the operator of shift in time from $t_{0}$ to
$t_{1}$ depends only on $t=t_{1}-t_{0}$: $U(t_{1},t_{0})=U(t)=\exp Kt$. We can
use (42) and write
$\begin{split}&U(t)=\sum_{q=1}^{n}\exp(\lambda_{q}t)|r_{q}\rangle\langle
l_{q}|\,,\\\ &(U(t))_{js}=\sum_{q=1}^{n}\exp(\lambda_{q}t)r_{qj}l_{qs}\\\
&\qquad\quad=\frac{1}{n}\sum_{q=1}^{n}\exp\left[t\left(\cos\frac{2\pi
q}{n}-1\right)\right]\cos\left((s-j)\frac{2\pi q}{n}+t\sin\frac{2\pi
q}{n}\right)\,.\end{split}$ (43)
This explicit formula allows us to compute all the necessary quantities
including the contraction coefficient $\delta_{U(t)}$ (4).
Now, let us produce the approximate formula for the same symmetric system by
mixers. First of all, let us represent the solution for the cycle by the path
summation formula. With the convention of cyclic enumeration, the set of paths
$\mathcal{I}_{i}$ started at $A_{i}$ is the sequence
$\mathcal{I}_{i}=\left\\{\begin{array}[]{l}A_{i}\xrightarrow{k_{i}}\,,\\\
A_{i}\xrightarrow{k_{i}}A_{i+1}\xrightarrow{k_{i+1}}\,,\\\
\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\\
A_{i}\xrightarrow{k_{i}}A_{i+1}\xrightarrow{k_{i+1}}A_{i+2}\xrightarrow{k_{i+2}}\ldots\xrightarrow{k_{i+j-1}}A_{i+j+1}\xrightarrow{k_{i+j}}\,,\\\
\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\end{array}\right\\}\,.$ (44)
Figure 4: Multi–sheeted representation of the path summation formula for a
cycle (46): a cycle (the base) is represented by an semi-infinite helix
produced by redirecting of reactions between sheets.
This sequence of paths corresponds to the multi–sheeted representation
presented in Fig. 4. First, we consider a infinite series of the copies of the
cycle. Each vertex of the extended system is numerated by two indexes:
$(A_{i},l)$, $i=1,2,\ldots,n$ (mod$n$), $l=1,2,3,\ldots$ is a natural number.
The reaction rate constants for copies are the same as for the initial
systems: $k_{(j,r)(i,l)}=k_{ji}\delta_{rl}$. This extended system obviously
satisfies the definition of the multi–sheeted extension of the cycle and in
its projection on the base we always have the kinetics of the cycle.
Let us select one number $i\in\\{1,\ldots,n\\}$ and recharge the reactions: we
annulate the “horizontal” reaction rate constant for
$(A_{i},l)\to(A_{i+1},l)$, $k_{(i+1,l)(i,l)}=0$, and instead of this reaction
take the reaction between levels, $(A_{i},l)\to(A_{i+1},l+1)$:
$k_{(i+1,l+1)(i,l)}=k_{i+1\,i}$ (see Fig. 4). This is also a multi–sheeted
extension of the cycle. Formula (26) for this multi–sheeted system allows us
to use integration of the infinite acyclic system (represented by the spiral
in Fig. 4)) instead of integration of the finite cyclic base system.
Now, let us put all $k_{i}=1$. For systems with constant coefficients we use
initial time moment $t_{0}=0$. For the set of paths $\mathcal{I}_{i}$ started
at $A_{i}$ the solution to the chain (11) with the initial conditions
$\varsigma_{i}(t_{0})=1$ and $\varsigma_{I}=0$ for $|I|>1$ is
$\varsigma_{I}(t)=\frac{t^{|I|-1}}{(|I|-1)!}e^{-t}\ .$ (45)
Obviously, $\sum_{I\in\mathcal{I}_{i}}\varsigma_{I}=1$. For concentration of
$A_{q}$, formula (17) gives
$u_{ji}(t)=e^{-t}\sum_{q=0}^{\infty}\frac{t^{qn+d_{ij}}}{(qn+d_{ij})!}\ ,$
(46)
where $d_{ij}$ is the length of the shortest oriented path from $A_{i}$ to
$A_{j}$ (here the length is the number of reactions and the trivial path from
$A_{i}$ to $A_{i}$ has the length zero).
For every two vertices $A_{i}$, $A_{j}$ we have only two mixers and both are
degenerated:
$A_{i}\xrightarrow{k}A_{i+1}\xrightarrow{k}\ldots\xrightarrow{k}A_{j}\xrightarrow{k}$,
length $j-i\mod n$ and
$A_{j}\xrightarrow{k}A_{j+1}\xrightarrow{k}\ldots\xrightarrow{k}A_{i}\xrightarrow{k}$,
length $i-j\mod n$.
Let us select one mixer
$A_{1}\xrightarrow{k}A_{2}\ldots\xrightarrow{k}A_{j}\xrightarrow{k}$ for
analysis. Initial conditions are: $c_{1}=1$, $c_{j}=-1$ and other
concentrations are equal to zero.
For this auxiliary chain with given initial conditions
$\begin{split}&c_{p}=\frac{t^{p-1}}{(p-1)!}e^{-t}\ \ (p=1,\ldots,j-1),\\\
&c_{j}=-e^{-t}\left(1-\frac{t^{j-1}}{(j-1)!}\right)\ .\end{split}$ (47)
The estimate (35) $\|G^{ij}(t)\|\leq 1-\int_{0}^{t}{\Pi_{S}^{+}(\tau)}\
{\mathrm{d}}\tau$ is valid until $c_{j}$ changes its sign. Hence, for $t$ we
have a boundary $t^{j-1}\leq(j-1)!$. The Stirling formula gives a convenient
estimate:
$\begin{split}&t^{j-1}\leq\sqrt{2\pi(j-1)}\left(\frac{j-1}{e}\right)^{j-1}\lesssim(j-1)!\\\
&t\leq t_{1}=\frac{j-1}{e}(2\pi(j-1))^{\frac{1}{2(j-1)}}\ .\end{split}$ (48)
Even a simpler estimate is $t<(j-1)/e$. If $t$ satisfies one of these
inequalities then concentration $c_{j}$ is negative and we can use the
estimate (35).
For this example,
$\begin{split}&\Pi_{S}^{+}(t)=c_{j-1}(t)=\frac{t^{j-2}}{(j-2)!}e^{-t}\
,\;\int_{0}^{t}{\Pi_{S}^{+}(\tau)}\
{\mathrm{d}}\tau=1-e^{-t}\sum_{p=0}^{j-2}\frac{t^{p}}{p!}\ ,\\\
&\|G^{ij}(t)\|\leq
e^{-t}\sum_{p=0}^{\min\\{d_{ji},d_{ij}\\}-1}\frac{t^{p}}{p!}\,,\;\delta_{U(t)}\leq
e^{-t}\sum_{p=0}^{\left[\frac{n}{2}\right]}\frac{t^{p}}{p!}\,,\end{split}$
(49)
where $\left[\frac{n}{2}\right]$ is the integer part of $n/2$. For $t>0$ this
estimate gives $\|G^{ij}(t)\|<1$ and $\delta_{U(t)}<1$ because
$\sum_{p=0}^{j-2}\frac{t^{p}}{p!}<e^{t}$. We can use the estimate
(LABEL:cycleestimeG) on an interval $[0,t_{1}]$, for example, on
$[0,\frac{j-1}{e}]$. Intersection of these intervals for all $i,j,i\neq j$ is
$[0,\frac{1}{e}]$ ($j\geq 2$). On this interval, the estimate
(LABEL:cycleestimeG) is valid for all $i,j$. For extension of such an estimate
for $t>\frac{1}{e}$ the submultiplicative property (5) can be used.
## 6 Ergodicity Boundary and Limitation of Ergodicity
In this Section we consider a reaction kinetic system (1) with constant
coefficients $k_{ji}>0$ for $(i,j)\in\mathcal{E}$.
Let us sort the values of kinetic parameters in decreasing order:
$k_{(1)}>k_{(2)}>\ldots>k_{(n)}$. The number in parenthesis is the number of
value in this order. Each of the constants $k_{(q)}$ is a reaction rate
constant $k_{ij}$ for some $i,j$ (and may be for several of them if values of
these constants coincide). Let us also suppose that the network is weakly
ergodic. We say that $k_{(r)},\,1\leq r\leq n$ is the ergodicity boundary [18]
if the network of reactions with parameters $k_{1},k_{2},\ldots,k_{r}$ is
weakly ergodic, but the network with parameters $k_{1},k_{2},\ldots,k_{r-1}$
is not. In other words, when eliminating reactions in decreasing order of
their characteristic times, starting with the slowest one, the ergodicity
boundary is the constant of the first reaction whose elimination breaks the
ergodicity of the reaction digraph.
Let $\mathcal{M}_{ij}$ ($i\neq j$) be a set of elementary mixers (29), (30)
between given $A_{i}$, $A_{j}$. For each $M\in\mathcal{M}_{ij}$ we can find a
cutting reaction rate constant, ${\rm cut}_{M}$:
$\begin{split}{\rm
cut}_{M}=\min\\{k_{i_{2}i_{1}},\ldots,k_{i_{r}i_{r-1}},k_{i_{r}i_{r+1}},\ldots,k_{i_{r+l-1}i_{r+l}}\\}\
\ \mbox{for (\ref{ElementaryMixer})}\ ;\\\ {\rm
cut}_{M}=\min\\{k_{i_{2}i_{1}},\ldots,k_{i_{r}i_{r-1}}\\}\ \ \mbox{for
(\ref{ElementaryMixerDegen})}\ .\end{split}$ (50)
Let us eliminate reactions in increasing orders of their constants (i.e. in
decreasing order of their characteristic times), starting with the smallest
one. To cut all elementary mixers between $A_{i},A_{j}$ ($i\neq j$), it is
necessary and sufficient to eliminate all $k_{pq}\leq{\rm cut}_{M}$ for all
$M\in\mathcal{M}_{ij}$. Therefore, for every pair $A_{i},A_{j}$ ($i\neq j$) we
can also introduce a cutting constant:
${\rm cut}_{ij}=\max_{M\in\mathcal{M}_{ij}}{\rm cut}_{M}\ .$
To destroy the weak ergodicity of the network $\mathcal{N}$ we have to cut al
least one pair $A_{i},A_{j}$ ($i\neq j$). The result can be formulates as the
following theorem.
###### Theorem 6
Theorem 6 The ergodicity boundary of a network $\mathcal{N}$ is the following
constant:
${\rm cut}_{\mathcal{N}}=\min_{i\neq j}{\rm cut}_{ij}\ .\ \ \ \ \ \ \ \square$
This boundary is a minimum (in pairs $A_{i},A_{j}$) of maxima (in mixers
$M\in\mathcal{M}_{ij}$) of minima (in constants).
Kinetic equations for elementary mixers (29), (30) allow explicit analytic
solutions. Nevertheless, explicit estimates in terms of cutting constants can
be also useful.
Let for an elementary mixer $M$ (29) $\kappa_{M}$ be the maximal sum of
constants of outgoing reactions:
$\kappa_{M}=\max\\{\kappa_{i_{p}}\ |\ p=i_{1},i_{2},\ldots,i_{r+l}\\},\ \
\kappa_{s}=\sum_{p,\ p\neq s}k_{ps}\ ,$
or for a degenerated elementary mixer $M$ (30)
$\kappa_{M}=\max\\{\kappa_{i_{p}}\ |\ p=i_{1},i_{2},\ldots,i_{r}\\}\ .$
Let us substitute all the constant for horizontal arrows in the elementary
mixer $M$ (29), (30) by $k={\rm cut}_{M}$, and all the constants for vertical
arrows ($i\neq i_{r}$) by $\kappa-k$, where $\kappa=\kappa_{M}$. This change
decreases the fluxes $\Pi^{\pm}$.
To find the estimate we have to solve the kinetic equation for a simple
uniform kinetic path:
$\setcounter{MaxMatrixCols}{11}\begin{CD}A_{1}@>{k}>{}>A_{2}@>{k}>{}>\ldots
@>{k}>{}>A_{s}@>{k}>{}>\\\ @V{}V{\kappa-k}V@V{}V{\kappa-k}V@V{}V{\kappa-k}V\\\
\end{CD}$ (51)
Similar to the simple cycle (47), we find
$c_{p}=\frac{(kt)^{p-1}}{(p-1)!}\exp(-\kappa t)\ \ (p=1,\ldots,s)\ ,$ (52)
the only difference is in exponents.
For the elementary mixers (29), (30) this formula gives
$\Pi^{+}(t)\geq k\frac{(kt)^{r-2}}{(r-2)!}\exp(-\kappa t)\ ,\ \ \Pi^{-}(t)\geq
k\frac{(kt)^{l-1}}{(l-1)!}\exp(-\kappa t)\ $
and the estimates from Theorems 4,5 (31), (35) become simple analytical
expressions after substitution of $\Pi^{\pm}$ by their estimates from below.
Let us find an universal estimate from below for $t_{1}$. It is
$\vartheta=\frac{1}{k+\kappa}\ .$
Indeed, in the degenerated elementary mixer (30) on the way from $A_{i}$ to
$A_{j}$ there exists at least one reaction with reaction rate constant $k$:
$A_{r}\to\ldots$. The integral flux through this reaction during the time
interval $[0,t]$ is
$\int_{0}^{t}kc_{r}(\tau)\ {\mathrm{d}}\tau\geq\int_{0}^{t}\Pi^{+}(\tau)\
{\mathrm{d}}\tau\ .$
The last inequality holds because all the flux in the mixer should go through
the reaction $A_{r}\to\ldots$ before it enters the last vertex. On the other
hand, $\int_{0}^{t}kc_{r}(\tau)\
{\mathrm{d}}\tau\leq\int_{0}^{t}k\exp(-k\tau)\ {\mathrm{d}}\tau$ (the last
integral corresponds to the case when all the concentration is collected at
the initial moment at $A_{r}$ and goes only through the reaction
$A_{r}\to\ldots$). Therefore,
$\int_{0}^{t}\Pi^{+}(\tau)\ {\mathrm{d}}\tau\leq 1-\exp(-k\tau)\ .$
From the condition (36) we find the estimate for $t_{1}$ from below:
$t_{1}\geq\tau_{1}$, where $\tau_{1}$ is solution to
$1-\exp(-k\tau)=\exp(-\kappa\tau)\ .$
We use convexity of exponential functions and substitute them in this equation
by linear approximation at point $\tau=0$: $\exp(-x)>1-x$ ($x>0$); this gives
us the estimate of $\tau_{1}$ from below:
$\tau_{1}<\vartheta=\frac{1}{k+\kappa}$.
For $t\in[0,\vartheta]$, $kt<1$ and
$1=\frac{(kt)^{0}}{0!}>\frac{(kt)^{1}}{1!}>\ldots>\frac{(kt)^{r}}{r!}>\ldots\
.$
For each mixer $M$ we introduce the length of mixer $d_{M}=\max\\{r-2,l-1\\}$
for (29) and $d_{M}=r-2$ for (30). In these notations, each mixer
$M\in\mathcal{M}_{ij}$ gives the estimate: for $t\in[0,\vartheta_{M}]$
$\|G^{ij}(t)\|\leq 1-\int_{0}^{t}{\rm cut}_{M}\frac{({\rm
cut}_{M}\tau)^{d_{M}}}{(d_{M})!}\exp(-\kappa_{M}\tau)\ {\mathrm{d}}\tau\ ,$
(53)
where
$\vartheta_{M}=\frac{1}{{\rm cut}_{M}+\kappa_{M}}\ .$
For each pair $i,j$ ($i\neq j$) we can select the “critical” elementary mixer
$M\in\mathcal{M}_{ij}$ with ${\rm cut}_{M}={\rm cut}_{ij}$ and put
$d_{ij}=d_{M}$, $\kappa_{ij}=\kappa_{M}$. If there are several critical
elementary mixers then we select one with minimal $d_{M}$, if there are
several such a mixers with minimal $d_{M}$ then we select one with minimal
$\kappa_{M}$. In this notation we have
$\|G^{ij}(t)\|\leq 1-\int_{0}^{t}{\rm cut}_{ij}\frac{({\rm
cut}_{ij}\tau)^{d_{ij}}}{(d_{ij})!}\exp(-\kappa_{ij}\tau)\ {\mathrm{d}}\tau\ $
(54)
for $t\in[0,\vartheta_{ij}]$, where
$\vartheta_{ij}=\frac{1}{{\rm cut}_{ij}+\kappa_{ij}}\ .$
Finally, for the whole network $\mathcal{N}$
${\rm cut}_{\mathcal{N}}=\min_{i,j,i\neq j}\\{{\rm cut}_{ij}\\},\
d_{\mathcal{N}}=\max_{i,j,i\neq j}\\{d_{ij}\\},\
\kappa_{\mathcal{N}}=\max_{i,j,i\neq j}\\{\kappa_{ij}\\},\
\vartheta_{\mathcal{N}}=\frac{1}{{\rm
cut}_{\mathcal{N}}+\kappa_{\mathcal{N}}}$
and for the contraction coefficient $\delta(t)$ (21) we obtain the estimate
$\begin{split}\delta(t)\leq&1-\int_{0}^{t}{\rm cut}_{\mathcal{N}}\frac{({\rm
cut}_{\mathcal{N}}\tau)^{d_{\mathcal{N}}}}{(d_{\mathcal{N}})!}\exp(-\kappa_{\mathcal{N}}\tau)\
{\mathrm{d}}\tau\\\ &=1-\left(\frac{{\rm
cut}_{\mathcal{N}}}{\kappa_{\mathcal{N}}}\right)^{d_{\mathcal{N}}+1}\left[1-\sum_{p=0}^{d_{\mathcal{N}}}\frac{(\kappa_{\mathcal{N}}t)^{p}}{p!}\exp(-\kappa_{\mathcal{N}}t)\right]\end{split}$
(55)
for $t\in[0,\vartheta_{\mathcal{N}}]$. For $t$ outside this interval, the
submultiplicative property (5) should be used.
## 7 Discussion
The kinetic path summation formula together with the multi–sheeted extension
of kinetics provide us with a factory of estimates. It is difficult to find,
who invented this approach.
The analysis of kinetic paths with selection of the most important (dominant)
paths allowed us to extract dominant systems from kinetic equations [11, 12].
A robust procedure for simplification of biochemical networks was created
[19]. This approach was developed into unified framework for hybrid
simplifications of Markov models of multiscale stochastic gene networks
dynamics [20]. Dominant subsystems were analyzed for dynamical models of
microRNA action on the protein translation process [21].
The multi–sheeted extension of kinetics provides us with a simple and useful
technique for estimation of relaxation processes in Master equation. This
method introduces an internal “microstructure” in the first order kinetic
systems. The kinetic path summation formula is a particular case of the
formula (26) (Proposition 2).
Indeed, let us construct the following multi–sheeted extension of the Master
equation. The set of components is $\mathcal{A}\times\mathcal{K}$, where
$\mathcal{K}=\\{0\\}\cup\mathcal{K}_{1}$ and $\mathcal{K}_{1}$ is the set of
all kinetic paths $I$ with lengths $|I|>1$ (non-degenerated paths). The
connections between sheets (redirected reactions) are:
$A_{i_{I^{-}},I^{-}}\xrightarrow{k_{I}}A_{i_{I},I}\ \ \mbox{instead of}\ \
A_{i_{I^{-}},I^{-}}\xrightarrow{k_{I}}A_{i_{I},I^{-}}\ .$
According to this rule, the reaction that continues the path $I^{-}$ to the
path $I$ is redirected and goes from the sheet $I^{-}$ to the sheet $I$. For a
degenerated $I^{-}$, we take $A_{i_{I^{-}},I^{-}}=A_{i_{I^{-}},0}$, this means
that all paths start on the zero sheet, and all reactions from this sheet lead
to other sheets: $A_{i}\to A_{j}$ transforms into $A_{i,0}\to
A_{j,\\{i,j\\}}$, where $\\{i,j\\}$ is a path of the length 2. Formula (26)
for this multi–sheeted structure coincide with the kinetic path summation
formula (17) (Theorem 2) for initial conditions $c_{i,0}=1$ and other
$c_{(j,I)}=0$.
This multi–sheeted extension may be considered as a generalization of the
Bethe lattices introduced by H. Bethe in 1935 [22]. For example, if in the
initial graph of reactions each vertex has the same number of outgoing edges
then the constructed multi–sheeted extension can be considered as a bundle of
the Bethe lattices, each of them starts from one point of the zeroth sheet.
For each starting point, $A_{(i,0)}$ the corresponding Bethe lattice
represents the “Green function” $u_{ji}(t,t_{0})$ for given $i$ and for all
possible $j$.
We produced the kinetic path summation formula for time–dependent kinetic
equations and applied this formula for evaluation of the ergodicity
coefficient. The evaluation of the contraction coefficient in the $l_{1}$ norm
is the main tool for studying of the relaxation in time–dependent Markov
processes since the seminal works of R. Dobrushin [15].
Another important context of this study is the analysis of the eigenvalues of
the stochastic matrices [23, 24] and, especially the analysis of these
eigenvalues for matrices with specified graph [25, 26]. In chemical kinetics,
evaluation of the eigenvalues through kinetic constants was given in series of
work by V.Cheresiz and G. Yablonskii [27, 28].
Various estimates of eigenvalues of $K$ could be produced from the estimates
of contraction (31), (35). The simplest one follows from (55):
$Re(\lambda)\leq\frac{\ln(\delta(\vartheta))}{\vartheta}<0\ .$ (56)
Several problems should be resolved to make the use of the path summation
formula more effective. Perhaps, the most important of them was mentioned in
the comment [29]). The amount of the kinetic path needed for accurate estimate
of the solution grows quickly in time for a sufficiently complex system.
Hence, we need either special tricks for the analysis of path sampling or
special asymptotic formulas for long paths instead of exact solutions.
Another possible approach to this problem is in the use of more complex
exactly solvable systems instead of paths. The set of reactions is solvable,
if there exists a linear transformation of coordinates $c\mapsto a$ such that
kinetic equations in new coordinates for all values of reaction constants have
the triangle form:
$\frac{{\mathrm{d}}a_{i}}{{\mathrm{d}}t}=f_{i}(a_{1},a_{2},...\,a_{i}).$ (57)
The algorithm for the analysis of reaction network solvability was developed
in [5] (see also [11]). The simplest examples of solvable networks give
acyclic graphs (reaction trees) and pairs of mutually inverse reactions. It
may be possible to decompose the complex system of transitions into a sequence
of solvable systems.
## References
* [1] S.R. Meyn, Control Techniques for Complex Networks, Cambridge University Press, Cambridge, 2007.
* [2] S.R. Meyn, R.L. Tweedie, Markov Chains and Stochastic Stability, 2nd Edition, Cambridge University Press, Cambridge, 2009.
* [3] N.G. Van Kampen, Stochastic processes in physics and chemistry, North–Holland, Amsterdam 1981.
* [4] J.C. Kuo, J. Wei, A lumping analysis in monomolecular reaction systems. Analysis of the approximately lumpable system. Ind. Eng. Chem. Fundam. 8 (1969) 124–133.
* [5] A.N. Gorban, V.I. Bykov, G.S. Yablonskii, Essays on chemical relaxation, Novosibirsk: Nauka, 1986.
* [6] S.X. Sun, Path Summation Formulation of the Master Equation, Phys. Rev. Lett. 96 (2006), 210602.
* [7] B. Harland, S.X. Sun, Path ensembles and path sampling in nonequilibrium stochastic systems, J. Chem. Phys. 127 (2007), 104103.
* [8] D.T. Gillespie, Exact Stochastic Simulation of Coupled Chemical Reactions, J. Phys. Chem. 81 (25) (1977), 2340–2361.
* [9] O. Flomenbom, J. Klafter, Closed-Form Solutions for Continuous Time Random Walks on Finite Chains, Phys. Rev. Lett. 95 (2005), 098105.
* [10] O. Flomenbom, R.J. Silbey, Path-probability density functions for semi-Markovian random walks, Phys. Rev E 76 (2007), 041101.
* [11] A.N. Gorban, O. Radulescu, Dynamic and static limitation in reaction networks, revisited, Advances in Chemical Engineering 34 (2008), 103–173. E-print: arXiv:physics/0703278 [physics.chem-ph]
* [12] A.N. Gorban, O. Radulescu, A.Y. Zinovyev, Asymptotology of chemical reaction networks, Chemical Engineering Science 65 (2010), 2310–2324. E-print: arXiv:0903.5072 [physics.chem-ph]
* [13] G.S. Yablonskii, V. I. Bykov, A.N. Gorban, V.I. Elokhin, Kinetic models of catalytic reactions (Series Comprehensive Chemical Kinetics, Vol. 32), Elsevier, Amsterdam, 1991.
* [14] O.N. Temkin, A.V. Zeigarnik, D.G. Bonchev, Chemical Reaction Networks: A Graph-Theoretical Approach, CRC Press, Boca Raton, FL, 1996.
* [15] R.L. Dobrushin, Central limit theorem for non-stationary Markov chains I, II, Theor. Prob. Appl. 1 (1956), 163–80, 329–383.
* [16] E. Seneta, Nonnegative Matrices and Markov Chains, Springer, New York, 1981.
* [17] P. Van Mieghem, Performance Analysis of Communications Networks and Systems, Cambridge University Press, Cambridge, 2006.
* [18] A.N. Gorban, O. Radulescu, Dynamical robustness of biological networks with hierarchical distribution of time scales, IET Syst. Biol., 1 (2007), 238–246. E-print: arXiv:q-bio/0701020 [q-bio.MN].
* [19] O. Radulescu, A.N. Gorban, A. Zinovyev, A. Lilienbaum, A. Robust simplifications of multiscale biochemical networks, BMC Systems Biology 2 (1) (2008), 86. http://www.biomedcentral.com/1752-0509/2/86
* [20] A. Crudu, A. Debussche and O. Radulescu, Hybrid stochastic simplifications for multiscale gene networks, BMC Systems Biology, 3 (2009), 89. http://www.biomedcentral.com/1752-0509/3/89/
* [21] A. Zinovyev, N. Morozova, N. Nonne, E. Barillot, A. Harel-Bellan, A.N. Gorban, Dynamical modeling of microRNA action on the protein translation process, BMC Systems Biology, 4 (2010), 13. E-print: arXiv:0911.1797 [q-bio.MN]
* [22] R.J. Baxter, Exactly solved models in statistical mechanics. Academic Press, New York, 1982.
* [23] N.A. Dmitriev, E.V. Dynkin, Characteristic roots of stochastic matrices, Izv. Akad. Nauk SSSR Ser Mat 10 (1946), 167–184. English translation in: Eleven Papers Translated from the Russian (American Mathematical Society Translations, ser. 2, v. 140, 1988, pp. 57–78.
* [24] F.I. Karpelevich, On the characteristic roots of matrices with nonnegative elements, Izv. Akad. Nauk SSSR Ser Mat, 15 (1951), 361-383. English translation in: Eleven Papers Translated from the Russian (American Mathematical Society Translations, ser. 2, v. 140, 1988, pp. 79–101.
* [25] C.R. Johnson, R.B. Kellog, A.B. Stephens, Complex eigenvalues of a nonnegative matrix with a specified graph. Linear Algebra and Appl. 20 (1978), 179–187.
* [26] C.R. Johnson, R.B. Kellog, A.B. Stephens, Complex eigenvalues of a nonnegative matrix with a specified graph. II, Linear and Multilinear Algebra 7 (1979), 129–143; 8 (1979/80), 171.
* [27] V.M. Cheresiz, G.S. Yablonskii, Estimation of relaxation times for chemical kinetic equations (linear case), React. Kinet. Catal. Lett. 22 (1983), 69–73.
* [28] G.S. Yablonskii, V.M. Cheresiz, Four types of relaxation in chemical kinetics (linear case), React. Kinet. Catal. Lett. 24 (1984), 49–53.
* [29] O. Flomenbom, J. Klafter, R.J. Silbey, Comment on “Path Summation Formulation of the Master Equation”, Phys. Rev. Lett. 97 (2006), 178901.
|
arxiv-papers
| 2010-06-21T17:27:17 |
2024-09-04T02:49:11.094272
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.N. Gorban",
"submitter": "Alexander Gorban",
"url": "https://arxiv.org/abs/1006.4128"
}
|
1006.4316
|
# Jacob’s ladders and the oscillations of the function $|\zeta(1/2+it)|^{2}$
around its mean-value; law of the almost exact equality of corresponding areas
Jan Moser Department of Mathematical Analysis and Numerical Mathematics,
Comenius University, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA
jan.mozer@fmph.uniba.sk
###### Abstract.
The oscillations of the function $Z^{2}(t),\ t\in[0,T]$ around the main part
$\sigma(T)$ of its mean-value are studied in this paper. It is proved that an
almost equality of the corresponding areas holds true. This result cannot be
obtained by methods of Balasubramanian, Heath-Brown and Ivic.
###### Key words and phrases:
Riemann zeta-function
## 1\. Introduction
### 1.1.
The Titchmarsh-Kober-Atkinson (TKA) formula
(1.1) $\int_{0}^{\infty}Z^{2}(t)e^{-2\delta t}{\rm
d}t=\frac{c-\ln(4\pi\delta)}{2\sin\delta}+\sum_{n=0}^{N}c_{n}\delta^{n}+\mathcal{O}(\delta^{N+1})$
(see [17], p. 131) remained as an isolated result for the period of 56 years.
We have discovered (see [5]) the nonlinear integral equation
(1.2) $\int_{0}^{\mu[x(T)]}Z^{2}(t)e^{-\frac{2}{x(T)}t}{\rm
d}t=\int_{0}^{T}Z^{2}(t){\rm d}t$
in which the essence of the TKA formula is encoded. Namely, we have shown in
[5] that the following almost exact expression for the Hardy-Littlewood
integral
(1.3) $\int_{0}^{T}Z^{2}(t){\rm
d}t=\frac{\varphi(T)}{2}\ln\frac{\varphi(T)}{2}+(c-\ln(2\pi))\frac{\varphi(T)}{2}+c_{0}+\mathcal{O}\left(\frac{\ln
T}{T}\right)$
takes place, where $\varphi(T)$ is the Jacob’s ladder, i.e. an arbitrary
solution to the nonlinear integral equation (1.2).
###### Remark 1.
Our formula (1.3) for the Hardy-Littlewood integral
(1.4) $\int_{1}^{T}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=\int_{1}^{T}Z^{2}(t){\rm d}t$
has been obtained after the time period of 90 years since this integral
appeared in 1918 with the first result
$\int_{1}^{T}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm d}t\sim
T\ln T$
(see [3], pp. 122, 151-156).
### 1.2.
Let us remind that
* (A)
The Good’s $\Omega$ \- theorem (see [2]) implies for the Balasubramanian
formula (see [1])
(1.5) $\int_{0}^{T}Z^{2}(t){\rm d}t\sim T\ln T+(2c-1-\ln 2\pi)T+R(T),\
R(T)=\mathcal{O}(T^{1/3+\epsilon})$
that
(1.6) $\limsup_{T\to\infty}|R(T)|=+\infty,$
i.e. the error term in (1.5) is unbounded at $T\to\infty$.
* (B)
In the case of our formula (1.3) the error term definitely tends to zero
(1.7) $\lim_{T\to\infty}r(T)=0;\qquad r(T)=\mathcal{O}\left(\frac{\ln
T}{T}\right),$
i.e. our formula is almost exact (see [5]).
###### Remark 2.
In this paper the geometric interpretation of (1.6) and (1.7) is obtained.
### 1.3.
For the mean-value of the function
$\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}=Z^{2}(t)$, where
$Z(t)=e^{i\vartheta(t)}\zeta\left(\frac{1}{2}+it\right),\quad\vartheta(t)=-\frac{t}{2}\ln\pi+\text{Im}\ln\Gamma\left(\frac{1}{4}+i\frac{t}{2}\right),$
we obtain from (1.3)
(1.8) $\frac{1}{T}\int_{0}^{T}Z^{2}(t){\rm
d}t=\frac{\varphi(T)}{2T}\ln\frac{\varphi(T)}{2}+(c-\ln
2\pi)\frac{\varphi(T)}{2T}+\frac{c_{0}}{T}+\mathcal{O}\left(\frac{\ln
T}{T}\right).$
Let
(1.9) $\sigma(T)=\frac{\varphi(T)}{2T}\ln\frac{\varphi(T)}{2}+(c-\ln
2\pi)\frac{\varphi(T)}{2T}+\frac{c_{0}}{T}$
denote the main part of the mean-value (1.8). In this paper the oscillation of
the values of the function $Z^{2}(t),\ t\in[0,T]$ around the main part
$\sigma(T)$ of its mean-value are studied.
###### Remark 3.
The main result of this paper is the following statement: the areas of the
figures corresponding to the parts of the graph of the function $Z^{2}(t),\
t\in[0,T]$ given by inequalities $Z^{2}(t)\geq\sigma(T)$ and
$Z^{2}(t)\leq\sigma(T)$, respectively, are almost exactly equal.
This paper is a continuation of the series [5]-[16].
## 2\. Result
### 2.1.
Let (see (1.9))
(2.1) $\begin{split}S^{+}(T)&=\\{t:\ Z^{2}(t)\geq\sigma(T),\ t\in[0,T]\\},\\\
S^{-}(T)&=\\{t:\ Z^{2}(t)<\sigma(T),\ t\in[0,T]\\}\end{split}$
and
(2.2) $\begin{split}\Pi^{+}(T)&=\\{(t,y):\ \sigma(T)\leq y\leq Z^{2}(t),\ t\in
S^{+}(T)\\},\\\ \Pi^{-}(T)&=\\{(t,y):\ Z^{2}(T)\leq y\leq\sigma(t),\ t\in
S^{-}(T)\\},\end{split}$
i.e. $\Pi^{+}$ is the figure that corresponds to the parts of the graph of
$y=Z^{2}(t),\ t\in[0,T]$ lying above the segment $y=\sigma(T)$ and similarly
$\Pi^{-}$ corresponds to the parts of the graph lying under that segment. Let
$m\\{\Pi^{+}(T)\\},\ m\\{\Pi^{-}(T)\\}$ denote measures of corresponding
figures, i.e.
(2.3)
$\begin{split}m\\{\Pi^{+}(T)\\}&=\int_{S^{+}(T)}\\{Z^{2}(t)-\sigma(T)\\}{\rm
d}t,\\\ m\\{\Pi^{-}(T)\\}&=\int_{S^{-}(T)}\\{\sigma(T)-Z^{2}(t)\\}{\rm
d}t.\end{split}$
The following theorem holds true.
###### Theorem.
First of all, we have the formula
(2.4) $m\\{\Pi^{+}(T)\\}=m\\{\Pi^{-}(T)\\}+\mathcal{O}\left(\frac{\ln
T}{T}\right)$
(see (1.3), (1.9), (2.1)-(2.3)). Next, the structure of the formula (2.4) is
as follows: there are the functions $\eta_{1}(T),\ \eta_{2}(T)$ that the
following formulae
(2.5)
$\begin{split}m\\{\Pi^{+}(T)\\}&=\frac{1+o(1)}{2\pi^{2}}\frac{T\ln^{4}T}{\eta_{1}-\eta_{2}}-\frac{\eta_{2}}{\eta_{1}-\eta_{2}}\mathcal{O}\left(\frac{\ln
T}{T}\right),\\\
m\\{\Pi^{-}(T)\\}&=\frac{1+o(1)}{2\pi^{2}}\frac{T\ln^{4}T}{\eta_{1}-\eta_{2}}-\frac{\eta_{1}}{\eta_{1}-\eta_{2}}\mathcal{O}\left(\frac{\ln
T}{T}\right)\end{split}$
hold true, and
(2.6) $AT^{2/3}\ln^{4}T<m\\{\Pi^{+}(T)\\},m\\{\Pi^{-}(T)\\}<AT\ln T.$
In addition to (2.6): on the Lindelöf hypothesis
(2.7) $m\\{\Pi^{+}(T)\\},m\\{\Pi^{-}(T)\\}>A(\epsilon)T^{1-\epsilon},$
and on Riemann hypothesis
(2.8) $m\\{\Pi^{+}(T)\\},m\\{\Pi^{-}(T)\\}>T^{1-\frac{A}{\ln\ln T}}.$
###### Corollary.
We have by (2.5), (2.6)
(2.9) $\eta_{1}(T)-\eta_{2}(T)>A\ln^{3}T.$
###### Remark 4.
Since from (2.4)
(2.10) $\lim_{T\to\infty}[m\\{\Pi^{+}(T)\\}-m\\{\Pi^{-}(T)\\}]=0$
follows then we have the almost exact equality of the areas
$m\\{\Pi^{+}(T)\\}$ and $m\\{\Pi^{-}(T)\\}$.
### 2.2.
In the case of the Balasubramanian formula (1.5) we have (comp. (1.3), (1.9))
$\sigma_{1}(T)=\ln T+2c-1-\ln 2\pi.$
Let
$S^{+}_{1}(T),S^{-}_{1}(T),\Pi^{+}_{1}(T),\Pi^{-}_{1}(T),m\\{\Pi^{+}_{1}(T)\\},m\\{\Pi^{-}_{1}(T)\\}$
correspond to $\sigma_{1}(T)$ similarly to (2.1)-(2.3). Then from (1.5) we
obtain
(2.11)
$m\\{\Pi^{+}_{1}(T)\\}=m\\{\Pi^{-}_{1}(T)\\}+\mathcal{O}(T^{1/3+\epsilon}),\
T\to\infty,$
and (see (1.6)
(2.12)
$\limsup_{T\to\infty}|m\\{\Pi^{+}_{1}(T)\\}-m\\{\Pi^{-}_{1}(T)\\}|=+\infty.$
###### Remark 5.
The following holds true:
* (A)
Our formula (1.3) which has been obtained by means of the Jacob’s ladders
leads to the almost exact equality of the areas (see (2.4), (2.10).
* (B)
The Balasubramanian formula (1.5) which has been obtained by means of
estimation of trigonometric sums leads to the formula (2.11) that possesses
quite large uncertainty (2.11) and this error term cannot be removed.
## 3\. Proof of Theorem
### 3.1.
We obtain from (1.3), (2.1)
(3.1) $\int_{S^{+}(T)}\\{Z^{2}(t)-\sigma(T)\\}{\rm
d}t+\int_{S^{-}(T)}\\{Z^{2}(t)-\sigma(T)\\}{\rm d}t=\mathcal{O}\left(\frac{\ln
T}{T}\right)$
and from (3.1) by (2.3) the formula
(3.2) $m\\{\Pi^{+}(T)\\}-m\\{\Pi^{-}(T)\\}=\mathcal{O}\left(\frac{\ln
T}{T}\right)$
follows, i.e. (2.4).
### 3.2.
Next, from the Ingham formula (see [4], p. 277, [17], p. 125)
(3.3) $\int_{0}^{T}Z^{4}(t){\rm
d}t=\frac{1}{2\pi^{2}}T\ln^{4}T+\mathcal{O}(T\ln^{3}T)$
we obtain (see (1.9)
(3.4) $\int_{0}^{T}\\{Z^{4}(t)-\sigma^{2}(T)\\}{\rm
d}t=\frac{1}{2\pi^{2}}T\ln^{4}T-T\sigma^{2}(T)+\mathcal{O}(T\ln^{3}T).$
Since ($\varphi(T)\sim T$)
$T\sigma^{2}(T)=\mathcal{O}\left\\{\frac{\varphi^{2}(T)}{T}\ln^{2}\frac{\varphi(T)}{2}\right\\}=\mathcal{O}(T\ln^{2}T),$
then from (3.4) the formula
(3.5) $\int_{0}^{T}\\{Z^{4}(t)-\sigma^{2}(T)\\}{\rm
d}t=\frac{1+o(1)}{2\pi^{2}}T\ln^{4}T$
follows.
### 3.3.
Since $Z^{4}(t)-\sigma^{2}(T)=(Z^{2}-\sigma)(Z^{2}+\sigma)$ and
$Z^{2}(t)-\sigma(T)$ is always of the same sign on $S^{+}(T)$ and on
$S^{-}(T)$, respectively, then from (3.5) we obtain (see (2.3))
(3.6)
$\eta_{1}(T)m\\{\Pi^{+}(T)\\}-\eta_{2}(T)m\\{\Pi^{-}(T)\\}=\frac{1+o(1)}{2\pi^{2}}T\ln^{4}T,$
where $\eta_{1}=\eta_{1}(T),\ \eta_{2}=\eta_{2}(T)$ are the mean-values of
$Z^{2}(t)+\sigma(T)$ relatively to the values of the functions
$Z^{2}(t)-\sigma(T)$ and $\sigma(T)-Z^{2}(t)$, respectively on the sets
$S^{+}(T)$ and $S^{-}(T)$, respectively. It is clear that
(3.7) $A\ln T<\eta_{1}(T),\eta_{2}(T)<AT^{1/3}$
(see (1.9); $|Z(t)|<t^{1/6}$). Next, $\eta_{1}(T)\not=\eta_{2}(T)$ is also
true. Since if $\eta_{1}=\eta_{2}$ then by (3.2), (3.6), (3.7) we would have
the contradiction. Hence, from the simple system of linear equations (3.2),
(3.6) we obtain (2.5).
### 3.4.
Since
(3.8) $0<\eta_{1}-\eta_{2}<\eta_{1}<AT^{1/3}$
(see (2.5), (3.7)) then we obtain from (2.5) the lower estimates in (2.6).
Next we have (see (1.9), (2.3)
$m\\{\Pi^{+}(T)\\},m\\{\Pi^{-}(T)\\}<\int_{0}^{T}\\{Z^{2}(t)-\sigma(T)\\}{\rm
d}t<AT\ln T$
i.e. the upper estimates in (2.6) hold true.
### 3.5.
Following the Lindelöf and the Riemann conjectures the estimates
$Z^{2}(t)<A(\epsilon)t^{\epsilon},\quad t^{\frac{A}{\ln\ln t}}$
take place correspondingly and then the conditional estimates (2.7) and (2.8)
follow.
I would like to thank Michal Demetrian for helping me with the electronic
version of this work.
## References
* [1] R. Balasubramanian, ‘An improvement on a theorem of Titchmarsh on the mean square of $|\zeta(1/2+it)|^{2}$‘, Proc. London Math. Soc. 3, 36 (1978), 540-575.
* [2] A. Good, ‘Ein $\Omega$ \- resultat für quadratische Mittel der Riemannschen Zetafunktion auf der kritische Linie‘, Invent. Math. 41, (1977), 233-251.
* [3] G.H. Hardy and J.E. Littlewood, ‘Contribution to the theory of the Riemann zeta-function and the theory of the distribution od Primes‘, Acta Math. 41, (1918), 119-195.
* [4] A.E. Ingham, ‘Mean-value theorems in the theory of the Riemann zeta-function‘, Proc. Lond. Math. Soc. (2), 27, (1926), 273-300.
* [5] J. Moser, ‘Jacob’s ladders and the almost exact asymptotic representation of the Hardy-Littlewood integral’, (2008), arXiv:0901.3973.
* [6] J. Moser, ‘Jacob’s ladders and the tangent law for short parts of the Hardy-Littlewood integral’, (2009), arXiv:0906.0659.
* [7] J. Moser, ‘Jacob’s ladders and the multiplicative asymptotic formula for short and microscopic parts of the Hardy-Littlewood integral’, (2009), arXiv:0907.0301.
* [8] J. Moser, ‘Jacob’s ladders and the quantization of the Hardy-Littlewood integral’, (2009), arXiv:0909.3928.
* [9] J. Moser, ‘Jacob’s ladders and the first asymptotic formula for the expression of the sixth order $|\zeta(1/2+i\varphi(t)/2)|^{4}|\zeta(1/2+it)|^{2}$’, (2009), arXiv:0911.1246.
* [10] J. Moser, ‘Jacob’s ladders and the first asymptotic formula for the expression of the fifth order $Z[\varphi(t)/2+\rho_{1}]Z[\varphi(t)/2+\rho_{2}]Z[\varphi(t)/2+\rho_{3}]\hat{Z}^{2}(t)$ for the collection of disconnected sets‘, (2009), arXiv:0912.0130.
* [11] J. Moser, ‘Jacob’s ladders, the iterations of Jacob’s ladder $\varphi_{1}^{k}(t)$ and asymptotic formulae for the integrals of the products $Z^{2}[\varphi^{n}_{1}(t)]Z^{2}[\varphi^{n-1}(t)]\cdots Z^{2}[\varphi^{0}_{1}(t)]$ for arbitrary fixed $n\in\mathbb{N}$‘ (2010), arXiv:1001.1632.
* [12] J. Moser, ‘Jacob’s ladders and the asymptotic formula for the integral of the eight order expression $|\zeta(1/2+i\varphi_{2}(t))|^{4}|\zeta(1/2+it)|^{4}$‘, (2010), arXiv:1001.2114.
* [13] J. Moser, ‘Jacob’s ladders and the asymptotically approximate solutions of a nonlinear diophantine equation‘, (2010), arXiv: 1001.3019.
* [14] J. Moser, ‘Jacob’s ladders and the asymptotic formula for short and microscopic parts of the Hardy-Littlewood integral of the function $|\zeta(1/2+it)|^{4}$‘, (2010), arXiv:1001.4007.
* [15] J. Moser, ‘Jacob’s ladders and the nonlocal interaction of the function $|\zeta(1/2+it)|$ with $\arg\zeta(1/2+it)$ on the distance $\sim(1-c)\pi(t)$‘, (2010), arXiv: 1004.0169.
* [16] J. Moser, ‘Jacob’s ladders and the $\tilde{Z}^{2}$ \- transformation of polynomials in $\ln\varphi_{1}(t)$‘, (2010), arXiv: 1005.2052.
* [17] E.C. Titchmarsh, ‘The theory of the Riemann zeta-function‘, Clarendon Press, Oxford, 1951.
|
arxiv-papers
| 2010-06-22T15:33:39 |
2024-09-04T02:49:11.108399
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jan Moser",
"submitter": "Michal Demetrian",
"url": "https://arxiv.org/abs/1006.4316"
}
|
1006.4402
|
# Simulating Concordant Computations
Bryan Eastin beastin@nist.gov National Institute of Standards and Technology,
Boulder, CO 80305
###### Abstract
A quantum state is called concordant if it has zero quantum discord with
respect to any part. By extension, a concordant computation is one such that
the state of the computer, at each time step, is concordant. In this paper, I
describe a classical algorithm that, given a product state as input, permits
the efficient simulation of any concordant quantum computation having a
conventional form and composed of gates acting on two or fewer qubits. This
shows that such a quantum computation must generate quantum discord if it is
to efficiently solve a problem that requires super-polynomial time
classically. While I employ the restriction to two-qubit gates sparingly, a
crucial component of the simulation algorithm appears not to be extensible to
gates acting on higher-dimensional systems.
The search for the origin of the computational power of quantum mechanics has
proven to be a recurring theme in quantum information theory. Primarily, this
search has focused on identifying the feature of quantum mechanics that
permits the efficient111The definition of “efficient” is taken from classical
computer science, where it refers to any computation that requires an amount
of resources (particularly time steps) scaling at most polynomially with the
problem size. solution of certain classically intractable problems. In
addition to being useful, computational speedups of this magnitude are
intriguing since, classically, no such improvement is to be found over rather
basic models of computation, e.g., the Turing machine.
Among the proposed sources of this quantum advantage, the most widely studied
is a kind of non-local correlation known as entanglement Horodecki et al.
(2009). The state of a composite system is entangled if it cannot be described
in terms of a, possibly uncertain, local assignment of states to individual
subsystems. Classically, non-trivial correlations indicate imperfect
information about the state of the system, but entanglement is possible for
quantum states of maximal knowledge, or pure states. At the extreme, an
entangled state of a composite system may be pure while the marginal state of
the component subsystems is maximally impure, or maximally mixed. In other
words, one may know everything possible about the state of a composite quantum
system without knowing anything about the state of the component subsystems.
As a distinctly non-classical property and a necessary resource for protocols
such as teleportation and quantum error correction, entanglement is a natural
suspect when investigating the power of quantum computing.
There are two kinds of evidence in favor of entanglement as the crucial
resource for achieving speedups that enable the efficient solution of a
classically intractable problem, a variety of speedup henceforth labeled
Promethean. First, there are proofs that pure-state quantum computations
generating only limited amounts of entanglement can be efficiently simulated
classically and are therefore incapable of solving any problem that cannot be
solved in polynomial time by a classical computer. An early result of this
sort was shown by Jozsa and Linden Jozsa and Linden (2003), who described a
method for efficiently simulating any quantum computation whose correlations
are approximately confined to regions of bounded size. Shortly thereafter,
Vidal proposed an efficient simulation algorithm for quantum computations
whose maximum Schmidt rank for any bipartition of the computer scales at most
as a polynomial Vidal (2003). These methods of simulation can each be applied
to quantum computations with either mixed or pure states, but in the former
case classical correlations, in addition to entanglement, are restricted. The
second kind of evidence for the importance of entanglement is its apparent
generation by all implementations of Shor’s quantum factoring algorithm. In
particular, a typical implementation of Shor’s algorithm has been shown to
generate entanglement that precludes its simulation by either Jozsa and
Linden’s or Vidal’s method Jozsa and Linden (2003); Orús and Latorre (2004).
To summarize, entanglement is necessary for obtaining Promethean speedups with
pure-state quantum computing, and there are indications that it may be
required for Shor’s algorithm.
Regarding mixed states, further, and contrary, evidence comes from the DQC1
model of quantum computation Knill and Laflamme (1998), where all but one of
the qubits in the computer is initially prepared in the maximally mixed state.
DQC1 is believed to be strictly less powerful than pure-state quantum
computing Knill and Laflamme (1998); A. Ambainis and Vazirani (2006), but it
nonetheless seems to be capable of providing Promethean speedups in, for
example, trace estimation. Datta, Flammia, and Caves have shown numerically
that trace estimation is possible even with a vanishing amount of entanglement
(as measured by the negativity of bipartite splittings) Datta et al. (2005).
Nevertheless, Datta and Vidal have shown that the Schmidt rank grows
exponentially for certain bipartitions of a quantum computer performing trace
estimation Datta and Vidal (2007), thereby demonstrating the existence of
correlations, though not necessarily entanglement, sufficient to thwart
Vidal’s simulation method. Based on these results, it seems probable that
Promethean speedups are possible even in the absence of entanglement.
But if entanglement is not the source of Promethean speedups in DQC1 then we
are left to ask what is. Among the proposed alternatives is a measure of non-
classical correlation known as quantum discord Zurek (2000). Datta, Shaji, and
Caves have shown that discord is indeed present in the trace-estimation
algorithm Datta et al. (2008), but it has never been proven to be necessary.
The work presented in this paper was motivated by the desire to show that
discord is necessary for Promethean speedups in mixed-state quantum
computations. Since, for pure states, discord reduces to a measure of
entanglement, this would amount to an extension of the result (described
above) about the utility of entanglement in pure-state quantum computing. To
this end, I considered the difficulty of simulating concordant computations,
i.e., those that generate no quantum discord, as suggested by Ref. Lanyon et
al. (2008).
Here, I describe an algorithm for efficiently simulating, using a classical
computer, any computation that does not generate discord and consists of a
sequence of one- and two-qubit unitary gates followed by single-qubit
measurements. Section I briefly introduces some notation and Sec. II covers
discord, concordance, and concordant computations and proves a few results
that are employed later. My simulation algorithm is described for quantum
computations in a conventional form in Sec. III and extensions to non-
conventional forms are discussed in Sec. IV. The conclusion contains a
discussion of open problems.
## I Notation
Unitary operators, projectors, and sets are denoted by capital roman letters
in math-italic, black-board, and calligraphic font, respectively, e.g., $U$,
${\mathbb{P}}$, and $\mathcal{A}$. For more generic functions on quantum
states I use capital Roman letters in math font. Throughout the paper, quantum
operators and states are given subscripts (which may be sets) to denote the
subsystems they act upon and/or to index the component corresponding to that
subsystem; all other identifying indices and labels are represented as
superscripts. Thus, the state of a composite system can be expressed as
$\rho_{\mathcal{A}\mathcal{B}}$, where $\mathcal{A}$ and $\mathcal{B}$ are
disjoint sets indexing the subsystems, and the marginal density operator of
part $\mathcal{B}$ of $\rho_{\mathcal{A}\mathcal{B}}$ is written as
$\rho_{\mathcal{B}}=\textup{{tr}}_{\mathcal{A}}(\rho_{\mathcal{A}\mathcal{B}})$,
where $\textup{{tr}}_{\mathcal{A}}$ is the trace over part $\mathcal{A}$.
Contrary to this example, I frequently omit the subscript when it would
specify the entire system. Whenever indicated, the time step is labeled by a
superscript. The symbols $\cup$, $\cap$, $\setminus$, and $\ominus$ are used
to denote the set-theoretic operations of union, intersection, difference, and
symmetric difference, and I denote the complement of a set $\mathcal{G}$ by
${\mathrel{\text{$\mathcal{G}$\hbox to0.0pt{\hss$\backslash$}}}}$. Vectors
over finite fields are denoted by placing a right arrow over a symbol, and the
subscripting of such vectors by a set represents the restriction of the vector
to the components indicated by the set, e.g.,
$\vec{i}_{\mathcal{G}}=\\{i_{k}:k\in\mathcal{G}\\}$. The support of an
operator is taken to mean the set of subsystems upon which the operator acts
nontrivially.
## II Concordance
The notion of a classical state frequently carries with it the idea of a
preferred basis. In a Stern-Gerlach experiment, for example, the resulting
superposition of different spins and locations is rarely considered as simply
representing a novel basis for classical particles. From this perspective, a
classical state is one selected from a preferred basis of orthogonal states,
where the basis for a composite system arises from the tensor product of the
preferred bases for the component subsystems. When the state of a system is
uncertain, we describe it using a probability distribution over known, or
pure, classical states.
A concordant state differs from this definition of classicality only in that
no preferred basis is specified; any set of orthogonal bases for the
subsystems may be used to determine the pure states allowed to the composite
system. I take a concordant computation, in turn, to be one in which the state
of the computer after any step is concordant. This usage of “concordant” seems
to have been coined by Andrew White, but it has not previously appeared in
publication. In the following subsections, I explicitly define concordant
states and computations as well as reviewing or proving some results used
later in the paper.
### II.1 Quantum discord
Quantum discord is a measure of non-classical correlations introduced by Zurek
Zurek (2000). Intuitively, it quantifies the amount of non-local disturbance
caused by measuring part of a quantum state. For a quantum state
$\rho_{\mathcal{A}\mathcal{B}}$, the quantum discord with respect to part
$\mathcal{B}$ can be defined as
$\displaystyle\begin{split}{\mathrm{D}}_{\mathcal{B}}(\rho_{\mathcal{A}\mathcal{B}})=\min_{\\{{\mathbb{P}}^{i}_{\mathcal{B}}\\}}&\left[{\mathrm{H}}\\!\\!\left(\rho_{\mathcal{A}\mathcal{B}}^{\\{{\mathbb{P}}^{i}_{\mathcal{B}}\\}}\right)-{\mathrm{H}}\\!\\!\left(\rho_{\mathcal{B}}^{\\{{\mathbb{P}}^{i}_{\mathcal{B}}\\}}\right)\right]\\\
&-\left[{\mathrm{H}}(\rho_{\mathcal{A}\mathcal{B}})-{\mathrm{H}}(\rho_{\mathcal{B}})\right]\end{split}$
where $\\{{\mathbb{P}}^{i}_{\mathcal{B}}\\}$ is a complete set of orthogonal
one-dimensional projectors (CSOOP) on part $\mathcal{B}$,
$\displaystyle\rho_{\mathcal{A}\mathcal{B}}^{\\{{\mathbb{P}}^{i}_{\mathcal{B}}\\}}$
$\displaystyle=\sum_{i}{\mathbb{P}}^{i}_{\mathcal{B}}\rho_{\mathcal{A}\mathcal{B}}{\mathbb{P}}^{i}_{\mathcal{B}}\;,$
and ${\mathrm{H}}(\rho)=-\textup{{tr}}(\rho\log_{2}\rho)$ is the Von Neumann
entropy, the quantum analog of Shannon entropy. This definition is somewhat
less general than that of Zurek, who did not insist on the minimization,
instead making quantum discord a function of the choice of projectors.
Ollivier and Zurek Ollivier and Zurek (2001) showed that
${\mathrm{D}}_{\mathcal{B}}(\rho_{\mathcal{A}\mathcal{B}})=0$ if and only if
$\displaystyle\rho_{\mathcal{A}\mathcal{B}}=\sum_{i}{\mathbb{P}}^{i}_{\mathcal{B}}\rho_{\mathcal{A}\mathcal{B}}{\mathbb{P}}^{i}_{\mathcal{B}}$
(1)
for some CSOOP $\\{{\mathbb{P}}^{i}_{\mathcal{B}}\\}$ on part $\mathcal{B}$,
or equivalently,
$\displaystyle\rho_{\mathcal{A}\mathcal{B}}=\sum_{i}\textup{{tr}}_{\mathcal{B}}(\rho_{\mathcal{A}\mathcal{B}}{\mathbb{P}}^{i}_{\mathcal{B}})\otimes{\mathbb{P}}^{i}_{\mathcal{B}}=\sum_{i}p_{i}\rho_{\mathcal{A}}^{{\mathbb{P}}^{i}_{\mathcal{B}}}\otimes{\mathbb{P}}^{i}_{\mathcal{B}}$
(2)
where
$p_{i}=\textup{{tr}}(\rho_{\mathcal{A}\mathcal{B}}{\mathbb{P}}^{i}_{\mathcal{B}})$,
$\rho_{\mathcal{A}}^{{\mathbb{P}}^{i}_{\mathcal{B}}}=\textup{{tr}}_{\mathcal{B}}(\rho_{\mathcal{A}\mathcal{B}}^{{\mathbb{P}}^{i}_{\mathcal{B}}})$,
and
$\displaystyle\rho_{\mathcal{A}\mathcal{B}}^{{\mathbb{P}}^{i}_{\mathcal{B}}}$
$\displaystyle={\mathbb{P}}^{i}_{\mathcal{B}}\rho_{\mathcal{A}\mathcal{B}}{\mathbb{P}}^{i}_{\mathcal{B}}/\textup{{tr}}(\rho_{\mathcal{A}\mathcal{B}}{\mathbb{P}}^{i}_{\mathcal{B}})\;.$
(3)
Lemma 1 shows that the set of projectors satisfying Eq. 1 is unique up to
degeneracy in part $\mathcal{B}$ of $\rho_{\mathcal{A}\mathcal{B}}$. The
notion of degeneracy on a part of a larger state is clarified by Definition 1.
###### Definition 1.
Two states are degenerate on part $\mathcal{B}$ of
$\rho_{\mathcal{A}\mathcal{B}}$ if the corresponding projectors
${\mathbb{P}}_{\mathcal{B}}$ and ${\mathbb{Q}}_{\mathcal{B}}$ satisfy
$\textup{{tr}}_{\mathcal{B}}(\rho_{\mathcal{A}\mathcal{B}}{\mathbb{P}}_{\mathcal{B}})=\textup{{tr}}_{\mathcal{B}}(\rho_{\mathcal{A}\mathcal{B}}{\mathbb{Q}}_{\mathcal{B}})$.
###### Lemma 1.
Given two CSOOPs on $\mathcal{B}$, $\\{{\mathbb{P}}^{i}_{\mathcal{B}}\\}$ and
$\\{{\mathbb{Q}}^{j}_{\mathcal{B}}\\}$, and a state
$\rho_{\mathcal{A}\mathcal{B}}=\sum_{i}{\mathbb{P}}^{i}_{\mathcal{B}}\rho_{\mathcal{A}\mathcal{B}}{\mathbb{P}}^{i}_{\mathcal{B}}$,
$\rho_{\mathcal{A}\mathcal{B}}=\sum_{j}{\mathbb{Q}}^{j}_{\mathcal{B}}\rho_{\mathcal{A}\mathcal{B}}{\mathbb{Q}}^{j}_{\mathcal{B}}$
if and only if
$\textup{{tr}}_{\mathcal{B}}(\rho_{\mathcal{A}\mathcal{B}}{\mathbb{P}}^{i}_{\mathcal{B}})=\textup{{tr}}_{\mathcal{B}}(\rho_{\mathcal{A}\mathcal{B}}{\mathbb{Q}}^{j}_{\mathcal{B}})$
for all ${\mathbb{P}}^{i}_{\mathcal{B}}{\mathbb{Q}}^{j}_{\mathcal{B}}\neq 0$.
###### Proof.
The forward implication follows from
$\displaystyle\begin{split}\textup{{tr}}_{\mathcal{B}}(\rho_{\mathcal{A}\mathcal{B}}{\mathbb{P}}^{i}_{\mathcal{B}}{\mathbb{Q}}^{j}_{\mathcal{B}})&=\sum_{h}\textup{{tr}}_{\mathcal{B}}({\mathbb{P}}^{h}_{\mathcal{B}}\rho_{\mathcal{A}\mathcal{B}}{\mathbb{P}}^{h}_{\mathcal{B}}{\mathbb{P}}^{i}_{\mathcal{B}}{\mathbb{Q}}^{j}_{\mathcal{B}})\\\
&=\textup{{tr}}_{\mathcal{B}}(\rho_{\mathcal{A}\mathcal{B}}{\mathbb{P}}^{i}_{\mathcal{B}}{\mathbb{Q}}^{j}_{\mathcal{B}}{\mathbb{P}}^{i}_{\mathcal{B}})=e_{ij}\textup{{tr}}_{\mathcal{B}}(\rho_{\mathcal{A}\mathcal{B}}{\mathbb{P}}^{i}_{\mathcal{B}})\\\
&=\sum_{h}\textup{{tr}}_{\mathcal{B}}({\mathbb{Q}}^{h}_{\mathcal{B}}\rho_{\mathcal{A}\mathcal{B}}{\mathbb{Q}}^{h}_{\mathcal{B}}{\mathbb{P}}^{i}_{\mathcal{B}}{\mathbb{Q}}^{j}_{\mathcal{B}})\\\
&=\textup{{tr}}_{\mathcal{B}}(\rho_{\mathcal{A}\mathcal{B}}{\mathbb{Q}}^{j}_{\mathcal{B}}{\mathbb{P}}^{i}_{\mathcal{B}}{\mathbb{Q}}^{j}_{\mathcal{B}})=e_{ij}\textup{{tr}}_{\mathcal{B}}(\rho_{\mathcal{A}\mathcal{B}}{\mathbb{Q}}^{j}_{\mathcal{B}})\end{split}$
where
$e_{ij}=\textup{{tr}}_{\mathcal{B}}({\mathbb{P}}^{i}_{\mathcal{B}}{\mathbb{Q}}^{j}_{\mathcal{B}})$.
The reverse implication follows from
$\displaystyle\begin{split}\rho_{\mathcal{A}\mathcal{B}}&=\sum_{i}\textup{{tr}}_{\mathcal{B}}(\rho_{\mathcal{A}\mathcal{B}}{\mathbb{P}}^{i}_{\mathcal{B}})\otimes{\mathbb{P}}^{i}_{\mathcal{B}}\\\
&=\sum_{i,j}\textup{{tr}}_{\mathcal{B}}(\rho_{\mathcal{A}\mathcal{B}}{\mathbb{P}}^{i}_{\mathcal{B}})\otimes({\mathbb{P}}^{i}_{\mathcal{B}}{\mathbb{Q}}^{j}_{\mathcal{B}})\\\
&=\sum_{i,j}\textup{{tr}}_{\mathcal{B}}(\rho_{\mathcal{A}\mathcal{B}}{\mathbb{Q}}^{j}_{\mathcal{B}})\otimes({\mathbb{P}}^{i}_{\mathcal{B}}{\mathbb{Q}}^{j}_{\mathcal{B}})\\\
&=\sum_{j}\textup{{tr}}_{\mathcal{B}}(\rho_{\mathcal{A}\mathcal{B}}{\mathbb{Q}}^{j}_{\mathcal{B}})\otimes{\mathbb{Q}}^{j}_{\mathcal{B}}\;.\end{split}$
∎
If the quantum discord of $\rho_{\mathcal{A}\mathcal{B}}$ is zero with respect
to both $\mathcal{A}$ and $\mathcal{B}$ then, by two applications of Eq. 1,
$\displaystyle\rho_{\mathcal{A}\mathcal{B}}=\sum_{i,j}p_{ij}{\mathbb{P}}^{i}_{\mathcal{A}}\otimes{\mathbb{P}}^{j}_{\mathcal{B}}$
(4)
for some CSOOPs $\\{{\mathbb{P}}^{i}_{\mathcal{A}}\\}$ and
$\\{{\mathbb{P}}^{j}_{\mathcal{B}}\\}$. For fixed
$\\{{\mathbb{P}}^{i}_{\mathcal{A}}\\}$, Lemma 1 shows that the set of
projectors $\\{{\mathbb{P}}^{j}_{\mathcal{B}}\\}$ satisfying Eq. 4 is unique
up to the degeneracy common to all
$\rho_{\mathcal{B}}^{{\mathbb{P}}^{i}_{\mathcal{A}}}$, that is, up to
degeneracy appearing in each of the subblocks of $\rho$ projected out by some
${\mathbb{P}}^{i}_{\mathcal{A}}$.
### II.2 Concordant states
The adjective “concordant” is intended to indicate a lack of quantum discord.
Because discord is an asymmetric, bipartite measure, however, it is not
completely obvious what this restriction ought to mean with regard to quantum
states, especially states of composite systems composed of more than two
subsystems. I choose to label a state as concordant if it has zero discord
with respect to any part. This is codified in the following definition.
###### Definition 2.
A state $\rho$ is concordant if ${\mathrm{D}}_{\mathcal{A}}(\rho)=0$ for any
strict subset $\mathcal{A}$ of the subsystems of $\rho$.
In particular, Def. 2 guarantees that ${\mathrm{D}}_{k}(\rho)=0$ for any $k$
labeling a single subsystem of some concordant state $\rho$. By Eq. 1, this
implies that, for any concordant state $\rho$, there exists a CSOOP
$\\{{\mathbb{P}}^{i}_{k}\\}$ for every subsystem $k$ such that
$\displaystyle\rho=\sum_{i}{\mathbb{P}}^{i}_{k}\rho{\mathbb{P}}^{i}_{k}\;.$
(5)
An equivalent form of the implication that often proves useful is
$\displaystyle\rho=\sum_{\vec{i}}{\mathbb{P}}^{\vec{i}}\rho{\mathbb{P}}^{\vec{i}}=\sum_{\vec{i}}p_{\vec{i}}{\mathbb{P}}^{\vec{i}}$
(6)
where ${\mathbb{P}}^{\vec{i}}=\prod_{k}{\mathbb{P}}^{i_{k}}_{k}$ and
$\\{{\mathbb{P}}^{i_{k}}_{k}\\}$ for fixed $k$ is a CSOOP for the $k$th
subsystem.
The reasoning above shows that Def. 2 implies Eq. 6, but conversely, any state
satisfying Eq. 6 clearly satisfies Def. 2. Thus, Eq. 6 can be taken as an
alternate definition of a concordant state. In words, a state is concordant if
there exists a product basis, that is, a basis arising from the tensor product
of local orthogonal bases, such that its density operator is diagonal.
### II.3 Concordant computations
In keeping with standard practice, I adopt a description of quantum
computation based on the quantum circuit model, where the evolution of the
state of a system is described by a sequence of operators. Most generally, the
operations applied can be chosen probabilistically, based, for example, on the
path the computation has taken thus far, as revealed by measurements. In this
model, it is natural to label a computation as concordant if the state of the
computer is concordant both initially and after each step of the evolution, a
notion formalized below.
###### Definition 3.
A quantum computation described by a sequence of operators
$\\{{\mathrm{G}}^{t}\\}$ acting on some input state $\rho^{0}$ is concordant
if each state
$\rho^{t}={\mathrm{G}}^{t}\circ\cdots\circ{\mathrm{G}}^{2}\circ{\mathrm{G}}^{1}(\rho^{0})$
is concordant for every path of the computation.
Being concordant, each computational state might be considered classical for
some choice of the classical basis, but a concordant computation is slightly
more general than a randomized classical computation in that the product
eigenbasis can change from one step to the next.
Definition 3 is problematic for questions of computational complexity since it
is possible to obscure the difficulty of an algorithm by employing very
complex operations or initial states. The specification of an arbitrary input
state $\rho^{0}$, for example, entails a quantity of real numbers exponential
in the number of subsystems, even if $\rho^{0}$ is concordant. (See Ref. Jozsa
and Linden (2003) for a careful treatment of the difficulties posed by the use
of real numbers.) I avoid these problems and simplify the following discussion
by initially considering only computations that are conventional, as defined
by Def. 4. In Sec. IV I discuss ways in which the restriction to conventional
computations can be relaxed.
###### Definition 4.
A conventional quantum computation consists of an input product state diagonal
in the standard basis, $\rho^{0}=\bigotimes_{k}\rho^{0}_{k}$, followed by a
sequence of unitary gates $\\{G^{t}\\}$, and concluded by single-subsystem
measurements determining the outcome of the computation. Each $\rho^{0}_{k}$
and $G^{t}$ (when restricted to its support) is required to be efficiently
computable.
The evolution of a concordant computation of the form given by Def. 4 is
particularly simple. Because the spectrum of a density operator is invariant
under conjugation by unitary operators, any unitary gate can be considered
simply as a change of eigenbasis for the density operator. For a concordant
computation, there is guaranteed to exist a product basis, both before and
after a gate, such that the density operator describing the state of the
computer is diagonal. Thus, the effect of any unitary operator can be, at
most, to change the product eigenbasis and permute the associated eigenvalues.
More specifically, Lemma 2 shows that a transformation between concordant
states induced by a unitary gate with support $\mathcal{G}$ is equivalent to a
change of product eigenbasis on $\mathcal{G}$ together with a permutation with
support $\mathcal{G}$ of the vectors indexing the eigenvalues. In general, the
unitary gate will not actually be a permutation followed by a change of
product eigenbasis but merely be equivalent to one for the given initial
state.
###### Lemma 2.
If $\sigma=G\rho G^{\dagger}$ where $G$ is a unitary operator with support
$\mathcal{G}$, $\rho$ and $\sigma$ are concordant, and
$\rho=\sum_{\vec{i}}p_{\vec{i}}{\mathbb{P}}^{\vec{i}}$ then
$\sigma=\sum_{\vec{j}}q_{\vec{j}}{\mathbb{P}}^{\vec{j}}_{{\mathrel{\text{$\mathcal{G}$\hbox
to0.0pt{\hss$\backslash$}}}}}{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}$ where
$q_{P\cdot\vec{i}}=p_{\vec{i}}$ for some permutation $P$ with support
$\mathcal{G}$.
###### Proof.
Since $\sigma$ is concordant there exists $\\{{\mathbb{Q}}^{\vec{j}}\\}$ such
that
$\displaystyle\sigma=\sum_{\vec{j}_{\mathcal{G}}}{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}\sigma{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}\;,$
where
${\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}=\prod_{k\in\mathcal{G}}{\mathbb{Q}}^{j_{k}}_{k}$
and likewise for subsequent similar projectors. Moreover,
$\displaystyle\begin{split}\sum_{\vec{i}_{\mathrel{\text{$\mathcal{G}$\hbox
to0.0pt{\hss$\backslash$}}}}}{\mathbb{P}}^{\vec{i}}_{\mathrel{\text{$\mathcal{G}$\hbox
to0.0pt{\hss$\backslash$}}}}\sigma{\mathbb{P}}^{\vec{i}}_{\mathrel{\text{$\mathcal{G}$\hbox
to0.0pt{\hss$\backslash$}}}}&=\sum_{\vec{i}_{\mathrel{\text{$\mathcal{G}$\hbox
to0.0pt{\hss$\backslash$}}}}}{\mathbb{P}}^{\vec{i}}_{\mathrel{\text{$\mathcal{G}$\hbox
to0.0pt{\hss$\backslash$}}}}G\rho
G^{\dagger}{\mathbb{P}}^{\vec{i}}_{\mathrel{\text{$\mathcal{G}$\hbox
to0.0pt{\hss$\backslash$}}}}\\\
&=G\sum_{\vec{i}_{\mathrel{\text{$\mathcal{G}$\hbox
to0.0pt{\hss$\backslash$}}}}}{\mathbb{P}}^{\vec{i}}_{\mathrel{\text{$\mathcal{G}$\hbox
to0.0pt{\hss$\backslash$}}}}\rho{\mathbb{P}}^{\vec{i}}_{\mathrel{\text{$\mathcal{G}$\hbox
to0.0pt{\hss$\backslash$}}}}G^{\dagger}=G\rho
G^{\dagger}=\sigma\;.\end{split}$
Thus, $\sigma$ can be written in the form
$\displaystyle\sigma=\sum_{\vec{j}}{\mathbb{P}}^{\vec{j}}_{{\mathrel{\text{$\mathcal{G}$\hbox
to0.0pt{\hss$\backslash$}}}}}{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}\sigma{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}{\mathbb{P}}^{\vec{j}}_{{\mathrel{\text{$\mathcal{G}$\hbox
to0.0pt{\hss$\backslash$}}}}}=\sum_{\vec{j}}q_{\vec{j}}{\mathbb{P}}^{\vec{j}}_{{\mathrel{\text{$\mathcal{G}$\hbox
to0.0pt{\hss$\backslash$}}}}}{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}\;.$
To see that the specified permutation exists, consider a graph $\Gamma$ where
the nodes correspond to the projectors ${\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}$
and $G{\mathbb{P}}^{\vec{i}}_{\mathcal{G}}G^{\dagger}$ and two nodes are
connected if their associated projectors are not orthogonal. Since
$\\{{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}\\}$ and
$\\{G{\mathbb{P}}^{\vec{i}}_{\mathcal{G}}G^{\dagger}\\}$ project onto two
eigenbases for the state
$\displaystyle{\sigma}^{{\mathbb{P}}^{\vec{j}}_{\mathrel{\text{$\mathcal{G}$\hbox
to0.0pt{\hss$\backslash$}}}}}_{\mathcal{G}}\propto\sum_{\vec{j}_{\mathcal{G}}}q_{\vec{j}}{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}=\sum_{\vec{j}_{\mathcal{G}}}p_{\vec{j}}G{\mathbb{P}}^{\vec{j}}_{\mathcal{G}}G^{\dagger}\;,$
projectors connected in $\Gamma$ are associated, by the uniqueness properties
of the spectral decomposition, with the same eigenvalue of
${\sigma}^{{\mathbb{P}}^{\vec{j}}_{\mathrel{\text{$\mathcal{G}$\hbox
to0.0pt{\hss$\backslash$}}}}}_{\mathcal{G}}$ and therefore with the same
eigenvalues of $\sigma$. Two spectral decompositions of the same density
operator are related by a unitary transformation, so each connected component
of $\Gamma$ includes an equal number of projectors from
$\\{{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}\\}$ and
$\\{G{\mathbb{P}}^{\vec{i}}_{\mathcal{G}}G^{\dagger}\\}$. Thus, it is possible
to assign $q_{\vec{j}}=p_{\vec{i}}$ where $\vec{j}=P\cdot\vec{i}$, $P$ is a
permutation such that $\vec{j}_{\mathrel{\text{$\mathcal{G}$\hbox
to0.0pt{\hss$\backslash$}}}}=\vec{i}_{\mathrel{\text{$\mathcal{G}$\hbox
to0.0pt{\hss$\backslash$}}}}$, and
$\\{{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}\\}$ and
$\\{G{\mathbb{P}}^{\vec{i}}_{\mathcal{G}}G^{\dagger}\\}$ are in the same
connected component of $\Gamma$. ∎
01 | For | each | subsystem $k$:
---|---|---|---
02 | | Choose $i_{k}$ according to the probability distribution ${\mathrm{Pr}}[i_{k}=w]=\left\langle w\left|\vphantom{w}{U^{0}_{k}}^{\dagger}\rho^{0}_{k}U^{0}_{k}\vphantom{w}\right|w\right\rangle$.
03 | $\vec{j}:=P\cdot\vec{i}$
04 | For each measured subsystem $k$:
05 | | Choose $h_{k}$ according to the probability distribution ${\mathrm{Pr}}[h_{k}=w]=\left\lvert\left\langle w\left|\vphantom{w}U^{s}_{k}\vphantom{j_{k}}\right|j_{k}\right\rangle\right\rvert^{2}$.
06 | Output $\vec{h}$.
Figure 1: Pseudocode for simulating a conventional concordant computation.
$U^{0}$ and $U^{s}$ are unitary product operators identifying the initial and
final product eigenbases respectively and $P$ is the permutation that acts on
$\rho^{0}$ equivalently to the specified sequence of unitary operators.
Pseudocode for converting a sequence of two-qubit unitary operators in a
concordant computation into an equivalent classical permutation and change of
basis is given in Fig. 2.
## III Simulating a conventional concordant computation
In the previous section I show that the transformation of one concordant state
to another by a unitary operator with support $\mathcal{G}$ is equivalent to a
permutation of eigenvalues together with a change of product eigenbasis on
$\mathcal{G}$. Combined with the fact that a density operator can be
considered as a probabilistic mixture of its eigenstates, this suggests the
following strategy for simulating a conventional concordant computation: Find
a change of product eigenbasis and permutation of the vectors labeling
eigenstates (and, therefore, the associated eigenvalues) equivalent to each
unitary gate in the computation, and then generate an output of the
computation by appropriately picking a vector labeling an eigenstate of the
input state, applying the derived permutations to the chosen vector, and
evaluating the final measurement on the indicated product state.
It is not immediately obvious that the described simulation is feasible
because the permutation and change of eigenbasis equivalent to each unitary
operator is dependent on the overall state of the computer. Nonetheless, the
following subsections provide detailed descriptions of the necessary
subcomponents of such a simulation for the special case of two-qubit unitary
gates, thereby proving Theorem 1. Section III.1 shows how a conventional
concordant computation can be simulated given the permutation and eigenbasis
change equivalent to each unitary operator. Section III.2 proves that it is
possible to efficiently determine a permutation and change of eigenbasis
equivalent to a unitary operator from the degeneracy of the pre-gate state.
Finally, Sec. III.3 explains how the relevant degeneracy can be found from the
previously applied permutations and an input product state, so long as the
computation contains only one- and two-qubit unitary gates. In addition to the
concordant-state condition given by Eq. 6, I employ an equivalent definition:
a state $\rho$ is concordant if and only if there exists a unitary product
operator $U=\bigotimes_{k}U_{k}$ such that $U^{\dagger}\rho U$ is diagonal in
the standard basis.
###### Theorem 1.
A conventional concordant computation with unitary operators having support on
only one or two qubits can be efficiently simulated by a classical computer.
01 | Stor | e th | e un | itar | y op | erat | or d | efining the initial product eigenbasis in $U$.
---|---|---|---|---|---|---|---|---
02 | $P:=I$ | | | | | | | (where $P$ is stored as a sequence of two-bit permutations)
03 | For each gate $G$ in the circuit:
04 | | If $G$ has support on only one qubit:
05 | | | $U:=GU$
06 | | Else if $G$ has support on some pair of qubits $\mathcal{G}=\\{k,l\\}$:
07 | | | For each permutation $Q$ which exchanges two states of the standard basis of part $\mathcal{G}$:
08 | | | | If $P^{\dagger}QP$ commutes with the initial density operator:
09 | | | | | The states exchanged by $Q$ are degenerate. Store this fact.
10 | | | Solve for $V$, and thus the new product eigenbasis, using the known degeneracy and the constraint
| | | that the post-gate state be diagonal in that basis.
11 | | | Pick a permutation $R$ such that $VRU^{\dagger}$ and $G$ transform the state identically.
12 | | | $P:=RP$
13 | | | $U:=V$
14 | Output $P$ and $U$.
Figure 2: Pseudocode for converting the sequence of unitary gates in a
conventional concordant computation composed of one- and two-qubit gates to an
equivalent permutation and change of basis.
### III.1 Simulation given many hints
Consider a conventional concordant computation for which the sequence of
unitary operators employed, $\\{G^{t}\\}$, is known to act equivalently to the
sequence $\left\\{U^{t}P^{t}{U^{t-1}}^{\dagger}\right\\}$ where each $P^{t}$
is a permutation (that is, a classical reversible gate) with the same support
as $G^{t}$ and each $U^{t}$ is a unitary product operator that transforms from
the standard basis to the product eigenbasis at time step $t$. Given this
information, the initial state $\rho^{0}$ must be of the form
$\displaystyle\rho^{0}$
$\displaystyle=\sum_{\vec{i}}p^{0}_{\vec{i}}U^{0}{\left|{\vec{i}}\right\rangle}{\left\langle{\vec{i}}\right|}{U^{0}}^{\dagger}$
(7)
where each ${\left|{\vec{i}}\right\rangle}$ is an element of the standard
basis. (By definition, $U^{0}$ is trivial for a conventional computation.) The
state of the computer after one step of the computation is
$\displaystyle\begin{split}\rho^{1}&=\sum_{\vec{i}}p^{0}_{\vec{i}}G^{1}U^{0}{\left|{\vec{i}}\right\rangle}{\left\langle{\vec{i}}\right|}{U^{0}}^{\dagger}{G^{1}}^{\dagger}\\\
&=\sum_{\vec{i}}p^{0}_{\vec{i}}U^{1}{U^{1}}^{\dagger}G^{1}U^{0}{\left|{\vec{i}}\right\rangle}{\left\langle{\vec{i}}\right|}{U^{0}}^{\dagger}{G^{1}}^{\dagger}U^{1}{U^{1}}^{\dagger}\\\
&=\sum_{\vec{i}}p^{0}_{\vec{i}}U^{1}P^{1}{\left|{\vec{i}}\right\rangle}{\left\langle{\vec{i}}\right|}{P^{1}}^{\dagger}{U^{1}}^{\dagger}\end{split}$
where $P^{1}$ is a permutation that acts identically to
${U^{1}}^{\dagger}G^{1}U^{0}$ on ${U^{0}}^{\dagger}\rho^{0}U^{0}$. Iterating
this process yields
$\displaystyle\rho^{s}$
$\displaystyle=\sum_{\vec{i}}p^{0}_{\vec{i}}U^{s}\left(\prod_{t=s}^{1}P^{t}\right){\left|{\vec{i}}\right\rangle}{\left\langle{\vec{i}}\right|}\left(\prod_{t=1}^{s}{P^{t}}^{\dagger}\right){U^{s}}^{\dagger}$
(8)
where each $P^{t}$ is a permutation that acts identically to
${U^{t}}^{\dagger}G^{t}U^{t-1}$ on ${U^{t-1}}^{\dagger}\rho^{t-1}U^{t-1}$.
The measurement statistics of a mixed state are identical to those of a
probabilistically chosen state in its decomposition where the probability is
given by the coefficient of the term associated with that state. Thus, the
expression for the final pre-measurement state shown in Eq. 8 suggests the
following simple technique for simulating the computation: Choose a single
vector $\vec{i}$ according to the probability distribution $p^{0}_{\vec{i}}$,
which can be done efficiently since $\rho^{0}$ is a product state. Apply the
permutation $\prod_{t=s}^{1}P^{t}$ to $\vec{i}$ to obtain a new vector
$\vec{j}$ identifying one component of the final pre-measurement state. And
last, for each measured subsystem $k$ choose a measurement outcome $h_{k}$
according to the probability distribution
$\displaystyle{\mathrm{Pr}}[h_{k}=w]=\left\lvert\left\langle
w\left|\vphantom{w}{U^{s}_{k}}\vphantom{j_{k}}\right|j_{k}\right\rangle\right\rvert^{2}\;.$
Fig. 1 presents pseudocode illustrating this method.
### III.2 Updating the product eigenbasis
In the $t$th step of a conventional concordant computation, the unitary gate
$G^{t}$ is applied to a concordant state $\rho^{t-1}$ to yield a concordant
state $\rho^{t}$. As explained in Sec. II.3, the effect of $G^{t}$ is
identical to that of a permutation $P^{t}$ of the vectors labeling eigenstates
followed by a change of product eigenbasis. Thus, if $U^{t-1}$ and $U^{t}$ are
unitary product operators that transform from the standard basis to the
product eigenbases at times $t-1$ and $t$, respectively, then
$\rho^{t}=G^{t}\rho^{t-1}{G^{t}}^{\dagger}=U^{t}P^{t}{U^{t-1}}^{\dagger}\rho^{t-1}U^{t-1}{P^{t}}^{\dagger}{U^{t}}^{\dagger}$
for some $P^{t}$ which permutes the elements of the standard basis. Moreover,
Lemma 2 shows that there exists a product eigenbasis for $\rho^{t}$ consistent
with $U^{t}$ such that $U^{t}_{k}=U^{t-1}_{k}$ for all $k$ not in
$\mathcal{G}^{t}$, the support of $G^{t}$, and additionally, that for such a
product eigenbasis there exists a permutation $P^{t}$ with support
$\mathcal{G}^{t}$.
The problem of finding $U^{t}_{k}$ for $k\not\in\mathcal{G}^{t}$ is addressed
by Lemma 3, which shows that the remaining components of a product eigenbasis
for $\rho^{t}$ can be calculated given one additional piece of information,
the degeneracy of part $\mathcal{G}^{t}$ of $\rho^{t-1}$. This calculation is
efficient in that it entails solving a system of equations whose number
depends only on the number of subsystems in $\mathcal{G}^{t}$ and their
dimension, not on the total number of subsystems in the computation. The
appropriate permutation is easily found from the eigenbases for $\rho^{t-1}$
and $\rho^{t}$; it is sufficient to pick any permutation mapping
eigenprojectors of $\rho^{t-1}$ to eigenprojectors of $\rho^{t}$ which are in
the same connected component of a graph $\Gamma$ defined as per Lemma 2.
(Remember that the permutation $P^{t}$ can be assumed to have support
$\mathcal{G}^{t}$, thereby limiting the size of the graph that must be
considered.) As indicated by Theorem 2, these results are sufficient to enable
the efficient simulation of concordant computations with most input states.
The question of arbitrary input states is taken up in the next section.
###### Lemma 3.
For $\rho=\sum_{\vec{i}}p_{\vec{i}}{\mathbb{P}}^{\vec{i}}$ and $\sigma=G\rho
G^{\dagger}$, where $G$ is a unitary gate with support $\mathcal{G}$,
$\\{{\mathbb{Q}}^{\vec{j}}\\}$ satisfies
$\sigma=\sum_{\vec{j}}{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}\sigma{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}$
if and only if
$\textup{{tr}}_{\mathcal{G}}(\rho{\mathbb{P}}^{\vec{i}}_{\mathcal{G}})=\textup{{tr}}_{\mathcal{G}}(\rho{\mathbb{P}}^{\vec{h}}_{\mathcal{G}})$
for all $\vec{h}$, $\vec{i}$, and $\vec{j}$ such that
$G{\mathbb{P}}^{\vec{h}}_{\mathcal{G}}G^{\dagger}{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}\neq
0$ and
$G{\mathbb{P}}^{\vec{i}}_{\mathcal{G}}G^{\dagger}{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}\neq
0$.
###### Proof.
$\displaystyle\sum_{\vec{i}}G{\mathbb{P}}^{\vec{i}}_{\mathcal{G}}G^{\dagger}\sigma
G{\mathbb{P}}^{\vec{i}}_{\mathcal{G}}G^{\dagger}=\sum_{\vec{i}}G{\mathbb{P}}^{\vec{i}}_{\mathcal{G}}\rho{\mathbb{P}}^{\vec{i}}_{\mathcal{G}}G^{\dagger}=G\rho
G^{\dagger}=\sigma\;,$
so by Lemma 1, $\\{{\mathbb{Q}}^{\vec{j}}\\}$ satisfies
$\sigma=\sum_{\vec{j}}{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}\sigma{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}$
if and only if $\textup{{tr}}_{\mathcal{G}}(\sigma
G{\mathbb{P}}^{\vec{h}}_{\mathcal{G}}G^{\dagger})=\textup{{tr}}_{\mathcal{G}}(\sigma{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}})$
for all
$G{\mathbb{P}}^{\vec{h}}_{\mathcal{G}}G^{\dagger}{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}\neq
0$. Given $\rho$, $\sigma$, $G$, $\\{{\mathbb{P}}^{\vec{i}}\\}$, and
$\\{{\mathbb{Q}}^{\vec{j}}\\}$ as defined, the condition
$\textup{{tr}}_{\mathcal{G}}(\sigma
G{\mathbb{P}}^{\vec{h}}_{\mathcal{G}}G^{\dagger})=\textup{{tr}}_{\mathcal{G}}(\sigma{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}})$
for all
$G{\mathbb{P}}^{\vec{h}}_{\mathcal{G}}G^{\dagger}{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}\neq
0$ is equivalent to
$\textup{{tr}}_{\mathcal{G}}(\rho{\mathbb{P}}^{\vec{h}}_{\mathcal{G}})=\textup{{tr}}_{\mathcal{G}}(\rho{\mathbb{P}}^{\vec{i}}_{\mathcal{G}})$
for all
$G{\mathbb{P}}^{\vec{h}}_{\mathcal{G}}G^{\dagger}{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}\neq
0$ and
$G{\mathbb{P}}^{\vec{i}}_{\mathcal{G}}G^{\dagger}{\mathbb{Q}}^{\vec{j}}_{\mathcal{G}}\neq
0$. ∎
Fig. 2 presents pseudocode for an algorithm calculating the necessary sequence
of permutations and basis changes.
###### Theorem 2.
A conventional concordant computation with an input product state that is
generic can be efficiently simulated by a classical computer.
###### Proof.
A generic product state has no degenerate eigenvalues, so the simulation
method as outlined thus far is sufficient for such input states. ∎
### III.3 Diagnosing the degeneracy
In order to update the product eigenbasis following the $t$th gate in a
conventional concordant computation, it is necessary to diagnose the
degeneracy of part $\mathcal{G}^{t}$ of $\rho^{t-1}$, where $\mathcal{G}^{t}$
is the support of $G^{t}$, the $t$th gate in the computation, and $\rho^{t-1}$
is the state of the computation at time $t-1$. This degeneracy can be found by
determining whether $\rho^{t-1}$ and $U^{t-1}Q{U^{t-1}}^{\dagger}$ commute for
each permutation $Q$ exchanging two eigenstates of the standard basis for the
subsystems in $\mathcal{G}^{t}$. As the simulation algorithm progresses,
permutations equivalent to each gate are found, so
$\rho^{t-1}=U^{t-1}P\rho^{0}P^{\dagger}{U^{t-1}}^{\dagger}$ where
$P=\prod_{r}^{t-1}P^{r}$ represents the sequence of (known) permutations up to
step $t-1$. Thus, one may equally well check whether
$\displaystyle\rho^{0}=P^{\dagger}QP\rho^{0}P^{\dagger}QP\;.$ (9)
I now restrict my attention to concordant computations composed of two-qubit
gates acting on a register of $n$ qubits. The permutation $P^{\dagger}QP$ is
an involution, i.e., it is self-inverse, and for the case of qubits and two-
qubit gates, it is affine when considered as a function on binary vectors.
Lemma 4 shows that such a permutation commutes with $\rho^{0}$ if and only if
Eq. 9 is satisfied for the pure product state corresponding to each of a
particular set of $n+1$ binary vectors. Consequently, the commutativity of
$\rho^{0}$ and $P^{\dagger}QP$, and therefore the degeneracy relevant to
updating the product eigenbasis, can be efficiently determined for concordant
computations composed of two-qubit gates.
###### Lemma 4.
A product state on qubits, $\rho=\bigotimes_{k}\rho_{k}$, such that $\rho$ is
diagonal in the standard basis and $e_{k}=\left\langle
1\left|\vphantom{1}\rho_{k}\vphantom{1}\right|1\right\rangle/\left\langle
0\left|\vphantom{0}\rho_{k}\vphantom{0}\right|0\right\rangle\leq 1$ for all
$k$ commutes with an affine involution $S$ if and only if
$\left\langle\vec{i}\left|\vphantom{\vec{i}}S\rho
S^{\dagger}-\rho\vphantom{\vec{i}}\right|\vec{i}\right\rangle=0$ for all
${\left|{\vec{i}}\right\rangle}$ such that $i_{k}=\updelta_{kl}$ or $i_{k}=0$.
###### Proof.
Throughout this proof, binary vectors labeling states are represented by the
set of indices identifying bits in the ${\left|{1}\right\rangle}$ state. Let
$\mathcal{S}$ be a version of $S$ that acts on such sets222For brevity I omit
brackets in the argument of this and other functions when the input is a
singleton, e.g., I write $\mathcal{S}(k)$ rather than $\mathcal{S}(\\{k\\})$..
In this representation, the affine linearity of $\mathcal{S}$ is expressed as
$\mathcal{S}(\mathcal{A}\ominus\mathcal{B})=\mathcal{S}(\mathcal{A})\ominus\mathcal{S}(\mathcal{B})\ominus\mathcal{K}$
for some fixed $\mathcal{K}$, while the fact that $\mathcal{S}$ is an
involution implies that $\mathcal{S}(\mathcal{S}(\mathcal{A}))=\mathcal{A}$.
Define $\mathcal{C}_{e}=\\{k:e_{k}=e\\}$ and
$f(\mathcal{B})=\prod_{k\in\mathcal{B}}e_{k}$. In terms of $f$ and
$\mathcal{S}$ the commutativity condition to be satisfied is
$\displaystyle f(\mathcal{B})=f(\mathcal{S}(\mathcal{B}))$ (10)
for any set of bits, $\mathcal{B}$.
The forward implication stated in this lemma is trivial. If Eq. 10 is
satisfied for any set $\mathcal{B}$ then it is obviously satisfied for any
singleton $\\{k\\}$ and for the empty set.
To demonstrate the reverse, I assume, for the remainder of the proof, that Eq.
10 is satisfied for the empty set and any singleton and seek to show that it
is satisfied in general. I organize what follows in terms of a sequence of
small points.
Point 0: $\mathcal{T}(\mathcal{B})=\mathcal{S}(\mathcal{B})\ominus\mathcal{K}$
is a linear involution, and $\mathcal{T}$ satisfies Eq. 11 if and only if
$\mathcal{S}$ satisfies Eq. 10.
$\mathcal{T}$ is linear since $\mathcal{S}$ is affine with constant
$\mathcal{K}$. Because $\mathcal{S}(\emptyset)=\mathcal{K}$ and $\mathcal{S}$
is an involution,
$\mathcal{S}(\mathcal{K})=\mathcal{S}(\mathcal{S}(\emptyset))=\emptyset$,
implying that $\mathcal{T}$ is an involution since
$\displaystyle\begin{split}\mathcal{T}(\mathcal{T}(\mathcal{B}))&=\mathcal{T}(\mathcal{S}(\mathcal{B})\ominus\mathcal{K})=\mathcal{S}(\mathcal{S}(\mathcal{B})\ominus\mathcal{K})\ominus\mathcal{K}\\\
&=\mathcal{S}(\mathcal{S}(\mathcal{B}))\ominus\mathcal{S}(\mathcal{K})\ominus\mathcal{K}\ominus\mathcal{K}=\mathcal{B}\;.\end{split}$
Furthermore, $\mathcal{K}\subseteq\mathcal{C}_{1}$ since if $\exists
k\in\mathcal{K}$ such that $k\not\in\mathcal{C}_{1}$ then $f(\mathcal{K})\leq
e_{k}<1=f(\emptyset)$. Consequently,
$f(\mathcal{B})=f(\mathcal{B}\ominus\mathcal{K})$, and thus Eq. 10 is
satisfied if and only if
$\displaystyle f(\mathcal{B})=f(\mathcal{T}(\mathcal{B}))$ (11)
Point 1: $\forall k\exists m\in\mathcal{T}(k)$ such that $k\in\mathcal{T}(m)$
Because $\mathcal{T}$ is a linear involution,
$\displaystyle
k=\mathcal{T}(\mathcal{T}(k))=\mathcal{T}\left(\mathop{\raisebox{-1.00006pt}{\Large\boldmath$\ominus$}}_{l\in\mathcal{T}(k)}\\{l\\}\right)=\mathop{\raisebox{-1.00006pt}{\Large\boldmath$\ominus$}}_{l\in\mathcal{T}(k)}\mathcal{T}(l)\;,$
so $\forall k\exists m\in\mathcal{T}(k)$ such that $k\in\mathcal{T}(m)$.
Point 2: $e_{l}\geq e_{k}\ \forall l\in\mathcal{T}(k)$
If $\exists l\in\mathcal{T}(k)$ such that $e_{l}<e_{k}$ then
$f(k)=e_{k}>e_{l}\geq f(\mathcal{T}(k))$, so $e_{l}\geq e_{k}\ \forall
l\in\mathcal{T}(k)$.
Point 3: $\exists m\in\mathcal{T}(k)$ such that $e_{m}=e_{k}$
By the previous two points $\exists m\in\mathcal{T}(k)$ such that
$k\in\mathcal{T}(m)$ and $e_{m}\geq e_{k}$, but this implies that
$e_{m}=e_{k}$ since, by Point 2, $k\in\mathcal{T}(m)$ implies $e_{k}\geq
e_{m}$.
Point 4: Each $k\in\mathcal{C}_{e}$ where $e>0$ is mapped by $\mathcal{T}$ to
a single $m\in\mathcal{C}_{e}$ together with (possibly) some elements of
$\mathcal{C}_{1}$.
By the previous point, $\exists m\in\mathcal{T}(k)$ such that
$m\in\mathcal{C}_{e_{k}}$, implying that
$\displaystyle
f(\mathcal{T}(k))=f(m)f(\mathcal{T}(k)\setminus\\{m\\})=e_{k}\prod_{l\in\mathcal{T}(k)\setminus\\{m\\}}e_{l}\;,$
which is equal to $f(k)=e_{k}$ only when $e_{k}=0$ or $e_{l}=1\ \forall
l\in\mathcal{T}(k)\setminus\\{m\\}$.
Point 5: Any two distinct elements $k,l\in\mathcal{C}_{e}$ where $e>0$ are
mapped by $\mathcal{T}$ to distinct elements of $\mathcal{C}_{e}$ together
with (possibly) some elements of $\mathcal{C}_{1}$.
If $\exists k,l\in\mathcal{C}_{e}$ with $k\neq l$ and $1>e>0$ such that
$\mathcal{T}(k)/\mathcal{C}_{1}=\mathcal{T}(l)/\mathcal{C}_{1}=\\{m\\}$ then
$k,l\in\mathcal{T}(m)$ since $k,l\not\in\mathcal{T}(o)$ for any
$o\in\mathcal{C}_{1}$, which contradicts the preceding point.
Point 6: If $\mathcal{B}\cap\mathcal{C}_{0}\neq\emptyset$ then
$\mathcal{T}(\mathcal{B})\cap\mathcal{C}_{0}\neq\emptyset$.
If $\exists\mathcal{B}$ such that
$\mathcal{B}\cap\mathcal{C}_{0}\neq\emptyset$ but
$\mathcal{T}(\mathcal{B})\cap\mathcal{C}_{0}=\emptyset$ then $\exists
l\in\mathcal{T}(\mathcal{B})$ such that $e_{l}>0$ and
$\mathcal{T}(l)\cap\mathcal{C}_{0}\neq\emptyset$, which contradicts my second
point.
Point 7: Eq. 11 is satisfied for any set $\mathcal{B}$.
If $\mathcal{B}\cap\mathcal{C}_{0}\neq\emptyset$ then
$\mathcal{T}(\mathcal{B})\cap\mathcal{C}_{0}\neq\emptyset$ so
$f(\mathcal{B})=f(\mathcal{T}(\mathcal{B}))=0$. Otherwise,
$\displaystyle\begin{split}f(\mathcal{B})=\prod_{l\in\mathcal{B}}f(l)&=\prod_{l\in\mathcal{B}}f(\mathcal{T}(l))=f\left(\mathop{\raisebox{-1.00006pt}{\Large\boldmath$\ominus$}}_{l\in\mathcal{B}}\mathcal{T}(l)\right)\\\
&=f\left(\mathcal{T}\left(\mathop{\raisebox{-1.00006pt}{\Large\boldmath$\ominus$}}_{l\in\mathcal{B}}\\{l\\}\right)\right)=f(\mathcal{T}(\mathcal{B}))\;,\end{split}$
where the middle equality follows from Point 5, which shows that
$\mathcal{T}(l)\cap\mathcal{T}(k)\subseteq\mathcal{C}_{1}$ for all $k$ and $l$
such that $k\neq l$ and $k,l\not\in\mathcal{C}_{0},\mathcal{C}_{1}$. ∎
## IV Extensions
Quantum computations, even those described in terms of quantum circuits,
frequently are not envisioned in the conventional form outlined by Def. 4. The
most common deviations are the inclusion of single-subsystem measurements
intermixed with the unitary operators and the introduction of new subsystems
during the course of the computation. Another possibility for concordant
computations is that the input state be a mixture of product states that is
not also a product of mixed states but that can be efficiently prepared due to
the mixture having few terms. Computations with these features can be
converted to conventional ones (allowing for some post selection to assist in
the generation of the desired input state), but, in general, the conversion
process preserves neither the concordance of the computation nor the maximal
support of its unitary operators. While subsystems introduced during the
course of a computation can equally well be introduced at its beginning, non-
terminal measurements and non-product-state inputs require special treatment.
### IV.1 Non-terminal measurements
It requires some effort to extend the simulation algorithm described in the
previous section to non-terminal measurements on single subsystems. Through
the first measurement, the simulation may proceed exactly as previously
explained, but subsequent to that, a more complex technique for diagnosing the
degeneracy is necessary since measurements introduce the possibility that the
degeneracy relevant to determining the permutation and change of eigenbasis
equivalent to a gate might be dependent on the outcome of the measurement
result. There seems to be a method of efficiently diagnosing the relevant
degeneracy when measurements are performed in the eigenbasis, but the more
general problem is one that I have not yet been able to solve.
### IV.2 Non-product-state inputs
Generically, Def. 4 excludes a very natural kind of mixed input state, namely,
the probabilistic mixture of a few pure product states. As it happens,
however, concordant computations with such input states are easy to simulate;
the state of the computer can simply be stored and updated explicitly. The
algorithm is the same as that described in Sec. III except that the degeneracy
is straightforward to evaluate since the state is explicitly known. Because
unitary operators do not change the rank of density matrix and projective
measurements can only decrease it, explicit storage of the state remains
practical throughout the simulation.
Effectively, a quantum computation on a low-rank input state becomes
complicated only because the eigenbasis becomes complicated. For a concordant
computation the eigenbasis remains manageable.
## V Conclusion
In summary, I have shown that conventional concordant computations composed
exclusively of gates acting on one or two qubits can be efficiently simulated
using a classical computer. As a consequence, such a computation must generate
quantum discord if it is to permit the efficient solution of a problem
requiring super-polynomial resources classically. A similar statement holds
for more general gate sets whenever the input state is either a generic
product state or a mixture of a few pure product states. These results lend
support to the idea that quantum discord is the appropriate generalization of
entanglement with regard to mixed-state quantum computation. That being said,
concordance is such a stringent property that it no doubt corresponds to the
case of zero quantum correlations for a variety of measures (including the
many flavors of discord), so this is far from the final word on the subject.
As has periodically been noted, it is also important to keep in mind that
there can be no single resource for quantum computing: If quantum computations
without property ${\mathscr{P}}$ can be efficiently simulated classically then
${\mathscr{P}}$ is a necessary resource for achieving a Promethean speedup.
Several possible directions for future research are suggested by previous work
on simulating quantum computations with restricted entanglement. The two most
prominent are investigating the performance of the simulation for
approximately concordant states and extending it to computations where discord
is restricted to blocks of qubits of bounded size. A block of qubits with
unrestricted correlations can be treated as a single quantum system, so
progress on the latter topic would likely require extending the simulation
method to qudits.
Though I specialize to qubits and two-qubit gates only in Sec. III.3, it is
doubtful whether my simulation method can be extended to more general gate
sets. Section III.3 depends crucially on the fact that permutations on one or
two bits of a vector are necessarily linear (or, from an alternate
perspective, that such permutations are Clifford gates) since this allows me
to determine whether Eq. 9 is satisfied by checking a small set of basis
vectors. On the other hand, permutations on systems of dimension greater than
two or on more than two bits need not be linear. Thus, directly generalizing
the method of simulation described in this paper requires a means of testing
Eq. 9 for an arbitrary sequence of permutations and input (mixed) product
state. This implies the ability to efficiently solve 3-SAT, an NP-Complete
problem, since $P$ in Eq. 9 can be chosen to implement a boolean formula, $Q$
to copy the result to an ancillary qubit, and $\rho^{0}$ to consist of
unbiased input qubits and maximally biased ancillary qubits, yielding
$\rho^{0}\neq P^{\dagger}QP\rho^{0}P^{\dagger}QP$ if and only if the boolean
formula is satisfied for some input. In other words, a direct extension of my
simulation method is effectively ruled out, though I am unable to exclude the
possibility that some more generally applicable method exists for simulating
concordant computations.
###### Acknowledgements.
I am grateful to Emanuel Knill, Anil Shaji, Carlton Caves, Vaibhav Madhok, and
Adam Meier for many productive discussions. This paper is a contribution by
the National Institute of Standards and Technology and, as such, is not
subject to U.S. copyright.
## References
* Horodecki et al. (2009) R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009), eprint arXiv:quant-ph/0702225.
* Jozsa and Linden (2003) R. Jozsa and N. Linden, Proc. R. Soc. A 459, 2011 (2003), eprint arXiv:quant-ph/0201143.
* Vidal (2003) G. Vidal, Phys. Rev. Lett. 91, 147902 (2003), eprint arXiv:quant-ph/0301063.
* Orús and Latorre (2004) R. Orús and J. I. Latorre, Phys. Rev. A 69, 052308 (2004), eprint arXiv:quant-ph/0311017.
* Knill and Laflamme (1998) E. Knill and R. Laflamme, Phys. Rev. Lett. 81, 5672 (1998), eprint arXiv:quant-ph/9802037.
* A. Ambainis and Vazirani (2006) L. S. A. Ambainis and U. Vazirani, Journal of the ACM 53, 507 (2006), eprint arXiv:quant-ph/0003136.
* Datta et al. (2005) A. Datta, S. T. Flammia, and C. M. Caves, Phys. Rev. A 72, 042316 (2005), eprint arXiv:quant-ph/0505213.
* Datta and Vidal (2007) A. Datta and G. Vidal, Phys. Rev. A 75, 042310 (2007), eprint arXiv:quant-ph/0611157.
* Zurek (2000) W. H. Zurek, Ann. Phys. 9, 855 (2000), eprint arXiv:quant-ph/0011039.
* Datta et al. (2008) A. Datta, A. Shaji, and C. M. Caves, Physical Review Letters 100, 050502 (2008), eprint arXiv:0709.0548.
* Lanyon et al. (2008) B. P. Lanyon, M. Barbieri, M. P. Almeida, and A. G. White, Phys. Rev. Lett. 101, 200501 (2008), eprint arXiv:0807.0668.
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|
arxiv-papers
| 2010-06-23T01:23:08 |
2024-09-04T02:49:11.116406
|
{
"license": "Public Domain",
"authors": "Bryan Eastin",
"submitter": "Bryan Eastin",
"url": "https://arxiv.org/abs/1006.4402"
}
|
1006.4470
|
# Generalized Mannheim Curves in Minkowski space-time $E_{1}^{4}$
Soley Ersoya , Murat Tosuna , Hiroo Matsudab
sersoy@sakarya.edu.tr , tosun@sakarya.edu.tr , matsuda@kanazawa-med.ac.jp
a Department of Mathematics, Sakarya University, Sakarya, TURKEY
b Department Mathematics, Kanazawa Medical University, Uchinada, Ishikawa,
920-02, JAPAN
###### Abstract
In this paper, the definition of generalized spacelike Mannheim curve in
Minkowski space-time $E_{1}^{4}$ is given. The necessary and sufficient
conditions for the generalized spacelike Mannheim curve are obtained. Also,
some characterizations of Mannheim curve are given.
Mathematics Subject Classification (2010): 53B30, 53A35, 53A04.
Keywords: Mannheim curve, Minkowski space-time
## 1 Introduction
The curves are a fundamental structure of differential geometry. An increasing
interest of the theory of curves makes a development of special curves to be
examined. A way to classification and characterization of curves is the
relationship between the Frenet vectors of the curves. For example, Saint
Venant proposed the question whether upon the surface generated by the
principal normal of a curve, a second curve can exist which has for its
principal normal of the given curve in 1845. This question was answered by
Bertrand in 1850. He showed that a necessary and sufficient condition for the
existence of such a second curve is that a linear relationship with constant
coefficients exists between the first and second curvatures of the given
original curve. The pairs of curves of this kind have been called Bertrand
partner curves or more commonly Bertrand curves [10], [14], [2]. There are
many works related with Bertrand curves in the Euclidean space and Minkowski
space, [15]–[3]. Also, generalized Bertrand curves in Euclidean 4- space are
defined and characterized in [6]. Another kind of associated curve have been
called Mannheim curve and Mannheim partner curve. The notion of Mannheim
curves was discovered by A. Mannheim in 1878. These curves in Euclidean
3-space are characterized in terms of the curvature and torsion as follows: A
space curve is a Mannheim curve if and only if its curvature $\kappa$ and
torsion $\tau$ satisfy the relation
$\kappa\left(s\right)=\alpha\left({{\kappa^{2}}\left(s\right)+{\tau^{2}}\left(s\right)}\right)$
for some constant $\alpha$. The articles concerning Mannheim curves are rather
few. In [13], a remarkable class of Mannheim curves is studied. General
Mannheim curves in the Euclidean 3-space are obtained in [11]. Mannheim
partner curves in Euclidean 3-space and Minkowski 3-space are studied and the
necessary and sufficient conditions for the Mannheim partner curves are
obtained in [5], [8]. Recently, Mannheim curves are generalized and some
characterizations and examples of generalized Mannheim curves in Euclidean
4-space ${E^{4}}$ are given by [7].
In this paper, we study the generalized spacelike Mannheim partner curves in
$4-$dimensional Minkowski space-time. We will give the necessary and
sufficient conditions for the generalized spacelike Mannheim partner curves.
## 2 Preliminaries
The basic concepts of the theory of curves in Minkowski space-time ${E^{4}}$
are briefly presented in this section. A more complete elementary treatment
can be found in [1]. Minkowski space-time ${E_{1}^{4}}$ is an Euclidean space
provided with the standard flat metric given by
$\left\langle{\,\,,\,\,}\right\rangle=-dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+dx_{4}^{2}$
where $\left({{x_{1}},\,{x_{2}},\,{x_{3}},\,{x_{4}}}\right)$ is a rectangular
coordinate system in ${E^{4}}$.
Since $\left\langle{\;,\;}\right\rangle$ is an indefinite metric, recall that
a vector ${\bf{v}}\in E_{1}^{4}$ can have one of the three causal characters;
it can be spacelike if $\left\langle{{\bf{v}},{\bf{v}}}\right\rangle>0$ or
${\bf{v}}={\bf{0}}$, timelike if
$\left\langle{{\bf{v}},{\bf{v}}}\right\rangle<0$ and null (lightlike) if
$\left\langle{{\bf{v}},{\bf{v}}}\right\rangle=0$ and ${\bf{v}}\neq{\bf{0}}$ .
Similarly, an arbitrary curve ${\bf{c}}={\bf{c}}\left(s\right)$ in ${E^{4}}$
can locally be spacelike, timelike or null (lightlike) if all of its velocity
vectors ${\bf{c^{\prime}}}\left(s\right)$ are, respectively, spacelike,
timelike or null. The norm of ${\bf{v}}\in E_{1}^{4}$ is given by
$\left\|{\bf{v}}\right\|=\sqrt{\left|{\left\langle{{\bf{v}},{\bf{v}}}\right\rangle}\right|}$.
If
$\left\|{{\bf{c^{\prime}}}\left(s\right)}\right\|=\sqrt{\left|{\left\langle{{\bf{c^{\prime}}}\left(s\right),{\bf{c^{\prime}}}\left(s\right)}\right\rangle}\right|}\neq
0$ for all $s\in L$, then $C$ is a regular curve in $E_{1}^{4}$. A spacelike
(timelike) regular curve $C$ is parameterized by arc-length parameter $s$
which is given by ${\bf{c}}:L\to E_{1}^{4}$, then the tangent vector
${\bf{c^{\prime}}}\left(s\right)$ along $C$ has unit length, that is,
$\left\langle{{\bf{c}}\left(s\right),{\bf{c}}\left(s\right)}\right\rangle=1\,,\,\quad\left({\left\langle{{\bf{c}}\left(s\right),{\bf{c}}\left(s\right)}\right\rangle=-1}\right)$
for all $s\in L$ .
Hereafter, curves are considered spacelike and regular ${C^{\infty}}$ curves
in $E_{1}^{4}$. Let
${{\bf{e}}_{1}}\left(s\right)={\bf{c^{\prime}}}\left(s\right)$ for all $s\in
L$, then the vector field ${{\bf{e}}_{1}}\left(s\right)$ is spacelike and it
is called spacelike unit tangent vector field on $C$.
The spacelike curve $C$ is called special spacelike Frenet curve if there
exist three smooth functions ${k_{1}}$, ${k_{2}}$, ${k_{3}}$ on $C$ and smooth
non-null frame field
$\left\\{{{{\bf{e}}_{1}},\,{{\bf{e}}_{2}},\,{{\bf{e}}_{3}},\,{{\bf{e}}_{4}}}\right\\}$
along the curve $C$. Also, the functions ${k_{1}},\,{k_{2}}$, and ${k_{3}}$
are called the first, the second, and the third curvature function on $C$,
respectively. For the ${C^{\infty}}$ special spacelike Frenet curve $C$, the
following Frenet formula is hold
$\left[\begin{array}[]{l}{{{\bf{e^{\prime}}}}_{1}}\\\
{{{\bf{e^{\prime}}}}_{2}}\\\ {{{\bf{e^{\prime}}}}_{3}}\\\
{{{\bf{e^{\prime}}}}_{4}}\\\
\end{array}\right]=\,\left[\begin{array}[]{l}\,\,\,0\,\,\,\,\,\,\,\,\,\,{k_{1}}\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,0\\\
{\mu_{1}}{k_{1}}\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,{k_{2}}\,\,\,\,\,\,\,0\\\
\,\,\,0\,\,\,\,\,\,\,{\mu_{2}}{k_{2}}\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,{k_{3}}\\\
\,\,\,0\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,{\mu_{3}}{k_{3}}\,\,\,\,0\\\
\end{array}\right]\left[\begin{array}[]{l}{{\bf{e}}_{1}}\\\
{{\bf{e}}_{\rm{2}}}\\\ {{\bf{e}}_{\rm{3}}}\\\ {{\bf{e}}_{4}}\\\
\end{array}\right]$
where ${\mu_{i}}=\mp 1,\,\,1\leq i\leq 3$, [1].
Due to characters of Frenet vectors of the spacelike curve $C$,
${\mu_{i}}\,\,\left({1\leq i\leq 3}\right)$ are defined as in the following
three subcases; Case 1: If ${{\bf{e}}_{4}}$ is timelike, then
${\mu_{i}},\,\,1\leq i\leq 3$ are
${\mu_{1}}={\mu_{2}}=-1\,\,,\,\,\,{\mu_{3}}=1$
where ${{\bf{e}}_{1}},\,{{\bf{e}}_{2}},\,{{\bf{e}}_{3}}$ and ${{\bf{e}}_{4}}$
are mutually orthogonal vector fields satisfying equations
$\left\langle{{{\bf{e}}_{1}}\,,\,{{\bf{e}}_{1}}}\right\rangle=\left\langle{{{\bf{e}}_{2}}\,,\,{{\bf{e}}_{2}}}\right\rangle=\left\langle{{{\bf{e}}_{3}}\,,\,{{\bf{e}}_{3}}}\right\rangle=1\,\,,\,\,\left\langle{{{\bf{e}}_{4}}\,,\,{{\bf{e}}_{4}}}\right\rangle=-1.$
Case 2: If ${{\bf{e}}_{3}}$ is timelike, then ${\mu_{i}},\,\,1\leq i\leq 3$
are
${\mu_{1}}=-1\,\,,\,\,{\mu_{2}}={\mu_{3}}=1$
where ${{\bf{e}}_{1}},\,{{\bf{e}}_{2}},\,{{\bf{e}}_{3}}$ and ${{\bf{e}}_{4}}$
are mutually orthogonal vector fields satisfying equations
$\left\langle{{{\bf{e}}_{1}}\,,\,{{\bf{e}}_{1}}}\right\rangle=\left\langle{{{\bf{e}}_{2}}\,,\,{{\bf{e}}_{2}}}\right\rangle=\left\langle{{{\bf{e}}_{4}}\,,\,{{\bf{e}}_{4}}}\right\rangle=1\,\,,\,\,\left\langle{{{\bf{e}}_{3}}\,,\,{{\bf{e}}_{3}}}\right\rangle=-1.$
Case 3: If ${{\bf{e}}_{2}}$ is timelike, then ${\mu_{i}},\,\,1\leq i\leq 3$
are
${\mu_{1}}={\mu_{2}}=1\,\,,\,\,{\mu_{3}}=-1$
where ${{\bf{e}}_{1}},\,{{\bf{e}}_{2}},\,{{\bf{e}}_{3}}$ and ${{\bf{e}}_{4}}$
are mutually orthogonal vector fields satisfying equations
$\left\langle{{{\bf{e}}_{1}}\,,\,{{\bf{e}}_{1}}}\right\rangle=\left\langle{{{\bf{e}}_{3}}\,,\,{{\bf{e}}_{3}}}\right\rangle=\left\langle{{{\bf{e}}_{4}}\,,\,{{\bf{e}}_{4}}}\right\rangle=1\,\,,\,\,\,\left\langle{{{\bf{e}}_{2}}\,,\,{{\bf{e}}_{2}}}\right\rangle=-1.$
For $s\in L$, the non-null frame field
$\left\\{{{{\bf{e}}_{1}},\,{{\bf{e}}_{2}},\,{{\bf{e}}_{3}},\,{{\bf{e}}_{4}}}\right\\}$
and curvature functions ${k_{1}}$ and ${k_{2}}$ are determined as follows
$\begin{array}[]{l}{1^{st}}\,\,\,\,\,\,{\rm{step}}\,\,\,\,\,\,{{\bf{e}}_{1}}\left(s\right)={\bf{c^{\prime}}}\left(s\right)\\\
{2^{nd}}\,\,\,\,{\rm{step}}\,\,\,\,\,\,\,{k_{1}}\left(s\right)=\left\|{{{{\bf{e^{\prime}}}}_{1}}\left(s\right)}\right\|>0\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\bf{e}}_{2}}\left(s\right)=\frac{1}{{{k_{1}}\left(s\right)}}{{{\bf{e^{\prime}}}}_{1}}\left(s\right)\\\
{3^{rd}}\,\,\,\,{\rm{step}}\,\,\,\,\,\,\,\,{k_{2}}\left(s\right)=\left\|{{{{\bf{e^{\prime}}}}_{2}}\left(s\right)-{\mu_{1}}{k_{1}}\left(s\right){{\bf{e}}_{1}}\left(s\right)}\right\|>0\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\bf{e}}_{3}}\left(s\right)=\frac{1}{{{k_{2}}\left(s\right)}}\left({{{{\bf{e^{\prime}}}}_{2}}\left(s\right)-{\mu_{1}}{k_{1}}\left(s\right){{\bf{e}}_{1}}\left(s\right)}\right)\\\
{4^{th}}\,\,\,{\rm{step}}\,\,\,\,\,\,\,\,\,\,{e_{4}}\left(s\right)=\varepsilon\frac{1}{{\left\|{{{{\bf{e^{\prime}}}}_{3}}\left(s\right)-{\mu_{2}}{k_{2}}\left(s\right){{\bf{e}}_{2}}\left(s\right)}\right\|}}\left({{{{\bf{e^{\prime}}}}_{3}}\left(s\right)-{\mu_{2}}{k_{2}}\left(s\right){{\bf{e}}_{2}}\left(s\right)}\right)\\\
\end{array}$
where $\varepsilon$ is taken $-1$ or $+1$ to make $+1$ the determinant of
$\left\\{{{{\bf{e}}_{1}},\,{{\bf{e}}_{2}},\,{{\bf{e}}_{3}},\,{{\bf{e}}_{4}}}\right\\}$,
that is, the non-null orthonormal frame field is of positive orientation. The
function ${k_{3}}$ is determined by
${k_{3}}\left(s\right)=\left\langle{{{{\bf{e^{\prime}}}}_{3}}\left(s\right)\,,\,{{\bf{e}}_{4}}\left(s\right)}\right\rangle\neq
0.$
So the function ${k_{3}}$ never vanishes.
In order to make sure that the spacelike curve $C$ is a special spacelike
Frenet curve, above steps must be checked, from ${1^{st}}$ step to ${4^{th}}$
step, for $s\in L$.
At each point of spacelike curve $C$, a line ${\ell_{1}}$ in the direction of
${{\bf{e}}_{2}}$ is called the first normal line, a line ${\ell_{2}}$ in the
direction of ${{\bf{e}}_{3}}$ is called the second normal line and a line
${\ell_{3}}$ in the direction of ${{\bf{e}}_{4}}$ is called the third normal
line.
Note that, according to three different case of spacelike curve $C$,
${\ell_{3}},\,{\ell_{2}}$ and ${\ell_{1}}$ can be timelike, respectively,
which are called second binormal, first binormal and principal normal line at
each point of the spacelike curve $C$.
## 3 Generalized spacelike Mannheim curves in $E_{1}^{4}$
In ${E^{4}}$ the Bertrand curves and Mannheim curves are generalized by [6]
and [7], respectively. In these regards, we have investigate generalization of
spacelike Mannheim curves Minkowski space in $E_{1}^{4}$.
###### Definition 3.1
A special spacelike curve $C$ in $E_{1}^{4}$ is a generalized spacelike
Mannheim curve if there exists a special spacelike Frenet curve ${C^{*}}$ in
$E_{1}^{4}$ such that the first normal line at each of $C$ is included in the
plane generated by the second normal line and the third normal line of
${C^{*}}$ at the corresponding point under $\phi$. Here $\phi$ is a bijection
from $C$ to ${C^{*}}$. The curve ${C^{*}}$ is called the generalized spacelike
Mannheim mate curve of $C$.
By the definition, a generalized Mannheim mate curve ${C^{*}}$ is given by
$\begin{array}[]{l}{{\bf{c}}^{*}}\left(s\right)={\bf{c}}\left(s\right)+\alpha\left(s\right){{\bf{e}}_{2}}\left(s\right),\,\,s\in
L\end{array}$ (3.1)
where $\alpha$ is a smooth function on $L$. Generally, the parameter $s$ isn’t
an arc-length of ${C^{*}}$. Let ${s^{*}}$ be the arc-length of ${C^{*}}$
defined by
${s^{*}}=\int\limits_{0}^{s}{\left\|{\frac{{d{{\bf{c}}^{*}}\left(s\right)}}{{ds}}}\right\|ds.}$
If a smooth function $f:L\to L$ is given by $f\left(s\right)={s^{*}}$, then
$\begin{array}[]{l}\,\,\,\,\frac{{d{{\bf{c}}^{*}}\left(s\right)}}{{ds}}=\,\,\,{{\bf{e}}_{1}}\left(s\right)+\alpha^{\prime}\left(s\right){{\bf{e}}_{2}}\left(s\right)+\alpha\left(s\right){\mu_{1}}{k_{1}}\left(s\right){{\bf{e}}_{1}}\left(s\right)+\alpha\left(s\right){k_{2}}\left(s\right){{\bf{e}}_{3}}\left(s\right)\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\;\,=\,\,\,\left({1+{\mu_{1}}\alpha\left(s\right){k_{1}}\left(s\right)}\right){{\bf{e}}_{1}}\left(s\right)+\alpha^{\prime}\left(s\right){{\bf{e}}_{2}}\left(s\right)+\alpha\left(s\right){k_{2}}\left(s\right){{\bf{e}}_{3}}\left(s\right).\\\
\end{array}$
for $\forall s\in L$. Thus, we have
$\begin{array}[]{l}f^{\prime}\left(s\right)=\frac{{d{s^{*}}}}{{ds}}=\left\|{\frac{{d{{\bf{c}}^{\bf{*}}}\left(s\right)}}{{ds}}}\right\|=\sqrt{\left|{{{\left({1+{\mu_{1}}\alpha\left(s\right){k_{1}}\left(s\right)}\right)}^{2}}+\varepsilon_{2}{{\left({\alpha^{\prime}\left(s\right)}\right)}^{2}}+\varepsilon_{3}{{\left({\alpha\left(s\right){k_{2}}\left(s\right)}\right)}^{2}}}\right|}\end{array}$
where
$\varepsilon_{i}=\left\\{\begin{array}[]{l}-1\,\,,\,\,\,{{\bf{e}}_{i}}\,\,{\rm{is}}\,{\rm{timelike}}\\\
\,\,\,\,1\,\,,\,\,\,\,{{\bf{e}}_{i}}\,{\rm{is}}\,{\rm{spacelike}}\\\
\end{array}\right.$, for $2\leq i\leq 4.$
This means that, in the Case 1, ${{\bf{e}}_{4}}$ is timelike and
$\begin{array}[]{l}f^{\prime}\left(s\right)=\sqrt{\left|{{{\left({1-\alpha\left(s\right){k_{1}}\left(s\right)}\right)}^{2}}+{{\left({\alpha^{\prime}\left(s\right)}\right)}^{2}}+{{\left({\alpha\left(s\right){k_{2}}\left(s\right)}\right)}^{2}}}\right|}\end{array}$
or in the Case 2, ${{\bf{e}}_{3}}$ is timelike and
$\begin{array}[]{l}f^{\prime}\left(s\right)=\sqrt{\left|{{{\left({1-\alpha\left(s\right){k_{1}}\left(s\right)}\right)}^{2}}+{{\left({\alpha^{\prime}\left(s\right)}\right)}^{2}}-{{\left({\alpha\left(s\right){k_{2}}\left(s\right)}\right)}^{2}}}\right|}\end{array}$
or in the Case 3, ${{\bf{e}}_{2}}$ is timelike and
$\begin{array}[]{l}f^{\prime}\left(s\right)=\sqrt{\left|{{{\left({1+\alpha\left(s\right){k_{1}}\left(s\right)}\right)}^{2}}-{{\left({\alpha^{\prime}\left(s\right)}\right)}^{2}}+{{\left({\alpha\left(s\right){k_{2}}\left(s\right)}\right)}^{2}}}\right|}.\end{array}$
The spacelike curve ${C^{*}}$ with arc-length parameter ${s^{*}}$ is
$\begin{array}[]{l}{{\bf{c}}^{*}}:\,{L^{*}}\to E_{1}^{4}\\\
\,\,\,\,\,\,\,\,\,\,\,{s^{*}}\,\,\to\,{{\bf{c}}^{*}}\left({{s^{*}}}\right).\\\
\end{array}$
For a bijection $\phi:\,C\to{C^{*}}$ defined by
$\phi\left({{\bf{c}}\left(s\right)}\right)={{\bf{c}}^{*}}\left({f\left(s\right)}\right),$
the reparametrization of ${C^{*}}$ is
$\begin{array}[]{l}{{\bf{c}}^{*}}\left({f\left(s\right)}\right)={\bf{c}}\left(s\right)+\alpha\left(s\right){{\bf{e}}_{2}}\left(s\right)\end{array}$
where $\alpha$ is a smooth function on $L$.
###### Theorem 3.1
If a special spacelike Frenet curve $C$ in $E_{1}^{4}$ is a generalized
spacelike Mannheim curve, then the first curvature function ${k_{1}}$ and the
second curvature function ${k_{2}}$ of $C$ satisfy the equality
$\begin{array}[]{l}{k_{1}}\left(s\right)=-\alpha\left({{\mu_{1}}k_{1}^{2}\left(s\right)+{\mu_{2}}k_{2}^{2}\left(s\right)}\right)\,\,,\,\,s\in
L\end{array}$ (3.2)
where $\alpha$ is a constant number and ${\mu_{1}}={\mu_{2}}=-1$ when
${{\bf{e}}_{4}}$ is timelike or ${\mu_{1}}=-1\,,\,\,{\mu_{2}}=1$ when
${{\bf{e}}_{3}}$ is timelike or ${\mu_{1}}={\mu_{2}}=1$ when ${{\bf{e}}_{2}}$
is timelike.
Proof. Let $C$ be a generalized spacelike Mannheim curve and ${C^{*}}$ be the
generalized spacelike Mannheim mate curve of $C$ with the diagram;
$\begin{array}[]{l}\,\,\,\,\,\,\,\,\,\,\,\,\mathop{\bf{c}}\limits_{\cdot\,\,\cdot}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\mathop{\bf{c}}\limits_{\cdot\,\,\cdot}^{*}}\\\
f:\,\,\,\,L\,\,\,\,\,\,\to\,\,\,\,\,{L^{*}}\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\downarrow\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\downarrow\\\
\phi\,:\,\,\,E_{1}^{4}\,\,\,\to\,\,\,E_{1}^{4}.\\\ \end{array}$
A smooth function $f$ is defined by
$f\left(s\right)=\int{\left\|{\frac{{d{{\bf{c}}^{*}}\left(s\right)}}{{ds}}}\right\|}ds={s^{*}}$
and ${s^{*}}$ is the arc-length parameter of ${C^{*}}$. Also $\phi$ is a
bijection which is defined by
$\phi\left({{\bf{c}}\left(s\right)}\right)={{\bf{c}}^{*}}\left({f\left(s\right)}\right).$
Thus, the spacelike curve ${C^{*}}$ is reparametrized by
$\begin{array}[]{l}{{\bf{c}}^{*}}\left({f\left(s\right)}\right)={\bf{c}}\left(s\right)+\alpha\left(s\right){{\bf{e}}_{2}}\left(s\right)\end{array}$
(3.3)
where $\alpha$ is a smooth function. By differentiating both sides of (3.3)
with respect to $s$
$\begin{array}[]{l}f^{\prime}\left(s\right){\bf{e}}_{1}^{*}\left({f\left(s\right)}\right)=\left({1+{\mu_{1}}\alpha\left(s\right){k_{1}}\left(s\right)}\right){{\bf{e}}_{1}}+\alpha^{\prime}\left(s\right){{\bf{e}}_{2}}\left(s\right)\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\alpha\left(s\right){k_{2}}\left(s\right){{\bf{e}}_{3}}\left(s\right)\\\
\end{array}$ (3.4)
is obtained.
On the other hand, since the first normal line at the each point of $C$ is
lying in the plane generated by the second normal line and the third normal
line of ${C^{*}}$ at the corresponding points under bijection $\phi$, the
vector field ${{\bf{e}}_{2}}\left(s\right)$ is given by
$\begin{array}[]{l}{{\bf{e}}_{2}}\left(s\right)=g\left(s\right){\bf{e}}_{3}^{*}\left({f\left(s\right)}\right)+h\left(s\right){\bf{e}}_{4}^{*}\left({f\left(s\right)}\right)\end{array}$
where $g$ and $h$ are some smooth functions on $L$. If we take into
consideration
$\begin{array}[]{l}\left\langle{{\bf{e}}_{1}^{*}\left({f\left(s\right)}\right),\,g\left(s\right){\bf{e}}_{3}^{*}\left({f\left(s\right)}\right)+h\left(s\right){\bf{e}}_{4}^{*}\left({f\left(s\right)}\right)}\right\rangle=0\end{array}$
and the equation (3.4), then we have $\alpha^{\prime}\left(s\right)=0$. So we
rewrite the equation (3.4) as
$\begin{array}[]{l}f^{\prime}\left(s\right){\bf{e}}_{1}^{*}\left({f\left(s\right)}\right)=\left({1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}\right){{\bf{e}}_{1}}\left(s\right)+\alpha{k_{2}}\left(s\right){{\bf{e}}_{3}}\left(s\right),\end{array}$
(3.5)
that is,
$\begin{array}[]{l}{\bf{e}}_{1}^{*}\left({f\left(s\right)}\right)=\frac{{\left({1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}\right)}}{{f^{\prime}\left(s\right)}}{{\bf{e}}_{1}}\left(s\right)+\frac{{\alpha{k_{2}}\left(s\right)}}{{f^{\prime}\left(s\right)}}{{\bf{e}}_{3}}\left(s\right)\end{array}$
where
$\begin{array}[]{l}f^{\prime}\left(s\right)=\sqrt{\left|{{{\left({1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}\right)}^{2}}+\varepsilon_{3}{{\left({\alpha{k_{2}}\left(s\right)}\right)}^{2}}}\right|}\,,\,\,\varepsilon_{3}=\left\\{\begin{array}[]{l}-1\,\,,\,\,{{\bf{e}}_{3}}\,\,{\rm{is\,\,timelike}}{\rm{,}}\\\
\,\,\,\,1\,\,\,,\,\,{{\bf{e}}_{3}}\,\,{\rm{is\,\,spacelike}}{\rm{.}}\\\
\end{array}\right.\end{array}$
By taking differentiation both sides of the equations (3.5) with respect to
$s$,
$\begin{array}[]{l}f^{\prime}\left(s\right)k_{1}^{*}\left({f\left(s\right)}\right){\bf{e}}_{2}^{*}\left({f\left(s\right)}\right)={\left({\frac{{1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}}{{f^{\prime}\left(s\right)}}}\right)^{\prime}}{{\bf{e}}_{1}}\left(s\right)\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\left({\frac{{\left({1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}\right){k_{1}}\left(s\right)+{\mu_{2}}\alpha{{\left({{k_{2}}\left(s\right)}\right)}^{2}}}}{{f^{\prime}\left(s\right)}}}\right){{\bf{e}}_{2}}\left(s\right)\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+{\left({\frac{{\alpha{k_{2}}\left(s\right)}}{{f^{\prime}\left(s\right)}}}\right)^{\prime}}{{\bf{e}}_{3}}\left(s\right)+\left({\frac{{\alpha{k_{2}}\left(s\right){k_{3}}\left(s\right)}}{{f^{\prime}\left(s\right)}}}\right){{\bf{e}}_{4}}\left(s\right)\\\
\end{array}$ (3.6)
is obtained for $s\in L$. Since
$\begin{array}[]{l}\left\langle{{\bf{e}}_{2}^{*}\left({f\left(s\right)}\right),\,g\left(s\right){\bf{e}}_{3}^{*}\left({f\left(s\right)}\right)+h\left(s\right){\bf{e}}_{4}^{*}\left({f\left(s\right)}\right)}\right\rangle=0,\end{array}$
then in the equation (3.6) the coefficient of ${{\bf{e}}_{2}}\left(s\right)$
vanishes, that is,
$\begin{array}[]{l}\left({1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}\right){k_{1}}\left(s\right)+{\mu_{2}}\alpha{\left({{k_{2}}\left(s\right)}\right)^{2}}=0.\end{array}$
Thus,
${k_{1}}\left(s\right)=-\alpha\left({{\mu_{1}}k_{1}^{2}\left(s\right)+{\mu_{2}}k_{2}^{2}\left(s\right)}\right)$
is satisfied. This completes the proof.
If we investigate the special cases separately, then we have
in the Case 1;
$\begin{array}[]{l}{k_{1}}\left(s\right)=\alpha\left({k_{1}^{2}\left(s\right)+k_{2}^{2}\left(s\right)}\right),\end{array}$
in the Case 2;
$\begin{array}[]{l}{k_{1}}\left(s\right)=\alpha\left({k_{1}^{2}\left(s\right)-k_{2}^{2}\left(s\right)}\right),\end{array}$
in the Case 3;
$\begin{array}[]{l}{k_{1}}\left(s\right)=-\alpha\left({k_{1}^{2}\left(s\right)+k_{2}^{2}\left(s\right)}\right).\end{array}$
###### Theorem 3.2
Let $C$ be a special spacelike Frenet curve in $E_{1}^{4}$ whose curvature
functions ${k_{1}}$ and ${k_{2}}$ are non-constant functions and satisfy the
equality
${k_{1}}\left(s\right)=-\alpha\left({{\mu_{1}}k_{1}^{2}\left(s\right)+{\mu_{2}}k_{2}^{2}\left(s\right)}\right)$,
where $\alpha$ is non-zero constant, for all $s\in L$. If the spacelike curve
${C^{*}}$ given by
$\begin{array}[]{l}{{\bf{c}}^{*}}\left(s\right)={\bf{c}}\left(s\right)+\alpha{{\bf{e}}_{2}}\left(s\right)\end{array}$
is a special spacelike Frenet curve, then ${C^{*}}$ is a generalized spacelike
Mannheim mate curve of $C$.
Proof . The arc-length parameter of ${C^{*}}$ is defined by
$\begin{array}[]{l}{s^{*}}=\int\limits_{0}^{s}{\left\|{\frac{{d{{\bf{c}}^{*}}\left(s\right)}}{{ds}}}\right\|}ds\end{array}$
for all $s\in L$. Under the assumptation of
$\begin{array}[]{l}{k_{1}}\left(s\right)=-\alpha\left({{\mu_{1}}k_{1}^{2}\left(s\right)+{\mu_{2}}k_{2}^{2}\left(s\right)}\right)\end{array}$
and after calculations for all cases, separately, we obtain
in the Case 1;
$\,\,\,\,f^{\prime}\left(s\right)=\sqrt{\left|{1-\alpha{k_{1}}\left(s\right)}\right|},$
in the Case 2;
$\,\,\,\,f^{\prime}\left(s\right)=\sqrt{\left|{1-\alpha{k_{1}}\left(s\right)}\right|},$
in the Case 3;
$\,\,\,\,f^{\prime}\left(s\right)=\sqrt{\left|{1+\alpha{k_{1}}\left(s\right)}\right|}.$
Thus, we can generalize
$\begin{array}[]{l}f^{\prime}\left(s\right)=\sqrt{\left|{1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}\right|}\end{array}$
for all $s\in L$.
By differentiating the equation
${{\bf{c}}^{*}}\left({f\left(s\right)}\right)={\bf{c}}\left(s\right)+\alpha{{\bf{e}}_{2}}\left(s\right)$
with respect to $s$, it is seen that
$\begin{array}[]{l}f^{\prime}\left(s\right){\bf{e}}_{1}^{*}\left({f\left(s\right)}\right)=\left({1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}\right){{\bf{e}}_{1}}\left(s\right)+\alpha{k_{2}}\left(s\right){{\bf{e}}_{3}}\left(s\right).\end{array}$
So, it is seen that
$\begin{array}[]{l}{\bf{e}}_{1}^{*}\left({f\left(s\right)}\right)=\left({\frac{{1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}}{{\sqrt{\left|{1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}\right|}}}{{\bf{e}}_{1}}\left(s\right)+\frac{{\alpha{k_{2}}\left(s\right)}}{{\sqrt{\left|{1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}\right|}}}{{\bf{e}}_{3}}\left(s\right)}\right)\end{array}$
(3.7)
for $s\in L$.
The differentiation of the last equation with respect to $s$ is
$\begin{array}[]{l}f^{\prime}\left(s\right)k_{1}^{*}\left({f\left(s\right)}\right){\bf{e}}_{2}^{*}\left({f\left(s\right)}\right)={\left({\sqrt{\left|{1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}\right|}}\right)^{\prime}}{{\bf{e}}_{1}}\left(s\right)\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\left({\frac{{\left({1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}\right){k_{1}}\left(s\right)+{\mu_{2}}\alpha
k_{2}^{2}\left(s\right)}}{{\sqrt{\left|{1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}\right|}}}}\right){{\bf{e}}_{2}}\left(s\right)\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+{\left({\frac{{\alpha{k_{2}}\left(s\right)}}{{\sqrt{\left|{1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}\right|}}}}\right)^{\prime}}{{\bf{e}}_{3}}\left(s\right)+\left({\frac{{\alpha{k_{2}}\left(s\right){k_{3}}\left(s\right)}}{{\sqrt{\left|{1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}\right|}}}}\right){{\bf{e}}_{4}}\left(s\right).\\\
\end{array}$ (3.8)
According to our assumption,
$\begin{array}[]{l}\frac{{\left({1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}\right){k_{1}}\left(s\right)+{\mu_{2}}\alpha
k_{2}^{2}\left(s\right)}}{{\sqrt{\left|{1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}\right|}}}=0\end{array}$
is hold. Thus, the coefficient of ${{\bf{e}}_{2}}\left(s\right)$ in the
equation (3.8) is zero. It is seen from the equation (3.8),
${\bf{e}}_{2}^{*}\left({f\left(s\right)}\right)$ is given by linear
combination of ${{\bf{e}}_{1}}\left(s\right),\;\,{{\bf{e}}_{3}}\left(s\right)$
and ${{\bf{e}}_{4}}\left(s\right)$. Also, from equation (3.7),
${\bf{e}}_{1}^{*}\left({f\left(s\right)}\right)$ is a linear combination of
${{\bf{e}}_{1}}\left(s\right)$ and ${{\bf{e}}_{3}}\left(s\right).$ Moreover,
${C^{*}}$ is a special spacelike Frenet curve that the vector
${{\bf{e}}_{2}}\left(s\right)$ is given by linear combination of
${\bf{e}}_{3}^{*}\left({f\left(s\right)}\right)$ and
${\bf{e}}_{4}^{*}\left({f\left(s\right)}\right)$.
Therefore, the first normal line $C$ lies in the plane generated by the second
normal line and third normal line of ${C^{*}}$ at the corresponding points
under a bijection $\phi$ which is defined by
$\phi\left({{\bf{c}}\left(s\right)}\right)={{\bf{c}}^{*}}\left({f\left(s\right)}\right)$.
Thus, the proof of the theorem is completed.
###### Remark 3.1
In 4-dimensional Minkowski space for a special spacelike Frenet curve $C$ with
curvature functions ${k_{1}}$ and ${k_{2}}$ satisfying
$\begin{array}[]{l}{k_{1}}\left(s\right)=-\alpha\left({{\mu_{1}}k_{1}^{2}\left(s\right)+{\mu_{2}}{\mu_{3}}k_{2}^{2}\left(s\right)}\right),\end{array}$
it is not clear that a smooth spacelike curve ${C^{*}}$ given by (3.1) is a
special Frenet curve. So, it is unknown whether the reverse of Theorem 3.1 is
true or false.
###### Theorem 3.3
Let $C$ be a spacelike special curve in $E_{1}^{4}$ with non-zero third
curvature function ${k_{3}}$. If there exists a spacelike special Frenet curve
${C^{*}}$ in $E_{1}^{4}$ such that the first normal line of $C$ is linearly
dependent with the third normal line of ${C^{*}}$ at the corresponding points
$\bf{c}\left(s\right)$ and ${\bf{c}^{*}}\left(s\right)$, respectively, under a
bijection $\phi:C\to{C^{*}}$, then the curvatures ${k_{1}}$ and ${k_{2}}$ of
$C$ are constant functions.
Proof. Let $C$ be a spacelike Frenet curve in $E_{1}^{4}$ with the Frenet
frame field
$\left\\{{{{\bf{e}}_{1}},\,{{\bf{e}}_{2}},\,{{\bf{e}}_{3}},\,{{\bf{e}}_{4}}}\right\\}$
and curvature functions ${k_{1}},\,{k_{2}}$ and ${k_{3}}$. Also, we assume
that ${C^{*}}$ be a spacelike special Frenet curve in $E_{1}^{4}$ with the
Frenet frame field
$\left\\{{{\bf{e}}_{1}^{*},\,{\bf{e}}_{2}^{*},\,{\bf{e}}_{3}^{*},\,{\bf{e}}_{4}^{*}}\right\\}$
and curvature functions $k_{1}^{*},\,k_{2}^{*}\,$ and $k_{3}^{*}$.
Let the first normal line of $C$ be linearly dependent with the third normal
line of ${C^{*}}$ at the corresponding points $C$ and ${C^{*}}$, respectively.
Then the parametrization of ${C^{*}}$ is
$\begin{array}[]{l}{{\bf{c}}^{*}}\left({f\left(s\right)}\right)={\bf{c}}\left(s\right)+\alpha\left(s\right){{\bf{e}}_{2}}\left(s\right)\end{array}$
(3.9)
for all $s\in L$. If ${s^{*}}$ is the arc-length parameter of ${C^{*}}$, then
$\begin{array}[]{l}{s^{*}}=\int\limits_{0}^{s}{\sqrt{\left|{{{\left({1+{\mu_{1}}\alpha{k_{1}}}\right)}^{2}}+\varepsilon_{2}\left({\alpha^{\prime}\left(s\right)}\right)+\varepsilon_{3}{{\left({\alpha\left(s\right){k_{2}}\left(s\right)}\right)}^{2}}}\right|}}ds\end{array}$
(3.10)
where
$\begin{array}[]{l}\varepsilon_{i}=\left\\{\begin{array}[]{l}-1\,\,,\,\,{{\bf{e}}_{i}}\,\,{\rm{is}}\,\,{\rm{timelike}}\\\
\,\,\,1\,\,\,,\,\,{{\bf{e}}_{i}}\,\,{\rm{is}}\,\,{\rm{spacelike}}\\\
\end{array}\right.,\,\,\,{\rm{for}}\,\,\,\,\,{\rm{}}2\leq i\leq 4\end{array}$
and
$\begin{array}[]{l}f:\,L\to{L^{*}}\\\
\,\,\,\,\,\,\,s\,\,\to\,f\left(s\right)={s^{*}}.\\\ \end{array}$
Moreover, $\phi:C\to{C^{*}}$ is a bijection given by
$\phi\left({{\bf{c}}\left(s\right)}\right)={{\bf{c}}^{*}}\left({f\left(s\right)}\right)$.
By differentiating the equation (3.9) with respect to $s$ and using Frenet
formulas, we have
$\begin{array}[]{l}f^{\prime}\left(s\right){\bf{e}}_{1}^{*}\left({f\left(s\right)}\right)=\left({1+{\mu_{1}}\alpha\left(s\right){k_{1}}\left(s\right)}\right){{\bf{e}}_{1}}\left(s\right)+\alpha^{\prime}\left(s\right){{\bf{e}}_{2}}\left(s\right)\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\alpha\left(s\right){k_{2}}\left(s\right){{\bf{e}}_{3}}\left(s\right).\\\
\end{array}$ (3.11)
Since
${\bf{e}}_{4}^{*}\left({f\left(s\right)}\right)=\mp{{\bf{e}}_{2}}\left(s\right)$,
then
$\begin{array}[]{l}\left\langle{f^{\prime}\left(s\right){\bf{e}}_{1}^{*}\left({f\left(s\right)}\right),\,{\bf{e}}_{4}^{*}\left({f\left(s\right)}\right)}\right\rangle=\left\langle{\left({1+{\mu_{1}}\alpha\left(s\right){k_{1}}\left(s\right)}\right){{\bf{e}}_{1}}\left(s\right)+\alpha^{\prime}\left(s\right){{\bf{e}}_{2}}\left(s\right)}\right.\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left.{+\alpha\left(s\right){k_{2}}\left(s\right){{\bf{e}}_{3}}\left(s\right),\,\mp{{\bf{e}}_{2}}\left(s\right)}\right\rangle,\\\
\end{array}$
that is,
$\begin{array}[]{l}0=\mp\alpha^{\prime}\left(s\right).\end{array}$
It is easily seen that $\alpha$ is a constant number from the last equation.
Thus, hereafter we can denote $\alpha\left(s\right)=\alpha$, for all $s\in L.$
From the equation (3.10), we get
$\begin{array}[]{l}f^{\prime}\left(s\right)=\sqrt{\left|{{{\left({1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}\right)}^{2}}+\varepsilon_{3}{{\left({\alpha{k_{2}}\left(s\right)}\right)}^{2}}}\right|}>0\end{array}$
where
$\begin{array}[]{l}\varepsilon_{3}=\left\\{\begin{array}[]{l}-1\,\,,\,\,{{\bf{e}}_{i}}\,\,{\rm{is}}\,\,{\rm{timelike}}\\\
\,\,\,\,1\,\,,\,\,{{\bf{e}}_{i}}\,\,{\rm{is}}\,\,{\rm{spacelike}}\\\
\end{array}\right.\,\,\,\,,\,{\rm{for}}\,\,\,\,\,2\leq i\leq 4.\end{array}$
Then, we rewrite the equation (3.11) as follows;
$\begin{array}[]{l}{\bf{e}}_{1}^{*}\left({f\left(s\right)}\right)=\left({\frac{{1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}}{{f^{\prime}\left(s\right)}}}\right){{\bf{e}}_{1}}\left(s\right)+\left({\frac{{\alpha{k_{2}}\left(s\right)}}{{f^{\prime}\left(s\right)}}}\right){{\bf{e}}_{3}}\left(s\right).\end{array}$
The differentiation of the last equation with respect to $s$ is
$\begin{array}[]{l}f^{\prime}\left(s\right)k_{1}^{*}\left({f\left(s\right)}\right){\bf{e}}_{2}^{*}\left({f\left(s\right)}\right)={\left({\frac{{1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}}{{f^{\prime}\left(s\right)}}}\right)^{\prime}}{{\bf{e}}_{1}}\left(s\right)\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\left({\frac{{{k_{1}}\left(s\right)+{\mu_{1}}\alpha
k_{1}^{2}\left(s\right)+{\mu_{2}}\alpha
k_{2}^{2}\left(s\right)}}{{f^{\prime}\left(s\right)}}}\right){{\bf{e}}_{2}}\left(s\right)\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+{\left({\frac{{\alpha{k_{2}}\left(s\right)}}{{f^{\prime}\left(s\right)}}}\right)^{\prime}}{{\bf{e}}_{3}}\left(s\right)+\left({\frac{{\alpha{k_{2}}\left(s\right){k_{3}}\left(s\right)}}{{f^{\prime}\left(s\right)}}}\right){{\bf{e}}_{4}}\left(s\right).\\\
\end{array}$ (3.12)
Since
$\left\langle{f^{\prime}\left(s\right)k_{1}^{*}\left({f\left(s\right)}\right){\bf{e}}_{2}^{*}\left({f\left(s\right)}\right),\,{\bf{e}}_{4}^{*}\left({f\left(s\right)}\right)}\right\rangle=0$
and
${\bf{e}}_{4}^{*}\left({f\left(s\right)}\right)=\mp{{\bf{e}}_{2}}\left(s\right)$
for all $s\in L$, we obtain
$\begin{array}[]{l}{k_{1}}\left(s\right)+{\mu_{1}}\alpha
k_{1}^{2}\left(s\right)+{\mu_{2}}\alpha k_{2}^{2}\left(s\right)=0\end{array}$
is satisfied. Then,
$\begin{array}[]{l}\alpha=-\frac{{{k_{1}}\left(s\right)}}{{{\mu_{1}}k_{1}^{2}\left(s\right)+{\mu_{2}}k_{2}^{2}\left(s\right)}}\end{array}$
(3.13)
is a non-zero constant number. Thus, from the equation (3.12), it is seen that
$\begin{array}[]{l}{\bf{e}}_{2}^{*}\left({f\left(s\right)}\right)=\frac{1}{{f^{\prime}\left(s\right)K\left(s\right)}}{\left({\frac{{1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}}{{f^{\prime}\left(s\right)}}}\right)^{\prime}}{{\bf{e}}_{1}}\left(s\right)+\frac{1}{{f^{\prime}\left(s\right)K\left(s\right)}}\left({\frac{{\alpha{k_{2}}\left(s\right)}}{{f^{\prime}\left(s\right)}}}\right){{\bf{e}}_{3}}\left(s\right)\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\frac{1}{{f^{\prime}\left(s\right)K\left(s\right)}}\left({\frac{{\alpha{k_{2}}\left(s\right){k_{3}}\left(s\right)}}{{f^{\prime}\left(s\right)}}}\right){{\bf{e}}_{4}}\left(s\right)\\\
\end{array}$
where $K\left(s\right)=k_{1}^{*}\left({f\left(s\right)}\right)$ for all $s\in
L$. By differentiating the last equation with respect to $s$, we obtain
$\begin{array}[]{l}f^{\prime}\left(s\right)\left[{{\mu_{1}}k_{1}^{*}\left({f\left(s\right)}\right){\bf{e}}_{1}^{*}\left({f\left(s\right)}\right)+k_{2}^{*}\left({f\left(s\right)}\right){{\bf{e}}_{3}^{*}}\left({f\left(s\right)}\right)}\right]={\left({\frac{1}{{f^{\prime}\left(s\right)K\left(s\right)}}{{\left({\frac{{1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}}{{f^{\prime}\left(s\right)}}}\right)}^{\prime}}}\right)^{\prime}}{{\bf{e}}_{1}}\left(s\right)\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\left({\frac{k_{1}{\left(s\right)}}{{f^{\prime}\left(s\right)K\left(s\right)}}{{\left({\frac{{1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}}{{f^{\prime}\left(s\right)}}}\right)}^{\prime}}+\frac{{{\mu_{2}}{k_{2}}\left(s\right)}}{{f^{\prime}\left(s\right)K\left(s\right)}}{{\left({\frac{{\alpha{k_{2}}\left(s\right)}}{{f^{\prime}\left(s\right)}}}\right)}^{\prime}}}\right){{\bf{e}}_{2}}\left(s\right)\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\left({{{\left({\frac{1}{{f^{\prime}\left(s\right)K\left(s\right)}}{{\left({\frac{{\alpha{k_{2}}\left(s\right)}}{{f^{\prime}\left(s\right)}}}\right)}^{\prime}}}\right)}^{\prime}}+\frac{{{\mu_{3}}{k_{3}}\left(s\right)}}{{f^{\prime}\left(s\right)K\left(s\right)}}\left({\frac{{\alpha{k_{2}}\left(s\right){k_{3}}\left(s\right)}}{{f^{\prime}\left(s\right)}}}\right)}\right){{\bf{e}}_{3}}\left(s\right)\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\left({{{\left({\frac{1}{{f^{\prime}\left(s\right)K\left(s\right)}}{{\left({\frac{{\alpha{k_{2}}\left(s\right){k_{3}}\left(s\right)}}{{f^{\prime}\left(s\right)}}}\right)}^{\prime}}}\right)}^{\prime}}+\frac{{{k_{3}}\left(s\right)}}{{f^{\prime}\left(s\right)K\left(s\right)}}{{\left({\frac{{\alpha{k_{2}}\left(s\right)}}{{f^{\prime}\left(s\right)}}}\right)}^{\prime}}}\right){{\bf{e}}_{4}}\left(s\right)\\\
\end{array}$
for all $s\in L$. If we take into consideration
$\begin{array}[]{l}\left\langle{f^{\prime}\left(s\right)\left({{\mu_{1}}k_{1}^{*}\left({f\left(s\right)}\right){\bf{e}}_{1}^{*}\left({f\left(s\right)}\right)+k_{2}^{*}\left({f\left(s\right)}\right){{\bf{e}}_{3}^{*}}\left({f\left(s\right)}\right)}\right)\,,\,\,{\bf{e}}_{4}^{*}\left({f\left(s\right)}\right)}\right\rangle=0\end{array}$
and
$\begin{array}[]{l}{\bf{e}}_{4}^{*}\left({f\left(s\right)}\right)=\mp{{\bf{e}}_{2}}\left(s\right),\end{array}$
then
$\begin{array}[]{l}{\mu_{1}}\alpha{k_{1}}\left(s\right){k_{1}}^{\prime}\left(s\right){f^{\prime}\left(s\right)}-{k_{1}}\left(s\right)\left({1+{\mu_{1}}\alpha{k_{1}}\left(s\right)}\right)f^{\prime\prime}\left(s\right)\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+{\mu_{2}}\alpha{k_{2}}\left(s\right){k_{2}}^{\prime}\left(s\right){f^{\prime}\left(s\right)}-{\mu_{2}}\alpha
k_{2}^{2}\left(s\right)f^{\prime\prime}\left(s\right)=0.\\\ \end{array}$
If we arrange the last equation, then we find
$\begin{array}[]{l}\alpha\left({{\mu_{1}}{k_{1}}\left(s\right){{k^{\prime}}_{1}}\left(s\right)+{\mu_{2}}{k_{2}}\left(s\right){{k^{\prime}}_{2}}\left(s\right)}\right)f^{\prime}\left(s\right)\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\left({{k_{1}}+\alpha\left({{\mu_{1}}k_{1}^{2}\left(s\right)+{\mu_{2}}k_{2}^{2}\left(s\right)}\right)}\right)f^{\prime\prime}\left(s\right)=0.\\\
\end{array}$ (3.14)
Moreover, the differentiation of the equation (3.13) with respect to $s$ is
$\begin{array}[]{l}{k^{\prime}_{1}}\left(s\right)+2\alpha\left({{\mu_{1}}{k_{1}}\left(s\right){{k^{\prime}}_{1}}\left(s\right)+{\mu_{2}}{k_{2}}\left(s\right){{k^{\prime}}_{2}}\left(s\right)}\right)=0.\end{array}$
From the above equation, we see
$\begin{array}[]{l}-\frac{{{{k^{\prime}}_{1}}\left(s\right)}}{2}=\alpha\left({{\mu_{1}}{k_{1}}\left(s\right){{k^{\prime}}_{1}}\left(s\right)+{\mu_{2}}{k_{2}}\left(s\right){{k^{\prime}}_{2}}\left(s\right)}\right).\end{array}$
(3.15)
If we substitute the equations (3.13) and (3.15) into the equation (3.14), we
obtain
$\begin{array}[]{l}-\frac{{{{k^{\prime}}_{1}}\left(s\right)}}{2}=0.\end{array}$
Finally, we find that the first curvature function is constant (that is,
positive constant).
Thus, from the equation (3.15) it is seen that the second curvature function
${k_{2}}$ is positive constant, too. This completes the proof.
In [9], a formula of parametric equation of Mannheim curve is given in
${E^{3}}$. Moreover, the parametric equation of generalized Mannheim curve in
${E^{4}}$ is obtained in [7]. The following theorem gives a parametric
representation of a generalized spacelike Mannheim curve with timelike second
binormal vector in $E_{1}^{4}$.
###### Theorem 3.4
Let be a spacelike special curve defined by
$\begin{array}[]{l}{\bf{c}}\left(u\right)=\left[{\begin{array}[]{*{20}{c}}{\alpha\int{f\left(u\right)\sinh
udu}}\\\ {\alpha\int{f\left(u\right)\cosh udu}}\\\
{\alpha\int{f\left(u\right)g\left(u\right)du}}\\\
{\alpha\int{f\left(u\right)h\left(u\right)du}}\\\
\end{array}}\right]\end{array}$
for $u\in I\subset\mathbb{R}$. Here $\alpha$ is a non-zero constant number,
$g:I\to\mathbb{R}$ and $h:I\to\mathbb{R}$ are any smooth functions and the
positive valued smooth function $f:I\to\mathbb{R}$ is given by
$\begin{array}[]{l}f\left(u\right)=\left({1+g^{2}\left(u\right)+h^{2}\left(u\right)}\right)^{{{-3}\mathord{\left/{\vphantom{{-3}2}}\right.\kern-1.2pt}2}}\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left|{-1-g^{2}\left(u\right)-h^{2}\left(u\right)+\dot{g}^{2}\left(u\right)+\dot{h}^{2}\left(u\right)+\left({\dot{g}\left(u\right)h\left(u\right)-g\left(u\right)\dot{h}\left(u\right)}\right)^{2}}\right|^{{{-5}\mathord{\left/{\vphantom{{-5}2}}\right.\kern-1.2pt}2}}\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\;\left|{\left({-1-g^{2}\left(u\right)-h^{2}\left(u\right)+\dot{g}^{2}\left(u\right)+\dot{h}^{2}\left(u\right)+\left({\dot{g}\left(u\right)h\left(u\right)-g\left(u\right)\dot{h}\left(u\right)}\right)^{2}}\right)^{3}}\right.\\\
\quad\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\left({1+g^{2}\left(u\right)+h^{2}\left(u\right)}\right)^{3}\left[{\left({g\left(u\right)-\ddot{g}\left(u\right)}\right)^{2}+\left({h\left(u\right)-\ddot{h}\left(u\right)}\right)^{2}}\right.\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left.{\,-\left({\left({g\left(u\right)\dot{h}\left(u\right)-\dot{g}\left(u\right)h\left(u\right)}\right)+\left({\dot{g}\left(u\right)\ddot{h}\left(u\right)-\ddot{g}\left(u\right)\dot{h}\left(u\right)}\right)}\right)^{2}+\left({g\left(u\right)\ddot{h}\left(u\right)-\ddot{g}\left(u\right)h}\right)\left(u\right)^{2}}\right|\\\
\quad\quad\quad\quad\quad\quad\;\;\quad\quad\quad\;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\\\
\end{array}$
for $u\in I$. Then the curvature functions ${k_{1}}$ and ${k_{2}}$ of $C$
satisfy
$\begin{array}[]{l}{k_{1}}\left(u\right)=\alpha\left({k_{1}^{2}\left(u\right)+k_{2}^{2}\left(u\right)}\right)\end{array}$
at the each point ${\bf{c}}\left(u\right)$ of $C$.
Proof. Let $C$ be a spacelike special curve defined by
$\begin{array}[]{l}{\bf{c}}\left(u\right)=\left[{\begin{array}[]{*{20}{c}}{\alpha\int{f\left(u\right)\sinh
udu}}\\\ {\alpha\int{f\left(u\right)\cosh udu}}\\\
{\alpha\int{f\left(u\right)g\left(u\right)du}}\\\
{\alpha\int{f\left(u\right)h\left(u\right)du}}\\\
\end{array}}\right]\quad,\quad u\in I\subset\mathbb{R}\end{array}$
where $\alpha$ is a non-zero constant number, $g$ and $h$ are any smooth
functions. $f$ is a positive valued smooth function. Thus, we obtain
$\begin{array}[]{l}{\bf{\dot{c}}}\left(u\right)=\left[{\begin{array}[]{*{20}{c}}{\alpha
f\left(u\right)\sinh u}\\\ {\alpha f\left(u\right)\cosh u}\\\ {\alpha
f\left(u\right)g\left(u\right)}\\\ {\alpha f\left(u\right)h\left(u\right)}\\\
\end{array}}\right]\quad,\quad u\in I\subset\mathbb{R}\end{array}$ (3.16)
where the subscript dot (.) denotes the differentiation with respect to $u$.
The arc-length parameter $s$ of $C$ is given by
$\begin{array}[]{l}s=\psi\left(u\right)=\int\limits_{{u_{0}}}^{u}{\left\|{{\bf{\dot{c}}}\left(u\right)}\right\|}du\end{array}$
where $\left\|{{\bf{\dot{c}}}\left(u\right)}\right\|=\alpha
f\left(u\right)\sqrt{1+{g^{2}}\left(u\right)+{h^{2}}\left(u\right)}.$
If $\varphi$ denotes the inverse function of $\psi:I\to L\subset\mathbb{R}$,
then $u=\varphi\left(s\right)$ and
$\begin{array}[]{l}\varphi^{\prime}\left(s\right)={\left\|{{{\left.{\frac{{d{\bf{c}}\left(u\right)}}{{du}}}\right|}_{u=\varphi\left(s\right)}}}\right\|^{-1}}\quad,\quad
s\in I\end{array}$
where the prime $\left({}^{\prime}\right)$ denotes the differentiation with
respect to $s$.
The unit tangent vector ${{\bf{e}}_{1}}\left(s\right)$ of the curve $C$ at the
each point ${\bf{c}}\left({\varphi\left(s\right)}\right)$ is given by
$\begin{array}[]{l}{{\bf{e}}_{1}}\left(s\right)={\left({1+{g^{2}}\left({\varphi\left(s\right)}\right)+{h^{2}}\left({\varphi\left(s\right)}\right)}\right)^{-{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}}\left[{\begin{array}[]{*{20}{c}}{\sinh\left({\varphi\left(s\right)}\right)}\\\
{\cosh\left({\varphi\left(s\right)}\right)}\\\
{g\left({\varphi\left(s\right)}\right)}\\\
{h\left({\varphi\left(s\right)}\right)}\\\ \end{array}}\right]\end{array}$
(3.17)
for all $s\in L$. Some simplifying assumptions are made for the sake of
brevity as follows;
$\begin{array}[]{l}\sinh:=\sinh\left({\varphi\left(s\right)}\right)\quad,\quad\quad\cosh:=\cosh\left({\varphi\left(s\right)}\right)\\\
f:=f\left({\varphi\left(s\right)}\right)\quad\quad\quad\,\,,\quad\quad
g:=g\left({\varphi\left(s\right)}\right)\quad\quad\quad,\quad
h:=h\left({\varphi\left(s\right)}\right),\\\
\dot{g}:=\dot{g}\left({\varphi\left(s\right)}\right)={\left.{\frac{{dg\left(u\right)}}{{du}}}\right|_{u=\varphi\left(s\right)}}\quad,\quad\dot{h}:=\dot{h}\left({\varphi\left(s\right)}\right)={\left.{\frac{{dh\left(u\right)}}{{du}}}\right|_{u=\varphi\left(s\right)}},\\\
\ddot{g}:=\ddot{g}\left({\varphi\left(s\right)}\right)={\left.{\frac{{{d^{2}}g\left(u\right)}}{{d{u^{2}}}}}\right|_{u=\varphi\left(s\right)}}\quad,\quad\ddot{h}:=\ddot{h}\left({\varphi\left(s\right)}\right)={\left.{\frac{{{d^{2}}h\left(u\right)}}{{d{u^{2}}}}}\right|_{u=\varphi\left(s\right)}},\\\
\varphi^{\prime}:=\varphi^{\prime}\left(s\right)={\left.{\frac{{d\varphi}}{{ds}}}\right|_{s}},\\\
A:=1+{g^{2}}+{h^{2}}\quad\,\,\,,\quad\quad B:=g\dot{g}+h\dot{h}\quad,\quad
C:={{\dot{g}}^{2}}+{{\dot{h}}^{2}},\\\
D:=g\ddot{g}+h\ddot{h}\quad\quad\quad,\quad\quad
E:=\dot{g}\ddot{g}+\dot{h}\ddot{h}\quad,\quad
F:={{\ddot{g}}^{2}}+{{\ddot{h}}^{2}}.\\\ \end{array}$
Then, we have
$\begin{array}[]{l}\dot{A}=2B\quad,\quad\dot{B}=C+D\quad,\quad\dot{C}=2E\quad,\quad\varphi^{\prime}={\alpha^{-1}}{f^{-1}}{A^{{{-1}\mathord{\left/{\vphantom{{-1}2}}\right.\kern-1.2pt}2}}}.\end{array}$
So, we rewrite the equation (3.17) as
$\begin{array}[]{l}{{\bf{e}}_{1}}:={{\bf{e}}_{1}}\left(s\right)={A^{-{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}}\left[{\begin{array}[]{*{20}{c}}{\sinh}\\\
{\cosh}\\\ g\\\ h\\\ \end{array}}\right].\end{array}$ (3.18)
By differentiating the last equation with respect to $s$, we find
$\begin{array}[]{l}{{\bf{e^{\prime}}}_{1}}=\varphi^{\prime}\left[{\begin{array}[]{*{20}{c}}{-\frac{1}{2}{A^{-{3\mathord{\left/{\vphantom{32}}\right.\kern-1.2pt}2}}}\dot{A}\sinh+{A^{-{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}}\cosh}\\\
{-\frac{1}{2}{A^{-{3\mathord{\left/{\vphantom{32}}\right.\kern-1.2pt}2}}}\dot{A}\cosh+{A^{-{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}}\sinh}\\\
{-\frac{1}{2}{A^{-{3\mathord{\left/{\vphantom{32}}\right.\kern-1.2pt}2}}}\dot{A}g+{A^{-{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}}\dot{g}}\\\
{-\frac{1}{2}{A^{-{3\mathord{\left/{\vphantom{32}}\right.\kern-1.2pt}2}}}\dot{A}h+{A^{-{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}}\dot{h}}\\\
\end{array}}\right],\end{array}$
that is,
$\begin{array}[]{l}{{\bf{e^{\prime}}}_{1}}=-\varphi^{\prime}{A^{-{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}}\left[{\begin{array}[]{*{20}{c}}{{A^{-1}}B\sinh-\cosh}\\\
{{A^{-1}}B\cosh-\sinh}\\\ {{A^{-1}}Bg-\dot{g}}\\\ {{A^{-1}}Bh-\dot{h}}\\\
\end{array}}\right].\end{array}$ (3.19)
From the last equation, we obtain
$\begin{array}[]{l}{k_{1}}:={k_{1}}\left(s\right)=\left\|{{{{\bf{e^{\prime}}}}_{1}}\left(s\right)}\right\|=\varphi^{\prime}{A^{-1}}{\left|{-A+AC-{B^{2}}}\right|^{{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}}.\end{array}$
(3.20)
By the fact that
${{\bf{e}}_{2}}\left(s\right)={\left({{k_{1}}\left(s\right)}\right)^{-1}}{{\bf{e^{\prime}}}_{1}}\left(s\right)$,
we have
$\begin{array}[]{l}{{\bf{e}}_{2}}:={{\bf{e}}_{2}}\left(s\right)=-{A^{{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}}{\left|{-A+AC-{B^{2}}}\right|^{{{-1}\mathord{\left/{\vphantom{{-1}2}}\right.\kern-1.2pt}2}}}\left[{\begin{array}[]{*{20}{c}}{{A^{-1}}B\sinh-\cosh}\\\
{{A^{-1}}B\cosh-\sinh}\\\ {{A^{-1}}Bg-\dot{g}}\\\ {{A^{-1}}Bh-\dot{h}}\\\
\end{array}}\right].\end{array}$
In order to get second curvature function ${k_{2}}$, we need to calculate
${k_{2}}\left(s\right)=\left\|{{{{\bf{e^{\prime}}}}_{2}}\left(s\right)-{\mu_{1}}{k_{1}}\left(s\right){{\bf{e}}_{1}}\left(s\right)}\right\|$.
It is seen from the above equation
$\left\langle{{{\bf{e}}_{2}}\left(s\right),{{\bf{e}}_{2}}\left(s\right)}\right\rangle=1$,
that is, ${{\bf{e}}_{2}}$ is spacelike. Thus, ${\mu_{1}}$ is equal to $-1$ and
${k_{2}}\left(s\right)=\left\|{{{{\bf{e^{\prime}}}}_{2}}\left(s\right)+{k_{1}}\left(s\right){{\bf{e}}_{1}}\left(s\right)}\right\|$.
After a long process of calculation, we have
$\begin{array}[]{l}{{\bf{e^{\prime}}}_{2}}+{k_{1}}{{\bf{e}}_{1}}=\varphi^{\prime}{A^{{{-3}\mathord{\left/{\vphantom{{-3}2}}\right.\kern-1.2pt}2}}}{\left|{-A+AC-{B^{2}}}\right|^{{{-3}\mathord{\left/{\vphantom{{-3}2}}\right.\kern-1.2pt}2}}}\left[{\begin{array}[]{*{20}{c}}{\left({P+Q}\right)\sinh-R\cosh}\\\
{\left({P+Q}\right)\cosh-R\sinh}\\\ {Pg-R\dot{g}+Q\ddot{g}}\\\
{Ph-R\dot{h}+Q\ddot{h}}\\\ \end{array}}\right]\end{array}$ (3.21)
where
$\begin{array}[]{l}P={\left({-A+AC-{B^{2}}}\right)^{2}}+\left({-A+AC-{B^{2}}}\right)\left({{B^{2}}-AC-
AD}\right)\\\ \,\,\,\,\,\,\,\,\,\,\,\,\,+AB\left({-B+AE-BD}\right),\\\
Q={A^{2}}\left({-A+AC-{B^{2}}}\right),\\\ R={A^{2}}\left({-B+AE-BD}\right).\\\
\end{array}$ (3.22)
If we simplify $P$ then we have
$\begin{array}[]{l}P={A^{2}}\left({1-C+BE+D-CD}\right).\end{array}$
Thus, we rewrite the equations (3.22) and (3.23) as
$\begin{array}[]{l}{{\bf{e^{\prime}}}_{2}}+{k_{1}}{{\bf{e}}_{1}}=\varphi^{\prime}{A^{{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}}{\left|{-A+AC-{B^{2}}}\right|^{{{-3}\mathord{\left/{\vphantom{{-3}2}}\right.\kern-1.2pt}2}}}\left[{\begin{array}[]{*{20}{c}}{\left({\tilde{P}+\tilde{Q}}\right)\sinh-\tilde{R}\cosh}\\\
{\left({\tilde{P}+\tilde{Q}}\right)\cosh-\tilde{R}\sinh}\\\
{\tilde{P}g-\tilde{R}\dot{g}+\tilde{Q}\ddot{g}}\\\
{\tilde{P}h-\tilde{R}\dot{h}+\tilde{Q}\ddot{h}}\\\
\end{array}}\right]\end{array}$ (3.23)
where
$\begin{array}[]{l}\tilde{P}=1-C+BE+D-CD,\\\ \tilde{Q}=-A+AC-{B^{2}},\\\
\tilde{R}=-B+AE-BD.\\\ \end{array}$ (3.24)
Consequently, from the equations (3.24) and (3.25), we find
$\begin{array}[]{l}{\left\|{{{{\bf{e^{\prime}}}}_{2}}+{k_{1}}{{\bf{e}}_{1}}}\right\|^{2}}={\left({\varphi^{\prime}}\right)^{2}}A{\left|{-A+AC-{B^{2}}}\right|^{-3}}\,\left|{{{\left({\tilde{P}+\tilde{Q}}\right)}^{2}}-{{\tilde{R}}^{2}}}\right.\\\
\quad\quad\quad\quad\quad\,+{{\tilde{P}}^{2}}\left({{g^{2}}+{h^{2}}}\right)+{{\tilde{R}}^{2}}\left({{{\dot{g}}^{2}}+{{\dot{h}}^{2}}}\right)+{{\tilde{Q}}^{2}}\left({{{\ddot{g}}^{2}}+{{\ddot{h}}^{2}}}\right)\\\
\quad\quad\quad\quad\quad\,\left.{-2\tilde{P}\tilde{R}\left({g\dot{g}+h\dot{h}}\right)-2\tilde{R}\tilde{Q}\left({\dot{g}\ddot{g}+\dot{h}\ddot{h}}\right)+2\tilde{P}\tilde{Q}\left({g\ddot{g}+h\ddot{h}}\right)}\right|.\\\
\end{array}$
If we substitute the abbreviations into the last equation, we get
$\begin{array}[]{l}{\left\|{{{{\bf{e^{\prime}}}}_{2}}+{k_{1}}{{\bf{e}}_{1}}}\right\|^{2}}={\left({\varphi^{\prime}}\right)^{2}}A{\left|{-A+AC-{B^{2}}}\right|^{-3}}\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left|{{{\tilde{P}}^{2}}A+2\tilde{P}\tilde{Q}+{{\tilde{Q}}^{2}}-{{\tilde{R}}^{2}}+{{\tilde{R}}^{2}}C}\right.\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left.{+{{\tilde{Q}}^{2}}F-2\tilde{P}\tilde{R}B-2\tilde{R}\tilde{Q}E+2\tilde{P}\tilde{Q}D}\right|.\\\
\end{array}$
After substituting the equation (3.24) into the last equation and simplifying
it, we have
$\begin{array}[]{l}k_{2}^{2}={\left\|{{{{\bf{e^{\prime}}}}_{2}}+{k_{1}}{{\bf{e}}_{1}}}\right\|^{2}}\\\
\quad\,={\left({\varphi^{\prime}}\right)^{2}}A{\left|{-A+AC-{B^{2}}}\right|^{-2}}\,\,\left|{\left({-A+AC-{B^{2}}}\right)\left({1+F}\right)}\right.\\\
\,\,\,\,\,\,\,\,\,\,\left.{+\left({1-C}\right){{\left({1+D}\right)}^{2}}+2BE\left({1+D}\right)-A{E^{2}}}\right|\,.\\\
\end{array}$
Moreover, from the equation (3.20) it is seen that
$\begin{array}[]{l}k_{1}^{2}={\left({\varphi^{\prime}}\right)^{2}}{A^{-2}}\left|{-A+AC-{B^{2}}}\right|.\end{array}$
The last two equation gives us
$\begin{array}[]{l}k_{1}^{2}+k_{2}^{2}={\left({\varphi^{\prime}}\right)^{2}}{A^{-2}}{\left|{-A+AC-{B^{2}}}\right|^{-2}}\left|{{{\left({-A+AC-{B^{2}}}\right)}^{3}}}\right.\\\
\quad\,\,\,\,\,\,\,\,\,\,\,\,\,\left.{+{A^{3}}\left({\left({-A+AC-{B^{2}}}\right)\left({1+F}\right)+\left({1-C}\right){{\left({1+D}\right)}^{2}}+2BE\left({1+D}\right)-A{E^{2}}}\right)}\right|.\\\
\end{array}$
By the fact
$\varphi^{\prime}={\alpha^{-1}}{f^{-1}}{A^{{{-1}\mathord{\left/{\vphantom{{-1}2}}\right.\kern-1.2pt}2}}}$,
we obtain
$\begin{array}[]{l}k_{1}^{2}+k_{2}^{2}={\alpha^{-2}}{f^{-2}}{A^{-3}}{\left|{-A+AC-{B^{2}}}\right|^{-2}}\left|{{{\left({-A+AC-{B^{2}}}\right)}^{3}}}\right.\\\
\quad\quad\quad\;\left.{\,\,+{A^{3}}\left({\left({-A+AC-{B^{2}}}\right)\left({1+F}\right)+\left({1-C}\right){{\left({1+D}\right)}^{2}}+2BE\left({1+D}\right)-A{E^{2}}}\right)}\right|.\\\
\end{array}$ (3.25)
and
$\begin{array}[]{l}{k_{1}}={\alpha^{-1}}{f^{-1}}{A^{-{3\mathord{\left/{\vphantom{32}}\right.\kern-1.2pt}2}}}{\left({-A+AC-{B^{2}}}\right)^{{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}}.\end{array}$
(3.26)
According to our assumption,
$\begin{array}[]{l}f={\left({1+{g^{2}}+{h^{2}}}\right)^{{{-3}\mathord{\left/{\vphantom{{-3}2}}\right.\kern-1.2pt}2}}}{\left|{-1-{g^{2}}-{h^{2}}+{{\dot{g}}^{2}}+{{\dot{h}}^{2}}+{{\left({\dot{g}h-g\dot{h}}\right)}^{2}}}\right|^{{{-5}\mathord{\left/{\vphantom{{-5}2}}\right.\kern-1.2pt}2}}}\\\
\,\,\,\,\,\,\,\,\,\,\,\left|{{{\left({-1-{g^{2}}-{h^{2}}+{{\dot{g}}^{2}}+{{\dot{h}}^{2}}+{{\left({\dot{g}h-g\dot{h}}\right)}^{2}}}\right)}^{3}}}\right.\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,-{\left({1+{g^{2}}+{h^{2}}}\right)^{3}}\,\left({{{\left({g-\ddot{g}}\right)}^{2}}+{{\left({h-\ddot{h}}\right)}^{2}}}\right.\\\
\quad\left.{\left.{\,\,\,\,\,\,\,\,\,-{{\left({\left({g\dot{h}-\dot{g}h}\right)+\left({\dot{g}\ddot{h}-\ddot{g}\dot{h}}\right)}\right)}^{2}}+{{\left({g\ddot{h}-\ddot{g}h}\right)}^{2}}}\right)}\right|,\\\
\end{array}$
we obtain
$\begin{array}[]{l}f={A^{-{3\mathord{\left/{\vphantom{32}}\right.\kern-1.2pt}2}}}{\left|{-A+AC-{B^{2}}}\right|^{{{-5}\mathord{\left/{\vphantom{{-5}2}}\right.\kern-1.2pt}2}}}\left|{{{\left({-A+AC-{B^{2}}}\right)}^{3}}}\right.\\\
\,\,\,\,\,\,\,\,\left.{\,+{A^{3}}\left({\left({1+F}\right)+\left({1-C}\right){{\left({1+D}\right)}^{2}}+2BE\left({1+D}\right)-A{E^{2}}}\right)}\right|.\\\
\end{array}$
If we substitute the above equations (3.25) and (3.26), we obtain
$\begin{array}[]{l}{k_{1}}=\alpha\left({k_{1}^{2}+k_{2}^{2}}\right).\end{array}$
The proof is completed.
In the above equation ${\mu_{1}}={\mu_{2}}=-1$ which is the special Case 1.
This formula is the parametric equation of generalized spacelike Mannheim
curve with timelike second binormal vector in the Minkowski space-time
$E_{1}^{4}$.
## References
* [1] B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, (1983).
* [2] D. J. Struik, Differential geometry, Second ed., Addison-Wesley, Reading, Massachusetts, (1961).
* [3] H. Balgetir, M. Bektaṣ, J. Inoguchi, Null Bertrand curves in Minkowski 3-space and their characterizations, Note Math., 23, no. 1, (2004).
* [4] H. Balgetir, M. Bektaṣ, M. Ergüt, Bertrand Curves for Nonnull Curves in 3-Dimensional Lorentzian Space, Hadronic Journal, 27, (2004).
* [5] H. Liu, F. Wang, Mannheim Partner curves in 3-space, Journal of Geometry, 88, 120-126, (2008).
* [6] H. Matsuda, S. Yorozu, Notes on Bertrand curves, Yokohama Math. J. 50, no. 1-2, 41-58, (2003).
* [7] H. Matsuda, S. Yorozu, On generalized Mannheim curves in Euclidean 4-space, (English), Nihonkai Math. J., 20, no. 1, 33-56, (2009).
* [8] K. Orbay, E. Kasap, On Mannheim Partner Curves in ${E^{3}}$, International Journal of Physical Sciences Vol. 4 (5), pp. 261-264, May, (2009).
* [9] L.P.A. Eisenhart, Treatise on the Differential Geometry of Curves and Surfaces, New York, Dover, (1960).
* [10] M.P. Do Carmo, Differential Geometry of Curves and Surfaces, Pearson Education, (1976).
* [11] O. Tigano, Sulla determinazione delle curve di Mannheim , Matematiche Catania 3, 25-29, (1948).
* [12] N. Ekmekci, K. Ilarslan, On Bertrand curves and their characterization, Differ. Geom. Dyn. Syst.(electronic), vol. 3, no. 2, (2001).
* [13] R. Blum, A remarkable class of Mannheim curves, Canad. Math. Bull. 9, 223-228, (1966).
* [14] W. Kuhnel, Differential geometry, Curves-surfaces-manifolds, Braunschweig, Wiesbaden, (1999).
* [15] Z. Nádeník, Bertrand curves in five-dimensional space, (Russian), Czechoslovak Mathematical Journal, vol. 2, issue 1, pp. 57-87, (1952).
|
arxiv-papers
| 2010-06-23T11:09:17 |
2024-09-04T02:49:11.126947
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Soley Ersoy, Murat Tosun, Hiroo Matsuda",
"submitter": "Soley Ersoy",
"url": "https://arxiv.org/abs/1006.4470"
}
|
1006.4617
|
# How are Feynman graphs resummed by the Loop Vertex Expansion?
Vincent Rivasseau, Zhituo Wang
Laboratoire de Physique Théorique, CNRS UMR 8627,
Université Paris XI, F-91405 Orsay Cedex, France
E-mail: rivass@th.u-psud.fr, zhituo.wang@th.u-psud.fr
###### Abstract
The purpose of this short letter is to clarify which set of pieces of Feynman
graphs are resummed in a Loop Vertex Expansion, and to formulate a conjecture
on the $\phi^{4}$ theory in non-integer dimension.
LPT-20XX-xx
MSC: 81T08, Pacs numbers: 11.10.Cd, 11.10.Ef
Key words: Feynman graphs, Combinatorics, Loop vertex expansion.
## 1 Introduction
In quantum field theory (hereafter QFT) any connected (i.e. interesting)
quantity is written as a sum of amplitudes for a certain category of connected
graphs
$S=\sum_{G\;\;{\rm connected}}{\cal{A}}_{G}$ (1)
but this formula is not a valid definition of $S$ since usually
$\sum_{G\;\;{\rm connected}}|{\cal{A}}_{G}|=\infty.$ (2)
This phenomenon, known since [1], is basically due to the very large number of
elements at order $n$ in the species [2] of Feynman graphs. Accordingly the
generating functional for the Feynman graphs species, namely the series
$\sum_{n}\frac{\lambda^{n}a_{n}}{n!}$, where $a_{n}$ is the number of Feynman
graphs at order $n$, has zero radius of convergence as power series in
$\lambda$. We call such a species a proliferating species. In zero space-time
dimension, quantum field theory reduces to this generating functional, hence
to graphs counting. In higher dimensions quantum field theory is in fact a
weighted such species, that is Feynman graphs have to be pondered with
weights, called Feynman amplitudes. For an introduction to the structure of
Feynman graphs, see [3]. Nevertheless these Feynman amplitudes tend to behave
as $K^{n}$ at order $n$ (at least in low dimensions). Hence the perturbation
series eg for the $\phi^{4}$ Euclidean Bosonic quantum field theory tends to
behave as $\sum_{n}(-\lambda)^{n}K^{n}n!$ and it has been proved to have zero
radius of convergence in one, two and three dimensions ([4, 5]). Nothing is
yet known for sure in dimension 4 but there are strong reasons to expect also
the renormalized Feynman series to diverge there as well (see [6] and
references therein).
In contrast Cayley’s theorem, which states that the total number of labeled
trees at order $n$ is $n^{n-2}$, implies that the species of trees is not
proliferating. This fact can be related to the local existence theorems for
flows in classical mechanics, since classical perturbation theory is indexed
by trees [7]. These theorems have no quantum counterpart, but constructive
theory can be seen as various recipes to replace the ordinary divergent
Feynman graph expansions by convergent ones, indexed by trees rather than
graphs [8]. It can therefore be considered a bridge between QFT and classical
mechanics, since it repacks the loops which are the fundamental feature of
QFT, and brings the expansion closer to those of classical mechanics.
Historically constructive theory used cumbersome non canonical tools borrowed
from lattice statistical mechanics, such as cluster expansions which did not
respect the rotational invariance of the underlying theory [9, 6]. The Loop
Vertex Expansion [10, 11] is a more canonical way to replace the ordinary
perturbative divergent expansion by a convergent one, which in principle
allows to compute quantities to arbitrary accuracy.
One of us (VR) was recently asked exactly which (pieces of) Feynman graphs are
resummed by this expansion. The answer is contained in the initial papers, but
perhaps not easy to extract. The purpose of this little note is therefore to
explain more explicitly exactly which pieces of which Feynman graphs of
different orders are combined together by the loop vertex expansion to create
a convergent expansion. This reshuffling is fully explicited up to third order
for the simplest of all possible examples, namely the $\phi^{4}_{0}$ quantum
field theory. Finally we also propose a conjecture, which, if true, would
allow to define QFT in non-integer dimensions of space-time.
## 2 Relative Tree Weights in a Graph
A graph may contain many (spanning) forests, and a forest can be extended into
many graphs with loops. So the relationship between graphs and their spanning
forests is not trivial.
The forest formula which we use [13, 14] can be viewed as a tool to associate
to any pair made of a graph $G$ and a spanning forest ${\cal{F}}\subset G$ a
unique rational number or weight $w(G,{\cal{F}})$ between 0 and 1, called the
relative weight of ${\cal{F}}$ in $G$.
The numbers $w(G,{\cal{F}})$ are multiplicative over disjoint unions 111And
also over vertex joints of graphs, just as in the universality theorem for the
Tutte polynomial.. Hence it is enough to give the formula for $(G,{\cal{F}})$
only when $G$ is connected and ${\cal{F}}={\cal{T}}$ is a spanning tree in
it222It is enough in fact to compute such weights for 1-particle irreducible
and 1-vertex-irreducible graphs, then multiply them in the appropriate way for
the general case..
The definition of these weights is
###### Definition 2.1.
$w(G,{\cal{T}})=\int_{0}^{1}\prod_{\ell\in{\cal{T}}}dw_{\ell}\prod_{\ell\not\in{\cal{T}}}x^{\cal{T}}_{\ell}(\\{w\\})$
(3)
where $x^{\cal{T}}_{\ell}(\\{w\\})$ is the infimum over the
$w_{\ell^{\prime}}$ parameters over the lines $\ell^{\prime}$ forming the
unique path in ${\cal{T}}$ joining the ends of $\ell$.
###### Lemma 2.1.
The relation
$\sum_{{\cal{F}}\subset G}w(G,{\cal{F}})=1$ (4)
holds for any connected graph $G$.
Proof It is a simple consequence of the forest formula [13, 14] applied to the
lines of the graph $G$.
### 2.1 Examples
For a fixed spanning tree inside a graph, we call loop lines the lines not in
the tree.
Consider the graph $G$ of Figure 1.
Figure 1: A first example
There are $5$ spanning trees inside this graph:
${\cal{T}}_{12}=\\{l_{1},l_{2}\\},\ {\cal{T}}_{13}=\\{l_{1},l_{3}\\},\
{\cal{T}}_{14}=\\{l_{1},l_{4}\\},\ {\cal{T}}_{23}=\\{l_{2},l_{3}\\},\
{\cal{T}}_{24}=\\{l_{2},l_{4}\\}.$
For example, for the tree ${\cal{T}}_{12}=\\{l_{2},l_{4}\\}$, the loop lines
are $l_{1}$ and $l_{3}$.
To take into account the weakening factors $x^{\cal{T}}_{\ell}(\\{w\\})$ of
(4) for each loop line $\ell$, it is convenient to decompose the integration
domain $[0,1]^{|{\cal{T}}|}$ into $|{\cal{T}}|!$ sectors corresponding to
complete orderings of the $w_{\ell}$ parameters for $\ell\in{\cal{T}}$.
Let us compute in this way the relative weights of the five trees of $G$.
First consider the contribution of the tree ${\cal{T}}_{12}$. In this case the
loop lines are $l_{3}$ and $l_{4}$. For each of them we have a factor
$\inf(w_{1}w_{2})$. Hence
$\displaystyle w(G,{\cal{T}}_{12})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\int_{0}^{1}dw_{1}dw_{2}[\inf(w_{1},w_{2})]^{2}$
$\displaystyle=$ $\displaystyle
2\int_{0}^{1}dw_{2}\int_{0}^{w_{2}}dw_{1}w_{1}^{2}=\frac{2}{12}=\frac{1}{6}.$
Next we consider the spanning tree ${\cal{T}}_{13}$. In this case the ”loop
lines“ are $l_{2}$ which connects the vertices $v_{1}$ and $v_{3}$ and $l_{4}$
which connects $v_{2}$ and $v_{3}$. So we have:
$\displaystyle w(G,{\cal{T}}_{13})$ $\displaystyle=$
$\displaystyle\int_{w_{1}<w_{3}}dw_{1}\int dw_{3}\inf(w_{1}w_{3})w_{3}$
$\displaystyle+$ $\displaystyle\int_{w_{3}<w_{1}}dw_{1}\int
dw_{3}\inf(w_{1}w_{3})w_{3}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dw_{3}\int_{0}^{w_{3}}dw_{1}w_{1}w_{3}+\int_{0}^{1}dw_{1}\int_{0}^{w_{1}}dw_{3}w_{3}^{2}=\frac{1}{8}+\frac{1}{12}=\frac{5}{24}.$
With the same method we find that
$w(G,{\cal{T}}_{14})=w(G,{\cal{T}}_{24})=w(G,{\cal{T}}_{23})=\frac{5}{24},$
(6)
and we have
$\sum_{{\cal{T}}\in G}w(G,{\cal{T}})=\frac{1}{6}+4.\frac{5}{24}=1.$ (7)
Let us treat a second example. Consider the graph $G^{\prime}$ of Fig. 2,
which has 6 edges:
$\\{l_{1},l_{2},l_{3},l_{4},l_{5},l_{6}\\}.$ (8)
To each edge $l_{i}$ we associate a factor $w_{i}$.
Figure 2: Example 2-the eye graph
There are 12 spanning trees:
$\displaystyle\\{l_{1},l_{2},l_{3}\\},\\{l_{1},l_{2},l_{4}\\},\\{l_{1},l_{3},l_{4}\\},\\{l_{2},l_{3},l_{4}\\},\\{l_{1},l_{2},l_{5}\\},\\{l_{1},l_{2},l_{6}\\},$
$\displaystyle\\{l_{3},l_{4},l_{5}\\},\\{l_{3},l_{4},l_{6}\\},\\{l_{1},l_{5},l_{4}\\},\\{l_{1},l_{6},l_{4}\\},\\{l_{3},l_{5},l_{2}\\},\\{l_{3},l_{6},l_{2}\\}.$
(9)
Let us compute the relative weight for each of these spanning trees in
$G^{\prime}$. First of all consider ${\cal{T}}_{123}=\\{l_{1},l_{2},l_{3}\\}$.
The other edges are drawn in dotted lines. See figure(3)
Figure 3: The spanning tree $\\{l_{1},l_{2},l_{3}\\}$
As is easily seen the corresponding loop lines are $l_{4}$, $l_{5}$ and
$l_{6}$. The weakening factor for $l_{5}$ and $l_{6}$ $\inf(w_{1},w_{3})$and
the weakening factor for $l_{4}$ is $\inf(w_{1},w_{2},w_{3})$. Therefore we
have
$\displaystyle
w(G^{\prime},{\cal{T}}_{123})=\int_{0<w_{1}<w_{2}<w_{3}<1}dw_{1}dw_{2}dw_{3}\inf(w_{1},w_{3})^{2}\inf(w_{1},w_{2},w_{3})$
$\displaystyle+$ $\displaystyle\rm{other}\ \rm{permutations}\ \rm{of}\
w_{1},w_{2},w_{3}$ $\displaystyle=$
$\displaystyle\int_{w_{1}<w_{2}<w_{3}}dw_{1}dw_{2}dw_{3}w_{1}^{3}+\int_{w_{2}<w_{3}<w_{1}}dw_{1}dw_{2}dw_{3}w_{3}^{2}w_{2}$
$\displaystyle+$
$\displaystyle\int_{w_{3}<w_{1}<w_{2}}dw_{1}dw_{2}dw_{3}w_{3}^{3}+\int_{w_{2}<w_{1}<w_{3}}dw_{1}dw_{2}dw_{3}w_{1}^{2}w_{2}$
$\displaystyle+$
$\displaystyle\int_{w_{3}<w_{2}<w_{1}}dw_{1}dw_{2}dw_{3}w_{3}^{3}+\int_{w_{1}<w_{3}<w_{2}}dw_{1}dw_{2}dw_{3}w_{1}^{3}.$
We compute only two of the integrals explicitly as others are obtained by
changing the names of variables.
$\int_{w_{1}<w_{2}<w_{3}}dw_{1}dw_{2}dw_{3}\
w_{1}^{3}=\int_{0}^{1}dw_{3}\int_{0}^{w_{3}}dw_{2}\int_{0}^{w_{2}}dw_{1}w_{1}^{3}=\frac{1}{120},$
(10) $\int_{w_{1}<w_{2}<w_{3}}dw_{1}dw_{2}dw_{3}\ w_{3}^{2}\
w_{2}=\frac{1}{60}.$ (11)
So we have
$w(G^{\prime},{\cal{T}}_{123})=\frac{1}{120}\times 4+\frac{1}{60}\times
2=\frac{1}{15}.$ (12)
The relative weights in $G^{\prime}$ of the spanning trees ${\cal{T}}_{124}$,
${\cal{T}}_{134}$ and ${\cal{T}}_{234}$ are the same.
Now we consider the tree $\\{l_{1},l_{2},l_{5}\\}$. (See figure 4).
Figure 4: The spanning tree $\\{l_{1},l_{2},l_{5}\\}$
To the loop line $l_{3}$ is associated a weakening factor $\inf(w_{1},w_{5})$.
To the loop line $l_{4}$ is associated a weakening factor $\inf(w_{2},w_{5})$.
To the loop line $l_{6}$ is associated a weakening factor $w_{5}$. So we have
$\displaystyle
w(G^{\prime},{\cal{T}}_{125})=\int_{w_{1}<w_{2}<w_{5}}dw_{1}dw_{2}dw_{5}\inf(w_{1},w_{5})\inf(w_{2},w_{5})w_{5}$
$\displaystyle+$ $\displaystyle\rm{other}\ \rm{permutations}\ \rm{of}\
w_{1},w_{2},w_{5}$ $\displaystyle=$
$\displaystyle\int_{w_{1}<w_{2}<w_{5}}dw_{1}dw_{2}dw_{5}w_{1}w_{2}w_{5}+\int_{w_{5}<w_{1}<w_{2}}dw_{1}dw_{2}dw_{5}w_{5}^{3}$
$\displaystyle+$
$\displaystyle\int_{w_{2}<w_{5}<w_{1}}dw_{1}dw_{2}dw_{5}w_{5}^{2}w_{2}+\int_{w_{2}<w_{1}<w_{5}}dw_{1}dw_{2}dw_{5}w_{1}w_{2}w_{5}$
$\displaystyle+$
$\displaystyle\int_{w_{1}<w_{5}<w_{2}}dw_{1}dw_{2}dw_{5}w_{5}^{2}w_{1}+\int_{w_{5}<w_{2}<w_{1}}dw_{1}dw_{2}dw_{5}w_{5}^{3}.$
We have
$\int_{w_{1}<w_{2}<w_{5}}dw_{1}dw_{2}dw_{5}w_{1}w_{2}w_{5}=\frac{1}{48},$ (14)
$\int_{w_{5}<w_{1}<w_{2}}dw_{1}dw_{2}dw_{5}w_{5}^{3}=\frac{1}{120},$ (15)
$\int_{w_{2}<w_{5}<w_{1}}dw_{1}dw_{2}dw_{5}w_{2}w^{2}_{5}=\frac{1}{60}.$ (16)
Similarly we get
$w(G^{\prime},{\cal{T}}_{125})=\frac{1}{120}\times 2+\frac{1}{60}\times
2+\frac{1}{48}\times 2=\frac{11}{120}.$ (17)
By the same method we find that this is also the relative weight of trees
${\cal{T}}_{126},{\cal{T}}_{345},{\cal{T}}_{346},{\cal{T}}_{125},{\cal{T}}_{145},{\cal{T}}_{146},{\cal{T}}_{235}$
and ${\cal{T}}_{236}$.
We can check again that
$\sum_{{\cal{T}}\in
G^{\prime}}w(G^{\prime},{\cal{T}})=4.\frac{1}{15}+8.\frac{11}{120}=1.$ (18)
## 3 Resumming Feynman Graphs
### 3.1 Naive Repacking
Consider the expansion (1) of a connected quantity $S$. The most naive way to
reorder Feynman perturbation theory according to trees rather than graphs is
to insert for each graph the relation (4)
$S=\sum_{G}A_{G}=\sum_{G}\sum_{{\cal{T}}\subset G}w(G,{\cal{T}}){\cal{A}}_{G}$
(19)
and exchange the order of the sums over $G$ and ${\cal{T}}$. Hence it writes
$S=\sum_{\cal{T}}{\cal{A}}_{\cal{T}},\quad{\cal{A}}_{\cal{T}}=\sum_{G\supset{\cal{T}}}w(G,{\cal{T}}){\cal{A}}_{G}.$
(20)
This rearranges the Feynman expansion according to trees, but each tree has
the same number of vertices as the initial graph. Hence it reshuffles the
various terms of a given, fixed order of perturbation theory. Remark that if
the initial graphs have say degree 4 at each vertex, only trees with degree
less than or equal to 4 occur in the rearranged tree expansion.
For Fermionic theories this is typically sufficient and one has for small
enough coupling
$\sum_{{\cal{T}}}|{\cal{A}}_{\cal{T}}|<\infty$ (21)
because Fermionic graphs mostly compensate each other at a fixed order by
Pauli’s principle; mathematically this is because these graphs form a
determinant and the size of a determinant is much less than what its
permutation expansion suggests. This is well known [15, 16, 17].
But this repacking fails for Bosonic theories, because the only compensations
there occur between graphs of different orders. Hence if we were to perform
this naive reshuffling, eg on the $\phi^{4}_{0}$ theory we would still have
$\sum_{T}|{\cal{A}}_{T}|=\infty.$ (22)
## 4 The Loop Vertex Expansion
The loop vertex expansion overcomes this difficulty by exchanging the role of
vertices and propagators before applying the forest formula. The corresponding
regrouping is completely different and each tree resums an infinite number of
pieces of the previous graphs. It relies on a technical tool (which physicists
call the intermediate field representation) which decomposes any interaction
of degree higher than three in terms of simpler three-body interactions. It is
particularly natural for 4-body interactions but can be generalized to higher
interactions as well [18].
This quite universal and powerful trick is linked to various deep physical and
mathematical tools, such as the color 1/N expansion and the Matthews-Salam and
Hubbard-Stratonovich methods in physics and the Kaufmann bracket of a knot and
many similar ideas in mathematics.
It is easy to describe the intermediate field method in terms of functional
integrals, as it is a simple generalization of the formula
$e^{-\lambda\phi^{4}/2}=\int
e^{-\sigma^{2}/2}e^{i\sqrt{\lambda}\sigma\phi^{2}}d\sigma.$ (23)
In this section we introduce the graphical procedure equivalent to this
formula.
In the case of a $\phi^{4}$ graph $G$ each vertex has exactly four half-lines
hence there are exactly three ways to pair these half-lines into two pairs.
Hence each fully labeled (vacuum) graph of order $n$ (with labels on vertices
and half-lines), which has $2n$ lines can be decomposed exactly into $3^{n}$
labeled graphs $G^{\prime}$ with degree 3 and two different types of lines
\- the $2n$ old ordinary lines
\- $n$ new dotted lines which indicate the pairing chosen at each vertex (see
Figure 5).
Figure 5: The extension and collapse for the order 1 graph
Such graphs $G^{\prime}$ are called the 3-body extensions of $G$ and we write
$G^{\prime}{\ \rm ext\ }G$ when $G^{\prime}$ is an extension of $G$. Let us
introduce for each such extension $G^{\prime}$ an amplitude
$A_{G^{\prime}}=3^{-n}A_{G}$ so that
$A_{G}=\sum_{G^{\prime}{\ \rm ext\ }G}A_{G^{\prime}}$ (24)
when $G^{\prime}$ is an extension of $G$.
Now the ordinary lines of any extension $G^{\prime}$ of any $G$ must form
cycles. These cycles are joined by dotted lines.
###### Definition 4.1.
We define the collapse $\bar{G}^{\prime}$ of such a $G^{\prime}$ graph as the
graph obtained by contracting each cycle to a ”bold” vertex (see Figure 5). We
write $\bar{G}^{\prime}{\ \rm coll\ }G^{\prime}$ if $\bar{G}^{\prime}$ is the
collapse of $G^{\prime}$, and define the amplitude of the collapsed graph
$\bar{G}^{\prime}$ as equal to that of $G^{\prime}$.
Remark that collapsed graphs, made of bold vertices and dotted lines, can have
now arbitrary degree at each vertex. Remark also that several different
extensions of a graph $G$ can have different collapsed graphs, see Figure 5.
Now the loop vertex expansion rewrites
$S=\sum_{G}A_{G}=\sum_{G^{\prime}{\ \rm ext\
}G}A_{G^{\prime}}=\sum_{\bar{G}^{\prime}{\ \rm coll\ }G^{\prime}{\ \rm ext\
}G}A_{\bar{G}^{\prime}}.$ (25)
Now we perform the tree repacking according to the graphs $\bar{G}^{\prime}$
with the $n$ dotted lines and not with respect to $G$. This is a completely
different repacking:
$A_{\bar{G}^{\prime}}=\sum_{\bar{\cal{T}}\in\bar{G}^{\prime}}w(\bar{G}^{\prime},\bar{\cal{T}})A_{\bar{G}^{\prime}},$
(26)
so that
$S=\sum_{G^{\prime}{\ \rm ext\
}G}A_{\bar{G}^{\prime}}=\sum_{\bar{\cal{T}}\in\bar{G}^{\prime}}A_{\bar{\cal{T}}},$
(27)
$A_{\bar{\cal{T}}}=\sum_{\bar{G}^{\prime}\supset\bar{\cal{T}}}w(\bar{G}^{\prime},\bar{\cal{T}})A_{\bar{G}^{\prime}}.$
(28)
The ”miracle” is that
###### Theorem 4.1.
For $\lambda$ small
$\sum_{\bar{\cal{T}}}|A_{\bar{\cal{T}}}|<\infty$ (29)
the result being the Borel sum of the initial perturbative series [12].
The proof of the theorem will not be recalled here (see [10, 11, 12]) but it
relies on the positivity property of the $x^{\cal{T}}_{\ell}(\\{w\\})$
symmetric matrix, and the representation of each $A_{\bar{\cal{T}}}$ amplitude
as an integral over a corresponding normalized Gaussian measure of a product
of resolvents bounded by 1. This convergence would not be true if we had
chosen naive $w({\cal{T}},G)$ barycentric weights such as 1/5 for each of the
five trees of the graph in Figure 1.
This method is valid for any $\phi^{4}$ model in any dimension with cutoffs
[11]. It is not limited to $\phi^{4}$ but works eg for any stable interaction
at the cost of introducing more intermediate particles until three body
elementary interactions are reached [18]. It also reproduces correctly the
large $N$ behaviour of $\phi^{4}$ matrix models, which was the key property
for which this expansion was found [10].
## 5 Examples
In this section we give the extension and collapse of the Feynman graphs for
$Z$ and $\log Z$ for the $\phi^{4}_{0}$ model up to order 3. We also recover
the combinatorics of those graphs through the ordinary functional integral
formula for the loop vertex expansion formula of [12].
The extension and collapse at order 1 was shown in Figure (6). In this case
the tree structure is easy. We find only the trivial ”empty” tree with one
vertex and no edge and the ”almost trivial” tree with two vertices and a
single edge. The weight for these trees is 1.
Figure 6: The extension and collapse for order 1 graph, with combinatoric
weight shown below, and the list of corresponding trees.
At second order we find one disconnected Feynman graph and two connected ones.
Only the connected ones survive in the expansion of $\log Z$.
Figure 7: The extension and collapse for order 2 graph and the number of
graphs.
The corresponding graphs and tree structures are shown in Figure (7) and
Figure(8). Using the loop vertex expansion formula we begin to see that graphs
that come from different order of the expansion of $\lambda$ are associated to
the same trees by the loop vertex expansion. Indeed we recover contributions
for the trivial and almost trivial trees of the previous figure. But we find
also a new contribution belonging to a tree with two edges.
Figure 8: The connected graphs and the tree structure from the Loop vertex
expansion. Figure 9: The order 3 vacuum graph and the number of graphs. Figure
10: The extension and collapse for order 3 graph. Figure 11: The graph
structure and combinatorics from the loop vertex expansion at order 3. The
symbols like 1122 means we have 4 loop vertices V, two of them have one
$\sigma$ field each and two of them have two $\sigma$ fields each, as we could
read directly from this figure. Figure 12: The tree structure of order 3
graphs.
At order three the computation becomes a bit more involved but the process is
clear. We could start from the ordinary Feynman graphs and get the graphs of
loop vertex expansion by extension and collapse. This is shown in Figure (10).
The number under each collapsed graph means the number of the corresponding
graphs, as in the previous case. The tree structure is shown in Figure(12). In
this figure the weight factor $w$ means always $w(G,{\cal{T}})$. We could also
get the graphs and combinatorics by using directly the loop vertex expansion,
namely we integrate the $\phi$ fields and consider only the Wick contractions
of the $\sigma$ fields. This is shown in the appendix and Figure (11). In this
process we expand both $\exp V$ and the vertex
$V=\mbox{tr}\log(1+2i\sqrt{2\lambda}\sigma)$. The interactions terms are then
the loop vertices $V$ with various attached $\sigma$ fields. This is shown on
the left hand of Figure (11). For example, the symbol $123$ means we consider
the $V^{3}/3!$ term in $\exp V$. We expand one of the $V$ to order
$\lambda^{1/2}$, namely with one $\sigma$ field attached, one to the order
$\lambda$, namely with 2 $\sigma$ fields attached and the third one to
$\lambda^{3/2}$, namely with 3 $\sigma$ fields. Then we contract the sigma
fields with respect to the Gaussian measure, obtaining all the contracted
graphs. The total number of $123$ graphs could be read directly from this
Gaussian integration. To get the combinatoric factor of each graph we need to
compute the relative weights of these graphs. This is shown in the following
example:
###### Example 5.1.
Figure 13: The example of ’123’ contractions.
We consider the $123$ case for example. This is shown more explicitly in
Figure(13). We use $a,b,\cdots,f$ to label the $\sigma$ fields attached to the
vertices. After the Wick contractions we get three different graphs $A$, $B$
and $C$. The number of possibilities to get $A$ is $3$, the number to get $B$
is $2\times 3=6$ and the number to get $C$ is also 6. So the relative weight
for graph $A$ is $3/(3+6+6)=1/5$ and the relative weights for $B$ and $C$ are
both $6/(3+6+6)=2/5$. As we could read directly from the loop vertex formula
that the total number of $123$ contraction graphs is $960$, we get finally the
combinatoric factor of graph $A$ to be $960\times 1/5=192$, and the
corresponding factors for graphs $B$ and $C$ are $960\times 2/5=384$. This
result agrees with the one coming from the Feynman graph computation.
From these examples we find that the structure of loop vertex expansion is
totally different from that of Feynman graph calculus. At each order of the
loop vertex expansion we combine terms in different orders of $\lambda$.
## 6 Non-integer Dimension
Let us now consider, eg for $0<D\leq 2$ the Feynman amplitudes for the
$\phi^{4}_{D}$ theory. They are given by the following convergent parametric
representation
$A_{D,G}=\int_{0}^{\infty}d\alpha\frac{e^{-m^{2}\sum_{\ell}\alpha_{\ell}}}{U_{G}^{D/2}}$
(30)
where $m$ is the mass and $U_{G}$ is the Kirchoff-Symanzik polynomial for $G$
$U_{G}=\sum_{{\cal{T}}\in G}\prod_{\ell\not\in{\cal{T}}}\alpha_{\ell}.$ (31)
All the previous decompositions working at the level of graphs, they are
independent of the space-time dimension. We can therefore repack the series of
Feynman amplitudes in non integer dimension to get the $D$ dimensional tree
amplitude:
$A_{D,\bar{\cal{T}}}=\sum_{G\supset\bar{\cal{T}}}w(\bar{\cal{T}},G)A_{D,G}$
(32)
We know that for $D=0$ and $D=1$ the loop vertex expansion is convergent.
Therefore it is tempting to conjecture , for instance at least for $D$ real
and $0\leq D<2$ (that is when no ultraviolet divergences require
renormalization)
###### Conjecture 6.1.
For $\lambda$ small
$\sum_{\bar{\cal{T}}}|A_{D,\bar{\cal{T}}}|<\infty$ (33)
the result being the Borel sum of the initial perturbative series.
Progress on this conjecture would be extremely interesting as it would allow
to bridge quantum field theories in various dimensions of space time, and ie
perhaps justify the Wilson-Fisher $4-\epsilon$ expansion that allows good
numerical approximate computations of critical indices in 3 dimensions.
We know however that when renormalization is needed, ie for $D\geq 2$, this
approach has to be completed with the introduction of the correct
counterterms. Presumably in this case the tree expansion should be adapted to
select optimal trees with respect to renormalization group scales. This is
work in progress.
An other possible approach to quantum field theory in non integer dimension,
also based on the forest formula but more radical, is proposed in [19].
## 7 Conclusion
The lessons we may draw from the Loop Vertex Expansion are
* •
Interactions should be decomposed into three body elementary interactions. The
corresponding fields might be more fundamental than the initial ones.
* •
Tree formulas solve the constructive problem ie resum perturbation theory at
the cost of loosing locality of the new vertices.
It may be also interesting to further understand why trees are so central both
in the parametric formulas (30) for single Feynman amplitudes and in the non-
perturbative treatment of the theory. The answer might imply a complete
refoundation of quantum field theory around the notion of trees, rather than
Feynman graphs or even functional integrals [19].
## 8 Appendix
In this Appendix we compute the weight of collapsed Feynman graphs using the
Loop Vertex Expansion.
For the $\phi^{4}_{0}$ model we have:
$Z=\frac{1}{\sqrt{2\pi}}\int d\phi
e^{-\frac{1}{2}\phi^{2}-\lambda\phi^{4}}=\frac{1}{\sqrt{2\pi}}\int d\sigma
e^{-\frac{1}{2}\sigma^{2}-\frac{1}{2}\log(1+2i\sqrt{2\lambda}\sigma)}.$ (34)
We define
$V=\frac{1}{2}\log(1+2i\sqrt{2\lambda}\sigma).$ (35)
In what follows we compute the vacuum graphs up to order $3$ in $\lambda$. We
expand $Z$ into powers of $V$:
$Z=\frac{1}{\sqrt{2\pi}}\int d\sigma
e^{-\frac{1}{2}\sigma^{2}}[1-V+\frac{1}{2!}V^{2}-\frac{1}{3!}V^{3}+\frac{1}{4!}V^{4}-\frac{1}{5!}V^{5}+\frac{1}{6!}V^{6}],$
(36)
and we have
$\displaystyle\log(1+2i\sqrt{2\lambda}\sigma)$ $\displaystyle=$ $\displaystyle
2\sqrt{2\lambda}i\sigma+4\lambda\sigma^{2}-\frac{16\sqrt{2}i}{3}\lambda^{3/2}\sigma^{3}-16\lambda^{2}\sigma^{4}$
(37) $\displaystyle+$
$\displaystyle\frac{128\sqrt{2}i}{5}\lambda^{5/2}\sigma^{5}+\frac{256}{3}\lambda^{3}\sigma^{6}.$
The first term
$\frac{1}{\sqrt{2\pi}}\int d\sigma e^{-\frac{1}{2}\sigma^{2}}1=1$ (38)
is trivial.
The order $V$ terms give:
$\displaystyle-\frac{1}{\sqrt{2\pi}}\int d\sigma
e^{-\frac{1}{2}\sigma^{2}}V=\frac{1}{\sqrt{2\pi}}\int d\sigma
e^{-\frac{1}{2}\sigma^{2}}[-2\lambda\sigma^{2}+8\lambda^{2}\sigma^{4}-\frac{128}{3}\lambda^{3}\sigma^{6}]$
(39) $\displaystyle=$ $\displaystyle-2\lambda+24\lambda^{2}-640\lambda^{3}.$
The $V^{2}$ terms give:
$\displaystyle\frac{1}{2!}\frac{1}{\sqrt{2\pi}}\int d\sigma
e^{-\frac{1}{2}\sigma^{2}}V^{2}=\frac{1}{2!}(\frac{1}{2})^{2}\frac{1}{\sqrt{2\pi}}\int
d\sigma
e^{-\frac{1}{2}\sigma^{2}}[-8\lambda\sigma^{2}+16\lambda^{2}\sigma^{4}$ (40)
$\displaystyle-$ $\displaystyle\frac{64\times
8}{9}\lambda^{3}\sigma^{6}+\frac{128}{3}\lambda^{2}\sigma^{4}-\frac{128\times
8}{5}\lambda^{3}\sigma^{6}-128\lambda^{3}\sigma^{6}]$ $\displaystyle=$
$\displaystyle-\lambda+22\lambda^{2}-\frac{320}{3}\lambda^{3}-624\lambda^{3}.$
The $V^{3}$ terms give:
$\displaystyle-\frac{1}{3!}\frac{1}{\sqrt{2\pi}}\int d\sigma
e^{-\frac{1}{2}\sigma^{2}}V^{3}=-\frac{1}{3!}(\frac{1}{2})^{3}\frac{1}{\sqrt{2\pi}}\int
d\sigma
e^{-\frac{1}{2}\sigma^{2}}[64\lambda^{3}\sigma^{6}-96\lambda^{2}\sigma^{4}$
(41) $\displaystyle+$ $\displaystyle
384\lambda^{3}\sigma^{6}+512\lambda^{3}\sigma^{6}]$ $\displaystyle=$
$\displaystyle 6\lambda^{2}-300\lambda^{3}.$
The $V^{4}$ terms give:
$\displaystyle\frac{1}{4!}(\frac{1}{2})^{4}\frac{1}{\sqrt{2\pi}}\int d\sigma
e^{-\frac{1}{2}\sigma^{2}}[64\lambda^{2}\sigma^{4}-\frac{2048}{3}\lambda^{3}\sigma^{6}-768\lambda^{3}\sigma^{6}]$
(42) $\displaystyle=$
$\displaystyle\frac{1}{2}\lambda^{2}-\frac{80}{3}\lambda^{3}-30\lambda^{3}.$
The $V^{5}$ terms give:
$\displaystyle-\frac{1}{5!}(\frac{1}{2})^{5}\frac{1}{\sqrt{2\pi}}\int d\sigma
e^{-\frac{1}{2}\sigma^{2}}\ 1280\lambda^{3}\sigma^{6}=-5\lambda^{3}.$ (43)
The $V^{6}$ term gives:
$\displaystyle-\frac{1}{6!}(\frac{1}{2})^{6}\frac{1}{\sqrt{2\pi}}\int d\sigma
e^{-\frac{1}{2}\sigma^{2}}\ 512\lambda^{3}\sigma^{6}=-\frac{1}{6}\lambda^{3}.$
(44)
So up to $3$rd order in $\lambda$ we recover
$Z=-3\lambda+\frac{105}{2}\lambda^{2}-\frac{10395}{6}\lambda^{3}=-4!!\lambda+\frac{8!!}{2!}\lambda^{2}-\frac{12!!}{3!}\lambda^{3},$
(45)
which of course coincide with the number of ordinary Wick contractions derived
by the regular $\lambda\phi^{4}$ Feynman expansion.
Acknowledgments We thank H. Knörrer for asking the question which lead to
writing this paper.
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* [24] J. Feldman, J. Magnen, V. Rivasseau and R. Sénéor, “Construction and Borel Summability of Infrared Phi**4 in Four Dimensions by a Phase Space Expansion,” Commun. Math. Phys. 109 (1987) 437.
* [25] H. Grosse and R. Wulkenhaar, “Renormalisation of phi**4 theory on noncommutative R**4 in the matrix base,” Commun. Math. Phys. 256, 305 (2005) [arXiv:hep-th/0401128].
* [26] V. Rivasseau, F. Vignes-Tourneret and R. Wulkenhaar, “Renormalization of noncommutative phi**4-theory by multi-scale analysis,” Commun. Math. Phys. 262, 565 (2006) [arXiv:hep-th/0501036].
* [27] M. Disertori, R. Gurau, J. Magnen and V. Rivasseau, “Vanishing of beta function of non commutative phi(4)**4 theory to all orders,” Phys. Lett. B 649 (2007) 95 [arXiv:hep-th/0612251].
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|
arxiv-papers
| 2010-06-23T19:09:20 |
2024-09-04T02:49:11.138140
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vincent Rivasseau, Zhituo Wang",
"submitter": "Zhituo Wang",
"url": "https://arxiv.org/abs/1006.4617"
}
|
1006.4643
|
11institutetext: Space Science Division, NASA-Ames, Moffett Field, CA 94035,
USA; Fermi Gamma Ray Space Telescope
# Cross-Analyzing Radio and $\gamma$-Ray Time Series Data:
Fermi Marries Jansky
Jeffrey D. Scargle On behalf of the Fermi/LAT Collaboration. This work is
funded by the NASA Applied Information Sciences Research Program and a Fermi
Guest Investigator grant with Jay Norris, James Chiang and Roger Blandford as
co-Investigators.
###### Abstract
A key goal of radio and $\gamma-$ray observations of active galactic nuclei is
to characterize their time variability in order to elucidate physical
processes responsible for the radiation. I describe algorithms for relevant
time series analysis tools – correlation functions, Fourier and wavelet
amplitude and phase spectra, structure functions, and time-frequency
distributions, all for arbitrary data modes and sampling schemes. For example
radio measurements can be cross-analyzed with data streams consisting of time-
tagged gamma-ray photons. Underlying these methods is the Bayesian block
scheme, useful in its own right to characterize local structure in the light
curves, and also prepare raw data for input to the other analysis algorithms.
One goal of this presentation is to stimulate discussion of these methods
during the workshop.
## 1 Introduction
Active galactic nuclei (AGN) are highly variable at all wavelengths. A major
fraction of their total luminosity fluctuates over time scales ranging from
the shortest for which statistically significant signals can be obtained, to
the longest time intervals over which data are available. Characterizing this
variability has yielded growing insight into the physical processes powering
the large AGN luminosities – a trend that will accelerate as observations,
data analysis, and theory proliferate.
This paper outlines time series methods for analysis of the disparate data
modes of radio, $\gamma-$ray, and other astronomical observations. The next
section introduces a data structure that generalizes data modes traditionally
used for time-sequential observations. This abstraction yields methods for
estimating, from arbitrary time series data, including heterogeneous mixtures
of data modes, all of the standard analysis functions:
$\bullet$
light curves
$\bullet$
autocorrelations
$\bullet$
Fourier power and phase spectra
$\bullet$
wavelet representations
$\bullet$
structure functions
$\bullet$
time-frequency distributions
As indicated in Table 1, for essentially arbitrary data modes these methods
yield amplitude and phase information for single or multiple time series
(auto- and cross- analysis, respectively) – if desired, conditional on
auxiliary variables.
Table 1: Time Series Analysis for All Data Modes | Auto | Cross | Amp | Phase | Condit.
---|---|---|---|---|---
Correlation | yes | yes | - | - | yes
Fourier | yes | yes | yes | yes | yes
Wavelet | yes | yes | yes | location | ?
Struct Fcn | yes | yes | - | - | yes
Time-Freq | yes | yes | yes | yes | yes
There are significant difficulties in the astrophysical interpretation of
these quantities. The methods described here are of use in some of these, such
as separation of observational errors from stochastic source variability (both
of which, unfortunately, are often called _noise_). But I do not discuss other
more difficult problems, which are probably beyond the scope of time series
analysis methods, such as assessing the importance of _cosmic variance_ ,
identifying causal or otherwise physically connected relationships in multi-
wavelength time series data, _etc_.
Subsequent sections discuss each of the above-listed functions and give sample
applications.
## 2 Abstract Data Cells
The time series algorithms to be described below can be applied to almost any
type of time-sequential astronomical data. This generality is facilitated by
identifying those features of the data modes that are necessary for analysis
algorithms.
Each individual act of measurement may yield a large set of data values
relevant to estimation of the signal amplitude, and its uncertainty, as a
function of time. Of these, two pieces of information, related to the
independent variable (time111In practice we always use a discrete time
representation, such as a micro-second scale computer clock tick, or the
finite time interval of signal averaging. of the measurement) and the
dependent variable (amplitude of the signal at that time), are necessary for
any time series algorithm. In radio astronomy the typical example is the
measurement of the flux of a source averaged over a short interval of time. In
$\gamma$-ray astronomy the typical example is the detection of individual
photons. The arrival time of the photon is obviously the timing quantity, but
what about the signal? One scheme is to represent an individual photon with a
delta-function in time. But more information can be extracted by incorporating
the time intervals222A method for analyzing event data based solely on inter-
event time intervals has been developed in ([Prahl 1996]). between photons.
Specifically, for each photon consider the interval starting half way back to
the previous photon and ending half way forward to the subsequent photon. This
interval, namely
$[{t_{n}-t_{n-1}\over 2},{t_{n+1}-t_{n}\over 2}]\ ,$ (1)
is the set of times closer to $t_{n}$ than to any other time,333These
intervals form the _V_ oronoi tessellation of the total observation interval.
See ([Okabe, Boots, Sugihara and Chiu 2000]) for a full discussion of this
construct, highly useful in spatial domains of 2, 3, or higher dimension; see
also ([Scargle 2001a, Scargle 2001c]). and has length equal to the average of
the two intervals connected by photon $n$, namely
$\Delta t_{n}={t_{n+1}-t_{n-1}\over 2}\ .$ (2)
Then the reciprocal
$x_{n}\equiv{1\over\Delta t_{n}}$ (3)
is taken as an estimate of the signal amplitude corresponding to observation
$n$. When the photon rate is large, the corresponding intervals are small.
Figure 1
Figure 1: Voronoi cell of a photon. Three successive photon detection times
are circles on the time axis. The vertical dotted lines underneath delineate
the time cell, and the light rectangle is the local estimate of the signal
amplitude. If the exposure at this time is less than unity, the length of the
data cell shrinks in proportion ($dt\rightarrow dt^{\prime}$), yielding a
larger estimate of the true event rate (darker rectangle). The height of the
rectangle is ${n/dt^{\prime}}$, where $n$ is the number of photons at exactly
the same time (almost always 1), or by the photon energy for a flux estimate.
demonstrates the data cell concept, including the simple modifications to
account for variable exposure and for weighting by photon energy.
Consider gaps in the data. By this we mean that there are portions of the
total observation interval during which the detection system is completely off
(exposure zero). This situation is readily handled by defining the start of
the data cell for the first photon detected after the gap at the end time of
the gap. Correspondingly the data cell for the last photon before the gap is
set at the start time of the gap. The statistical nature of independent events
assures that this procedure rigorously estimates the true photon rate at the
edge of the gap. Of course, no information is available about the signal
during such a gap, and the various algorithms deal with gaps accordingly.
Now we consider data modes generally. Three common examples are: (a)
measurements of a quasi-continuous physical variable (eg. radio astronomy flux
measurements) (b) the time of occurrence of discrete events (e.g. photons) and
(c) counts of events in bins. The signal of interest is the time dependence of
the measured quantity in case (a), or of the event rate in case (b). Case (c)
is actually very similar to (b), but is often described as density estimation
or determination of the event distribution function. In all cases it is useful
to introduce the concept of _cells_ to represent the measurements. Letting
${\bf x_{n}}$ be the estimate of the signal amplitude for a cell at time
$t_{n}$, a data set of $N$ sequential observations is denoted
$C_{n}\equiv\\{{\bf x}_{n},t_{n}\\},\ \ \ n=1,2,\dots,N.$ (4)
The specific meaning of the quantities ${\bf x}_{n}$ depends on the type of
data. For example, in the three cases mentioned above the array ${\bf x}$
contains (a) the sum or average over the measurement interval of an extensive
or intensive quantity, respectively, plus one or more quantifiers of
measurement uncertainty, (b) coordinates of events, such as photon arrival
times, and (c) sizes and locations of the bins, and the count of events in
them.
A major reason for constructing this abstract data representation is that it
unifies all data modes into a common format that makes construction of
universal algorithms easy. As we will see in the next sections, even mixtures
of data types – either in the sense of cross-analyzing two very different data
types, or mixing data within a single time series – can be handled.
## 3 Light Curve Analysis: Bayesian Blocks
The simplest and most direct way to study variability is to construct a
representation of the intensity of the source as a function of time. More can
be done than just plotting the intensity measurements as a function of the
time of the measurement. Smoothing, interpolation, gap filling, etc. are all
techniques meant to enhance one’s understanding of the variability. Here we
discuss a different procedure, namely construction of a simple, generic, non-
parametric model of the data that as much as possible shows the actual
variability of the source, and minimizes the effect of observation errors. The
model adopted is the simplest possible non-parametric representation of time
series data, namely a piece-wise constant model. Details of this approach are
given in ([Scargle, Norris, Jackson and Chiang 2010]); the improved algorithm
given there replaces the approximate one described in ([Scargle 1998]).
The Bayesian Blocks algorithm finds the best partition of the observation
interval into blocks, such that the source intensity is modeled as varying
from block to block, but constant within each block. This is just a step-
function representation of the data. The meaning of the “best” model is the
one that maximizes a measure of goodness-of-fit function described in detail
in ([Scargle, Norris, Jackson and Chiang 2010]). Another change since the
earlier reference is the use of a very simple maximum likelihood fitness
function, preferable to the Bayesian posterior previously used because it is
invariant to a scale change in the time variable, thus eliminating a parameter
from the analysis.
Figure 2 shows the Bayesian
Figure 2: Bayesian Block representations of the lightcurves of 3C273 and
3C279, two AGN in the OVRO/Fermi project. The co-aligned times are in days,
relative to an arbitrary zero point; amplitudes are on a common relative
scale. Binned LAT data is shown for comparison, but the BB representation is
based on the photon data only.
blocks analysis of two AGN in the OVRO/Fermi joint program. The data shown are
from somewhat earlier in the program, where the overlap between the to
instruments was not huge. Also these were just the first and third objects in
the long list of observed sources, and were not particularly selected for
being highly variable cases.
## 4 Correlation Functions
Figure 3: Summation schemes for autocorrelation functions. The points
represent data cells, derived from measured values (as in radio astronomy) or
time-tagged events (as in Fermi photon data). Top: Summation over data with
arbitrary spacing in the Edelson and Krolik algorithm. From each point average
over all points within a bin $d\tau$ distant by $\tau$; $\tau$ is binned, but
$t$ is not. Bottom: Standard lag summation over evenly spaced data. From each
point (except near the ends) there is another point distant by exactly $\tau=$
an integer multiple of $\Delta t$.
A rather underutilized technique for studying correlated variability of two
observables (such as time series for different wavelengths) is to construct a
scatter plot of one against the other. If done carefully, this approach allows
study of joint probability distributions for the two variables; these contain
more statistical information than correlation functions or any of the other
functions discussed here. The challenges of this approach include the
difficulty of depicting the all-important time-sequence connecting the points
in the scatter plot, and the need to consider plotting lagged versions of the
variables, for a number of values of the lag. The understanding that comes
from careful study of scatter plots most often makes it worthwhile to conquer
these difficulties.
But probably the most used tool for studying statistical variability
properties of a single time series is the auto-correlation function (ACF) or,
for studying relations between the variability in two or more sets of
simultaneous time series, the cross-correlation function (CCF). The meaning of
the latter can be understood by modeling one time series as a lagged version
of the other, and evaluating the posterior distribution of the lag $\tau$,
yielding
$P(\tau)\sim e^{R_{X,Y}(\tau)\over K}\ ,$ (5)
where K is a constant and $R_{X,Y}(\tau)$ is the cross-correlation function
defined below ([Scargle 2001b]).
Concentrating on the CCF, of which the ACF is really a special case, and
following the notation and definitions of ([Papoulis 1965, Papoulis 1977]), we
have this definition of the _cross-correlation function_ of two real processes
${\bf x}(t)$ and ${\bf y}(t)$
$R_{xy}(t_{1},t_{2})=<\\{\ {\bf x}(t_{1}){\bf y}(t_{2})\ \\}>$ (6)
Assuming the processes are stationary, the time dependence is on only the
difference $\tau\equiv t_{2}-t_{2}$ and we have
$R_{xy}(\tau)=<\\{\ {\bf x}(t){\bf y}(t+\tau)\ \\}>$ (7)
The symbol $<>$ means the _expected value_ , informally to be thought of as an
average over realizations of the underlying random process $X$. In data
analysis this theoretical quantity is typically not known, and must be
therefore be estimated from the data at hand, _e.g._
$E[X(t)Y(t+\tau)]\equiv{1\over N(\tau)}\sum_{n}x_{n}y_{n+\tau}\ $ (8)
where $x_{n}$ and $y_{n}$ are the samples of the variable $X,Y$444Caution: It
is common to center the processes about their means, to yield the _cross-
covariance_ and _auto-covaraince_ functions. Such mean-removal can have
unfortunate consequences, such as distortion of the low-frequency power
spectrum. In addition, the nomenclature is not completely standard. Various
terms are used for the cases where the means of the processes have been
subtracted off, and/or the resulting function normalized to unity at
$\tau=0$., and $N(\tau)$ is the number of terms for which the sum can be
taken.
Figure 3 is a cartoon of the lag relationships for correlation functions of
evenly spaced data (bottom), as well as a solution to the difficulty posed by
unevenly spaced time samples in general, and event data in particular. For a
given sample or event at $t_{n}$ there will in general not be a corresponding
one at $t_{n}+\tau$, no matter what restriction is placed on $\tau$.
For this problem an ingenious if straightforward algorithm ([Edelson and
Krolik 1988]) is in wide use. The basic idea is to pre-define a set of bins in
the variable $\tau$ in order to construct a histogram of the corresponding
time separations $\tau=t_{m}-t_{n}$, weighted by the corresponding
$x_{n}y_{m}$ product. To be more specific, and modifying slightly Edelson and
Krolik’s formulas for our case (including not subtracting the process means),
define for all measured pairs $(x_{n},y_{m})$ the quantity
$UDFC_{nm}={x_{n}y_{m}\over\sqrt{(\sigma_{x}^{2}-e_{x}^{2})(\sigma_{y}^{2}-e_{y}^{2})}}\
,$ (9)
(for Unbinned Discrete Correlation Function) where $\sigma_{x}$ is the
standard deviation of the $X$-observations, $e_{x}$ is the $X$-measurement
error, and similarly for $Y$. The estimate of the correlation function is then
$R_{xy}(\tau)={1\over N_{\tau}}\sum UDCF_{nm}$ (10)
where the sum is over the pairs, $N_{\tau}$ in number, for which $t_{m}-t_{n}$
lies in the corresponding $\tau$-bin.
There has been some confusion over the rationale for the denominator in eq.
(9) (“ … to preserve the proper normalization”) and how to estimate it. The
quantity $(\sigma_{x}^{2}-e_{x}^{2})$ is in principle the difference between
the total observed variance and that ascribed to observational errors. How
they are estimated from source and calibration data, and other instrumental
considerations, no doubt varies from case to case. Edelson and Krolik discuss
potential corruption by correlated observational errors. I recommend following
their advice to exclude the terms $n=m$ from eq. (10) only for
autocorrelations, and then only if it is really necessary. These terms yield a
spike in the autocorrelation function at $\tau=0$, which can be a convenient
visual assessment of the importance of the observational variance; it can be
easily removed if needed. For CCFs it makes no sense to remove these terms,
absent observational errors correlated between the two observables.
Figure 4: Autocorrelation functions for the same two AGN as in Figure 2 for
radio and $\gamma$-ray data. Solid line with dark error band: OVRO 15 GhZ;
Dotted line with light error band: Fermi LAT.
Auto- and cross- correlation involving photon event data is a simple matter of
inserting the quantity in eq. (3) into eq. (9). Since essentially any time
series data mode yields at least surrogates for $t_{n}$ and $x_{n}$, the same
is true in general. Figure 4 shows autocorrelation functions computed in this
way, for the same AGNs shown in Figure 2 and Figure 5 shows the corresponding
cross-correlation functions.
Figure 5: Cross-correlation functions for the same two AGN as in Figure 2, for
radio and $\gamma$-ray data.
## 5 Fourier Power and Phase Spectra
Perhaps the most used time series analysis technique in astronomy is
estimation of the Fourier power spectrum, mainly with the goal of detecting
and then characterizing periodic signals hidden in noisy data, but also for
analyzing non-periodic signals such as quasi periodic oscillations and
colored, or “${1\over f}$,” noise. There are methods for direct estimation of
Fourier power ([Scargle 1982]) and phase ([Scargle 1989]) spectra from time
series data. However, it is often more convenient to make use of the well-
known result that the power spectrum is the Fourier transform of the ACF
computed as described above in §4. The sliding window power spectra depicted
in §8 were computed in this way.
## 6 Wavelet Representations
It is relatively straightforward to compute the wavelet transform for any time
series that can be put into the standard data cell representation. The wavelet
shape (in this case the piecewise constant Haar wavelet) is integrated against
the empirical signal amplitude assigned by the data cells. Figure 6 shows the
scalegrams, or wavlet power spectra ([Scargle _et al._ 1993]), for the same
AGN data as in Figure 4. There is not enough data to yield much detail in
these spectral representations, but the rough power law characteristic of
${1\over f}$ processes can be seen, as well as the noise floor for the LAT
data.
Figure 6: Wavelet Power (scalegrams) for OVRO and LAT data on 3c273 and 3c279,
with the Haar Wavelet. $log_{10}$ of the power plotted against $log_{2}$ of
time scale in days.
## 7 Structure Functions
Another concept in wide usage is the structure function. For the most part its
auto- and cross- versions are a repackaging of the same information contained
in the corresponding correlations. This point has recently been emphasized by
([Emmanoulopoulos, McHardy and Uttley 2010]). In addition to summarizing some
of the caveats and problems associated with structure functions, these authors
give a formal proof of the exact relation between structure functions and the
corresponding auto- and cross-correlation functions. In addition, the
literature contains a number of claims for the superiority of the structure
function that seem unwarranted, especially in view of the relation just
mentioned. An example is the misconception that structure functions are
somehow immune from sampling effects, including aliasing. Finally, some
analysts believe that at short timescales the structure function always
becomes flat; the actual generic behavior can be derived from eq. (A10) of
([Emmanoulopoulos, McHardy and Uttley 2010]); the normalized structure
function satisfies
$NSF(\tau)=2[1-ACF(\tau)]\rightarrow C\tau^{2}$ (11)
for $\tau\rightarrow 0$, since autocorrelation functions are even in $\tau$.
In practice this dependence may _seem_ flat compared to steeper behavior at
intermediate time scales, transitioning to the typical asymptotic loss of
correlation at large time scale expressed as $NSF(\tau)\rightarrow 2$,
correctly assessed as flat.
A few other points perhaps favor the use of structure functions (beyond the
fact that they have been widely used in the past, and therefore arguably
should be computed if only for comparison with previous work). When the
structure and correlation functions are estimated from actual data, this
equivalence result quoted above does not hold exactly. There can in fact be
significant departures from the theoretical relations in Appendix A of
([Emmanoulopoulos, McHardy and Uttley 2010]), due to end effects always
present for finite data streams. In addition, when measuring slope of powerlaw
relationships it can be slightly more convenient to fit polynomials to the
typical shape of a structure function than to the corresponding correlation
function or power spectrum.
## 8 Time-Frequency Distributions
The term _time-frequency distribution_ refers to techniques for studying the
time-evolution of the power spectrum of time series. This concept must deal
with the fact that the spectrum is a property of the entire time interval, so
that estimating it locally in time results in the need for trading off time
resolution against frequency resolution. See ([Flandrin 1999]) for a complete
exposition of these issues.
There are many algorithms for computing time-frequency distributions, but
little has been done for the case of event data, one exception being the
approach described in ([Galleani, Cohen, Nelson, and Scargle (2001)]).
Although there are advanced techniques based on the Wigner-Ville distribution,
Cohen’s class of distribution, and others, in many applications the sliding
window power spectrum is of considerable use. The idea is simple: compute the
power spectrum of a subsample of the data within a restricted time-interval,
small compared to the total interval. Information on the time dependence
results from the fact that the window is slid along the observation interval.
Information on frequency dependence is contained in the power spectrum. The
tradeoff of time- and frequency resolution is mediated by the length of the
time window: a short window yields high time resolution and low spectral
resolution, and _vice versa_ for a long window. Implementation of this
approach is straightforward through use of the techniques in §4 and 5.
Figure 7: Time-frequency distribution for 4 AGN data sets: 3c 120 x-ray data
from Chatterjee et al. (2009, ApJ, 704, 1689) provided by Alan Marscher;
optical, R magnitude data on OJ 278, by Villforth C., Nilsson K., Heidt J., et
al., 2010, MNRAS, 402, 2087, provided by Ivan Agudo, 37 GhZ observations of 3c
454 and 3c 279 from the Metsahovi Radio Observatory, provided by Anne
Lahteenmaki.
Figure 7 shows sliding window power spectra computed, in this way, from time
series data on four AGN provided by other authors at this workshop. These
time-frequency distributions can show spectral details that are washed out in
a power spectrum of the whole interval. In these cases there is little
evidence for periodicities of any kind. Note that these are preliminary
results, with no attempt being made to adjust the size of the window.
## 9 Conclusions
Rather than regurgitating the discussion above, I end with a few practical
suggestions. They may seem obvious or trivial, but I have found them
surprisingly useful in practice.
When addressing time series data in the form of eq. (4), the first step should
be to study the time intervals $t_{n+1}-t_{n}$; in particular compute, plot,
and study the their distribution with suitably constructed histograms. (Even
if the provider of the data swears the times are evenly spaced, check it!)
This often reveals many defects in the data, such as duplicate entries and
observations out of order. The outliers of the distribution signal
peculiarities, perhaps expected (such as known sampling irregularities,
regularities, or semi-regularities) but often unexpected surprises. Figure 8
shows examples from the data for which time-frequency distributions were shown
above.
Figure 8: Sampling histograms: the distributions of the time intervals between
the samples for the data in Figure 7.
The reader is invited to see what conclusions can be deduced from these
distributions.
Don’t subtract the mean value! Or at least do so with attention to its
effects. Too often time series data are detrended without careful
consideration of the resulting effects on the estimated functions. Mean
removal is a special case of detrending.
While there are some cases where the distinction between stationary and non-
stationary processes is important, with limited data it is difficult or
impossible to make this distinction in practice. For different reasons, the
distinction between linearity and non-linearity is best left to the realm of
physical models rather than data analysis. Linearity is a property of physical
processes, and mathematical definitions ([Priestly 1988, Tong 1990]) may or
may not connect meaningfully to physical concepts.
Finally, in thinking about AGN variability in general it is useful to think in
terms of the mathematical concept of doubly stochastic (or Cox) processes.
Essentially, this is a picture in which there are two distinct random
processes: the intrinsic variability of the source (truly random, periodic,
quasi-periodic, _etc_.) and the observation process. The latter is random due
to observational errors from photon counting, detector noise, background
variability, _etc._ It is a major data analysis challenge to cleanly separate
out the observational process to reveal the true variability of the
astronomical source.
###### Acknowledgements.
For various contributions I am indebted to Brad Jackson, many members of the
Fermi Gamma Ray Space Telescope Collaboration, especially Jay Norris, Jim
Chiang, and Roger Blandford, and Tony Readhead, Joey Richards, Walter Max-
Moerbeck and others in the Caltech Owens Valley Radio Observatory group, and
to Alan Marscher, Ivan Agudo, Anne Lahteenmaki, and Sascha Trippe for kindly
providing data sets.
## References
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{
"license": "Public Domain",
"authors": "Jeffrey D. Scargle",
"submitter": "Jeffrey D. Scargle",
"url": "https://arxiv.org/abs/1006.4643"
}
|
1006.4742
|
arxiv-papers
| 2010-06-24T10:43:00 |
2024-09-04T02:49:11.153203
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Minwook Kwon, Zhou Du, Jinwook Kim, Mingyu Yoon, Jinhwan Koh",
"submitter": "Zhou Du",
"url": "https://arxiv.org/abs/1006.4742"
}
|
|
1006.4787
|
# Curvature estimates for the level sets of spatial quasiconcave solutions to
a class of parabolic equations
Chuanqiang Chen Department of Mathematics
University of Science and Technology of China
Hefei 230026, Anhui Province, CHINA. cqchen@mail.ustc.edu.cn and Shujun Shi
Department of Mathematics
University of Science and Technology of China
Hefei 230026, Anhui Province, CHINA
and School of Mathematical Sciences
Harbin Normal University
Harbin 150025, Heilongjiang Province, CHINA. shjshi@mail.ustc.edu.cn
###### Abstract.
We prove a constant rank theorem for the second fundamental form of the
spatial convex level surfaces of solutions to equations
$u_{t}=F(\nabla^{2}u,\nabla u,u,t)$ under a structural condition, and give a
geometric lower bound of the principal curvature of the spatial level
surfaces.
2000 Mathematics Subject Classification: 45B99, 35K10.
Keywords and phrases: curvature estimates, level sets, constant rank theorem,
spatial quasiconcave solutions.
Research of the first author was supported by Grant 10871187 from the National
Natural Science Foundation of China. Research of the second author was
supported in part by the Science Research Program from the Education
Department of Heilongjiang Province (11551137).
## 1\. Introduction
In this paper, we consider the convexity and principal curvature estimates of
the spatial level surfaces of the spatial quasiconcave solutions to a class of
parabolic equations under some structural conditions. A continuous function
$u(x,t)$ on $\Omega\times[0,T]$ is called spatial quasiconcave if its level
sets $\\{x\in\Omega|u(x,t)\geq c\\}$ are convex for each constant $c$ and any
fixed $t\in[0,T]$.
The convexity of the level sets of the solutions to elliptic partial
differential equations has been studied extensively. For instance, Ahlfors [1]
contains the well-known result that level curves of Green function on simply
connected convex domain in the plane are the convex Jordan curves. In 1956,
Shiffman [20] studied the minimal annulus in $\mathbb{R}^{3}$ whose boundary
consists of two closed convex curves in parallel planes $P_{1},P_{2}$. He
proved that the intersection of the surface with any parallel plane $P$,
between $P_{1}$ and $P_{2}$, is a convex Jordan curve. In 1957, Gabriel [9]
proved that the level sets of the Green function on a 3-dimensional bounded
convex domain are strictly convex. In 1977, Lewis [14] extended Gabriel’s
result to $p$-harmonic functions in higher dimensions. Caffarelli-Spruck [7]
generalized the Lewis [14] results to a class of semilinear elliptic partial
differential equations. Motivated by the result of Caffarelli-Friedman [6],
Korevaar [13] gave a new proof on the results of Gabriel and Lewis by applying
the deformation process and the constant rank theorem of the second
fundamental form of the convex level sets of $p$-harmonic function. A survey
of this subject is given by Kawohl [12]. For more recent related extensions,
please see the papers by Bianchini-Longinetti-Salani [4], Bian-Guan [2], Xu
[23] and Bian-Guan-Ma-Xu [3].
There is also an extensive literature on the curvature estimates of the level
sets of the solutions to elliptic partial differential equations. For
2-dimensional harmonic function and minimal surface with convex level curves,
Ortel-Schneider [19], Longinetti [15] and [16] proved that the curvature of
the level curves attains its minimum on the boundary (see Talenti [21] for
related results). Longinetti also studied the precise relation between the
curvature of the convex level curves and the height of 2-dimensional minimal
surface in [16]. Ma-Ou-Zhang [17] got the Gaussian curvature estimates of the
convex level sets on higher dimensional harmonic function, and Wang-Zhang [22]
got the similar curvature estimates of some quasi-linear elliptic equations
under certain structure condition [4]. Both of their test functions involved
the Gaussian curvature of the boundary and the norm of the gradient on the
boundary. Furthermore, for the $p$-harmonic function with strictly convex
level sets, Ma-Zhang [18] obtained that the curvature function introduced in
it is concave with respect to the height of the $p$-harmornic function. For
the principal curvature estimates in higher dimension, in terms of the
principal curvature of the boundary and the norm of the gradient on the
boundary, Chang-Ma-Yang [8] obtained the lower bound estimates of principal
curvature for the strictly convex level sets of higher dimensional harmonic
functions and solutions to a class of semilinear elliptic equations under
certain structure condition [4]. Recently, in Guan-Xu [11], they got a lower
bound for the principal curvature of the level sets of solutions to a class of
fully nonlinear elliptic equations in convex rings under the general structure
condition [4] via the approach of constant rank theorem.
Naturally, we hope to give a characterization about the convexity and
curvature of the level surfaces of the solutions to the corresponding
parabolic equations. Borell [5] showed the same property in [9] and [14] for
the solution of the corresponding heat conduction problem with zero initial
data. In this paper, we will consider the following parabolic equations
(1.1) $\frac{{\partial u}}{{\partial t}}=F(\nabla^{2}u,\nabla
u,u,t),\quad~{}\text{in}~{}\Omega\times(0,T],$
where $\Omega$ is a domain in $\mathbb{R}^{n}$, and $\nabla^{2}u$, $\nabla u$
are the spatial Hessian and spatial gradient of $u(x,t)$ respectively. Let
$\mathcal{S}^{n}$ denote the space of real symmetric $n\times n$ matrices,
$\Lambda\subset\mathcal{S}^{n}$ an open set, $\mathbb{S}^{n-1}$ a unit sphere
and $F=F(r,p,u,t)$ a $C^{2,1}$ function in
$\Lambda\times\mathbb{R}^{n}\times\mathbb{R}\times[0,T]$. We will assume that
$F$ satisfies the following conditions: there are $\gamma_{0}>0$ and
$c_{0}\in\mathbb{R}$,
(1.2) $F^{\alpha\beta}:=\left(\frac{\partial F}{\partial
r_{\alpha\beta}}(r,p,u,t)\right)>0,\quad\forall\;(r,p,u,t)\in\Lambda\times\mathbb{R}^{n}\times(-\gamma_{0}+c_{0},\gamma_{0}+c_{0})\times[0,T],$
and for each $(\theta,u)\in\mathbb{S}^{n-1}\times\mathbb{R}$ fixed,
(1.3) $F(s^{2}A,s\theta,u,t)\text{ is locally concave in }(A,s)\text{ for each
fixed }t.$
Now we state our theorems.
###### Theorem 1.1.
Suppose $u\in C^{3,1}(\Omega\times[0,T])$ is a spatial quasiconcave solution
to parabolic equation (1.1) such that $(\nabla^{2}u(x,t),\nabla
u(x,t),u(x,t))\in\Lambda\times\mathbb{R}^{n}\times(-\gamma_{0}+c_{0},\gamma_{0}+c_{0})$
for each $(x,t)\in\Omega\times[0,T]$. Suppose that, $F$ satisfies conditions
(1.2) and (1.3), $\nabla u\neq 0$ and the spatial level sets
$\\{x\in\Omega|u(x,t)\geq c\\}$ of $u$ are connected and locally convex for
all $c\in(-\gamma_{0}+c_{0},\gamma_{0}+c_{0})$ for some $\gamma_{0}>0$. Then
the second fundamental form of spatial level surfaces
$\\{x\in\Omega|u(x,t)=c\\}$ has the same constant rank for all
$c\in(-\gamma_{0}+c_{0},\gamma_{0}+c_{0})$. Moreover, let $l(t)$ be the
minimal rank of the second fundamental form in $\Omega$, then $l(s)\leqslant
l(t)$ for all $s\leqslant t\leqslant T$.
Inspired by [11], we also consider to establish a geometric lower bound for
the principal curvature of the spatial level surfaces of solutions to
parabolic equation on the convex rings as follows,
(1.4) $\left\\{\begin{array}[]{lcl}\frac{{\partial u}}{{\partial
t}}=F(\nabla^{2}u,\nabla u,u,t)&\text{in}&{\Omega\times(0,T]},\\\
u(x,0)=u_{0}(x)&\text{in}&\Omega,\\\
u(x,t)=0&\text{on}&\partial\Omega_{0}\times(0,T],\\\
u(x,t)=1&\text{on}&\partial\Omega_{1}\times(0,T],\end{array}\right.$
where $\Omega=\Omega_{0}\backslash\overline{\Omega_{1}}$, $\Omega_{0}$,
$\Omega_{1}$ are two convex domains with
$\overline{\Omega_{1}}\subset\Omega_{0}$, $F(\nabla^{2}u_{0},\nabla
u_{0},u_{0},0)>0$ and $u_{0}$ is quasiconcave and satisfies
(1.5) $\left\\{\begin{array}[]{lcl}u_{0}=0&\text{on}&\partial\Omega_{0},\\\
u_{0}=1&\text{on}&\partial\Omega_{1}.\end{array}\right.$
We denote $\kappa_{s}(x,t)$ the smallest principal curvature of the spatial
level set $\Sigma^{u(x_{0},t)}=\\{x\in\Omega|u(x,t)=u(x_{0},t)\\}$ at $(x,t)$.
For each $(x_{0},t)$, set
(1.6)
$\kappa^{u(x_{0},t)}=\mathop{\inf}\limits_{x\in\Sigma^{u(x_{0},t)}}\kappa_{s}(x,t).$
We will assume that there exists $\lambda>0$, such that
(1.7) $(F^{\alpha\beta}(\nabla^{2}u,\nabla
u,u,t))\geq\lambda(\delta_{\alpha\beta}),\quad\forall(x,t)\in\overline{\Omega}\times[0,T].$
###### Theorem 1.2.
Suppose $u\in C^{3,1}(\Omega\times[0,T])$ is a spatial quasiconcave solution
to parabolic equation (1.4), and $F$ satisfies conditions (1.7) and (1.3),
$\nabla u\neq 0$, then
(1.8) $\kappa^{u(x,t)}\geq\min\\{\kappa^{0},\kappa^{1}e^{-A}\\}e^{Au(x,t)}$
for some universal constant $A$ depending only on $\left\|F\right\|_{C^{2}}$,
$n$, $\lambda$,
$\mathop{\min}\limits_{(x,t)\in\overline{\Omega}\times[0,T]}\left|{\nabla
u}\right|$, $\left\|u\right\|_{C^{3}}$. Moreover, if $"="$ holds for some
$u(x,t)\in(0,1)$, then the $"="$ holds for all $u(x,t)\in[0,1]$.
Theorem 1.1 and Theorem 1.2 may be looked as some parabolic versions for
Theorem 1.1 in [3] and Theorem 1.5 in [11] respectively. The main idea to
prove the main theorems in this paper can be found in the two literatures.
The rest of the paper is organized as follows. In section 2, we prove Theorem
1.1. In section 3, we prove Theorem 1.2.
Acknowledgement The authors would like to express sincere gratitude to Prof.
Xi-Nan Ma for his encouragement and many suggestions in this subject.
## 2\. Proof of Theorem 1.1
Suppose $u(x,t)\in C^{3,1}(\Omega\times[0.T])$, and $u_{n}\neq 0$ for any
fixed $(x,t)\in\Omega\times[0,T]$. It follows that the upward inner normal
direction of the spatial level sets $\\{x\in\Omega|u(x,t)=c\\}$ is
(2.1) $\displaystyle\vec{n}=\frac{|u_{n}|}{|\nabla
u|u_{n}}(u_{1},u_{2},...,u_{n-1},u_{n}),$
where $\nabla u=(u_{1},u_{2},...,u_{n-1},u_{n})$ is the spatial gradient of
$u$.
The second fundamental form $II$ of the spatial level surface of function $u$
with respect to the upward normal direction (2.1) is
(2.2)
$b_{ij}=-\frac{|u_{n}|(u_{n}^{2}u_{ij}+u_{nn}u_{i}u_{j}-u_{n}u_{j}u_{in}-u_{n}u_{i}u_{jn})}{|\nabla
u|u_{n}^{3}}.$
Set
(2.3)
$h_{ij}=u_{n}^{2}u_{ij}+u_{nn}u_{i}u_{j}-u_{n}u_{j}u_{in}-u_{n}u_{i}u_{jn},$
we may write
(2.4) $b_{ij}=-\frac{|u_{n}|h_{ij}}{|\nabla u|u_{n}^{3}}.$
Note that if $\Sigma^{c,t}=\\{x\in\Omega|u(x,t)=c\\}$ is locally convex, then
the second fundamental form of $\Sigma^{c,t}$ is semipositive definite with
respect to the upward normal direction (2.1). Let $a(x,t)=(a_{ij}(x,t))$ be
the symmetric Weingarten tensor of $\Sigma^{c,t}=\\{x\in\Omega|u(x,t)=c\\}$,
then $a$ is semipositive definite. As computed in [3], if $u_{n}\neq 0$, and
the Weingarten tensor is
(2.5) $a_{ij}=-\frac{|u_{n}|}{|\nabla
u|{u_{n}}^{3}}\left\\{h_{ij}-\frac{u_{i}u_{l}h_{jl}}{W(1+W)u_{n}^{2}}-\frac{u_{j}u_{l}h_{il}}{W(1+W)u_{n}^{2}}+\frac{u_{i}u_{j}u_{k}u_{l}h_{kl}}{W^{2}(1+W)^{2}u_{n}^{4}}\right\\}.$
With the above notations, at the point $(x,t)$ where $u_{n}(x,t)=|\nabla
u(x,t)|>0,\,u_{i}(x,t)=0$, $i=1,\cdots,n-1$, $a_{ij,k}$ is commutative, that
is, they satisfy the Codazzi property $a_{ij,k}=a_{ik,j},\;\forall i,j,k\leq
n-1$.
### 2.1. Calculations on the test function
Since Theorem 1.1 is of local feature, we may assume level surface
$\Sigma^{c,t}=\\{x\in\Omega|u(x,t)=c\\}$ is connected for each
$c\in(c_{0}-\gamma_{0},c_{0}+\gamma_{0})$. Suppose $a(x,t_{0})$ attains
minimal rank $l=l(t_{0})$ at some point $z_{0}\in\Omega$. We may assume
$l\leqslant n-2$, otherwise there is nothing to prove. And we assume $u\in
C^{3,1}(\Omega\times[0,T])$ and $u_{n}>0$ in the rest of this paper. So there
is a neighborhood $\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta]$ of
$(z_{0},t_{0})$, such that there are $l$ ”good” eigenvalues of $(a_{ij})$
which are bounded below by a positive constant, and the other $n-1-l$ ”bad”
eigenvalues of $(a_{ij})$ are very small. Denote $G$ be the index set of these
”good” eigenvalues and $B$ be the index set of ”bad” eigenvalues. And for any
fixed point $(x,t)\in\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta]$, we may
express $(a_{ij})$ in a form of (2.5), by choosing
$e_{1},\cdots,e_{n-1},e_{n}$ such that
(2.6) $|\nabla u(x,t)|=u_{n}(x,t)>0\ \mbox{and}(u_{ij}),i,j=1,..,n-1,\mbox{is
diagonal at}\ (x,t).$
Without loss of generality we assume $u_{11}\geq u_{22}\geq\cdots\geq
u_{n-1n-1}$. So, at $(x,t)\in\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta)$,
from (2.5), we have the matrix $(a_{ij}),i,j=1,..,n-1,$ is also diagonal, and
without loss of generality we may assume $a_{11}\geq a_{22}\geq...\geq
a_{n-1,n-1}$. There is a positive constant $C>0$ depending only on
$\|u\|_{C^{4}}$ and $\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta]$, such that
$a_{11}\geq a_{22}\geq...\geq a_{ll}>C$ for all
$(x,t)\in\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta)$. For convenience we
denote $G=\\{1,\cdots,l\\}$ and $B=\\{l+1,\cdots,n-1\\}$ be the ”good” and
”bad” sets of indices respectively. If there is no confusion, we also denote
(2.7) $G=\\{a_{11},...,a_{ll}\\}$ and $B=\\{a_{l+1,l+1},...,a_{n-1,n-1}\\}$.
Note that for any $\delta>0$, we may choose
$\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta]$ small enough such that
$a_{jj}<\delta$ for all $j\in B$ and
$(x,t)\in\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta]$.
For each $c$, let $a=(a_{ij})$ be the symmetric Weingarten tensor of
$\Sigma^{c,t}$. Set
(2.10) $\displaystyle p(a)=\sigma_{l+1}(a_{ij}),\quad q(a)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{llr}\frac{\sigma_{l+2}(a_{ij})}{\sigma_{l+1}(a_{ij})},&\mbox{if}\;\sigma_{l+1}(a_{ij})>0&\\\
0,&\mbox{otherwise}.&\end{array}\right.$
Theorem 1.1 is equivalent to say $p(a)\equiv 0$ (defined in (2.10) ) in
$\mathcal{O}\times(t_{0}-\delta,t_{0}]$. Since we are dealing with general
fully nonlinear equation (1.1), as in the case for the convexity of solutions
in [2], there are technical difficulties to deal with $p(a)$ alone. A key idea
in [2] is the introduction of function $q$ as in (2.10) and explore some
crucial concavity properties of $q$. We consider function
(2.11) $\phi(a)=p(a)+q(a),$
where $p$ and $q$ as in (2.10). We will use notion $h=O(f)$ if $|h(x,t)|\leq
Cf(x,t)$ for $(x,t)\in\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta]$ with
positive constant $C$ under control.
To get around $p=0$, for $\varepsilon>0$ sufficiently small, we instead
consider
(2.12) $\phi_{\varepsilon}(a)=\phi(a_{\varepsilon}),$
where $a_{\varepsilon}=a+\varepsilon I.$ We will also denote
$G_{\varepsilon}=\\{a_{ii}+\varepsilon,i\in G\\},$
$B_{\varepsilon}=\\{a_{ii}+\varepsilon,i\in B\\}.$
To simplify the notations, we will drop subindex $\varepsilon$ with the
understanding that all the estimates will be independent of $\varepsilon.$ In
this setting, if we pick $\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta]$ small
enough, there is $C>0$ independent of $\varepsilon$ such that
(2.13) $\phi(a(x,t))\geq C\varepsilon,\quad\sigma_{1}(B)\geq
C\varepsilon,~{}\quad~{}\mbox{ for all}\
(x,t)\in\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta].$
In what follows, we will use $i,j,\cdots$ as indices run from $1$ to $n-1$ and
use the Greek indices $\alpha,\beta,\cdots$ as indices run from $1$ to $n$.
Denote
$\displaystyle F^{\alpha\beta}=\frac{{\partial F}}{{\partial
u_{\alpha\beta}}},F^{p_{\alpha}}=\frac{{\partial F}}{{\partial
u_{\alpha}}},F^{u}=\frac{{\partial F}}{{\partial u}},F^{t}=\frac{{\partial
F}}{{\partial t}},$ $\displaystyle
F^{\alpha\beta,\gamma\eta}=\frac{{\partial^{2}F}}{{\partial
u_{\alpha\beta}\partial
u_{\gamma\eta}}},F^{\alpha\beta,p_{\gamma}}=\frac{{\partial^{2}F}}{{\partial
u_{\alpha\beta}\partial
u_{\gamma}}},F^{\alpha\beta,u}=\frac{{\partial^{2}F}}{{\partial
u_{\alpha\beta}\partial u}},$ $\displaystyle
F^{p_{\alpha}p_{\beta}}=\frac{{\partial^{2}F}}{{\partial u_{\alpha}\partial
u_{\beta}}},F^{p_{\alpha},u}=\frac{{\partial^{2}F}}{{\partial
u_{\alpha}\partial u}},F^{u,u}=\frac{{\partial^{2}F}}{{\partial u}^{2}}.$
We also denote
(2.14) $\mathcal{H}_{\phi}=\sum_{i,j\in B}|\nabla a_{ij}|+\phi.$
###### Lemma 2.1.
For any fixed $(x,t)\in\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta]$, with the
coordinate chosen as in (2.6) and (2.7),
(2.15) $\phi_{t}=-u_{n}^{-3}\sum_{j\in
B}\left[\sigma_{l}(G)+\frac{{\sigma}^{2}_{1}(B|j)-{\sigma}_{2}(B|j)}{{\sigma}^{2}_{1}(B)}\right][u_{n}^{2}u_{jjt}-2u_{n}u_{jn}u_{jt}]+O(\mathcal{H}_{\phi})$
and
$\displaystyle\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}\phi_{\alpha\beta}$
$\displaystyle=$ $\displaystyle u_{n}^{-3}\sum_{j\in
B}\left[\sigma_{l}(G)+\frac{{\sigma}^{2}_{1}(B|j)-{\sigma}_{2}(B|j)}{{\sigma}^{2}_{1}(B)}\right][-u_{n}^{2}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta
jj}+2u_{n}u_{nj}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta j}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad+4u_{n}u_{nj}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta
j}-6u_{nj}^{2}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta}]$
$\displaystyle+2u_{n}^{-3}\sum_{j\in B,i\in
G}\left[\sigma_{l}(G)+\frac{{\sigma}^{2}_{1}(B|j)-{\sigma}_{2}(B|j)}{{\sigma}^{2}_{1}(B)}\right]\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}\frac{1}{u_{ii}}[u_{n}u_{ij\alpha}-2u_{i\alpha}u_{jn}][u_{n}u_{ij\beta}-2u_{i\beta}u_{jn}]$
$\displaystyle-\frac{1}{{\sigma}^{3}_{1}(B)}\sum_{\alpha,\beta=1}^{n}\sum_{i\in
B}F^{\alpha\beta}[{\sigma}_{1}(B)a_{ii,\alpha}-a_{ii}\sum_{j\in
B}a_{jj,\alpha}][{\sigma}_{1}(B)a_{ii,\beta}-a_{ii}\sum_{j\in B}a_{jj,\beta}]$
$\displaystyle-\frac{1}{{\sigma}_{1}(B)}\sum_{\alpha,\beta=1}^{n}\sum_{i\neq
j\in B}F^{\alpha\beta}a_{ij,\alpha}a_{ij,\beta}+O(\mathcal{H}_{\phi}).$
Proof: For any fixed point
$(x,t)\in\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta]$, choose a coordinate
system as in (2.6) so that $|\nabla u|=u_{n}>0$ and the matrix $(a_{ij}(x,t))$
is diagonal for $1\leq i,j\leq n-1$ and nonnegative. From the definition of
$\phi$,
(2.16) $\displaystyle
a_{jj}=-\frac{h_{jj}}{u^{3}_{n}}=-\frac{u_{jj}}{u_{n}}=O(\mathcal{H}_{\phi}),\forall
j\in B,$
and
(2.17) $\displaystyle\phi_{t}$ $\displaystyle=$ $\displaystyle\sum_{j\in
B}\left[\sigma_{l}(G)+\frac{{\sigma}^{2}_{1}(B|j)-{\sigma}_{2}(B|j)}{{\sigma}^{2}_{1}(B)}\right]a_{jj,t}+O(\mathcal{H}_{\phi})$
$\displaystyle=$ $\displaystyle-u_{n}^{-3}\sum_{j\in
B}\left[\sigma_{l}(G)+\frac{{\sigma}^{2}_{1}(B|j)-{\sigma}_{2}(B|j)}{{\sigma}^{2}_{1}(B)}\right][u_{n}^{2}u_{jj,t}-2u_{n}u_{jn}u_{jt}]+O(\mathcal{H}_{\phi})$
Using relationship (2.16), we have
(2.18) $\displaystyle\phi_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\sum_{j\in
B}\left[\sigma_{l}(G)+\frac{{\sigma}^{2}_{1}(B|j)-{\sigma}_{2}(B|j)}{{\sigma}^{2}_{1}(B)}\right]\Big{[}a_{jj,\alpha\beta}-2\sum_{i\in
G}\frac{a_{ij,\alpha}a_{ij,\beta}}{a_{ii}}\Big{]}$
$\displaystyle-\frac{1}{{\sigma}^{3}_{1}(B)}\sum_{i\in
B}\Big{[}{\sigma}_{1}(B)a_{ii,\alpha}-a_{ii}\sum_{j\in
B}a_{jj,\alpha}\Big{]}\Big{[}{\sigma}_{1}(B)a_{ii,\beta}-a_{ii}\sum_{j\in
B}a_{jj,\beta}\Big{]}$ $\displaystyle-\frac{1}{{\sigma}_{1}(B)}\sum_{i\neq
j\in B}a_{ij,\alpha}a_{ij,\beta}+O(\mathcal{H}_{\phi}).$
So far, we have followed standard calculations as in [10, 3, 2]. Since
$u_{k}=0$ for $k=1,\cdots,n-1$, from (2.5),
(2.19) $\displaystyle
u_{n}u_{ij\alpha}=-u_{n}^{2}a_{ij,\alpha}+u_{nj}u_{i\alpha}+u_{ni}u_{j\alpha}+u_{n\alpha}u_{ij},\quad\forall\;i,j\leq
n-1,$
and for each $j\in B$,
(2.20) $\displaystyle a_{jj,\alpha\beta}$ $\displaystyle=$
$\displaystyle-\frac{1}{u_{n}^{3}}h_{jj,\alpha\beta}+O(\mathcal{H}_{\phi})$
$\displaystyle=$
$\displaystyle-\frac{1}{u_{n}^{3}}[u_{n}^{2}u_{jj\alpha\beta}+2u_{nn}u_{j\alpha}u_{j\beta}+2u_{n\alpha}u_{nj}u_{j\beta}+2u_{n\beta}u_{nj}u_{j\alpha}$
$\displaystyle\qquad\quad-2u_{n}u_{nj}u_{\alpha\beta
j}-2u_{n}u_{j\alpha}u_{nj\beta}-2u_{n}u_{j\beta}u_{nj\alpha}]+O(\mathcal{H}_{\phi}).$
Hence, for $j\in B$,
(2.21)
$\displaystyle\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}a_{jj,\alpha\beta}=$
$\displaystyle\sum_{\alpha,\beta=1}^{n}\frac{F^{\alpha\beta}}{u_{n}^{3}}[-u_{n}^{2}u_{\alpha\beta
jj}-4u_{n\alpha}u_{nj}u_{j\beta}+4u_{n}u_{j\alpha}u_{nj\beta}$
$\displaystyle\qquad\qquad\quad+2u_{n}u_{nj}u_{\alpha\beta
j}-2u_{nn}u_{j\alpha}u_{j\beta}]+O(\mathcal{H}_{\phi}).$
Using the fact that $\sum_{\alpha=1}^{n}F^{\alpha
n}u_{n\alpha}=(\sum_{\alpha,\beta=1}^{n}-\sum_{\beta=1}^{n-1}\sum_{\alpha=1}^{n})F^{\alpha\beta}u_{\alpha\beta}$,
$\forall j\in B$,
$\displaystyle\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{n\alpha}u_{j\beta}=u_{nj}(\sum_{\alpha,\beta=1}^{n}-\sum_{\beta=1}^{n-1}\sum_{\alpha=1}^{n})F^{\alpha\beta}u_{\alpha\beta}+O(\mathcal{H}_{\phi}),$
$\displaystyle\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{j\alpha}u_{nj\beta}=u_{nj}(\sum_{\alpha,\beta=1}^{n}-\sum_{\alpha=1}^{n-1}\sum_{\beta=1}^{n})F^{\alpha\beta}u_{\alpha\beta
j}+O(\mathcal{H}_{\phi}),$
and
$\displaystyle-2u_{nn}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{j\alpha}u_{j\beta}=-2u_{nn}F^{nn}u_{nj}^{2}+O(\mathcal{H}_{\phi})$
$\displaystyle=$
$\displaystyle-2u_{nj}^{2}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta}+4u_{nj}^{2}\sum_{\alpha=1}^{n-1}F^{\alpha
n}u_{n\alpha}+2u_{nj}^{2}\sum_{\alpha,\beta=1}^{n-1}F^{\alpha\beta}u_{\alpha\beta}+O(\mathcal{H}_{\phi}).$
Put above to (2.21),
(2.22) $\displaystyle\sum_{j\in
B}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{n}^{3}a_{jj,\alpha\beta}$
$\displaystyle=$ $\displaystyle-u_{n}^{2}\sum_{j\in
B}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta jj}+6u_{n}\sum_{j\in
B}u_{nj}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta j}$
$\displaystyle-6\sum_{j\in
B}u_{nj}^{2}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta}-4u_{n}\sum_{j\in
B}u_{nj}\sum_{\alpha=1}^{n-1}\sum_{\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta
j}$ $\displaystyle+8\sum_{j\in B}u_{nj}^{2}\sum_{\alpha=1}^{n-1}F^{\alpha
n}u_{n\alpha}+6\sum_{j\in
B}u_{nj}^{2}\sum_{\alpha,\beta=1}^{n-1}F^{\alpha\beta}u_{\alpha\beta}+O(\mathcal{H}_{\phi}).$
By (2.19), for $j\in B$,
(2.23) $\displaystyle
u_{n}\sum_{\alpha=1}^{n-1}\sum_{\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta
j}=u_{n}\sum_{\alpha=1}^{n}\bigg{(}\sum_{i\in B}F^{\alpha
i}u_{ij\alpha}+\sum_{i\in G}F^{\alpha i}u_{ij\alpha}\bigg{)}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{n}\sum_{i\in G}F^{\alpha
i}(-u_{n}^{2}a_{ij,\alpha}+u_{i\alpha}u_{jn}+u_{j\alpha}u_{in})$
$\displaystyle+\sum_{\alpha=1}^{n}\sum_{i\in B}F^{\alpha
i}(u_{i\alpha}u_{jn}+u_{j\alpha}u_{in})+O(\mathcal{H}_{\phi})$
$\displaystyle=$ $\displaystyle-u_{n}^{2}\sum_{\alpha=1}^{n}\sum_{i\in
G}F^{\alpha i}a_{ij,\alpha}+u_{nj}\sum_{i\in
G}F^{ii}u_{ii}+2u_{nj}(\sum_{i=1}^{n-1}F^{ni}u_{ni})+O(\mathcal{H}_{\phi}).$
(2.22) and (2.23) yield
(2.24)
$\displaystyle\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{n}^{3}a_{jj,\alpha\beta}$
$\displaystyle=$ $\displaystyle-
u_{n}^{2}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta
jj}+2u_{n}u_{nj}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta j}$
$\displaystyle+4u_{n}u_{nj}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta
j}-6u_{nj}^{2}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta}$
$\displaystyle+4u_{n}^{2}u_{nj}\sum_{\alpha=1}^{n}\sum_{i\in G}F^{\alpha
i}a_{ij,\alpha}+2u_{nj}^{2}\sum_{i\in G}F^{ii}u_{ii}+O(\mathcal{H}_{\phi}).$
So,
$\displaystyle\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}\phi_{\alpha\beta}$
$\displaystyle=$ $\displaystyle u_{n}^{-3}\sum_{j\in
B}\left[\sigma_{l}(G)+\frac{{\sigma}^{2}_{1}(B|j)-{\sigma}_{2}(B|j)}{{\sigma}^{2}_{1}(B)}\right][-u_{n}^{2}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta
jj}+2u_{n}u_{nj}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta j}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad+4u_{n}u_{nj}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta
j}-6u_{nj}^{2}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta}]$
$\displaystyle-2\sum_{j\in B,i\in
G}\left[\sigma_{l}(G)+\frac{{\sigma}^{2}_{1}(B|j)-{\sigma}_{2}(B|j)}{{\sigma}^{2}_{1}(B)}\right]\left[\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}\frac{a_{ij,\alpha}a_{ij,\beta}}{a_{ii}}-2\frac{u_{nj}}{u_{n}}\sum_{\alpha=1}^{n}F^{\alpha
i}a_{ij,\alpha}-\frac{u_{nj}^{2}}{u_{n}^{3}}F^{ii}u_{ii}\right]$
$\displaystyle-\frac{1}{{\sigma}^{3}_{1}(B)}\sum_{\alpha,\beta=1}^{n}\sum_{i\in
B}F^{\alpha\beta}[{\sigma}_{1}(B)a_{ii,\alpha}-a_{ii}\sum_{j\in
B}a_{jj,\alpha}][{\sigma}_{1}(B)a_{ii,\beta}-a_{ii}\sum_{j\in B}a_{jj,\beta}]$
$\displaystyle-\frac{1}{{\sigma}_{1}(B)}\sum_{\alpha,\beta=1}^{n}\sum_{i\neq
j\in B}F^{\alpha\beta}a_{ij,\alpha}a_{ij,\beta}+O(\mathcal{H}_{\phi}).$
In fact, for any $i\in G,j\in B$,
(2.25)
$\displaystyle\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}\frac{a_{ij,\alpha}a_{ij,\beta}}{a_{ii}}-2\frac{u_{nj}}{u_{n}}\sum_{\alpha=1}^{n}F^{\alpha
i}a_{ij,\alpha}-\frac{u_{nj}^{2}}{u_{n}^{3}}F^{ii}u_{ii}$ $\displaystyle=$
$\displaystyle-\frac{1}{u_{n}^{3}}[\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}\frac{h_{ij,\alpha}h_{ij,\beta}}{h_{ii}}-2\frac{u_{nj}}{u_{n}}\sum_{\alpha=1}^{n}F^{\alpha
i}h_{ij,\alpha}+u_{nj}^{2}F^{ii}u_{ii}]$ $\displaystyle=$
$\displaystyle-\frac{1}{u_{n}^{3}}\left\\{\sum_{\alpha,\beta=1}^{n-1}F^{\alpha\beta}\frac{1}{u_{n}^{2}u_{ii}}[u_{n}^{2}u_{ij\alpha}-u_{n}u_{i\alpha}u_{jn}][u_{n}^{2}u_{ij\beta}-u_{n}u_{i\beta}u_{jn}]\right.$
$\displaystyle\qquad\qquad+2\sum_{\alpha=1}^{n-1}F^{\alpha
n}\frac{1}{u_{n}^{2}u_{ii}}[u_{n}^{2}u_{ij,\alpha}-u_{n}u_{i\alpha}u_{jn}][u_{n}^{2}u_{ijn}-2u_{n}u_{in}u_{jn}]$
$\displaystyle\qquad\qquad+F^{nn}\frac{1}{u_{n}^{2}u_{ii}}[u_{n}^{2}u_{ijn}-2u_{n}u_{in}u_{jn}][u_{n}^{2}u_{ijn}-2u_{n}u_{in}u_{jn}]$
$\displaystyle\qquad\qquad-2\sum_{\alpha=1}^{n-1}F^{\alpha
i}\frac{1}{u_{n}^{2}u_{ii}}[u_{n}^{2}u_{ij\alpha}-2u_{n}u_{i\alpha}u_{jn}][u_{n}u_{ii}u_{nj}]$
$\displaystyle\qquad\qquad-2F^{ni}\frac{1}{u_{n}^{2}u_{ii}}[u_{n}^{2}u_{ijn}-2u_{n}u_{in}u_{jn}][u_{n}u_{ii}u_{nj}]$
$\displaystyle\qquad\qquad\left.+F^{ii}\frac{1}{u_{n}^{2}u_{ii}}(u_{n}u_{ii}u_{nj})^{2}\right\\}$
$\displaystyle=$
$\displaystyle-\frac{1}{u_{n}^{3}}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}\frac{1}{u_{n}^{2}u_{ii}}[u_{n}^{2}u_{ij\alpha}-2u_{n}u_{i\alpha}u_{jn}][u_{n}^{2}u_{ij\beta}-2u_{n}u_{i\beta}u_{jn}].$
Obviously, we can get
(2.26)
$\displaystyle\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}\frac{a_{ij,\alpha}a_{ij,\beta}}{a_{ii}}-2\frac{u_{nj}}{u_{n}}\sum_{\alpha=1}^{n}F^{\alpha
i}a_{ij,\alpha}-\frac{u_{nj}^{2}}{u_{n}^{3}}F^{ii}u_{ii}\geq 0,$
this is the Claim in [3].
From the above formulas, Lemma 2.1 holds. ∎
### 2.2. Proof of Theorem 1.1
We start this section with a discussion on structure condition (1.3). For any
function $F(r,p,u,t)$, denote $F^{\alpha\beta}=\frac{\partial F}{\partial
r_{\alpha\beta}},F^{p_{l}}=\frac{\partial F}{\partial u_{l}},\cdots$ as
partial derivatives of $F$ with respect to corresponding arguments.
###### Lemma 2.2.
If $F$ satisfies condition (1.3), then
(2.27) $\displaystyle Q(V,V)$ $\displaystyle=$ $\displaystyle
F^{\alpha\beta,\gamma\eta}X_{\alpha\beta}X_{\gamma\eta}+2F^{\alpha\beta,p_{l}}\theta_{l}X_{\alpha\beta}Y+F^{p_{k},p_{l}}\theta_{k}\theta_{l}Y^{2}$
$\displaystyle+4s^{-1}F^{\alpha\beta}X_{\alpha\beta}Y-6F^{\alpha\beta}A_{\alpha\beta}Y^{2}$
$\displaystyle\leqslant$ $\displaystyle 0,$
for every
$(X_{\alpha\beta},Y)=((s^{2}\widetilde{X}_{\alpha\beta}+2sA_{\alpha\beta}\widetilde{Y}),\widetilde{Y})$,
with any
$\widetilde{V}=((\widetilde{X}_{\alpha\beta}),\widetilde{Y})\in\mathcal{S}^{n}\times\mathbb{R}$,
where $F^{\alpha\beta,rs},F^{\alpha\beta,u_{l}},etc.$ are evaluated at
$(s^{2}A,s\theta,u,t)$, and the Einstein summation convention is used.
Proof: Denoting $\widetilde{F}(A,s)=F(s^{2}A,s\theta,u,t),$ condition (1.3)
implies that $\widetilde{F}(A,s)$ is locally concave, that is,
(2.28)
$\displaystyle\widetilde{F}^{\alpha\beta,\gamma\eta}\widetilde{X}_{\alpha\beta}\widetilde{X}_{\gamma\eta}+2\widetilde{F}^{\alpha\beta,s}\widetilde{X}_{\alpha\beta}\widetilde{Y}+\widetilde{F}^{s,s}\widetilde{Y}^{2}\leq
0,$
for any
$\widetilde{V}=((\widetilde{X}_{\alpha\beta}),\widetilde{Y})\in\mathcal{S}^{n}\times\mathbb{R}$.
At $(A,s)$,
$\displaystyle\widetilde{F}^{\alpha\beta,\gamma\eta}=F^{\alpha\beta,\gamma\eta}s^{2}\cdot
s^{2},$
$\displaystyle\widetilde{F}^{\alpha\beta,s}=F^{\alpha\beta,\gamma\eta}s^{2}\cdot
2sA_{\gamma\eta}+F^{\alpha\beta,p_{l}}s^{2}\cdot\theta_{l}+F^{\alpha\beta}2s,$
$\displaystyle\widetilde{F}^{s,s}=F^{\alpha\beta,\gamma\eta}2sA_{\alpha\beta}\cdot
2sA_{\gamma\eta}+2F^{\alpha\beta,p_{l}}2sA_{\alpha\beta}\cdot\theta_{l}+F^{p_{k},p_{l}}\theta_{k}\cdot\theta_{l}+F^{\alpha\beta}2A_{\alpha\beta}.$
Set
(2.29) $\displaystyle
X_{\alpha\beta}=s^{2}\widetilde{X}_{\alpha\beta}+2sA_{\alpha\beta}\widetilde{Y},$
(2.30) $\displaystyle Y=\widetilde{Y},$
so (2.28) is equivalent to
$\displaystyle
F^{\alpha\beta,\gamma\eta}X_{\alpha\beta}X_{\gamma\eta}+2F^{\alpha\beta,p_{l}}\theta_{l}X_{\alpha\beta}Y+F^{u_{k},p_{l}}\theta_{k}\theta_{l}Y^{2}$
$\displaystyle+4s^{-1}F^{\alpha\beta}s^{2}\widetilde{X}_{\alpha\beta}\widetilde{Y}+2F^{\alpha\beta}A_{\alpha\beta}\widetilde{Y}^{2}$
$\displaystyle=$ $\displaystyle
F^{\alpha\beta,\gamma\eta}X_{\alpha\beta}X_{\gamma\eta}+2F^{\alpha\beta,p_{l}}\theta_{l}X_{\alpha\beta}Y+F^{p_{k},p_{l}}\theta_{k}\theta_{l}Y^{2}$
$\displaystyle+4s^{-1}F^{\alpha\beta}X_{\alpha\beta}Y-6F^{\alpha\beta}A_{\alpha\beta}Y^{2}$
$\displaystyle\leqslant$ $\displaystyle 0.$
Therefore, (2.27) follows from above, and Lemma 2.2 holds. ∎
Theorem 1.1 is a direct consequence of the following proposition and the
strong maximum principle.
###### Proposition 2.3.
Suppose that the function $F,u$ satisfy assumptions in Theorem 1.1. If the
second fundamental form $b_{ij}$ of $\Sigma^{c,t_{0}}$ attains minimum rank
$l=l(t_{0})$ at certain point $x_{0}\in\Omega$, then there exist a
neighborhood $\mathcal{O}\times(t_{0}-\delta_{0},t_{0}+\delta_{0}]$ of
$(x_{0},t_{0})$ and a positive constant $C$ independent of $\phi$ (defined in
(2.11)), such that
(2.31)
$\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}\phi_{\alpha\beta}(x,t)-\phi_{t}\leq
C(\phi+|\nabla\phi|),~{}~{}\forall~{}(x,t)\in\mathcal{O}\times(t_{0}-\delta_{0},t_{0}+\delta_{0}].$
Proof: Let $u\in C^{3,1}(\Omega\times[0,T])$ be a spatial quasiconcave
solution of equation (1.1) and $(u_{ij})\in\mathcal{S}^{n}.$ Let $l=l(t_{0})$
be the minimum rank of the second fundamental forms $h_{ij}$ of
$\Sigma^{c,t_{0}}$ ($l\in\\{0,1,...,n-1\\}$) for every $c$ in
$(-\gamma_{0}+c_{0},\gamma_{0}+c_{0})$, suppose the minimum rank $l$ arrives
at point $x_{0}\in\Sigma^{c,t_{0}}$. We work on a small open neighborhood
$\mathcal{O}\times(t_{0}-\delta_{0},t_{0}+\delta_{0}]$ of $(x_{0},t_{0})$. We
may assume $l\leq n-2$. Lemma 2.1 implies $\phi\in
C^{1,1}(\mathcal{O}\times(t_{0}-\delta_{0},t_{0}+\delta_{0}]),$ $\phi(x,t)\geq
0,~{}\qquad~{}\phi(x_{0},t_{0})=0$. For $\epsilon>0$ sufficient small, let
$\phi_{\epsilon}$ defined as in (2.11) and (2.12), we need to verify (2.31)
for each point $(x,t)\in\mathcal{O}\times(t_{0}-\delta_{0},t_{0}+\delta_{0}]$.
For each fixed $(x,t)$, choose a local coordinate $e_{1},\cdots,e_{n-1},e_{n}$
such that (2.6) and (2.7) are satisfied. We want to establish differential
inequality (2.31) for $\phi_{\varepsilon}$ defined in (2.12) with constant $C$
independent of $\varepsilon$. Note that we will omit the subindex
$\varepsilon$ with the understanding that all the estimates are independent of
$\varepsilon$.
By Lemma 2.1,
(2.32)
$\displaystyle\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}\phi_{\alpha\beta}-\phi_{t}$
$\displaystyle\leq$ $\displaystyle-u_{n}^{-3}\sum_{j\in
B}\left[\sigma_{l}(G)+\frac{{\sigma}^{2}_{1}(B|j)-{\sigma}_{2}(B|j)}{{\sigma}^{2}_{1}(B)}\right]\Big{[}u_{n}^{2}(\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{jj\alpha\beta}-u_{jjt})$
$\displaystyle-2u_{n}u_{jn}(\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{j\alpha\beta}-u_{jt})-4u_{n}u_{jn}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{j\alpha\beta}+6u_{jn}^{2}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta}\Big{]}$
$\displaystyle-\frac{1}{{\sigma}^{3}_{1}(B)}\sum_{\alpha,\beta=1}^{n}\sum_{i\in
B}F^{\alpha\beta}[{\sigma}_{1}(B)a_{ii,\alpha}-a_{ii}\sum_{j\in
B}a_{jj,\alpha}][{\sigma}_{1}(B)a_{ii,\beta}-a_{ii}\sum_{j\in B}a_{jj,\beta}]$
$\displaystyle-\frac{1}{{\sigma}_{1}(B)}\sum_{\alpha,\beta=1}^{n}\sum_{i\neq
j,i,j\in B}F^{\alpha\beta}a_{ij,\alpha}a_{ij,\beta}+O(\mathcal{H}_{\phi}).$
For each $j\in B$, differentiating equation (1.1) in $e_{j}$ direction at $x$,
(2.33) $u_{jt}=\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta
j}+F^{u_{n}}u_{jn}+O(\mathcal{H}_{\phi}),$
and
(2.34) $\displaystyle u_{jjt}$ $\displaystyle=$
$\displaystyle\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta
jj}+\sum_{\alpha,\beta,r,s=1}^{n}F^{\alpha\beta,rs}u_{\alpha\beta
j}u_{rsj}+2\sum_{\alpha,\beta,l=1}^{n}F^{\alpha\beta,u_{l}}u_{\alpha\beta
j}u_{lj}$
$\displaystyle+2\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta,u}u_{j\alpha\beta}u_{j}+\sum_{l,s=1}^{n}F^{u_{l},u_{s}}u_{lj}u_{sj}-2\sum_{l=1}^{n}F^{u_{l},u}u_{lj}u_{j}$
$\displaystyle+F^{u,u}u_{j}^{2}+\sum_{l=1}^{n}F^{u_{l}}u_{ljj}+F^{u}u_{jj}.$
It follows from (2.19) that, at $(x,t)$
(2.35) $\displaystyle\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta
jj}-u_{jjt}$ $\displaystyle=$
$\displaystyle-\sum_{\alpha,\beta,r,s=1}^{n}F^{\alpha\beta,rs}u_{\alpha\beta
j}u_{rsj}-2\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta,u_{n}}u_{j\alpha\beta}u_{nj}$
$\displaystyle-F^{u_{n},u_{n}}u^{2}_{jn}-2\frac{F^{u_{n}}}{u_{n}}u^{2}_{jn}+O(\mathcal{H}_{\phi}).$
Since $u_{\alpha\beta jj}=u_{jj\alpha\beta}$, (2.33) and (2.35) yield
(2.36) $\displaystyle F^{\alpha\beta}\phi_{\alpha\beta}-\phi_{t}$
$\displaystyle\leq$ $\displaystyle\sum_{j\in
B}u_{n}^{-3}\left[\sigma_{l}(G)+\frac{{\sigma}^{2}_{1}(B|j)-{\sigma}_{2}(B|j)}{{\sigma}^{2}_{1}(B)}\right]\left\\{\Big{[}\sum_{\alpha,\beta,r,s=1}^{n}F^{\alpha\beta,rs}u_{\alpha\beta
j}u_{rsj}\right.$
$\displaystyle\qquad\qquad\qquad\qquad+2\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta,u_{n}}u_{j\alpha\beta}u_{jn}+F^{u_{n},u_{n}}u_{jn}^{2}\Big{]}u_{n}^{2}$
$\displaystyle\qquad\qquad\qquad\qquad+\left.4u_{jn}u_{n}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{j\alpha\beta}-6u_{jn}^{2}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta}\right\\}$
$\displaystyle-\frac{1}{{\sigma}^{3}_{1}(B)}\sum_{\alpha,\beta=1}^{n}\sum_{i\in
B}F^{\alpha\beta}[{\sigma}_{1}(B)a_{ii,\alpha}-a_{ii}\sum_{j\in
B}a_{jj,\alpha}][{\sigma}_{1}(B)a_{ii,\beta}-a_{ii}\sum_{j\in B}a_{jj,\beta}]$
$\displaystyle-\frac{1}{{\sigma}_{1}(B)}\sum_{\alpha,\beta=1}^{n}\sum_{i\neq
j,i,j\in B}F^{\alpha\beta}a_{ij,\alpha}a_{ij,\beta}+O(\mathcal{H}_{\phi}).$
For each $j\in B$, set
(2.37) $\displaystyle S_{j}=$
$\displaystyle\Big{[}\sum_{\alpha,\beta,r,s=1}^{n}F^{\alpha\beta,rs}u_{j\alpha\beta}u_{rsj}+2\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta,u_{n}}u_{j\alpha\beta}u_{jn}+F^{u_{n},u_{n}}u_{jn}^{2}\Big{]}u_{n}^{2}$
$\displaystyle+$ $\displaystyle
4\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{j\alpha\beta}u_{jn}u_{n}-6\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta}u_{jn}^{2}$
For each $j\in B$, set
(2.38) $\displaystyle X_{\alpha\beta}=u_{\alpha\beta
j}u_{n},\forall(\alpha,\beta),$ (2.39) $\displaystyle Y=u_{jn}u_{n}.$
In the coordinate system (2.6),
$(\nabla^{2}u(x),\nabla u(x),u(x),t)=(\nabla^{2}u,(0,...,0,|\nabla u|),u,t).$
Equalize it to $(s^{2}A,s\theta,u,t)$, the components of $\widetilde{V}$
defined in Lemma 2.2 are
$\displaystyle\widetilde{X}_{\alpha\beta}=\frac{u_{\alpha\beta
j}}{u_{n}}-\frac{2u_{\alpha\beta}u_{jn}}{u^{2}_{n}},\quad\forall(\alpha,\beta),$
$\displaystyle\widetilde{Y}=u_{jn}u_{n}.$
For $j\in B$, Lemma 2.2 implies
(2.40) $S_{j}\leq 0.$
Condition (1.2) implies
(2.41) $(F^{\alpha\beta})\geq\delta_{0}I,\;\quad\mbox{for some $\delta_{0}>0$,
and $\forall x\in\mathcal{O}$.}$
Set
$V_{i\alpha}={\sigma}_{1}(B)a_{ii,\alpha}-a_{ii}\sum_{j\in B}a_{jj,\alpha}.$
Combining (2.36), (2.40) and (2.41),
(2.42) $\displaystyle F^{\alpha\beta}\phi_{\alpha\beta}\leq
C(\phi+\sum_{i,j\in B}|\nabla a_{ij}|)-\delta_{0}[\frac{\sum_{i\neq j\in
B,\alpha=1}^{n}a^{2}_{ij\alpha}}{\sigma_{1}(B)}+\frac{\sum_{i\in
B,\alpha=1}^{n}V_{i\alpha}^{2}}{{\sigma}^{3}_{1}(B)}].$
By Lemma 3.3 in [2], for each $M\geq 1$, for any
$M\geq|\gamma_{i}|\geq\frac{1}{M}$, there is a constant $C$ depending only on
$n$ and $M$ such that, $\forall\alpha$,
(2.43) $\sum_{i,j\in B}|a_{ij\alpha}|\leq
C(1+\frac{1}{\delta_{0}^{2}})(\sigma_{1}(B)+|\sum_{i\in
B}\gamma_{i}a_{ii\alpha}|)+\frac{\delta_{0}}{2}[\frac{\sum_{i\neq j\in
B}|a_{ij\alpha}|^{2}}{\sigma_{1}(B)}+\frac{\sum_{i\in
B}V_{i\alpha}^{2}}{\sigma_{1}^{3}(B)}].$
Taking
$\gamma_{i}=\sigma_{l}(G)+\frac{{\sigma}^{2}_{1}(B|i)-{\sigma}_{2}(B|i)}{{\sigma}^{2}_{1}(B)}$
for each $i\in B$, the Newton-MacLaurine inequality implies
$\sigma_{l}(G)+1\geq\sigma_{l}(G)+\frac{{\sigma}^{2}_{1}(B|j)-{\sigma}_{2}(B|j)}{{\sigma}^{2}_{1}(B)}\geq\sigma_{l}(G),\quad\forall
j\in B.$
Therefore we conclude from Lemma 2.1 and (2.43) that $\sum_{i,j\in B}|\nabla
a_{ij}|$ can be controlled by the rest terms on the right hand side in (2.42)
and $\phi+|\nabla\phi|$. The proof is complete. ∎
## 3\. Proof of Theorem 1.2
In this section, through modifying the proof of Theorem 1.1, we will give a
proof of Theorem 1.2. Also it is a parabolic equation case corresponding to
[11].
Suppose that $u(x,t)$ is a spatial quasiconcave solution of (1.4), and assume
that level surface $\Sigma^{u(x_{0},t)}=\\{x\in\Omega|u(x,t)=u(x_{0},t)\\}$ is
connected for each $(x_{0},t)\in\mathcal{O}\times[0,T]$.
Set
(3.1) $\widetilde{a}=a-\eta_{0}gI,\quad\eta_{0}\geqslant 0,\quad
g(x,t)=e^{Au(x,t)},$
where $A>0$ is a constant to be determined. We want to show $\widetilde{a}$ is
of constant rank. Theorem 1.1 corresponds to the case $\eta_{0}=0$. If
$\min\\{\kappa^{0},\kappa^{1}\\}=0$, there is nothing to prove instead of
utilizing Theorem 1.1. We will assume $\min\\{\kappa^{0},\kappa^{1}\\}>0$ in
the rest of the paper. Denote $\kappa_{s}(x,t)$ and
$\widetilde{\kappa}_{s}(x,t)$ be the minimum eigenvalue of matrix $a(x)$ and
$\widetilde{a}(x)$ respectively. Since the spatial level sets are strictly
convex, and $\overline{\Omega}$ is compact, $a$ is strictly positive definite.
That is, $\kappa_{s}(x,t)$ has a positive lower bound.
For a positive constant $A$ to be determined, increasing $\eta_{0}$ from 0,
such that $\widetilde{a}$ is degenerate at some points, i.e. $\widetilde{a}$
is semi-positive with the rank is not full. (1.8) follows easily if this
happens only on the boundary. We want to show that, if the degeneracy happens
at an interior point of $\Omega$, then $\widetilde{a}$ is degenerate through
out $\Omega$ with the same rank. This implies that the ”=” holds in (1.8) and
Theorem 1.2 is proved.
Therefore, the main task is to prove constant rank theorem for
$\widetilde{a}$. Suppose $\widetilde{a}(x,t_{0})$ attains minimal rank
$l=l(t_{0})$ at some point $z_{0}\in\Omega$. We may assume $l\leqslant n-2$,
otherwise there is nothing to prove. And we assume $u\in C^{3,1}$ and
$u_{n}>0$ in the rest of this paper. So there is a neighborhood
$\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta]$ of $(z_{0},t_{0})$, such that
there are $l$ ”good” eigenvalues of $(\widetilde{a}_{ij})$ which are bounded
below by a positive constant, and the other $n-1-l$ ”bad” eigenvalues of
$(\widetilde{a}_{ij})$ are very small. Denote $G$ be the index set of these
”good” eigenvalues and $B$ be the index set of ”bad” eigenvalues. And for any
fixed point $(x,t)\in\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta]$, we may
express $(\widetilde{a}_{ij})$ in a form of (3.1) and (2.5), by choosing
$e_{1},\cdots,e_{n-1},e_{n}$ such that
(3.2) $|\nabla u(x,t)|=u_{n}(x,t)>0\ \mbox{and}\ (u_{ij}),i,j=1,..,n-1,\
\mbox{is diagonal at}\ (x,t).$
Without loss of generality, we assume $u_{11}\geq u_{22}\geq\cdots\geq
u_{n-1,n-1}$. So, at $(x,t)\in\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta)$,
from (2.5), we have the matrix $(a_{ij}),i,j=1,..,n-1$, is also diagonal. And
without loss of generality we may assume $a_{11}\geq a_{22}\geq...\geq
a_{n-1,n-1}$, then
$\widetilde{a}_{11}\geq\widetilde{a}_{22}\geq...\geq\widetilde{a}_{n-1,n-1}$.
There is a positive constant $C>0$ depending only on $\|u\|_{C^{4}}$ and
$\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta]$, such that
$\widetilde{a}_{11}\geq\widetilde{a}_{22}\geq...\geq\widetilde{a}_{ll}>C$ for
all $(x,t)\in\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta)$. For convenience we
denote $G=\\{1,\cdots,l\\}$ and $B=\\{l+1,\cdots,n-1\\}$ be the ”good” and
”bad” sets of indices respectively. If there is no confusion, we also denote
(3.3) $G=\\{\widetilde{a}_{11},...,\widetilde{a}_{ll}\\}$ and
$B=\\{\widetilde{a}_{l+1,l+1},...,\widetilde{a}_{n-1,n-1}\\}$.
Note that for any $\delta>0$, we may choose
$\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta]$ small enough such that
$\widetilde{a}_{jj}<\delta$ for all $j\in B$ and
$(x,t)\in\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta]$.
For each $(x,t)$, let $a=(a_{ij})$ be the symmetric Weingarten tensor of
$\Sigma^{u(x,t)}$. Set
(3.6) $\displaystyle p(\widetilde{a})=\sigma_{l+1}(\widetilde{a}_{ij}),\quad
q(\widetilde{a})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{llr}\frac{\sigma_{l+2}(\widetilde{a}_{ij})}{\sigma_{l+1}(\widetilde{a}_{ij})},&\mbox{if}\;\sigma_{l+1}(\widetilde{a}_{ij})>0,&\\\
0,&\mbox{otherwise}.&\end{array}\right.$
Theorem 1.2 is equivalent to say $p(\widetilde{a})\equiv 0$ (defined in (3.6)
) in $\mathcal{O}\times(t_{0}-\delta,t_{0}]$. As in the description of the
proof of Theorem 1.1, we should consider the function
(3.7) $\phi(\widetilde{a})=p(\widetilde{a})+q(\widetilde{a}),$
where $p$ and $q$ as in (3.6). We will use notion $h=O(f)$ if $|h(x,t)|\leq
Cf(x,t)$ for $(x,t)\in\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta]$ with
positive constant $C$ under control.
To get around $p=0$, for $\varepsilon>0$ sufficiently small, we instead
consider
(3.8) $\phi_{\varepsilon}(\widetilde{a})=\phi(\widetilde{a}_{\varepsilon}),$
where $a_{\varepsilon}=\widetilde{a}+\varepsilon I.$ We will also denote
$G_{\varepsilon}=\\{\widetilde{a}_{ii}+\varepsilon,i\in G\\},$
$B_{\varepsilon}=\\{\widetilde{a}_{ii}+\varepsilon,i\in B\\}.$
To simplify the notations, we will drop subindex $\varepsilon$ with the
understanding that all the estimates will be independent of $\varepsilon.$ In
this setting, if we pick $\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta]$ small
enough, there is $C>0$ independent of $\varepsilon$ such that
(3.9) $\phi(\widetilde{a}(x,t))\geq C\varepsilon,\quad\sigma_{1}(B)\geq
C\varepsilon,~{}\quad~{}\rm{for~{}all~{}}(x,t)\in\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta].$
We also denote
(3.10) $\mathcal{H}_{\phi}=\sum_{i,j\in B}|\nabla\widetilde{a}_{ij}|+\phi.$
###### Lemma 3.1.
For any fixed $(x,t)\in\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta]$, with the
coordinate chosen as in (3.2) and (3.3),
$\displaystyle\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}\phi_{\alpha\beta}-\phi_{t}$
$\displaystyle=$ $\displaystyle u_{n}^{-3}\sum_{j\in
B}\left[\sigma_{l}(G)+\frac{{\sigma}^{2}_{1}(B|j)-{\sigma}_{2}(B|j)}{{\sigma}^{2}_{1}(B)}\right][-u_{n}^{2}(\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta
jj}-u_{jjt})+2u_{n}u_{nj}(\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta
j}-u_{jt})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad+4u_{n}u_{nj}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta
j}-6u_{nj}^{2}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta}]$
$\displaystyle+2u_{n}^{-3}\sum_{j\in B,i\in
G}\left[\sigma_{l}(G)+\frac{{\sigma}^{2}_{1}(B|j)-{\sigma}_{2}(B|j)}{{\sigma}^{2}_{1}(B)}\right]\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}\frac{1}{u_{ii}}[u_{n}u_{ij\alpha}-2u_{i\alpha}u_{jn}][u_{n}u_{ij\beta}-2u_{i\beta}u_{jn}]$
$\displaystyle+\eta_{0}g\left[-A^{2}F^{nn}u_{n}^{2}+AO(1)+O(1)\right]$
$\displaystyle-\frac{1}{{\sigma}^{3}_{1}(B)}\sum_{\alpha,\beta=1}^{n}\sum_{i\in
B}F^{\alpha\beta}[{\sigma}_{1}(B)\widetilde{a}_{ii,\alpha}-\widetilde{a}_{ii}\sum_{j\in
B}\widetilde{a}_{jj,\alpha}][{\sigma}_{1}(B)\widetilde{a}_{ii,\beta}-\widetilde{a}_{ii}\sum_{j\in
B}\widetilde{a}_{jj,\beta}]$
$\displaystyle-\frac{1}{{\sigma}_{1}(B)}\sum_{\alpha,\beta=1}^{n}\sum_{i\neq
j\in
B}F^{\alpha\beta}\widetilde{a}_{ij,\alpha}\widetilde{a}_{ij,\beta}+O(\mathcal{H}_{\phi}).$
Proof: For any fixed $(x,t)\in\mathcal{O}\times(t_{0}-\delta,t_{0}+\delta]$,
we choose the coordinate as in (3.2) such that $|\nabla u(x)|=u_{n}(x)>0$ and
the matrix $(\widetilde{a}_{ij}(x))$ is diagonal for $1\leq i,j\leq n-1$ and
nonnegative. From the definition of $p$,
(3.11) $\displaystyle
a_{jj}=-\frac{h_{jj}}{u^{3}_{n}}=-\frac{u_{jj}}{u_{n}}=O(\mathcal{H}_{\phi}),\forall
j\in B,$
and
(3.12) $\phi_{t}=\sum_{j\in
B}\left[\sigma_{l}(G)+\frac{{\sigma}^{2}_{1}(B|j)-{\sigma}_{2}(B|j)}{{\sigma}^{2}_{1}(B)}\right]\widetilde{a}_{jjt}+O(\mathcal{H}_{\phi}).$
Using relationship (3.11), we have
(3.13) $\displaystyle\phi_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\sum_{j\in
B}\left[\sigma_{l}(G)+\frac{{\sigma}^{2}_{1}(B|j)-{\sigma}_{2}(B|j)}{{\sigma}^{2}_{1}(B)}\right]\Big{[}\widetilde{a}_{jj,\alpha\beta}-2\sum_{i\in
G}\frac{\widetilde{a}_{ij,\alpha}\widetilde{a}_{ij,\beta}}{\widetilde{a}_{ii}}\Big{]}$
$\displaystyle-\frac{1}{{\sigma}^{3}_{1}(B)}\sum_{i\in
B}\Big{[}{\sigma}_{1}(B)\widetilde{a}_{ii,\alpha}-\widetilde{a}_{ii}\sum_{j\in
B}\widetilde{a}_{jj,\alpha}\Big{]}\Big{[}{\sigma}_{1}(B)\widetilde{a}_{ii,\beta}-\widetilde{a}_{ii}\sum_{j\in
B}\widetilde{a}_{jj,\beta}\Big{]}$
$\displaystyle-\frac{1}{{\sigma}_{1}(B)}\sum_{i\neq j\in
B}\widetilde{a}_{ij,\alpha}\widetilde{a}_{ij,\beta}+O(\mathcal{H}_{\phi}).$
So,
(3.14)
$\displaystyle\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}[\widetilde{a}_{jj,\alpha\beta}-2\sum_{i\in
G}\frac{\widetilde{a}_{ij,\alpha}\widetilde{a}_{ij,\beta}}{\widetilde{a}_{ii}}]-\widetilde{a}_{jj,t}$
$\displaystyle=$
$\displaystyle\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}a_{jj,\alpha\beta}-a_{jj,t}+\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}(-\eta_{0}g_{\alpha\beta})+\eta_{0}g_{t}$
$\displaystyle-2\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}\sum_{i\in
G}\frac{\widetilde{a}_{ij,\alpha}\widetilde{a}_{ij,\beta}}{\widetilde{a}_{ii}}.$
From the definition of $a_{ij}$, and $u_{k}=0$ for $k=1,\cdots,n-1$, we can
get
(3.15) $\displaystyle
u_{n}u_{ij\alpha}=-u_{n}^{2}a_{ij,\alpha}+u_{nj}u_{i\alpha}+u_{ni}u_{j\alpha}+u_{n\alpha}u_{ij}$
and
(3.16) $\displaystyle u_{n}^{3}a_{jj,\alpha\beta}$ $\displaystyle=$
$\displaystyle-u_{n}^{2}u_{jj\alpha\beta}+2u_{n}u_{nj}u_{\alpha\beta
j}-2u_{n}(u_{n\beta}u_{jj\alpha}+u_{n\alpha}u_{jj\beta})$
$\displaystyle+2u_{n}(u_{j\alpha}u_{nj\beta}+u_{j\beta}u_{nj\alpha})+2u_{nj}(u_{n\alpha}u_{j\beta}+u_{n\beta}u_{j\alpha})-2u_{nn}u_{j\alpha}u_{j\beta}$
$\displaystyle-2(u_{n\alpha}u_{n\beta}+u_{n}u_{\alpha\beta
n})u_{jj}-2\eta_{0}gu_{j\alpha}u_{j\beta}u_{n}-3\eta_{0}u_{n}^{2}(u_{n\alpha}g_{\beta}+u_{n\beta}g_{\alpha})$
$\displaystyle-\eta_{0}g(3u_{n}^{2}u_{n\alpha\beta}+6u_{n\alpha}u_{n\beta}u_{n}+\sum\limits_{i=1}^{n-1}{u_{i\alpha}u_{i\beta}u_{n}})+O(\mathcal{H}_{\phi}).$
Direct calculation and (3.15), we can get
(3.17) $\displaystyle-
a_{jj,t}+\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}(-\eta_{0}g_{\alpha\beta})}+\eta_{0}g_{t}$
$\displaystyle=$
$\displaystyle\frac{1}{{u_{n}^{3}}}[u_{n}^{2}u_{jjt}-2u_{n}u_{nj}u_{jt}]$
$\displaystyle+\eta_{0}g[-A^{2}F^{nn}u_{n}^{2}-A(\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}u_{\alpha\beta}}-u_{t})+\frac{{u_{nt}}}{{u_{n}}}].$
From (3.16),
$\displaystyle\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}a_{jj,\alpha\beta}}$
$\displaystyle=$
$\displaystyle\sum\limits_{\alpha\beta=1}^{n}{\frac{{F^{\alpha\beta}}}{{u_{n}^{3}}}[-u_{n}^{2}u_{jj\alpha\beta}+2u_{n}u_{nj}u_{\alpha\beta
j}}$
$\displaystyle-4u_{nj}u_{n\alpha}u_{j\beta}+4u_{nj}u_{n\alpha}u_{j\beta}-2u_{nn}u_{j\alpha}u_{j\beta}$
$\displaystyle-2\eta_{0}u_{n}^{2}u_{n\alpha}g_{\beta}-\eta_{0}g(u_{n}^{2}u_{n\alpha\beta}+2u_{j\alpha}u_{j\beta}u_{n}+\sum\limits_{i=1}^{n-1}{u_{i\alpha}u_{i\beta}u_{n}})]+O(\mathcal{H}_{\phi}),$
so, as in [11], we can get
$\displaystyle
u_{n}^{3}\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}a_{jj,\alpha\beta}}$
$\displaystyle=$ $\displaystyle-
u_{n}^{2}\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}u_{jj\alpha\beta}}+2u_{n}u_{nj}\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}u_{\alpha\beta
j}}$
$\displaystyle+4u_{n}u_{nj}\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}u_{\alpha\beta
j}}-6u_{nj}^{2}\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}u_{\alpha\beta}}$
$\displaystyle+4u_{n}^{2}\sum\limits_{\alpha=1}^{n}{\sum\limits_{i\in
G}{F^{\alpha i}a_{ij,\alpha}}}+2u_{nj}^{2}\sum\limits_{i\in G}{F^{ii}u_{ii}}$
$\displaystyle+2u_{nj}^{2}\sum\limits_{i\in
B}{F^{ii}u_{ii}}-12u_{jn}u_{jj}\sum\limits_{\alpha=1}^{n}{F^{j\alpha}u_{n\alpha}}+4u_{n}u_{jj}\sum\limits_{\alpha=1}^{n}{F^{j\alpha}u_{jn\alpha}}-2u_{nn}F^{jj}u_{jj}^{2}$
$\displaystyle-\eta_{0}g(u_{n}^{2}u_{n\alpha\beta}+2u_{j\alpha}u_{j\beta}u_{n}+\sum\limits_{i=1}^{n-1}{u_{i\alpha}u_{i\beta}u_{n}})$
$\displaystyle-2\eta_{0}\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}u_{n\alpha}g_{\beta}}u_{n}+4\eta_{0}\sum\limits_{\alpha=1}^{n}{F^{j\alpha}g_{\alpha}u_{jn}u_{n}^{2}}+O(\mathcal{H}_{\phi}),$
$\displaystyle=$ $\displaystyle-
u_{n}^{2}\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}u_{jj\alpha\beta}}+2u_{n}u_{nj}\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}u_{\alpha\beta
j}}$
$\displaystyle+4u_{n}u_{nj}\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}u_{\alpha\beta
j}}-6u_{nj}^{2}\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}u_{\alpha\beta}}$
$\displaystyle+4u_{n}^{2}\sum\limits_{\alpha=1}^{n}{\sum\limits_{i\in
G}{F^{\alpha i}a_{ij,\alpha}}}+2u_{nj}^{2}\sum\limits_{i\in G}{F^{ii}u_{ii}}$
$\displaystyle+\eta_{0}g\left[AO(1)+O(1)\right]+O(\mathcal{H}_{\phi}).$
Also, with the similar computations (2.25) in the Lemma 2.1,
(3.19)
$\displaystyle\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}\frac{{\widetilde{a}_{ij,\alpha}\widetilde{a}_{ij,\beta}}}{{\widetilde{a}_{ii}}}}-\frac{1}{{u_{n}^{3}}}[2u_{n}^{2}u_{nj}\sum\limits_{\alpha=1}^{n}{F^{\alpha
i}a_{ij,\alpha}}+u_{nj}^{2}F^{ii}u_{ii}]$ $\displaystyle=$
$\displaystyle\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}\frac{{a_{ij,\alpha}a_{ij,\beta}}}{{a_{ii}}}}+\eta_{0}g\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}\frac{{a_{ij,\alpha}a_{ij,\beta}}}{{a_{ii}\widetilde{a}_{ii}}}}$
$\displaystyle-\frac{1}{{u_{n}^{3}}}[2u_{n}^{2}u_{nj}\sum\limits_{\alpha=1}^{n}{F^{\alpha
i}a_{ij,\alpha}}+u_{nj}^{2}F^{ii}u_{ii}]$ $\displaystyle=$
$\displaystyle\eta_{0}g\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}\frac{{a_{ij,\alpha}a_{ij,\beta}}}{{a_{ii}\widetilde{a}_{ii}}}}$
$\displaystyle-\frac{1}{{u_{n}^{3}}}\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}\frac{1}{{u_{ii}}}}[-u_{n}u_{ij\alpha}+u_{nj}u_{i\alpha}+u_{ni}u_{j\alpha}][-u_{n}u_{ij\beta}+u_{nj}u_{i\beta}+u_{ni}u_{j\beta}]$
$\displaystyle-\frac{1}{{u_{n}^{3}}}[2u_{nj}\sum\limits_{\alpha=1}^{n}{F^{\alpha
i}(-u_{n}u_{ij\alpha}+u_{nj}u_{i\alpha}+u_{ni}u_{j\alpha})}+u_{nj}^{2}F^{ii}u_{ii}]$
$\displaystyle=$
$\displaystyle\eta_{0}g\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}\frac{{a_{ij,\alpha}a_{ij,\beta}}}{{a_{ii}\widetilde{a}_{ii}}}}$
$\displaystyle-\frac{1}{{u_{n}^{3}}}\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}\frac{1}{{u_{ii}}}}[-u_{n}u_{ij\alpha}+2u_{nj}u_{i\alpha}][-u_{n}u_{ij\beta}+2u_{nj}u_{i\beta}]$
$\displaystyle-\frac{1}{{u_{n}^{3}}}u_{jj}[\sum\limits_{\alpha=1}^{n-1}{F^{\alpha
j}\frac{2}{{u_{ii}}}}u_{ni}(-u_{n}u_{ij\alpha}+u_{nj}u_{i\alpha})+F^{ii}\frac{1}{{u_{ii}}}u_{jj}u_{ni}^{2}$
$\displaystyle\qquad\qquad+2F^{jn}\frac{1}{{u_{ii}}}u_{ni}(-u_{n}u_{ijn}+2u_{nj}u_{in})+F^{ij}u_{ni}u_{nj}]$
$\displaystyle=$
$\displaystyle-\frac{1}{{u_{n}^{3}}}\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}\frac{1}{{u_{ii}}}}[-u_{n}u_{ij\alpha}+2u_{nj}u_{i\alpha}][-u_{n}u_{ij\beta}+2u_{nj}u_{i\beta}]$
$\displaystyle+\eta_{0}g\left[\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}\frac{{a_{ij,\alpha}a_{ij,\beta}}}{{a_{ii}\widetilde{a}_{ii}}}}+O(1).\right]$
From the above calculations, the proof is complete. ∎
Theorem 1.2 is a direct consequence of the following proposition and the
strong maximum principle.
###### Proposition 3.2.
Suppose that the function $F,u$ satisfy assumptions in Theorem 1.2. If the
second fundamental form $b_{ij}$ of $\Sigma^{u(x,t_{0})}$ attains minimum rank
$l=l(t_{0})$ at certain point $x_{0}\in\Omega$, then there exist a
neighborhood $\mathcal{O}\times(t_{0}-\delta_{0},t_{0}+\delta_{0}]$ of
$(x_{0},t_{0})$ and a positive constant $C$ independent of $\phi$ (defined in
(3.7)), such that
(3.20)
$\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}\phi_{\alpha\beta}(x,t)-\phi_{t}\leq
C(\phi+|\nabla\phi|)+\eta_{0}g\left[-A^{2}F^{nn}u_{n}^{2}+AO(1)+O(1)\right]$
holds for any $(x,t)\in\mathcal{O}\times(t_{0}-\delta_{0},t_{0}+\delta_{0}]$.
Proof: Since
(3.21) $u_{t}=F(\nabla^{2}u,\nabla u,u,t),$
for each $j\in B$, differentiating the above equation in $e_{j}$ direction at
$x$,
(3.22) $u_{jt}=\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta
j}+F^{u_{n}}u_{jn}+O(\mathcal{H}_{\phi})$
and
(3.23) $\displaystyle u_{jjt}$ $\displaystyle=$
$\displaystyle\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta
jj}+\sum_{\alpha,\beta,r,s=1}^{n}F^{\alpha\beta,rs}u_{\alpha\beta
j}u_{rsj}+2\sum_{\alpha,\beta,l=1}^{n}F^{\alpha\beta,u_{l}}u_{\alpha\beta
j}u_{lj}$
$\displaystyle+2\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta,u}u_{j\alpha\beta}u_{j}+\sum_{l,s=1}^{n}F^{u_{l},u_{s}}u_{lj}u_{sj}-2\sum_{l=1}^{n}F^{u_{l},u}u_{lj}u_{j}$
$\displaystyle+F^{u,u}u_{j}^{2}+\sum_{l=1}^{n}F^{u_{l}}u_{ljj}+F^{u}u_{jj}.$
It follows from (3.11)and (3.15) that, at $(x,t)$
(3.24)
$\displaystyle\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}u_{\alpha\beta
j}}-u_{jt}=-F^{p_{n}}u_{nj}+\eta_{0}gF^{p_{j}}u_{n}+O(\mathcal{H}_{\phi})$
and
$\displaystyle\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta}u_{\alpha\beta
jj}}-u_{jjt}$ $\displaystyle=$
$\displaystyle-\sum\limits_{\alpha\beta\gamma\eta=1}^{n}{F^{\alpha\beta,\gamma\eta}u_{\alpha\beta
j}}u_{\gamma\eta
j}-2\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta,p_{n}}u_{\alpha\beta
j}u_{nj}}-F^{p_{n},p_{n}}u_{nj}u_{nj}-2\frac{{F^{p_{n}}}}{{u_{n}}}u_{nj}^{2}$
$\displaystyle+\eta_{0}g[-AF^{p_{n}}u_{n}^{2}]$
$\displaystyle+\eta_{0}g[2\sum\limits_{\alpha\beta=1}^{n}{F^{\alpha\beta,p_{j}}u_{\alpha\beta
j}u_{n}}+F^{p_{j},p_{j}}u_{jj}u_{n}+2F^{p_{n},p_{j}}u_{nj}u_{n}+F^{p_{n}}u_{n}+2F^{p_{j}}u_{jn}+F^{p_{l}}u_{nl}]$
$\displaystyle+O(\mathcal{H}_{\phi}).$
From lemma 3.1,
$\displaystyle F^{\alpha\beta}\phi_{\alpha\beta}-\phi_{t}$ $\displaystyle=$
$\displaystyle\sum_{j\in
B}u_{n}^{-3}\left[\sigma_{l}(G)+\frac{{\sigma}^{2}_{1}(B|j)-{\sigma}_{2}(B|j)}{{\sigma}^{2}_{1}(B)}\right]\left\\{\Big{[}\sum_{\alpha,\beta,r,s=1}^{n}F^{\alpha\beta,rs}u_{\alpha\beta
j}u_{rsj}\right.$
$\displaystyle\qquad\qquad\qquad\qquad+2\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta,u_{n}}u_{j\alpha\beta}u_{jn}+F^{u_{n},u_{n}}u_{jn}^{2}\Big{]}u_{n}^{2}$
$\displaystyle\qquad\qquad\qquad\qquad+\left.4u_{jn}u_{n}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{j\alpha\beta}-6u_{jn}^{2}\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}u_{\alpha\beta}\right\\}$
$\displaystyle+2u_{n}^{-3}\sum_{j\in B,i\in
G}\left[\sigma_{l}(G)+\frac{{\sigma}^{2}_{1}(B|j)-{\sigma}_{2}(B|j)}{{\sigma}^{2}_{1}(B)}\right]\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}\frac{1}{u_{ii}}[u_{n}u_{ij\alpha}-2u_{i\alpha}u_{jn}][u_{n}u_{ij\beta}-2u_{i\beta}u_{jn}]$
$\displaystyle+\eta_{0}g\left[-A^{2}F^{nn}u_{n}^{2}+AO(1)+O(1)\right]$
$\displaystyle-\frac{1}{{\sigma}^{3}_{1}(B)}\sum_{\alpha,\beta=1}^{n}\sum_{i\in
B}F^{\alpha\beta}[{\sigma}_{1}(B)\widetilde{a}_{ii,\alpha}-\widetilde{a}_{ii}\sum_{j\in
B}\widetilde{a}_{jj,\alpha}][{\sigma}_{1}(B)\widetilde{a}_{ii,\beta}-\widetilde{a}_{ii}\sum_{j\in
B}\widetilde{a}_{jj,\beta}]$
$\displaystyle-\frac{1}{{\sigma}_{1}(B)}\sum_{\alpha,\beta=1}^{n}\sum_{i\neq
j\in
B}F^{\alpha\beta}\widetilde{a}_{ij,\alpha}\widetilde{a}_{ij,\beta}+O(\mathcal{H}_{\phi}).$
So, following the argument in the proof of Proposition 2.3, we get,
(3.25)
$\sum_{\alpha,\beta=1}^{n}F^{\alpha\beta}\phi_{\alpha\beta}(x,t)-\phi_{t}\leq
C(\phi+|\nabla\phi|)+\eta_{0}g\left[-A^{2}F^{nn}u_{n}^{2}+AO(1)+O(1)\right].$
The proof is completed. ∎
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|
arxiv-papers
| 2010-06-24T13:21:59 |
2024-09-04T02:49:11.158385
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chuanqiang Chen, Shujun Shi",
"submitter": "Chuanqiang Chen",
"url": "https://arxiv.org/abs/1006.4787"
}
|
1006.4804
|
# The General Solutions of Linear ODE and Riccati Equation by Integral
Serieslabel1
Yimin Yan yanyimin@foxmail.com
###### Abstract
This paper gives out the general solutions of variable coefficients Linear ODE
and Riccati equation by way of integral series $\mathcal{E}(X)$ and
$\mathcal{F}(X)$. Such kinds of integral series are the generalized form of
exponential function, and keep the properties of convergent and reversible.
###### keywords:
Linear ODE,Riccati equation,integral series, general solution,variable
coefficients
proofProof
[label1]Many thanks to Prof.Qiyan Shi’s guidance.
## 1 Introduction
It is a classical problem to solve the n-th order Linear ODE :
${\frac{d^{n}}{d{x}^{n}}}u+a_{1}(x){\frac{d^{n-1}}{d{x}^{n-1}}}u+a_{2}(x){\frac{d^{n-2}}{d{x}^{n-2}}}u+\cdots+a_{n}(x)u=f(x)$
(1)
which is equivalent to
${\frac{d}{dx}}U=AU+F$ (2)
with
$\left\\{\begin{aligned}
U&={\left[\begin{array}[]{cccc}{\frac{d^{n-1}}{d{x}^{n-1}}}u&{\frac{d^{n-2}}{d{x}^{n-2}}}u&\cdots&u\end{array}\right]}^{T}\\\
F&=\left[\begin{array}[]{cccc}f\left(x\right)&0&\cdots&0\end{array}\right]^{T}\\\
A(x)&=\left[\begin{array}[]{ccccc}-a_{{1}}&-a_{{2}}&-a_{{3}}&\cdots&-a_{{n}}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr 1&0&0&\cdots&0\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr 0&1&0&\cdots&0\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\cdots&\cdots&\cdots&\cdots&\cdots\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr 0&0&\cdots&1&0\end{array}\right]\\\ \end{aligned}\right.$ (3)
As we all known,
1. 1.
if $\big{\\{}a_{n}(x)\big{\\}}$ are all constants, Eq.(1) could be solved by
method of eigenvalue ( Euler), or by exponential function in matrix form
$U=e^{A\cdot x}\cdot C+e^{A\cdot x}\cdot\int_{0}^{x}e^{-A\cdot s}\cdot
F\left(s\right){ds}$
_where C is a $n\times 1$ constant matrix ._
2. 2.
if $\big{\\{}a_{n}(x)\big{\\}}$ are some variable coefficients, such as some
special functions [Wang, , P337,206]
$\frac{d^{2}y}{dx^{2}}+\frac{1}{x}\frac{dy}{dx}+\big{(}1-\frac{n^{2}}{x^{2}}\big{)}y=0$
(Bessel Equation)
$(1-x^{2})\frac{d^{2}y}{dx^{2}}-2x\frac{dy}{dx}+n(n+1)y=0$ (Legendre Equation)
special function theory answers them.
But when it comes to the general circumstances, the existing methods meet
difficulties in dealing with Eq.(2) , because of the variable coefficients. In
order to overcome it, two functions are invited :
### 1.1 Definition
$\left\\{\begin{aligned}
\mathcal{E}\big{[}X(x)\big{]}=&I+\int_{0}^{x}\\!X\left(t\right){dt}+\int_{0}^{x}\\!X\left(t\right)\int_{0}^{t}\\!X\left(s\right){ds}{dt}+\int_{0}^{x}\\!X\left(t\right)\int_{0}^{t}\\!X\left(s\right)\int_{0}^{s}\\!X\left(\xi\right){d\xi}{ds}{dt}+\cdots\\\
\mathcal{F}\big{[}X(x)\big{]}=&I+\int_{0}^{x}\\!X\left(t\right){dt}+\int_{0}^{x}\\!\int_{0}^{t}\\!X\left(s\right){ds}X\left(t\right){dt}+\int_{0}^{x}\\!\int_{0}^{t}\\!\int_{0}^{s}\\!X\left(\xi\right){d\xi}\
X(s){ds}\ X\left(t\right){dt}+\cdots\\\ \end{aligned}\right.$ (4)
It will be seen that such definition is reasonable and necessary. Clearly,
when $X(x)$ and $\int_{0}^{x}\\!X(t)dt$ are exchangeable, then
$\mathcal{E}\big{[}X(x)\big{]}=e^{\int_{0}^{x}\\!X(t)dt}=\mathcal{F}\big{[}X(x)\big{]}$
Besides, $\mathcal{E}(X)$ and $\mathcal{F}(X)$ extend some main properties of
the exponential functions, such as convergent , reversible and determinant
(see Theorem 3.1). In addition, a $n\times m$ matrix
$A(x)=\big{(}a_{ij}(x)\big{)}_{nm}$ is bounded and integral in [0,b] means
that all its element $a_{ij}(x)$ are bounded and integral in [0,b].
## 2 Main Results
###### Theorem 2.1
the general solution of the Linear ODE (2) is:
$U=\mathcal{E}\big{[}A(x)\big{]}\cdot
C+\mathcal{E}\big{[}A(x)\big{]}\cdot\int_{0}^{x}\mathcal{F}\big{[}-A(s)\big{]}\cdot
F\left(s\right){ds}$ (5)
where C is a $n\times 1$ constant matrix .
###### Theorem 2.2
For the bounded and integrable matrix , $A(x)=(a_{ij})_{nn}$,
$B(x)=(b_{ij})_{mm}$, $P(x)=(p_{ij})_{mn}$, $Q(x)=(q_{ij})_{nm}$, in [0,b],
the general solution of Riccati equation
${\frac{d}{dx}}W+WPW+WB-AW-Q=0$ (6)
is
$W=W_{1}\cdot W^{-1}_{2}$ (7)
where
$\left[\begin{array}[]{c}W_{{1}}\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
W_{{2}}\end{array}\right]=\mathcal{E}\biggl{(}\left[\begin{array}[]{cc}A&Q\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
P&B\end{array}\right]\biggl{)}\cdot\left[\begin{array}[]{c}W\mid_{x=0}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr I\end{array}\right]$ (8)
or the other equivalent form:
$W=U^{-1}_{2}\cdot U_{1}$ (9)
where
$\begin{array}[]{ll}\left[\begin{array}[]{cc}U_{1}&U_{2}\end{array}\right]=\left[\begin{array}[]{cc}I&W\mid_{x=0}\end{array}\right]\cdot\mathcal{F}\biggl{(}\left[\begin{array}[]{cc}-B&P\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
Q&-A\end{array}\right]\biggl{)}\end{array}$ (10)
## 3 Solutions of Linear ODE
### 3.1 Properties of $\mathcal{E}(X)$ and $\mathcal{F}(X)$
From the Definition(4), it holds that
$\left\\{\begin{aligned}
{\frac{d}{dx}}\mathcal{E}\biggl{[}X(x)\biggl{]}&=X\cdot\mathcal{E}\biggl{[}X(x)\biggl{]}\\\
{\frac{d}{dx}}\mathcal{F}\biggl{[}X(x)\biggl{]}&=\mathcal{F}\biggl{[}X(x)\biggl{]}\cdot
X\\\ \end{aligned}\right.$ (11)
Now, we will see more explicit properties of $\mathcal{E}(X)$ and
$\mathcal{F}(X)$.
###### Theorem 3.1 (Properties of $\mathcal{E}(X)$ and $\mathcal{F}(X)$)
If $X(x)$ is bounded and integrable, it holds that
1. 1.
$\mathcal{E}(X)$ and $\mathcal{F}(X)$ are convergent;
2. 2.
$\det\mathcal{E}(X)=\det\mathcal{F}(X)=\det
e^{\int_{0}^{x}\\!X(t){dt}}=e^{\int_{0}^{x}\\!trX(t){dt}}=e^{tr\int_{0}^{x}\\!X(t){dt}}$
(12)
3. 3.
$\mathcal{E}(X)$ and $\mathcal{F}(X)$ are reversible, and
$\mathcal{F}(X)\mathcal{E}(-X)=\mathcal{E}(-X)\mathcal{F}(X)=I$ (13)
###### Proof 3.1.
1. 1.
Firstly, $\mathcal{E}(A)$ is convergent,since
$\bigl{\\{}a_{k}(x)\bigl{\\}}^{n}_{k=1}$ are bounded in [0,b]:
$\exists M>0$, s.t. $|a_{k}(x)|<M$, $\forall x\in[0,b]$, $k=1,2,\cdots,n$
So
1. (a)
$\displaystyle\big{\|}\int_{0}^{x}\\!A\left(t\right){dt}\big{\|}=max\left|\int_{0}^{x}\\!a_{{k}}\left(t\right){dt}\right|<M|x|$
2. (b)
$\displaystyle\big{\|}\int_{0}^{x}\\!A\left(t\right)\int_{0}^{t}\\!A\left(s\right){ds}{dt}\big{\|}=max\left|\sum_{i}\int_{0}^{x}\\!a_{{k}}\left(t\right)\int_{0}^{t}\\!a_{{i}}\left(s\right){ds}{dt}\right|<nM^{2}\left|\int_{0}^{x}\\!\int_{0}^{t}\\!1{ds}{dt}\right|<\frac{n}{2!}(M|x|)^{2}$
3. (c)
$\displaystyle\big{\|}\int_{0}^{x}\\!A\left(t\right)\int_{0}^{t}\\!A\left(s\right)\int_{0}^{s}\\!A\left(\xi\right){d\xi}{ds}{dt}\big{\|}=max\left|\sum_{i,j}\int_{0}^{x}\\!a_{{k}}\left(t\right)\int_{0}^{t}\\!a_{{i}}\left(s\right)\int_{0}^{s}\\!a_{{j}}\left(\xi\right){d\xi}{ds}{dt}\right|$
$\displaystyle<$ $\displaystyle
n^{2}M^{3}\left|\int_{0}^{x}\\!\int_{0}^{t}\\!\int_{0}^{s}\\!{d\xi}{ds}{dt}\right|<\frac{n^{2}}{3!}(M|x|)^{3}$
4. (d)
$\cdots\cdots$
It follows that
$\displaystyle\|\mathcal{E}(A)\|$ $\displaystyle<$ $\displaystyle
1+\frac{1}{n}\biggl{[}nM|x|+\frac{1}{2!}(nMx)^{2}+\frac{1}{3!}(nMx)^{3}+\cdots\biggl{]}=1+\frac{1}{n}e^{nM|x|}$
Clearly, $\mathcal{E}(A)$ is convergent.
Similarly, $\mathcal{F}(X)$ is also convergent.
2. 2.
_$\forall n\times n$ matrix $A(x)$, if $trA(x)$ is bounded and integral , then
_
$\det\mathcal{E}(A(x))=e^{\int_{0}^{x}\\!trA(t){dt}}=e^{tr\int_{0}^{x}\\!A(t){dt}}$
(14)
which is a special case of Abel’s formulaChen : _If W and B are $n\times n$
matrixes , s.t._
${\frac{d}{dx}}W=BW$ (15)
then,
$\det W=e^{trB}$ (16)
Here we just take $2\times 2$ matrix for verification:
Let
$Y(x)=\mathcal{E}\biggl{[}A(x)\biggl{]}=\left[\begin{array}[]{cc}y_{{1,1}}&y_{{1,2}}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr y_{{2,1}}&y_{{2,2}}\end{array}\right]$,
$A(x)=\left[\begin{array}[]{cc}a_{{1,1}}&a_{{1,2}}\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr a_{{2,1}}&a_{{2,2}}\end{array}\right]$
so ${\frac{d}{dx}}Y=A\cdot Y$means that
${\frac{d}{dx}}\left[\begin{array}[]{cc}y_{{1,1}}&y_{{1,2}}\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr
y_{{2,1}}&y_{{2,2}}\end{array}\right]=\left[\begin{array}[]{cc}a_{{1,1}}&a_{{1,2}}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
a_{{2,1}}&a_{{2,2}}\end{array}\right]\cdot\left[\begin{array}[]{cc}y_{{1,1}}&y_{{1,2}}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr y_{{2,1}}&y_{{2,2}}\end{array}\right]$
(17)
it follows
$\displaystyle{\frac{d}{dx}}(\det Y)$
$\displaystyle=\det\left[\begin{array}[]{cc}{\frac{d}{dx}}y_{{1,1}}&{\frac{d}{dx}}y_{{1,2}}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
y_{{2,1}}&y_{{2,2}}\end{array}\right]+\det\left[\begin{array}[]{cc}y_{{1,1}}&y_{{1,2}}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr{\frac{d}{dx}}y_{{2,1}}&{\frac{d}{dx}}y_{{2,2}}\end{array}\right]$
$\displaystyle=det\left[\begin{array}[]{cc}{a_{1,1}y_{1,1}+a_{1,2}y_{2,1}}&{a_{1,1}y_{1,2}+a_{1,2}y_{2,2}}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
y_{{2,1}}&y_{{2,2}}\end{array}\right]+\det\left[\begin{array}[]{cc}y_{{1,1}}&y_{{1,2}}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
a_{2,1}y_{1,1}+a_{2,2}y_{2,1}&a_{2,1}y_{1,2}+a_{2,2}y_{2,2}\end{array}\right]$
$\displaystyle=a_{1,1}\det\left[\begin{array}[]{cc}y_{{1,1}}&y_{{1,2}}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
y_{{2,1}}&y_{{2,2}}\end{array}\right]+a_{2,2}\det\left[\begin{array}[]{cc}y_{{1,1}}&y_{{1,2}}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr y_{{2,1}}&y_{{2,2}}\end{array}\right]$
$\displaystyle=\biggl{[}a_{1,1}+a_{2,2}\biggl{]}detY=trA\cdot detY$
Thus, Abel’s formula holds and $\mathcal{E}(A(x))$ is reversible.
By the times:
$\det\mathcal{F}(X)=e^{\int_{0}^{x}\\!trX(t){dt}}=e^{tr\int_{0}^{x}\\!X(t){dt}}$
(18)
so, all we need to proof is
$\det e^{\int_{0}^{x}\\!X(t){dt}}=e^{\int_{0}^{x}\\!trX(t){dt}}$ (19)
Because $e^{\int_{0}^{x}\\!X(t){dt}}$ no longer satisfies Abel’s formula (one
reason is $X$ and $\int_{0}^{x}\\!X(t){dt}$ are unnecessarily exchangeable ) ,
we seek the other approach:
$\forall n\times n$ matrix A, $\exists n\times n$ reversible matrix P , s.t.
$P^{-1}AP=diag\\{J_{1},J_{2},\cdots,J_{s}\\}:=J$
_J is A’s Jordan matrix, $J_{i}$ is the Jordan block with eigenvalue
$\lambda_{i}(x)$._
It follows that
$e^{J_{i}}=e^{\lambda_{i}(x)}\left[\begin{array}[]{cccccc}1&1&\frac{1}{2!}&\frac{1}{3!}&\cdots&\cdots\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr 0&1&1&\frac{1}{2!}&\cdots&\cdots\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr 0&0&1&1&\cdots&\cdots\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr 0&0&0&0&\cdots&1\end{array}\right]$ (20)
So,
$P^{-1}e^{A}P=e^{P^{-1}AP}=e^{J}=diag\\{e^{J_{1}},e^{J_{2}},\cdots,e^{J_{s}}\\}$
Therefore
$\det e^{A}=\det e^{J}=e^{trJ}=e^{trA}$
which yields
$\det e^{\int_{0}^{x}\\!X(t){dt}}=e^{\int_{0}^{x}\\!trX(t){dt}}$
3. 3.
_Notice that $\forall n\times n$ matrix A, there exists a companion matrix
$A^{*}$,s.t._
$A\cdot A^{*}=A^{*}\cdot A=\texttt{det}A\cdot I$ (21)
_so, if $\det A\neq 0$ , A is invertible.
_
Therefore, $\mathcal{E}(X)$ and $\mathcal{F}(X)$ are invertible.
Furthermore, it holds that
$\mathcal{F}(X)\mathcal{E}(-X)=\mathcal{E}(-X)\mathcal{F}(X)=I$ (22)
Because:
1. (a)
$\displaystyle{\frac{d}{dx}}\biggl{[}\mathcal{F}(X)\mathcal{E}(-X)\biggl{]}={\frac{d}{dx}}\mathcal{F}(X)\cdot\mathcal{E}(-X)+\mathcal{F}(X)\cdot{\frac{d}{dx}}\mathcal{E}(-X)=\mathcal{F}(X)X\cdot\mathcal{E}(-X)-\mathcal{F}(X)\cdot
X\mathcal{E}(-X)=0$
So,
$\displaystyle\mathcal{F}(X)\mathcal{E}(-X)$
$\displaystyle=const.=\big{[}\mathcal{F}(X)\mathcal{E}(-X)\big{]}\big{|}_{x=0}=I$
2. (b)
Due to the special property(21) of matrix, Eq.(22) is obtained.
### 3.2 Proof of Theorem2.1
###### Proof 3.2.
According to Definition(4) and Theorem 3.1 , it follows
$\left\\{\begin{aligned}
{\frac{d}{dx}}\mathcal{E}\big{[}A(x)\big{]}&=&A(x)\cdot\mathcal{E}\big{[}A(x)\big{]}\\\
{\frac{d}{dx}}G(x)&=&A(x)\cdot G(x)+F\\\ \end{aligned}\right.$ (23)
_where_
$G(x)=\mathcal{E}\big{[}A(x)\big{]}\cdot\int_{0}^{x}\mathcal{F}\big{[}-A(s)\big{]}\cdot
F\left(s\right){ds}$
because
$\displaystyle{\frac{d}{dx}}G(x)=$
$\displaystyle{\frac{d}{dx}}\mathcal{E}\big{[}A(x)\big{]}\cdot\int_{0}^{x}\mathcal{F}\big{[}-A(s)\big{]}\cdot
F\left(s\right){ds}+\mathcal{E}\big{[}A(x)\big{]}\cdot\mathcal{F}\big{[}-A(x)\big{]}\cdot
F(x)$ $\displaystyle=$ $\displaystyle
A(x)\cdot\mathcal{E}\big{[}A(x)\big{]}\cdot\int_{0}^{x}\mathcal{F}\big{[}-A(s)\big{]}\cdot
F\left(s\right){ds}+F$
Clearly $U(x)=\mathcal{E}\big{[}A(x)\big{]}\cdot
C+\mathcal{E}\big{[}A(x)\big{]}\cdot\int_{0}^{x}\mathcal{F}\big{[}-A(s)\big{]}\cdot
F\left(s\right){ds}$ is convergent.
Moreover, since $\mathcal{E}(A)$ is reversible, $U(x)$ is the general solution
of Eq.(2).
###### Theorem 2.
Assume that $A(x)=(a_{ij})_{n\times n}$, $B(x)=(b_{ij})_{m\times m}$,
$P(x)=(p_{ij})_{n\times m}$ are bounded and integrable matrixes , and $U(x)$
is the desired $n\times m$ matrix. The Linear ODE :
${\frac{d}{dx}}U=A(x)U+UB(x)+P(x)$ (24)
has general solutions
$U(x)=\mathcal{E}(A)\biggl{[}\int_{0}^{x}\\!\mathcal{F}\big{(}-A(t)\big{)}P(t)\mathcal{E}\big{(}-B(t)\big{)}{dt}+C\biggl{]}\mathcal{F}(B)$
(25)
_where C is $n\times m$ constant matrix._
###### Proof 3.3.
Let $U=\mathcal{E}(A)\cdot W\cdot\mathcal{F}(B)$, then
${\frac{d}{dx}}U=A(x)U+UB(x)+\mathcal{E}(A){\frac{d}{dx}}W\cdot\mathcal{F}(B)$
(26)
So Eq.(24) could be reduced to
$\mathcal{E}(A){\frac{d}{dx}}W\cdot\mathcal{F}(B)=P$ (27)
or,
${\frac{d}{dx}}W=\mathcal{F}(-A)\cdot P\cdot\mathcal{E}(-B)$ (28)
It’s obviously that
$W(x)=\int_{0}^{x}\\!\mathcal{F}\big{[}-A(t)\big{]}P(t)\cdot\mathcal{E}\big{[}-B(t)\big{]}{dt}+C$
(29)
_C is $n\times m$ constant matrix ._
## 4 Solutions of Riccati equation
In mathematical investigation of the dynamics of a system, the introduction of
a nonlinearity always leads to some form of the Riccati equation Watkins :
${\frac{d}{dx}}y+a(x)y^{2}+b(x)y+c(x)=0$ (30)
But it is usually the case that not even one solution of the Riccati equation
is known. In the following text, we try to give out solutions of Riccati
equation in matrix form:
${\frac{d}{dx}}W+WPW+WB-AW-Q=0$ (31)
_where $A(x)=(a_{ij})_{nn}$, $B(x)=(b_{ij})_{mm}$, $P(x)=(p_{ij})_{mn}$,
$Q(x)=(q_{ij})_{nm}$ _.
### 4.1 Proof of Theorem.2.2
###### Proof 4.1.
1. 1.
Firstly , define[Polyanin, , Ch 0.1.4]
$W_{2}:=\mathcal{E}(PW+B)$ (32)
so $W_{2}$ is reversible, if $PW+B$ is bounded;
meanwhile,
${\frac{d}{dx}}W_{2}=(PW+B)W_{2}$ (33)
Secondly, let $W_{1}:=WW_{2}$, so
$\begin{array}[]{ll}{\frac{d}{dx}}W_{1}&={\frac{d}{dx}}W\cdot
W_{2}+W\cdot{\frac{d}{dx}}W_{2}={\frac{d}{dx}}W\cdot
W_{2}+W\cdot\biggl{[}PW+B\biggl{]}W_{2}=\biggl{[}{\frac{d}{dx}}W+WPW+WB\biggl{]}W_{2}\end{array}$
(34)
so, with Eq.(31) and Definition (32), it holds
${\frac{d}{dx}}W_{1}=AW_{1}+QW_{2}$ (35)
Take the relationship (33) and (35) into consideration,
${\frac{d}{dx}}\left[\begin{array}[]{c}W_{{1}}\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr W_{{2}}\end{array}\right]=\left[\begin{array}[]{cc}A&Q\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
P&B\end{array}\right]\cdot\left[\begin{array}[]{c}W_{{1}}\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr W_{{2}}\end{array}\right]$ (36)
we can solve $W_{1}$ and $W_{2}$.
On the other hand, according to Definition (32) , it’s obviously that
$W_{2}|_{x=0}=\mathcal{E}(PW+B)|_{x=0}=I$ (37)
so it goes without saying that
$W_{1}|_{x=0}=(WW_{2})|_{x=0}=W|_{x=0}$ (38)
We immediately obtain
$\left[\begin{array}[]{c}W_{{1}}\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
W_{{2}}\end{array}\right]=\mathcal{E}\biggl{(}\left[\begin{array}[]{cc}A&Q\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
P&B\end{array}\right]\biggl{)}\cdot\left[\begin{array}[]{c}W\mid_{x=0}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr I\end{array}\right]$ (39)
Therefore $W=W_{1}\cdot W^{-1}_{2}$ is the solution of Eq.(31).
2. 2.
Similarly, we can get
$\begin{array}[]{ll}{\frac{d}{dx}}\left[\begin{array}[]{cc}U_{1}&U_{2}\end{array}\right]=\left[\begin{array}[]{cc}I&W\mid_{x=0}\end{array}\right]\cdot\left[\begin{array}[]{cc}-B&P\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr Q&-A\end{array}\right]\end{array}$ (40)
so, $W=U^{-1}_{2}\cdot U_{1}$ is also the solution of Eq.(31).
3. 3.
But the two solutions are equivalence! That is,
$W_{1}\cdot W^{-1}_{2}=U^{-1}_{2}U_{1}$ (41)
or
$U_{2}\cdot W_{1}-U_{1}\cdot W_{2}=0$ (42)
Because, according to Eq.(36) and Eq.(40)
$\begin{array}[]{ll}&{\frac{d}{dx}}\biggl{[}U_{2}\cdot W_{1}-U_{1}\cdot
W_{2}\biggl{]}={\frac{d}{dx}}U_{2}\cdot
W_{1}+U_{2}\cdot{\frac{d}{dx}}W_{1}-{\frac{d}{dx}}U_{1}\cdot
W_{2}-U_{1}\cdot{\frac{d}{dx}}W_{2}\\\ =&\biggl{[}U_{1}P-U_{2}A\biggl{]}\cdot
W_{1}+U_{2}\cdot\biggl{[}AW_{1}+QW_{2}\biggl{]}-\biggl{[}U_{2}Q-U_{1}B\biggl{]}\cdot
W_{2}-U_{1}\cdot\biggl{[}PW_{1}+BW_{2}\biggl{]}=0\end{array}$ (43)
As a result,
$\displaystyle U_{2}\cdot W_{1}-U_{1}\cdot W_{2}=const.=\big{[}U_{2}\cdot
W_{1}-U_{1}\cdot W_{2}\big{]}\big{|}_{x=0}=0$ (44)
which implied that two solutions are equivalence.
4. 4.
Uniqueness. If Eq.(31) has more than one solution,such as $X(x),Y(x)$, under
the same initial condition,i.e. $X(0)=Y(0)$. Let $W(x)=X(x)-Y(x)$. So it is
clear that what we need to prove is equitant to show
$\left\\{\begin{aligned} &{\frac{d}{dx}}W+WPW+WB-AW=0\\\
&W|_{x=0}=0\end{aligned}\right.$ (45)
has uniqueness solution $W(x)=0$.
Take advantage the proof steps we have established: according to step(39) and
(35),
> Any solution of Eq.(45), such as $W(x)$, it is reasonable to define
>
> $W_{2}=\mathcal{E}(PW+B),\qquad\quad W_{1}=W\cdot W_{2}$
>
> It follows that $W_{2}$ is bounded ,
>
> ${\frac{d}{dx}}W_{1}=AW_{1}$ (46)
>
> and
>
> $W_{1}=\mathcal{E}[A]\cdot W_{1}|_{x=0}=\mathcal{E}[A]\cdot W|_{x=0}=0$ (47)
>
> Therefore, $W(x)=W_{1}\cdot W_{2}^{-1}=0$
### 4.2 Simplify solutions of Riccati equation by particular solution
In the research of Riccati equation, particular solution plays crucial
important role. Too much of works have been done. The first important result
in the analysis of the Riccati equation is that if one solution is known then
a whole family of solutions can be found Watkins .
###### Theorem 1.
The same conditions as theorem 2.2, Riccati equation
${\frac{d}{dx}}W+WPW+WB-AW-Q=0$ (48)
has the unique solution
$\displaystyle
W=Y+\mathcal{E}\big{(}A-YP\big{)}\cdot\big{(}W|_{x=0}\big{)}\cdot\biggl{[}I+\int_{0}^{x}\\!R(t)\left(t\right){dt}\cdot(W\mid_{x=0})\biggl{]}^{-1}\cdot\mathcal{F}\big{(}-[B+PY]\big{)}$
(49)
where
$\left\\{\begin{aligned} Y&\hbox{ is solution of Eq.(\ref{r.ps}) when
}W|_{x=0}=0,\hbox{i.e. }Y|_{x=0}=0\\\
R&:=\mathcal{F}\big{(}-[B+PY]\big{)}\cdot
P\cdot\mathcal{E}\big{(}A-YP\big{)}\\\ \end{aligned}\right.$ (50)
###### Proof 4.2.
1. 1.
According to Theorem 2.2 , Eq.(48) has solutions. Take any one of it, such as
$Y$, and let
$V=W-Y$ (51)
It follows that
$\begin{array}[]{ll}VPV&=(W-Y)P(W-Y)=\big{(}WPW-YPY\big{)}-(W-Y)PY-
YP(W-Y)=\big{(}WPW-YPY\big{)}-VPY-YPV\\\
&\stackrel{{\scriptstyle{Eq.(\ref{r.ps})}}}{{=}}\Big{(}[-{\frac{d}{dx}}W-WB+AW+Q]-[-{\frac{d}{dx}}Y-YB+AY+Q]\Big{)}-VPY-
YPV\\\ &=\Big{(}-{\frac{d}{dx}}V+AV-VB\Big{)}-VPY-
YPV=-{\frac{d}{dx}}V+(A-YP)V-V(B+PY)\end{array}$ (52)
That is,
${\frac{d}{dx}}V+VPV+V(B+PY)-(A-YP)V=0$ (53)
2. 2.
Obviously, $\mathcal{E}\Big{(}A-YP\Big{)}$ and
$\mathcal{F}\Big{(}-[B+PY]\Big{)}$ are reversible , we may let
$V=\mathcal{E}\Big{(}A-YP\Big{)}\cdot U\cdot\mathcal{F}\Big{(}-[B+PY]\Big{)}$
(54)
Now Eq.(53) could be transformed into
$\biggl{[}\mathcal{E}\Big{(}A-YP\Big{)}\cdot{\frac{d}{dx}}U\cdot\mathcal{F}\Big{(}-[B+PY]\Big{)}+(A-YP)V-V(B+PY)\biggl{]}+VPV+V(B+PY)-(A-YP)V=0$
(55)
or,
${\frac{d}{dx}}U+U\biggl{[}\mathcal{F}\Big{(}-[B+PY]\Big{)}\cdot
P\cdot\mathcal{E}\Big{(}A-YP\Big{)}\cdot\biggl{]}U=0$ (56)
3. 3.
Let
$R:=\mathcal{F}\Big{(}-[B+PY]\Big{)}\cdot P\cdot\mathcal{E}\Big{(}A-YP\Big{)}$
(57)
According to Theorem 2.2, $U$ has solution
$U=W_{1}\cdot W^{-1}_{2}$ (58)
where
$\displaystyle\left[\begin{array}[]{c}W_{{1}}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr
W_{{2}}\end{array}\right]=\mathcal{E}\biggl{(}\left[\begin{array}[]{cc}0&0\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
R&0\end{array}\right]\biggl{)}\cdot\left[\begin{array}[]{c}U\mid_{x=0}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
I\end{array}\right]=\biggl{(}I+\int_{0}^{x}\\!{\left[\begin{array}[]{cc}0&0\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
R&0\end{array}\right]{dt}\biggl{)}}\cdot\left[\begin{array}[]{c}U\mid_{x=0}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
I\end{array}\right]=\left[\begin{array}[]{c}U\mid_{x=0}\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr I+\int_{0}^{x}\\!R(t)\left(t\right){dt}\cdot
U\mid_{x=0}\end{array}\right]$ (59)
Now, Let’s consider how to choose Y , so that both $W$ and $U\mid_{x=0}$ are
as simple as possible. It’s clear that
when $Y|_{x=0}=0$, $U|_{x=0}=Y|_{x=0}=W|_{x=0}$
In this case,
$U=W|_{x=0}\cdot\biggl{[}I+\int_{0}^{x}\\!R(t)\left(t\right){dt}\cdot(W\mid_{x=0})\biggl{]}^{-1}$
(60)
It should be noticed that
$\biggl{[}I+\int_{0}^{x}\\!R(t)\left(t\right){dt}\cdot(W\mid_{x=0})\biggl{]}$
is reversible, otherwise
$I+\int_{0}^{x}\\!R(t)\left(t\right){dt}\cdot(W\mid_{x=0})\equiv 0$ (61)
which is clearly impossible.
According to transformation(54), the solution of Eq.(48) is
$\displaystyle W=Y+V=Y+\mathcal{E}\Big{(}A-YP\Big{)}\cdot
W|_{x=0}\cdot\biggl{[}I+\int_{0}^{x}\\!R(t)\left(t\right){dt}\cdot(W\mid_{x=0})\biggl{]}^{-1}\cdot\mathcal{F}\Big{(}-[B+PY]\Big{)}$
(62)
_where $Y(x)\equiv 0$ , if and only if $Q(x)\equiv 0$._
## 5 Acknowledgments
Thanks Prof.Qiyan Shi’s enthusiastic instruction and precious advice on the
thesis . The work is also supported by Prof.Youdong Zeng; thanks for his many
helpful discussions and suggestions on this paper. Besides, thanks Prof.Guowei
Chen for many valuable personal communications and guidance concerning the
school work.
## References
* [1] Zhuxi Wang & Dunren Guo, An Introduction to Special Functions, Peking University Press.
* [2] Andrei D. Polyanin and Valentin F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Chapman and Hall/CRC, 2nd Edition(0.1.4), 2002\.
* [3] Gongning Chen, The Theory and Application of Matrix, Science Press(Beijing), 2007\.
* [4] Thayer Watkins, Silicon Valley & Tornado Alley, The Solution of the Riccati Equation, applet-magic.com .
|
arxiv-papers
| 2010-06-24T14:40:47 |
2024-09-04T02:49:11.166821
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yimin Yan",
"submitter": "Yimin Yan",
"url": "https://arxiv.org/abs/1006.4804"
}
|
1006.4869
|
# Automorphism Groups on Tropical Curves:
Some Cohomology Calculations
David Joyner, Amy Ksir, and Caroline Grant Melles David Joyner, Mathematics
Department, United States Naval Academy, Annapolis, MD 21402 wdj@usna.edu Amy
Ksir, Mathematics Department, United States Naval Academy, Annapolis, MD 21402
ksir@usna.edu Caroline Grant Melles, Mathematics Department, United States
Naval Academy, Annapolis, MD 21402 cgg@usna.edu
(Date: 2010-10-15)
###### Abstract.
Let $X$ be an abstract tropical curve and let $G$ be a finite subgroup of the
automorphism group of $X$. Let $D$ be a divisor on $X$ whose equivalence class
is $G$-invariant. We address the following question: is there always a divisor
$D^{\prime}$ in the equivalence class of $D$ which is $G$-invariant? Our main
result is that the answer is “yes” for all abstract tropical curves. A key
step in our proof is a tropical analogue of Hilbert’s Theorem 90.
###### 2010 Mathematics Subject Classification:
14T05, 14H37
## 1\. Introduction
We begin by defining an abstract tropical curve $X$ in terms of star-shaped
sets, as a generalization of a metric graph in which all leaves have infinite
length. Our definition is based on papers of Zhang [Z], Baker and Rumely [BR],
and Haase, Musiker, and Yu [HMY]. See also Mikhalkin and Zharkov [MZ], Baker
and Faber [BF], and Richter-Gebert, Sturmfels, and Theobald [RST]. We define
rational functions, divisors, and divisor classes in this setting, following
the conventions of Mikhalkin and Zharkov [MZ], Gathmann and Kerber [GK], and
Haase, Musiker, and Yu [HMY]. We note that the automorphism group of an
abstract tropical curve $X$ is necessarily finite unless $X$ is homeomorphic
to a circle or a closed interval.
In Section 3 we review basic definitions of group cohomology and set up two
long exact sequences which will be used to prove our main results. These long
exact sequences give relationships among the cohomology groups of $G$ with
coefficients in the real numbers $\mathbb{R}$, the group $M(X)$ of rational
functions on $X$, the group ${\rm Prin}(X)$ of principal divisors on $X$, the
group ${\rm Div}(X)$ of divisors on $X$, and the Picard group ${\rm Pic}(X)$
of classes of linearly equivalent divisors on $X$.
In Section 4 we use methods similar to those used in the classical case in
Goldstein, Guralnick, and Joyner [GGJ] to show that if $G$ is a finite
subgroup of the automorphism group of $X$ then
1. (1)
$H^{1}(G,\mathbb{R})=0$,
2. (2)
$H^{1}(G,M(X))=0$ (Tropical Analogue of Hilbert’s Theorem 90),
3. (3)
$H^{2}(G,\mathbb{R})=0$, and
4. (4)
$H^{1}(G,{\rm Prin}(X))=0$ (a direct consequence of the vanishing of
$H^{1}(G,M(X))$ and $H^{2}(G,\mathbb{R})$).
The vanishing of $H^{1}(G,\mathbb{R})$ implies that every $G$-invariant
principal divisor is the image of a $G$-invariant rational function. The
vanishing of $H^{1}(G,{\rm Prin}(X))$ gives our main result, which is that
every $G$-invariant divisor class contains a $G$-invariant divisor.
In Section 5 we give two additional results on group cohomology for abstract
tropical curves. We show that if $G$ is a finite subgroup of the automorphism
group of $X$ then $H^{1}(G,{\rm Div}(X))=0$ and
$H^{2}(G,M(X)\otimes\mathbb{Q})=0$. It would be interesting to know whether
$H^{2}(G,M(X))$ vanishes, since this would be a tropical analogue of Tsen’s
Theorem.
We conclude in Section 6 with some remarks on invariance in degree 0.
## 2\. Background on Abstract Tropical Curves
Let $\mathbb{T}$ be the tropical semiring
$\mathbb{T}=\mathbb{R}\cup\\{-\infty\\}$
with the tropical operations
$x\oplus y=\max\\{x,y\\}$
and
$x\odot y=x+y$
(so tropical multiplication is classical addition). We follow the conventions
of Mikhalkin [M1], using max rather than min for tropical addition.
Note that there is no inverse for tropical addition, but that $-\infty$ is a
neutral element for tropical addition since
$-\infty\oplus x=\max\\{-\infty,x\\}=x$
for any $x$ in $\mathbb{T}$.
Similarly, $0$ is a neutral element for tropical multiplication since
$0\odot x=0+x=x$
for any $x$ in $\mathbb{T}$. Every element $x$ of $\mathbb{T}$ except
$-\infty$ has an inverse $-x$ under tropical multiplication.
The topology on $\mathbb{T}$ will be taken to be the topology generated by all
open sets of $\mathbb{R}$ plus all sets of the form
$[-\infty,b)=\\{-\infty\\}\cup(-\infty,b)$ for $b\in\mathbb{R}$. In this
topology, the set $[-\infty,b]$ is compact.
For convenience, we sometimes omit the tropical operators. For example, a
tropical polynomial
$\sum_{i=0}^{n}a_{i}x^{i},$
with $a_{i}\in\mathbb{T}$ for all $i$, means
$\max\\{a_{i}+ix\\}.$
Thus a tropical polynomial on $\mathbb{R}$ is a piecewise linear function with
nonnegative integer slopes, except when it is identically $-\infty$, i.e.,
except when $a_{i}=-\infty$ for all $i$.
A tropical polynomial in two variables may be used to define a tropical curve
embedded in $\mathbb{R}^{2}$, whose support is the nonlinear locus of the
polynomial. Embedded tropical curves may also be defined in $\mathbb{R}^{n}$
and in tropical projective space $\mathbb{T}{\mathbb{P}}^{n}$. See, e.g.,
Mikhalkin [M2] and [M3], Richter-Gebert, Sturmfels, and Theobald [RST], Speyer
and Sturmfels [SS], and Maclagan and Sturmfels [MS]. In this paper, however,
we are concerned with abstract tropical curves, rather than embedded curves.
There are several ways to define an abstract tropical curve. We define an
abstract tropical curve in terms of star-shaped sets, as a generalization of a
metric (or metrized) graph in which all leaves have infinite length. Our
definition is based on papers of Zhang [Z], Baker and Rumely [BR], and Haase,
Musiker, and Yu [HMY]. See also Mikhalkin and Zharkov [MZ], Mikhalkin [M1],
and Baker and Faber [BF].
$\bullet$$n=1$ | $\bullet$$n=2$ | $\bullet$$n=3$
---|---|---
Figure 1. Star-shaped set having $n$ arms.
###### Definition 1.
[Star-shaped set]
A star-shaped set is a set of the form
$S(n,r)=\\{z\in\mathbb{C}:z=te^{\frac{2\pi ik}{n}}\mbox{ for some
}t\in[0,r)\mbox{ and }k\in\mathbb{Z}\\}$
where $n$ is a positive integer and $r$ is a positive real number. For a fixed
$k\in\mathbb{Z}$ the subset $\\{z\in\mathbb{C}:z=te^{\frac{2\pi ik}{n}}\mbox{
for some }t\in[0,r)\\}$ is called an arm; the number of distinct arms is $n$.
The point at which $z=0$ is called the center of the star-shaped set. We give
each arm of $S(n,r)$ the metric induced from the Euclidean metric on
$\mathbb{C}$; we give $S(n,r)$ as a whole the path metric and the metric
topology.
###### Definition 2.
[Metric topological graph]
Let $X$ be a compact connected topological space such that each point $P\in X$
has a neighborhood homeomorphic to a star-shaped set $S(n_{p},r_{P})$, where
the homeomorphism takes $P$ to the center of the star-shaped set. The positive
integer $n_{P}$, which is the number of arms of $S(n_{P},r_{P})$, is called
the valence of $P$. Let $X_{0}$ be $X\setminus\\{P\in X:n_{P}=1\\}$, i.e., $X$
with its 1-valent points removed. A metric topological graph is a topological
space $X$ as above, with a metric space structure on $X_{0}$ so that each
point $P\in X_{0}$ has a neighborhood isometric to $S(n_{P},r_{P})$ for some
integer $n_{P}$ and some positive real number $r_{P}$.
Note that by compactness, there will be at most finitely many points $P\in X$
with valence $n_{P}\neq 2$.
###### Definition 3.
[Model of a metric topological graph]
Suppose that $X$ is a metric topological graph. Let $V$ be any finite nonempty
subset of $X$ such that $V$ contains all of the points with valence $n_{P}\neq
2$. Then $X\setminus V$ is homeomorphic to a finite disjoint union of open
intervals. For a given $X$, such a choice of $V$ gives rise to a graph
$G(X,V)$ with $V$ as the vertex set and the connected components of
$X\setminus V$ as the edge set. This graph is called a model for $X$. Unless
$X$ is homeomorphic to a circle, we can take $V$ to be $\\{P\in X:n_{P}\neq
2\\}$; we will call the associated graph the minimal graph for $X$. For any
model of $X$, an edge adjacent to a 1-valent vertex is called a leaf; the
other edges are called inner edges.
###### Definition 4.
[Abstract tropical curve]
Let $X$ be a metric topological graph such that, in every model, all inner
edges have finite length and all leaves have infinite length. An abstract
tropical curve is such a metric topological graph, with a positive integer
multiplicity associated to each edge of its minimal graph, or, in the case of
a circle, a multiplicity associated to the circle itself.
We will call 1-valent vertices of an abstract tropical curve infinite points.
All other points are called finite points. We note that the topology near a
1-valent point is not the metric topology, because the leaf with its endpoints
is compact but has infinite length. Note also that if $P$ is a 1-valent point,
then there is a homeomorphism $\iota$ from an interval $[-\infty,b)$ in
$\mathbb{T}$, where $b\in\mathbb{R}$, to a neighborhood of $P$ in $X$, such
that $\iota$ takes $-\infty$ to $P$ and such that the restriction of $\iota$
to $(-\infty,b)$ is an isometry.
###### Remark 1.
Given a finite graph $G$ with
1. (1)
a finite length associated to each inner edge,
2. (2)
infinite length associated to each leaf, and
3. (3)
a positive integer multiplicity associated to each edge,
there is a tropical curve (as defined above) with $G$ as a model.
###### Definition 5.
[Automorphisms of abstract tropical curves]
An automorphism $g:X\rightarrow X$ of an abstract tropical curve $X$ will be
defined to be a map such that
1. (1)
$g$ is a homeomorphism on the underlying topological space of $X$,
2. (2)
$g$ is an isometry on $X_{0}$, and
3. (3)
$g$ preserves multiplicities.
###### Remark 2.
If $X$ is not homeomorphic to a circle, then $g$ will be a graph automorphism
on the minimal graph for $X$, taking vertices to vertices and edges to edges.
The automorphisms of $X$ form a group, Aut($X$). In the classical case,
Hurwitz’s automorphism theorem gives a bound on the number of automorphisms of
a smooth complex projective algebraic curve of genus $g>1$. In the tropical
case, we note the following bound.
###### Theorem 1.
If an abstract tropical curve $X$ has a minimal graph with only one edge, or
is homeomorphic to a circle, then Aut($X$) contains an infinite number of
translations. Otherwise, the automorphism group Aut($X$) of $X$ is finite, and
moreover if $l$ is the number of leaves of the minimal model for $X$ and $i$
is the number of inner edges, then ${\rm Aut}(X)$ is contained in the product
of symmetric groups $S_{l}\times S_{2i}$.
###### Proof.
In the case where $X$ has a minimal graph with only one edge, or is
homeomorphic to a circle, a translation satisfies all three conditions to be
an automorphism. In any other case, each leaf must have a finite endpoint, and
any automorphism of $X$ will map a leaf to another leaf, with the finite
endpoint mapping to the finite endpoint and the infinite endpoint mapping to
the infinite endpoint. For each pair of leaves, there is exactly one way to do
this preserving the metric on $X_{0}$. Similarly, an automorphism of $X$ must
map an inner edge of the minimal graph isometrically to another inner edge of
the minimal graph, with the same length and multiplicity. For each such pair
of edges, there are two such isometries. $\Box$
###### Remark 3.
The tropical projective line $\mathbb{T}{\mathbb{P}}^{1}$ is a single edge of
infinite length plus its endpoints, and the circle is a genus 1 tropical
curve. See Mikhalkin [M1] for more details.
###### Example 1.
Let $n$ be an integer greater than $1$, and let $\Gamma_{n}$ be the abstract
tropical curve consisting of $n$ leaves, with their endpoints, emanating from
a single $n$-valent point. Then ${\rm Aut}(\Gamma_{n})=S_{n}$.
Let $X$ be an abstract tropical curve and let $f$ be a continuous real-valued
function on $X_{0}$. Let $P$ be a point in $X_{0}$ and let
$\iota:S(n_{P},r_{P})\rightarrow U_{P}$ be an isometry from a star-shaped set
to a neighborhood of $P$, taking the center of $S(n_{P},r_{P})$ to $P$. We
will say that $f$ is piecewise linear at $P$ if $f\circ\iota$ is piecewise
linear on each arm of the star-shaped set. In other words, for each
$k\in\\{1,\ldots,n_{P}\\}$, the composition $[0,r_{P})\to\mathbb{R}$ given by
$t\mapsto f(\iota(te^{\frac{2\pi ik}{n_{P}}}))$ is piecewise linear. If $f$ is
piecewise linear at every point $P\in X_{0}$, we will say that it is piecewise
linear on $X$. A point of $X_{0}$ at which $f$ is not linear is called a
singular point of $f$. If $f$ is not locally constant at a point $P$ of
valence $n_{P}>2$, then $P$ is a singular point of $f$. The slope of $f$, on
any open set on which $f$ is linear, is well-defined up to sign. We will say
that $f$ is piecewise linear with integer slope if $f$ is piecewise linear and
has integer slope on any open set on which it is linear.
Recalling that tropical polynomials on $\mathbb{R}$ (if not identically
$-\infty$) are piecewise linear functions with nonnegative integer slope, and
that tropical division corresponds to classical subtraction, we define
rational functions as follows.
###### Definition 6.
[Rational functions on an abstract tropical curve]
A rational function on an abstract tropical curve $X$ is a continuous real-
valued function on $X_{0}$, the abstract tropical curve minus its 1-valent
points, which is piecewise linear with integer slope and which has only
finitely many singular points. Note that a rational function does not have to
be defined at the 1-valent points.
Note also that for the purposes of this paper, we do not include functions
which are identically equal to $-\infty$ in the set of rational functions.
Let $M(X)$ denote the set of all rational functions on $X$. Note that $M(X)$
forms a group with identity element $0$ under tropical multiplication
(classical addition).
Automorphisms of $X$ act on $M(X)$ via their action on $X$. If $g$ is an
automorphism of $X$ and $f$ is a rational function on $X$, then $gf$ is the
rational function given by
$gf(P)=f(g^{-1}(P))$
for every point $P$ in the abstract tropical curve without infinite points
$X_{0}$, i.e., $gf=f\circ g^{-1}:X_{0}\rightarrow\mathbb{R}$.
###### Definition 7.
[Divisors on abstract tropical curves]
A divisor on an abstract tropical curve $X$ is a finite formal sum of the form
$D=\sum_{P\in X}a_{P}P$
where, for each $P$, $a_{P}$ is an integer, and all but finitely many are $0$.
The collection of all divisors on $X$ forms a group ${\rm Div}(X)$ under
addition, i.e., the free group over $\mathbb{Z}$ generated by the points of
$X$.
###### Definition 8.
[Order of a rational function $f$ at a point $P$ of $X$]
Let $f$ be a rational function on an abstract tropical curve $X$. Essentially,
the order of $f$ at a point $P$ of $X$ is the weighted sum of all slopes of
$f$ in the direction outward from $P$, for all edges emanating from $P$, where
each edge is weighted according to its multiplicity. We state this condition
more explicitly below.
First, consider the case in which $P$ is not an infinite point, i.e., which is
not $1$-valent. Then there is an isometry $\iota$ from a star-shaped set
$S(n_{P},r_{P})$ to a neighborhood of $P$, taking the center of
$S(n_{P},r_{P})$ to $P$. Since $f$ is a rational function, we can restrict the
neighborhood and choose a smaller $r_{P}$, if necessary, so that $f$ is linear
on each arm of $S(n_{P},r_{P})$. Thus for each integer
$k\in\\{1,...,n_{P}\\}$, the composition $[0,r_{P})\to\mathbb{R}$ given by
$t\mapsto f(\iota(te^{\frac{2\pi ik}{n_{P}}}))$ is linear, with integer slope,
i.e.,
$f(\iota(te^{\frac{2\pi ik}{n_{P}}}))=\lambda(k)t+b$
for some integer $\lambda(k)$ and real number $b$. We define the order of $f$
at $P$ to be
$\text{ord}_{P}(f)=\sum_{k=1}^{n_{P}}m(k)\lambda(k),$
where $m(k)$ is the multiplicity of the edge which contains the image under
$\iota$ of $te^{\frac{2\pi ik}{n_{P}}}$, $0<t<r_{P}$.
Now suppose that $P$ is a 1-valent point. Then there is an isometry $\iota$
from the interval $(-\infty,b)$ in $\mathbb{R}$ to a punctured neighborhood of
$P$. Again, since $f$ is a rational function, we can restrict the neighborhood
and choose a smaller $b$, if necessary, so that $f\circ\iota$ is linear with
integer slope $\lambda$. In this case we define
$\text{ord}_{P}(f)=m\lambda,$
where again $m$ is the multiplicity of the edge adjacent to $P$.
If a rational function $f$ is linear at a point $P$, then
$\text{ord}_{P}(f)=0$, so that there are only a finite number of points $P$ at
which $\text{ord}_{P}(f)\neq 0$ since $f$ has only finitely many singular
points.
###### Definition 9.
[Principal divisors on an abstract tropical curve $X$]
Let $f$ be an element of $M(X)$, i.e., $f$ is a rational function on the
abstract tropical curve $X$. We define the divisor determined by $f$ to be
$\text{div}(f)=(f)=\sum_{P\in X}\text{ord}_{P}(f)P.$
We call such divisors principal. The set of all principal divisors forms a
subgroup ${\rm Prin}(X)$ of ${\rm Div}(X)$.
We will say that the degree of a divisor $D=\sum a_{P}P$ is $\sum a_{P}$. Note
that the degree of a principal divisor is always 0, since if $P$ and $Q$ are
endpoints of a segment on which $f$ is linear, the slopes of $f$ emanating
from $P$ and $Q$ are the negative of one another. (In the special case $P=Q$,
i.e., if $f$ is linear on a loop, then $f$ must be constant on the loop, so
the slopes emanating from $P=Q$ on the loop are zero.) Note also that the
degree map is a homomorphism from ${\rm Div}(X)$ to $\mathbb{Z}$.
###### Definition 10.
[Linear equivalence of divisors]
Divisors $D$ and $D^{\prime}$ are said to be linearly equivalent if there is a
rational function $f$ such that
$D=D^{\prime}+(f).$
###### Example 2.
Let $\Gamma_{n}$ be the abstract tropical curve consisting of $n$ leaves, with
their endpoints, emanating from a single $n$-valent point $O$. Let $P$ be any
other point on $\Gamma_{n}$. Then $P$ and $O$ are linearly equivalent as
divisors, because there is a rational function with slope 1 on the path from
$O$ to $P$ and constant everywhere else.
The map div is a group homomorphism
$\text{div}:M(X)\rightarrow{\rm Div}(X)$
from the group of rational functions on $X$ under tropical multiplication to
the group of divisors under addition since
$\text{div}(f_{1}\odot
f_{2})=\text{div}(f_{1}+f_{2})=\text{div}(f_{1})+\text{div}(f_{2}).$
The image of the map div is the group ${\rm Prin}(X)$ of principal divisors.
The quotient group
${\rm Pic}(X)={\rm Div}(X)/{\rm Prin}(X)$
is called the Picard group. The elements of the Picard group are called
divisor classes. The divisor class of a divisor $D$ is denoted $[D]$ and
consists of all divisors which are linearly equivalent to $D$.
###### Example 3.
Let $\Gamma_{n}$ be as in Example 2. Every degree $d$ divisor on $\Gamma_{n}$
is linearly equivalent to $dO$, by Example 2. Therefore
${\rm Pic}(\Gamma_{n})\cong\mathbb{Z}.$
## 3\. Background on Group Cohomology
Let $X$ be an abstract tropical curve and let $G$ be a finite subgroup of the
automorphism group Aut($X$) of $X$. Recall that if $X$ has a minimal graph
with only one edge, or is homeomorphic to a circle, then the automorphism
group Aut($X$) contains an infinite number of translations. Otherwise,
Aut($X$) is finite, so every subgroup $G$ of Aut($X$) is necessarily finite.
We review some background material on group cohomology which we will need.
Group cohomology may also be defined in terms of the Ext functor (see, e.g.,
Rotman [R] p. 870). For further information on group cohomology, we refer to
Serre [S], ch. VII, or the survey by Joyner [J].
Let $A$ be a $\mathbb{Z}[G]$-module. We can view $\mathbb{Z}$ as another
$\mathbb{Z}[G]$-module, via the trivial action of $G$ on $\mathbb{Z}$. The
$0$th cohomology group of $G$ with coefficients in $A$ is
$H^{0}(G,A)=\text{Hom}_{G}(\mathbb{Z},A),$
and is isomorphic to the group $A^{G}$ of $G$-invariant elements of $A$. The
covariant functor of $G$-invariants, $A\longmapsto H^{0}(G,A)\cong A^{G}$ is
left exact.
The $1$-cocycles on $G$ with coefficients in $A$ are defined by
$Z^{1}(G,A)=\\{\phi:G\to A\ |\ \forall g_{1},g_{2}\in G,\
\phi(g_{1})+g_{1}\phi(g_{2})=\phi(g_{1}g_{2})\\},$
the $1$-coboundaries by
$B^{1}(G,A)=\\{\phi:G\to A\ |\ \exists f\in A{\ :\ }\forall g\in G,\
\phi(g)=gf-f\\},$
and the $1$-cohomology by
$H^{1}(G,A)=Z^{1}(G,A)/B^{1}(G,A).$
(It is straightforward to check that $B^{1}(G,A)\subset Z^{1}(G,A)$.)
The $2$-cocycles on $G$ with coefficients in $A$ are defined by
$\begin{array}[]{l}Z^{2}(G,A)=\\{\phi:G\times G\to A\ |\ \forall
g_{1},g_{2},g_{3}\in G,\\\ \qquad\qquad\qquad\qquad
g_{1}\phi(g_{2},g_{3})-\phi(g_{1}g_{2},g_{3})+\phi(g_{1},g_{2}g_{3})-\phi(g_{1},g_{2})=0\\},\end{array}$
the $2$-coboundaries111It is straightforward to check that $B^{2}(G,A)\subset
Z^{2}(G,A)$. by
$\begin{array}[]{l}B^{2}(G,A)=\\{\phi:G\times G\to A\ |\ \exists\psi:G\to A{\
:}\\\ \qquad\qquad\qquad\qquad\forall g_{1},g_{2}\in G,\
\phi(g_{1},g_{2})=\psi(g_{1})+g_{1}\psi(g_{2})-\psi(g_{1}g_{2})\\},\end{array}$
and the $2$-cohomology by
$H^{2}(G,A)=Z^{2}(G,A)/B^{2}(G,A).$
Now we wish to apply this general theory to the case of abstract tropical
curves. We will describe two short exact sequences. Lemma 1 below is the
tropical analogue of the well-known short exact sequence
$1\rightarrow F^{\times}\rightarrow F(X)^{\times}\rightarrow{\rm
Prin}(X)\rightarrow 0,$
for an irreducible non-singular algebraic curve $X$ over an algebraically
closed field $F$, where $F^{\times}$ denotes the field minus its zero element
and $F(X)^{\times}$ denotes the rational functions on $X$ which are not
identically 0. In the tropical case we replace $F^{\times}$ by
$\mathbb{T}^{\times}=\mathbb{R}$ and $F(X)^{\times}$ by $M(X)$.
We note that $\mathbb{R}$, $M(X)$, and ${\rm Div}(X)$ may be viewed as
$\mathbb{Z}[G]$-modules. The action of $G$ on $\mathbb{R}$ is the trivial
action. The action of $G$ on $M(X)$ is given by $gf(P)=f(g^{-1}P)$, for $g\in
G$, $f\in M(X)$, and $P\in X$. The action of $G$ on ${\rm Div}(X)$ is the
obvious one, i.e., if $D=\sum a_{P}P$ and $g\in G$, then $gD=\sum a_{P}gP$. We
note that the actions of $G$ on $M(X)$ and ${\rm Div}(X)$ are compatible,
since if $f\in M(X)$ and $g\in G$, then
$\displaystyle\text{div}(gf)$ $\displaystyle=\sum_{P\in X}\text{ord}_{P}(gf)P$
$\displaystyle=\sum_{Q\in X}\text{ord}_{gQ}(f\circ g^{-1})gQ$
$\displaystyle=\sum_{Q\in X}\text{ord}_{Q}(f)gQ$ $\displaystyle=g\
\text{div}(f).$
Thus the map $\text{div}:M(X)\rightarrow{\rm Div}(X)$ is a
$\mathbb{Z}[G]$-module homomorphism.
###### Lemma 1.
There is a short exact sequence of $\mathbb{Z}[G]$-modules,
$0\rightarrow\mathbb{R}\rightarrow M(X)\rightarrow{\rm Prin}(X)\rightarrow 0.$
###### Proof.
The order of a rational function $f$ at a point is the sum of the outgoing
slopes. For $f$ to be in the kernel of the map $M(X)\rightarrow{\rm Prin}(X)$,
the sum of its outgoing slopes at each point must be equal to $0$.
For $f$ to have order $0$ at every $1$-valent point, $f$ must be constant on a
punctured open neighborhood of each $1$-valent point (i.e., on a neighborhood
of the vertex minus the vertex itself). Removing these open sets gives us a
compact set $Y$ on which $f$ is continuous. Therefore, $f$ must take a minimum
somewhere on $Y$. But at the point where the minimum is attained, all outgoing
slopes are greater than or equal to $0$. Since the slopes sum to 0, they must,
in fact, all be $0$. Therefore, $f$ must be constant. $\Box$
By the definition of the Picard group, we have a short exact sequence of
$\mathbb{Z}[G]$-modules.
$0\rightarrow{\rm Prin}(X)\rightarrow{\rm Div}(X)\rightarrow{\rm
Pic}(X)\rightarrow 0.$
From Lemma 1 and the short exact sequence for ${\rm Pic}(X)$ above, we obtain
long exact sequences
(1) $\begin{split}0\rightarrow H^{0}(G,\mathbb{R})\rightarrow
H^{0}(G,M(X))\rightarrow H^{0}(G,{\rm Prin}(X))\rightarrow\\\
H^{1}(G,\mathbb{R})\rightarrow H^{1}(G,M(X))\rightarrow H^{1}(G,{\rm
Prin}(X))\rightarrow\\\ H^{2}(G,\mathbb{R})\rightarrow
H^{2}(G,M(X))\rightarrow H^{2}(G,{\rm Prin}(X))\rightarrow\ldots\end{split}$
and
(2) $\begin{split}0\rightarrow H^{0}(G,{\rm Prin}(X))\rightarrow H^{0}(G,{\rm
Div}(X))\rightarrow H^{0}(G,{\rm Pic}(X))\rightarrow\\\ H^{1}(G,{\rm
Prin}(X))\rightarrow H^{1}(G,{\rm Div}(X))\rightarrow H^{1}(G,{\rm
Pic}(X))\rightarrow\\\ H^{2}(G,{\rm Prin}(X))\rightarrow H^{2}(G,{\rm
Div}(X))\rightarrow H^{2}(G,{\rm Pic}(X))\rightarrow\ldots.\end{split}$
## 4\. Proof of Main Result
Let $X$ be an abstract tropical curve and let $G$ be a finite subgroup of the
automorphism group of $X$. In order to prove our main result, Theorem 3, we
will compute various terms of the long exact sequences (1) and (2).
###### Lemma 2.
$H^{1}(G,\mathbb{R})=0.$
###### Proof.
Since the action of $G$ on $\mathbb{R}$ is trivial, the condition on
1-cocycles reduces to
$Z^{1}(G,\mathbb{R})=\\{\phi:G\rightarrow\mathbb{R}\mid\forall g_{1},g_{2}\in
G,\phi(g_{1})+\phi(g_{2})=\phi(g_{1}g_{2})\\}.$
This means that $\phi$ is a homomorphism from the finite group $G$ to
$\mathbb{R}$, so $\phi$ must be the zero map. $\Box$
###### Corollary 1.
The following is a short exact sequence
$0\rightarrow\mathbb{R}\to M(X)^{G}\rightarrow{\rm Prin}(X)^{G}\rightarrow 0.$
In particular, every $G$-invariant principal divisor is the divisor of a
$G$-invariant rational function.
###### Proof.
Apply Lemma 2 to the long exact sequence (1). $\Box$
In the case of an algebraic curve, $H^{1}(G,F(X)^{\times})=1$, by Hilbert’s
Theorem 90 (see, e.g., Rotman [R] 10.128 and 10.129). The following theorem is
a tropical analogue of Hilbert’s Theorem 90.
###### Theorem 2.
Let $X$ be an abstract tropical curve and let $G$ be a finite subgroup of the
automorphism group of $X$. Then
$H^{1}(G,M(X))=0,$
where $M(X)$ is the group of rational functions on $X$ under tropical
multiplication (classical addition).
###### Proof.
Pick $\phi\in Z^{1}(G,M(X))$. Let $f$ be the tropical sum
$f=-{\sum_{g\in G}}^{\text{trop}}\phi(g),$
i.e., if $P\in X$,
$f(P)=-\max_{g\in G}\\{\phi(g)(P)\\},$
which is the negative of the tropical average of $\phi$ over $G$.
We compute, for $h\in G$,
$\displaystyle hf(P)$ $\displaystyle=-\max\\{h\phi(g)(P)\\}$
$\displaystyle=-\max\\{-\phi(h)(P)+\phi(hg)(P)\\}$
$\displaystyle=\phi(h)(P)+f(P).$
Therefore every cocycle is a coboundary. $\Box$
###### Lemma 3.
$H^{2}(G,\mathbb{R})=0.$
###### Proof.
Since the action of $G$ on $\mathbb{R}$ is trivial,
$\begin{array}[]{l}Z^{2}(G,\mathbb{R})=\\{\phi:G\times G\to\mathbb{R}\ |\
\forall g_{1},g_{2},h\in G,\\\
\qquad\qquad\qquad\qquad\phi(g_{2},h)-\phi(g_{1}g_{2},h)+\phi(g_{1},g_{2}h)-\phi(g_{1},g_{2})=0\\}.\end{array}$
Given $\phi\in Z^{2}(G,\mathbb{R})$, define $\psi:G\rightarrow\mathbb{R}$ by
the classical sum
$\psi(g)=\frac{1}{\mid G\mid}\sum_{h\in G}\phi(g,h).$
Then for any $g_{1},g_{2}\in G$ we have
$\displaystyle\psi(g_{1})+g_{1}\psi(g_{2})-\psi(g_{1}g_{2})$
$\displaystyle=\psi(g_{1})+\psi(g_{2})-\psi(g_{1}g_{2})$
$\displaystyle=\frac{1}{\mid G\mid}\sum_{h\in
G}\left(\phi(g_{1},h)+\phi(g_{2},h)-\phi(g_{1}g_{2},h)\right)$
$\displaystyle=\frac{1}{\mid G\mid}\sum_{h\in
G}\left(\phi(g_{1},g_{2}h)+\phi(g_{2},h)-\phi(g_{1}g_{2},h)\right)$
$\displaystyle=\phi(g_{1},g_{2}).$
Therefore every 2-cocycle is a 2-coboundary, so $H^{2}(G,\mathbb{R})=0$.
$\Box$
###### Corollary 2.
$H^{1}(G,{\rm Prin}(X))=0.$
###### Proof.
Apply Proposition 2 and Lemma 3 to the long exact sequence (1). $\Box$
The following theorem is our main result and implies that the answer to the
question raised in the introduction is “yes” for all abstract tropical curves.
###### Theorem 3.
Let $X$ be an abstract tropical curve and let $G$ be a finite subgroup of the
automorphism group of $X$. Then the map
${\rm Div}(X)^{G}\to{\rm Pic}(X)^{G}$
is surjective, i.e., every $G$-invariant divisor class contains a
$G$-invariant divisor.
###### Proof.
Apply Corollary 2 to the long exact sequence (2). $\Box$
## 5\. Further Results on Group Cohomology
of Abstract Tropical Curves
Let $X$ be an abstract tropical curve and let $G$ be a finite subgroup of the
automorphism group of $X$. Proposition 1 below is analogous to a result for
algebraic curves which is proven in Goldstein, Guralnick, and Joyner [GGJ]
using Shapiro’s Lemma. The proof below is similar but more direct.
###### Proposition 1.
$H^{1}(G,{\rm Div}(X))=0.$
###### Proof.
For each $P\in X$, let $G_{P}$ be the stabilizer subgroup of $G$ given by
$G_{P}=\\{g\in G|gP=P\\}$. If $h_{1}$ and $h_{2}$ are elements of $G$ whose
left cosets $\tilde{h_{1}}$ and $\tilde{h_{2}}$ in $G/G_{P}$ are equal, then
$h_{1}P=h_{2}P$. Therefore it makes sense to define, for the left coset
$\tilde{h}$ of any element $h\in G$, $\tilde{h}P=hP$. Let
$L_{P}=\oplus_{\tilde{h}\in G/G_{P}}\mathbb{Z}[\tilde{h}P].$
Let $GX$ be the set of all orbits of points in $X$ and let $GX/G$ be a
complete set of representatives in $X$ of these orbits. Then ${\rm Div}(X)$ is
the direct sum of the subgroups $L_{P}$ for $P$ in $GX/G$.
Using the characterization of group cohomology as an Ext functor (see, e.g.,
Rotman [R] p. 870) and the fact that Ext preserves direct products in its
second argument (see, e.g., Rotman [R] p. 854), it follows that if
$H^{1}(G,L_{P})=0$ for all $P$ in $GX/G$, then $H^{1}(G,{\rm Div}(X))=0$.
Next we show that $L_{P}$ is isomorphic to the co-induced group
$L^{\prime}=\text{Coind}_{G_{P}}^{G}(\mathbb{Z})$ given by
$L^{\prime}=\\{f:G\rightarrow\mathbb{Z}\ |\ f(gh)=f(h)\ \text{for all}\ g\in
G_{P}\ \text{and}\ h\in G\\}.$
Each divisor in $L_{P}$ may be written in the form $\sum_{\tilde{h}\in
G/G_{P}}a(\tilde{h})\tilde{h}P$, where $a(\tilde{h})$ is an integer for each
$\tilde{h}$. Given such a divisor, we define a function
$f:G\rightarrow\mathbb{Z}$ by $f(h)=a(\tilde{h^{-1}})$. It is easily checked
that $f\in L^{\prime}$. If $f\in L^{\prime}$, and if
$\tilde{h_{1}}=\tilde{h_{2}}$, for some $h_{1}$, and $h_{2}$ in $G$, then
$f(h_{1}^{-1})=f(h_{2}^{-1})$, so we may define $a(\tilde{h})=f(h^{-1})$ and
the corresponding divisor $\sum_{\tilde{h}\in G/G_{P}}a(\tilde{h})\tilde{h}P$
in $L_{P}$.
The action of $G$ on $L^{\prime}$ is given by $gf(h)=f(hg)$ for $g$ and $h$ in
$G$. This action is consistent with the action of $G$ on $L_{P}$ and thus
$L_{P}$ and $L^{\prime}$ are isomorphic as $\mathbb{Z}[G]$-modules.
We will show that every 1-cocycle of $G$ in $L^{\prime}$ is a 1-coboundary.
Suppose that $\phi:G\rightarrow L^{\prime}$ is in $Z^{1}(G,L^{\prime})$. Let
$f$ be the map $f:G\rightarrow\mathbb{Z}$ given by
$f(h)=-\phi(h^{-1})(h)$
for $h\in G$. First we will show that $f\in L^{\prime}$ and then that
$\phi(k)=kf-f$ for all $k\in G$, so that $\phi\in B^{1}(G,L^{\prime})$.
Suppose that $g\in G_{P}$ and $h\in G$. We have
$\displaystyle f(gh)$ $\displaystyle=-\phi(h^{-1}g^{-1})(gh)$
$\displaystyle=-h^{-1}\phi(g^{-1})(gh)-\phi(h^{-1})(gh)\qquad\text{since
$\phi\in Z^{1}(G,L^{\prime})$}$
$\displaystyle=-\phi(g^{-1})(g)-\phi(h^{-1})(gh)\qquad\text{by the action of
$G$ on $L^{\prime}$}$
$\displaystyle=-\phi(g^{-1})(g)-\phi(h^{-1})(h)\qquad\text{because
$\phi(h^{-1})\in L^{\prime}$ and $g\in G_{P}$}$ $\displaystyle=f(g)+f(h).$
In particular, the restriction of $f$ to $G_{P}$ is a homomorphism from
$G_{P}$ to $\mathbb{Z}$, so $f$ must be $0$ on $G_{P}$, since $G_{P}$ is
finite. Therefore $f(gh)=f(h)$ for all $g\in G_{P}$ and $h\in G$, so $f$ is in
$L^{\prime}$.
Now we check that $\phi(k)=kf-f$ for all $k\in G$. For all $h$, $k$, and $l$
in $G$,
$\displaystyle\phi(k)(h)$
$\displaystyle=-k\phi(l)(h)+\phi(kl)(h)\qquad\text{since $\phi\in
Z^{1}(G,L^{\prime})$}$ $\displaystyle=-\phi(l)(hk)+\phi(kl)(h)\qquad\text{by
the action of $G$ on $L^{\prime}$.}$
Letting $l=k^{-1}h^{-1}$ gives
$\displaystyle\phi(k)(h)$
$\displaystyle=-\phi(k^{-1}h^{-1})(hk)+\phi(h^{-1})(h)$
$\displaystyle=f(hk)-f(h)$ $\displaystyle=kf(h)-f(h).$
Hence $\phi$ is in $B^{1}(G,L^{\prime})$, so
$H^{1}(G,L^{\prime})=H^{1}(G,L_{P})=0$. $\Box$
In the case of an algebraic curve, $H^{2}(G,F^{\times}(X)))=1$ by Tsen’s
theorem (a function field over an algebraically closed field is a $C^{1}$
field; see the Corollaries on pages 96 and 109 of Shatz [Sh], or §4 and §7 of
chapter X in Serre [S]). An analogue of Tsen’s theorem for tropical curves
would be the computation of $H^{2}(G,M(X))$. Such an analogue, if it exists,
would be very interesting. A partial result is as follows.
###### Lemma 4.
$H^{2}(G,M(X)\otimes\mathbb{Q})=0.$
###### Proof.
We will show that every 2-cocycle of $G$ in $M(X)\otimes\mathbb{Q}$ is a
2-coboundary. Suppose that $\phi\in Z^{2}(G,M(X)\otimes\mathbb{Q})$. Since
(tropical) $\mid G\mid$-th roots exist in $M(X)\otimes\mathbb{Q}$, we may
define a map $\psi:G\rightarrow M(X)\otimes\mathbb{Q}$ by the classical sum
$\psi(g)=\frac{1}{\mid G\mid}\sum_{h\in G}\phi(g,h).$
Then for $g_{1},g_{2}\in G$ we have
$\displaystyle\psi(g_{1})$ $\displaystyle+g_{1}\psi(g_{2})-\psi(g_{1}g_{2})$
$\displaystyle=\frac{1}{\mid G\mid}\sum_{h\in
G}\left(\phi(g_{1},h)+g_{1}\phi(g_{2},h)-\phi(g_{1}g_{2},h)\right)$
$\displaystyle=\frac{1}{\mid G\mid}\sum_{h\in
G}\left(\phi(g_{1},h)+\phi(g_{1}g_{2},h)-\phi(g_{1},g_{2}h)+\phi(g_{1},g_{2})-\phi(g_{1}g_{2},h)\right)$
$\displaystyle=\phi(g_{1},g_{2})+\frac{1}{\mid G\mid}\sum_{h\in
G}\phi(g_{1},h)-\frac{1}{\mid G\mid}\sum_{h\in G}\phi(g_{1},g_{2}h)$
$\displaystyle=\phi(g_{1},g_{2}).$
Hence $\phi$ is in $B^{2}(G,M(X)\otimes\mathbb{Q})$, so
$H^{2}(G,M(X)\otimes\mathbb{Q})=0$. $\Box$
## 6\. Invariance in Degree 0
Let $X$ be an abstract tropical curve and let $G$ be a finite subgroup of the
automorphism group of $X$. Let ${\rm Pic}^{0}(X)$ be the subgroup of ${\rm
Pic}(X)$ consisting of all degree $0$ divisors, i.e., the ${\rm Pic}^{0}(X)$
is the Jacobian variety of $X$.
###### Remark 4.
Consider the short exact sequence
$0\rightarrow{\rm Prin}(X)\rightarrow{\rm Div}^{0}(X)\rightarrow{\rm
Pic}^{0}(X)\rightarrow 0.$
Note that the map
${\rm Div}^{0}(X)^{G}\rightarrow{\rm Pic}^{0}(X)^{G}$
is surjective, as a trivial consequence of our main result. Thus every
$G$-invariant degree zero divisor class contains a $G$-invariant degree zero
divisor. The classical curve case is more complicated.
###### Remark 5.
Also, by Corollary 2, the map
$H^{1}(G,{\rm Div}^{0}(X))\rightarrow H^{1}(G,{\rm Pic}^{0}(X))$
is an injection.
Acknowledgement: The authors would like to thank the anonymous referee for
helpful suggestions.
## References
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http://arxiv.org/abs/math/0407428.
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http://arxiv.org/abs/math/0407427.
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http://front.math.ucdavis.edu/0706.0549) 2009.
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http://arxiv.org/abs/math/0601041.
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|
arxiv-papers
| 2010-06-24T20:11:51 |
2024-09-04T02:49:11.174527
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "David Joyner, Amy Ksir, and Caroline Grant Melles",
"submitter": "Amy Ksir",
"url": "https://arxiv.org/abs/1006.4869"
}
|
1006.4881
|
# Modeling Reactive Wetting when Inertial Effects are Dominant
Daniel Wheeler daniel.wheeler@nist.gov James A. Warren William J. Boettinger
Metallurgy Division, Materials Science and Engineering Laboratory, National
Institute of Standards and Technology, Gaithersburg, MD 20899, USA
###### Abstract
Recent experimental studies of molten metal droplets wetting high temperature
reactive substrates have established that the majority of triple-line motion
occurs when inertial effects are dominant. In light of these studies, this
paper investigates wetting and spreading on reactive substrates when inertial
effects are dominant using a thermodynamically derived, diffuse interface
model of a binary, three-phase material. The liquid-vapor transition is
modeled using a van der Waals diffuse interface approach, while the solid-
fluid transition is modeled using a phase field approach. The results from the
simulations demonstrate an $O\left(t^{-\nicefrac{{1}}{{2}}}\right)$ spreading
rate during the inertial regime and oscillations in the triple-line position
when the metal droplet transitions from inertial to diffusive spreading. It is
found that the spreading extent is reduced by enhancing dissolution by
manipulating the initial liquid composition. The results from the model
exhibit good qualitative and quantitative agreement with a number of recent
experimental studies of high-temperature droplet spreading, particularly
experiments of copper droplets spreading on silicon substrates. Analysis of
the numerical data from the model suggests that the extent and rate of
spreading is regulated by the spreading coefficient calculated from a force
balance based on a plausible definition of the instantaneous interface
energies. A number of contemporary publications have discussed the likely
dissipation mechanism in spreading droplets. Thus, we examine the dissipation
mechanism using the entropy-production field and determine that dissipation
primarily occurs in the locality of the triple-line region during the inertial
stage, but extends along the solid-liquid interface region during the
diffusive stage.
reactive wetting
## I Introduction
Characterizations of metal alloys wetting and spreading on dissolving
substrates typically assume that inertial effects are not dominant or that the
majority of dissipation is due to viscous forces Warren et al. (1998);
Villanueva et al. (2008); Su et al. (2009); Villanueva et al. (2009). In many
respects this seems an entirely reasonable approach since the majority of
experiments do not capture the early time behavior when inertial effects are
dominant, but focus on the late-stage spreading when chemical-diffusion
dominates and substrate dissolution occurs. Typically, experimental studies
measure only slow spreading on the order of seconds or even minutes for
millimeter-sized metal droplets consistent with diffusion-dominated spreading
Saiz et al. (1998); Warren et al. (1998); Voitovitch et al. (1999); Saiz and
Tomsia (2004). However, using improved techniques, a number of recent
experiments N. et al. (1998); Saiz and Tomsia (2004); Protsenko et al. (2008);
Yin et al. (2009) capture the rapid early-stage spreading and demonstrate that
the spreading duration is consistent with the inertial time scale Saiz et al.
(2007). The variations in experimental findings can be attributed to
differences in substrate temperature, composition of the vapor phase
influencing substrate oxidation, contact mechanisms between the substrate and
molten droplet, camera shutter speed, as well as other factors Saiz and Tomsia
(2004). An often important aspect of managing these factors is arresting the
formation of a substrate ridge on which the triple line becomes attached,
which can retard spreading considerably Saiz et al. (1998).
The spreading droplet is often characterized in terms of a velocity versus
contact angle relationship where the velocity is scaled using the
instantaneous Capillary number, $\operatorname{Ca}^{*}=U^{*}\nu/\gamma$, where
$U^{*}$ is the instantaneous spreading speed, $\nu$ is the liquid viscosity
and $\gamma$ is the liquid-vapor interface energy. Saiz et al. postulated that
the dissipation mechanism may not be due to viscous forces as previously
understood Saiz et al. (2000); Saiz and Tomsia (2004). Clearly, in cases where
the dissipation mechanism is not due to viscous effects, $\operatorname{Ca}$
is no longer a useful quantity for characterizing the spreading and an
alternative parameter is required. An effective “triple-line friction” derived
from molecular kinetics theory is suggested by Saiz et al. that is independent
of viscosity but still dependent on interface energy and the contact angle. A
number of recent experimental studies Saiz et al. (2007) clearly show that a
large proportion of the spreading is characterized entirely by the inertial
time scale ($t_{i}=\sqrt{\rho R_{0}^{3}/\gamma}$, where $\rho$ is the liquid
density and $R_{0}$ is the drop radius) with $U\sim t^{-\nicefrac{{1}}{{2}}}$,
which is much faster than typical viscous spreading laws Biance et al. (2004).
Furthermore, molecular dynamics studies of Ag-Ni and Ag-Cu systems seem to
confirm the $t^{-\nicefrac{{1}}{{2}}}$ dependence of the spreading rate even
for relatively small droplets Webb III et al. (2005); Sun and Webb III (2009).
This paper employs a diffuse interface method in order to analyze the issues
surrounding the inertial spreading regime and dissipation mechanism discussed
above. The diffuse interface approach implicitly includes a wide range of
phenomena and as such does not require a posited relationship between
spreading rate and contact angle JACQMIN (2000). Villanueva et al. Villanueva
et al. (2009) used a diffuse interface method to model reactive wetting and
clearly identified two separate spreading regimes: an initial viscous regime
and a subsequent diffusive regime Villanueva et al. (2008). The viscous regime
demonstrated excellent agreement with standard viscous spreading laws. Further
work by these authors Villanueva et al. (2009) employed the same model to
examine the effects of dissolution on spreading by first recovering the non-
dissolutive hydrodynamic limit as a base state. In the viscous regime they
found the spreading to be independent of the diffusion coefficient, but
accelerated in the diffusive regime as the diffusion coefficient is increased.
This paper outlines a similar process using the initial liquid concentration
to vary the driving force for dissolution, while maintaining a constant
diffusion coefficient. The general consensus of the literature is that
inertial spreading occurs more slowly in systems that exhibit dissolution than
in immiscible systems that do not exhibit dissolution Yin (2006); Warren et
al. (1998). However, this is contradicted by a number of experiments for
saturated and pure liquids that show that the spreading can be on a similar
time scale under certain experimental conditions Protsenko et al. (2008); Saiz
and Tomsia (2004).
The work of Villanueva et al. Villanueva et al. (2009) considers droplets that
do not exhibit inertial effects due to the small drop size, which is limited
by the requirement of having a narrow interface ($\approx 1$ n m ). In
contrast to reference Villanueva et al. (2009), this work sacrifices the
realistic interface width in an attempt to model a system that exhibits
inertial effects. Due to the drop size restrictions, the inertial time scale
used in Villanueva et al. is $t_{i}\approx$$6e-11$\text{\,}\mathrm{s}$$ and
the capillary time scale, $t_{c}=\nu
R_{0}/\gamma\approx$$2e-11$\text{\,}\mathrm{s}$$. At these values, the extent
of spreading during the inertial stage is limited and the characteristic
inertial effects are suppressed by viscous forces. The Ohnesorge number, given
by $\operatorname{Oh}=t_{c}/t_{i}$, quantifies the relative importance of
inertial and viscous effects. Typically, millimeter-sized metal droplets are
highly inertial in nature with $\operatorname{Oh}\approx$1e-3$$.
Characteristic inertial effects, such as triple-line position oscillations and
large droplet curvature variations, are reduced for $\operatorname{Oh}>0.01$
and eliminated for $\operatorname{Oh}>1$ Schiaffino and Sonin (1997). In
Villanueva et al., $\operatorname{Oh}\approx 0.3$ and in this work
$\operatorname{Oh}\approx$6e-3$$.
Jacqmin makes an extensive study of the role of the diffuse interface method,
specifically for a Cahn-Hilliard–van der Waals system (CHW), in relieving the
stress singularity that occurs for classical sharp interface methods JACQMIN
(2000). Since the interface is diffuse, the CHW does not require an explicit
alteration to the no-slip boundary condition to allow for triple-line slip.
Jacqmin demonstrates that the CHW has the same far field and macroscopic
behavior as classical hydrodynamic models of slip. Thus, in diffuse interface
models that include hydrodynamics there is no need to define a slip length.
The interface width determines both an effective slip length and the
concentration profiles within the diffuse interface associated with
adjustments to adsorption and desorption; these factors affect the evolution
of the system in subtle ways. There is no exact expression relating interface
width and the effective slip length, however, $\lambda=\delta/2R_{0}$ is
suggested as a good rule of thumb in Ding and Spelt DING and SPELT (2007),
where $\lambda$ is the dimensionless effective slip length for a diffuse
interface model. It is claimed that the slip length can be as large as
$50\text{\,}\mathrm{nm}$ Schneemilch et al. (1998), which is close to the
chosen interface width in the present work, although the drop radius is only
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The slip length is found by Ding
and Spelt to influence the onset of oscillations that occur when the droplet
transitions from the inertial stage to the diffusive stage. The critical value
of $\operatorname{Re}^{*}$ for which oscillations occur is reduced with
decreasing $\lambda$. Hocking and Davis Hocking and Davis (2002) have
demonstrated that there is no simple relationship between contact angle and
velocity when the approach to equilibrium becomes oscillatory, which seems to
be the case in a number of experimental and numerical studies of millimeter-
sized droplets N. et al. (1998); Protsenko et al. (2008); DING and SPELT
(2007); Schiaffino and Sonin (1997).
The code used for the numerical analysis in this paper is developed using the
FiPy PDE solver Guyer et al. (2009). Details of how to install FiPy as well as
the reactive wetting code used here are given on the FiPy web site rea . The
numerical analysis and figures presented in this paper can be reproduced with
the open source tools available. The underlying linear solvers and parallel
capabilities are provided by the Trilinos tool suite Heroux et al. (2005).
In the following section the governing equations are presented followed by a
discussion of the associated dimensionless parameters in section III. Results
from the numerical solution of the governing equations outlined are presented
in section IV. Section V analyzes the results in the context of previous work
and ends with a discussion of the dissipation mechanism. Section VI presents
the conclusions. Appendix A derives the governing equations presented in
section II, while appendix B presents details of the numerical methods.
## II Governing Equations
In this section, the final forms of the governing equations are presented
along with the associated thermodynamic parameters and functions. The full
derivation of the governing equations is described in appendix A. The system
consists of a three phase (solid, liquid and vapor) binary alloy. The liquid-
vapor system is modeled as a two component van der Waals fluid, while the
solid-fluid system is modeled with a phase field description. The density
field acts as the order parameter for the liquid-vapor transition. Thus, the
system is fully characterized by the spatio-temporal evolution of the mass
density of component 1, $\rho_{1}$, the mass density of component 2,
$\rho_{2}$, the phase field, $\phi$, as well as the barycentric velocity field
$\vec{u}$, as determined through the momentum equation. The three dimensional
equations are reduced to two dimensions by imposing cylindrical symmetry about
$r=0$. The initial configuration consists of a spherical droplet with a radius
of $1$ $\mu$m tangent to a solid substrate surrounded by a vapor. The
incompressible approximation is not made in this work for numerical reasons
outlined in appendix B; all the phases are compressible. The solid is modeled
as a very viscous fluid as in previous phase field reactive wetting studies
Villanueva et al. (2008, 2009). As the total mass density,
$\rho=\rho_{1}+\rho_{2}$, appears so frequently in the equations, it is more
convenient to use $\rho$ and $\rho_{2}$ as the independent density variables.
For economy in notation, we write spatial derivatives
$\partial_{i}\equiv\partial/\partial x_{i}$,
$\partial_{i}^{2}\equiv\partial^{2}/\partial x_{i}^{2}$ and require that
repeated indices are summed, unless otherwise indicated. Note that although
the equations are solved with cylindrical symmetry, the equations are
presented in the following Cartesian forms:
#### II.0.1 Continuity
$\frac{\partial\rho}{\partial t}+\partial_{j}\left(\rho u_{j}\right)=0.$ (1)
#### II.0.2 Diffusion
$\frac{\partial\rho_{2}}{\partial
t}+\partial_{j}\left(\rho_{2}u_{j}\right)=\partial_{j}\left(\frac{M}{T}\partial_{j}\left(\mu_{2}^{NC}-\mu_{1}^{NC}\right)\right).$
(2)
#### II.0.3 Phase
$\frac{\partial\phi}{\partial
t}+u_{j}\partial_{j}\phi=\epsilon_{\phi}M_{\phi}\partial_{j}^{2}\phi-\frac{M_{\phi}}{T}\frac{\partial
f}{\partial\phi}$ (3)
#### II.0.4 Momentum
$\begin{split}\frac{\partial\left(\rho u_{i}\right)}{\partial
t}+\partial_{j}\left(\rho
u_{i}u_{j}\right)&=\partial_{j}\left(\nu\left[\partial_{j}u_{i}+\partial_{i}u_{j}\right]\right)\\\
&-\rho_{1}\partial_{i}\mu_{1}^{NC}-\rho_{2}\partial_{i}\mu_{2}^{NC}+\left(\epsilon_{\phi}T\partial_{j}^{2}\phi-\frac{\partial
f}{\partial\phi}\right)\partial_{i}\phi\end{split}$ (4)
where $u_{i}$ is a velocity component, $T$ is the temperature and
$M=\mathrm{bar}{M}\rho_{1}\rho_{2}/\rho^{2}$ is the chemical mobility, which
is proportional to the diffusivity, $D$, as outlined in Eq. (14). The values
of $\mathrm{bar}{M}$ and $\nu$ vary from the solid to the fluid phases with
the interpolation scheme chosen to be
$\mathrm{bar}{M}=\mathrm{bar}{M}_{s}^{\psi}\mathrm{bar}{M}_{f}^{1-\psi}$ (5)
and
$\nu=\nu_{s}^{\psi}\nu_{f}^{1-\psi}$ (6)
where $\psi=\phi^{a}$ with $a=4$. The values used in the simulations for
$\mathrm{bar}{M}_{s}$, $\mathrm{bar}{M}_{f}$, $\nu_{s}$ and $\nu_{f}$ are in
Table 1. The choice of $a$ is discussed in subsection V.3. The free energy per
unit volume is postulated to have the form Plischke and Bergersen (1994),
$f=p\left(\phi\right)f_{s}+\left(1-p\left(\phi\right)\right)f_{f}+W\phi^{2}\left(1-\phi\right)^{2}$
where $W$ is the phase field barrier height and
$p(\phi)=\phi^{3}(10-15\phi+6\phi^{2})$ represents a smoothed step function
common in phase field models Boettinger et al. (2002). The free energies per
unit volume in the separate fluid and solid phases are given by,
$f_{f}=\frac{e_{1}\rho_{1}^{2}}{m^{2}}+\frac{e_{12}\rho_{1}\rho_{2}}{m^{2}}+\frac{e_{2}\rho_{2}^{2}}{m^{2}}+\frac{RT}{m}\left[\rho_{1}\ln{\rho_{1}}+\rho_{2}\ln{\rho_{2}}-\rho\ln{\left(m-\mathrm{bar}{v}\rho\right)}\right]$
(7)
and
$f_{s}=\frac{A_{1}\rho_{1}}{m}+\frac{A_{2}\rho_{2}}{m}+\frac{RT}{m}\left(\rho_{1}\ln\rho_{1}+\rho_{2}\ln\rho_{2}-\rho\ln\rho\right)+\frac{B}{\rho
m}\left(\rho_{s}^{\text{ref}}-\rho\right)^{2}$ (8)
where $m$ is the molecular weight (assumed to be equal for each component), R
is the gas constant, $\mathrm{bar}{v}$ is the exclusion volume due to the
finite size of the atoms, $B$ is the solid compressibility,
$\rho_{s}^{\text{ref}}$ is a reference density for the solid and the
$e_{i}\rho_{i}/m$ are the free energy contributions per unit mole due to
intermolecular attraction in the van der Waals model. The $A_{1}$ and $A_{2}$
are temperature dependent parameters related to the heat of fusion between the
solid and fluid phases. Along with the free energy, the specification of the
pressure and the non-classical chemical potentials are required to fully
define the system,
$\displaystyle P$ $\displaystyle=$ $\displaystyle\rho_{1}\frac{\partial
f}{\partial\rho_{1}}+\rho_{2}\frac{\partial f}{\partial\rho_{2}}-f$ (9)
$\displaystyle\mu_{1}^{NC}$ $\displaystyle=$ $\displaystyle\frac{\partial
f}{\partial\rho_{1}}-\epsilon_{1}T\partial_{j}^{2}\rho_{1}$ (10)
$\displaystyle\mu_{2}^{NC}$ $\displaystyle=$ $\displaystyle\frac{\partial
f}{\partial\rho_{2}}-\epsilon_{2}T\partial_{j}^{2}\rho_{2}$ (11)
where $\epsilon_{1}$ and $\epsilon_{2}$ are free energy gradient coefficients.
The parameter values for Eqs. (1) to (11) are presented in Table 1. The
corresponding isothermal phase diagram for the molar fraction of component 1
verses the molar volume is displayed in figure 1.
Eqs. (1)–(4) are solved using a cell-centered, collocated finite-volume (FV)
scheme. The solution algorithm uses a fully coupled Krylov solver with Picard
non-linear updates using $\rho_{1}$, $\rho_{2}$, $\phi$ and $\vec{u}$ as the
independent variables. Further discussion of the numerical approach is given
in appendix B.
## III Dimensionless Equations and Timescales
It is useful for the purposes of analysis and completeness to clearly present
the various dimensionless numbers and time scales that arise from solving Eqs.
(1) (2) (3) and (4) in the context of spreading droplets. The dimensionless
forms of Eqs. (2) and (4) are given by,
$\frac{\partial\rho_{2}}{\partial
t}+\partial_{j}\left(u_{j}\rho_{2}\right)=\frac{1}{\operatorname{Pe}}\partial_{j}\left(\frac{\rho_{1}\rho_{2}}{\rho^{2}}\partial_{j}\left(\mu_{2}-\mu_{1}-\operatorname{Q}\partial_{k}^{2}\left(\rho_{2}-\rho_{1}\right)\right)\right)$
(12)
and
$\frac{\partial\left(\rho u_{i}\right)}{\partial t}+\partial_{j}\left(\rho
u_{i}u_{j}\right)=\frac{1}{\operatorname{Re}}\partial_{j}\left(\partial_{j}u_{i}+\partial_{i}u_{j}\right)-\frac{1}{\operatorname{Ma}^{2}}\partial_{i}P+\frac{1}{\operatorname{We}}\left(\rho_{1}\partial_{i}\partial_{j}^{2}\rho_{1}+\rho_{2}\partial_{i}\partial_{j}^{2}\rho_{2}-\tilde{\epsilon}_{\phi}\partial_{i}\phi\partial_{j}^{2}\phi\right)$
(13)
where the variables and operators are now dimensionless (the analysis of Eqs.
(1) and (3) is not particularly revealing and is omitted). For completeness,
all the time scales referred to in this paper are displayed in table 2 as a
prerequisite for presenting the dimensionless numbers in table 3. It should be
noted that in table 2, $U^{*}=U^{*}\left(t\right)$ is the instantaneous
spreading speed and $U$ is a fixed spreading speed posited a priori.
The time scale $t_{\text{diff}}$ represents the time required for the solid-
liquid interface to move a distance $\delta$ due to diffusion mediated melting
or freezing. The expression for $t_{\text{diff}}=\delta^{2}/4K^{2}D_{f}$ is
determined using an error function based similarity solution (see Boettinger
and McFadden ) where $K$ is the solution to
$K+\left(\frac{X_{1}^{l}-X_{1}^{l,\text{equ}}}{X_{1}^{s}-X_{1}^{l}}\right)\frac{\exp{\left(-K^{2}\right)}}{1-\text{erf}\left(K\right)}\frac{1}{\sqrt{\pi}}=0$
and the chemical diffusion coefficient in the fluid, $D_{f}$, is defined by
$D_{f}=\frac{\mathrm{bar}{M}_{f}R}{m\rho_{l}^{\text{equ}}}=$$9.58e-10$\text{\,}{\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-1}$$
(14)
If we substitute $R_{0}$ for $\delta$ in the expression for $t_{\text{diff}}$,
a rough estimate is obtained for complete equilibration of the system. Since
$t_{\text{diff}}\gg t_{i}$, the motion of the solid-liquid interface is
negligible for a simulation that is both computationally feasible and
adequately resolves the inertial time scale. The motion of the solid interface
due to dissolution is controlled by both diffusion ($t_{\text{diff}}$) and
boundary kinetics (represented by $t_{\phi}$). Here $t_{\phi}\ll t_{i}$, thus
dissolution will be limited by diffusion rather than boundary kinetics.
Additionally, solid interface motion due to hydrodynamic effects is negligible
because the solid viscosity is chosen such that $t_{s}\gg t_{i}$ where $t_{s}$
represents the time scale for discernible motion of the solid.
Table 3 presents the dimensionless numbers in terms of their constituent time
scales where appropriate. Note that there are now two separate expressions for
both the Reynolds number and the Capillary number based on $U$ and $U^{*}$. By
making an informed choice for the value of $U$, estimates are obtained for the
likely values of the dimensionless numbers when using $U^{*}$. Here,
$U=R_{0}/t_{i}=$$5.08e1$\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$ is
selected based on the spreading rate for a system that is dominated by
inertial effects. The values of $\operatorname{Oh}$, $\operatorname{Re}$ and
$\operatorname{Pe}$ in table 3 all indicate that the interface energy and
inertial forces dominate over viscous and diffusive forces. Since
$\operatorname{We}=1$, the interface energy and inertial forces are of
approximately equivalent magnitude. Small values of $\operatorname{Oh}$ are
representative of many experimental systems of technical interest: for
example, $\operatorname{Oh}\approx$2e-3$$ for a millimeter sized droplet of
copper and $\operatorname{Oh}\approx$2e-2$$ for a micrometer-sized drop of
lead.
Other dimensionless numbers (included for completeness) in table 3 include the
Mach number, $\operatorname{Ma}$, which requires a definition for the speed of
sound in the liquid, given by Landau and Lifshitz (1987),
$c=\left.\sqrt{\frac{\partial
P}{\partial\rho}}\right|_{\rho_{l}^{\text{equ}}}=$$8.89e2$\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$
and $\operatorname{Q}$, which represents the ratio between interface and
internal forces in the liquid droplet, but has not been identified in the
literature by the authors.
## IV Results
In this section, we explore the rate and extent of droplet spreading based on
variations in the initial liquid concentration and the Ohnesorge number. The
initial liquid concentration determines the driving force for dissolution,
while manipulating the Ohnesorge number influences the impact of inertial
effects on spreading. The results presented here will provide the basis for
comparison with other authors’ work in section V.
The extent of dissolution is established by decreasing the initial value of
the liquid concentration, $X_{1}^{l}$, requiring the solid to dissolve in
order to restore $X_{1}^{l}$ to its equilibrium value, $X_{1}^{l,\text{equ}}$.
Explicitly, we set
$\begin{split}X_{1}^{l}\left(t=0\right)&=\left(1-\xi\right)X_{1}^{l,\text{equ}}\\\
\rho_{l}\left(t=0\right)&=\rho_{l}^{\text{equ}}\end{split}$ (15)
where $\xi$ defines a measure of the magnitude of the driving force for
dissolution ($\xi<0$ induces freezing). When $\xi=0$, the system has no
potential for dissolution, similar to pure hydrodynamic spreading where
surface tension forcing dominates and interface motion is due only to
convection as phase change is negligible. In this limit, comparisons can be
made with simpler spreading models and power laws. In addition to the
hydrodynamic case ($\xi=0$), simulations were conducted with values of
$\xi=0.5$ and $\xi=0.9$.
Figure 2 demonstrates the highly inertial nature of the spreading dynamics.
Upon initiation of the simulation, pressure waves appear at the interface
regions and travel through the interior of the droplet, but then disperse
quickly. Simultaneously, triple-line motion begins with a rapid change in the
local contact angle, but without any discernible motion elsewhere on the drop
interface. This initiates the most noticeable feature of the spreading: a
capillary wave propagates from the triple line along the liquid-vapor
interface, initiating the onset of the triple-line motion and progressing to
the top of the droplet, causing a rapid rise in the drop height. The wave then
travels back to the triple-line location while the droplet completes the
majority of the spreading, with both events having a duration that corresponds
to $\approx 2t_{i}$. During this interval, the triple-line motion is monotonic
and without interruption. On return to the triple-line location, the wave
induces a reversal in the triple-line motion. Subsequent waves induce further
reversals in the triple-line motion and the drop height with a period of
$\approx 2t_{i}$. The amplitude of the oscillations diminishes in the manner
of an under-damped oscillator, completing approximately 5 or 6 full cycles
before ceasing entirely. Subsequently, very slow monotonic spreading occurs
with the liquid-vapor interface appearing to have almost constant curvature.
Figures 3 and 4 display the scaled radial position of the triple-line,
$r_{tl}/R_{0}$, against the scaled time for varying values of $\xi$ and
$\operatorname{Oh}$. The two intervals of fast and slow monotonic spreading
can clearly be seen as well as the intervening period of oscillatory spreading
as discussed in the previous paragraph. Increasing $\xi$ reduces the extent of
spreading slightly, while increasing $\operatorname{Oh}$ eliminates the
oscillations entirely and considerably reduces the spreading rate. In each of
these cases, the actual amount of substrate dissolution is negligible (the
solid-fluid interface moves less than $\delta/5$) due to the large disparity
between the dissolutive and inertial time scales as discussed in section III.
In figure 3, at early times ($t<0.1t_{i}$), the value of $\xi$ has no impact
on the spreading, but at later times ($t>0.1t_{i}$) the curves diverge. When
$t>10t_{i}$, the curves stop diverging and seem to remain at a fixed distance
apart. Increasing $\xi$ not only results in a slight reduction in the extent
of spreading, but also results in a notable reduction in the amplitude of the
oscillations. These factors indicate that there is a seemingly modest decrease
in the driving force for spreading with increasing $\xi$. In figure 4, the
$\operatorname{Oh}=5.7\times 10^{-1}$ curve diverges from the other curves at
very early times and has a greatly diminished spreading rate. Eventually, the
curves become coincident at late times when the spreading is free of
observable inertial manifestations for all values of $\operatorname{Oh}$.
In order to compare with other models, the radial position results presented
in figure 4 are presented using a scaled spreading velocity in figure 5. The
spreading velocity is scaled using a Reynolds number,
$\lambda\operatorname{Re}^{*}$ ($\lambda=\delta/2R_{0}$), based on the
interface width, $\delta$, rather than using a standard Reynolds number based
on the initial drop radius, $R_{0}$ DING and SPELT (2007). The spreading
velocity data used in figure 5 is smoothed to remove noise on the order of a
grid spacing, the details of which are described in appendix B. The sign
changes in the blue curve, when $t_{i}<t<10t_{i}$, correspond to the triple-
line oscillations seen in figure 4. The oscillations lie between intervals
with monotonically decreasing spreading velocity. The
$\operatorname{Oh}=$5.7e-1$$ (yellow) curve exhibits a fairly steady decrease
in velocity and then a much sharper reduction when $t\approx 10t_{i}$, which
corresponds to a slope change in frigure 4. Note that the $\operatorname{Oh}$
values for simulations presented in figure 5 are manipulated by changing the
value of $\nu_{f}$ only, and thus, a corresponding figure with no scaling for
the spreading velocity would show only slight differences between the vertical
positions of the curves.
Figure 6 displays the apparent contact angle, $\theta$, against the Capillary
number for $\operatorname{Oh}=$5.7e-3$$ and $\xi=0$ demonstrating the
convergence of $\theta$ to the nominal flat-interface, equilibrium contact
angle, $\theta^{\text{equ}}$. The angle $\theta$ is calculated using
techniques similar to those described in Villanueva et al. Villanueva et al.
(2009). Although $\theta$ exhibits a hysteresis loop, it remains relatively
steady during the period of oscillatory spreading and only varies by
$\approx$0.03\pi\text{\,}\mathrm{rad}$$ for the largest oscillation.
## V Discussion
### V.1 Comparison with other models
At early times, the flow is dominated by inertia and comparisons with theories
of spreading on flat, non-reactive substrates are fruitful. Indeed, an
analytical spreading rate for the inertial regime can be derived, see Biance
et al. Biance et al. (2004), and is given by $t^{-\nicefrac{{1}}{{2}}}$. In
figure 5, the slope of this power law (black dashed line) shows reasonable
agreement with the $\operatorname{Oh}=$5.7e-3$$ (blue curve) during the
inertial regime. The vertical position of the black dashed line is selected to
enable easy comparison with the blue curve.
In the work of Ding and Spelt DING and SPELT (2007), phase field and level set
models of a spreading droplet are compared for a range of Ohnesorge numbers
($$7.1e-3$\leq\operatorname{Oh}\leq$2.8e-1$$) making it a useful study for
comparing with our work. The black dotted curve in figure 5 is a digitized
curve of the lowest value of $\operatorname{Oh}$ simulated in Ding and Spelt.
This particular simulation is selected for display here as it manifests the
most pronounced oscillations. They simulate droplets with an initial contact
angle of $\pi/3\text{\,}\mathrm{rad}$ and an equilibrium contact angle of
$\pi/18\text{\,}\mathrm{rad}$ using an effective dimensionless slip length of
$\lambda=0.01$ ($\lambda=0.05$ in our work). Despite these differences, the
overall motion of the droplets agrees well qualitatively for droplets with
similar Ohnesorge numbers, although triple line motion was not seen to reverse
direction in their work. In figure 6, the contact angle experiences a
hysteresis loop in a similar fashion to the work of Ding and Spelt, which is
reproduced in the black dotted curve.
It has been conjectured DING and SPELT (2007); Hocking and Davis (2002) that
the value of $\lambda\operatorname{Re}^{*}$ controls whether or not the
spreading becomes oscillatory. In the simulations presented here,
$\lambda\operatorname{Re}^{*}$ varies between 1 and 10 for the lowest value of
$\operatorname{Oh}$, but this is harder to determine for experimental systems.
Hydrodynamic analysis of experimental data results in a slip length that can
vary substantially for different materials (typically between
$1\text{\,}\mathrm{nm}$ and $100\text{\,}\mathrm{nm}$ Saiz and Tomsia (2004)).
Using these bounds, a typical millimeter sized metal drop results in
$0.01<\lambda\operatorname{Re}^{*}<1$ assuming a spreading rate of
$1\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ (in this work the spreading
rate is $\approx$50\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$). It is
interesting to note that for values of $\lambda\operatorname{Re}^{*}<0.1$, no
oscillatory motion was seen in the work by Ding and Spelt DING and SPELT
(2007). In Schiaffino and Sorin Schiaffino and Sonin (1997) it is
experimentally determined that the transition between under-damped
oscillations to over-damped decay (no oscillations) occurs as
$\operatorname{Oh}$ increases above $1\times 10^{-2}$. This is seemingly
confirmed in figure 4 where the curve that corresponds to
$\operatorname{Oh}=5.7\times 10^{-3}$ has multiple oscillations, while the
curve for $\operatorname{Oh}=5.7\times 10^{-1}$ has no oscillations.
### V.2 Comparison with experiments
In figures 7 and 8 the triple-line radial position results from the present
work for $\operatorname{Oh}=$5.7e-3$$ are compared with experimental results
from Saiz and Tomsia Saiz and Tomsia (2004); Saiz et al. (2007) and Protsenko
et al. Protsenko et al. (2008). These experiments are conducted at a high
temperature ($1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$) and exhibit fast
spreading, which is either absent or undocumented in many other reactive
wetting experiments Eustathopoulos . In Saiz and Tomsia, the experimental
results are for Au and Cu droplets with an initial radius of
$1\text{\,}\mathrm{mm}$ spreading on Ni and Mo substrates, respectively, while
in Protsenko et al. the experiments are for Cu droplets of a similar size
spreading on Si substrates. The reasonable quantitative agreement between the
experimental and simulation results in figures 7 and 8 (within $\approx 20\%$
for the Cu-Mo combination) suggests that the spreading in the experimental
systems is predominantly inertial in nature Saiz et al. (2007).
The Cu on Mo spreading in figure 7 indicates oscillatory behavior at the end
of the inertial regime, although there are only a handful of data points
supporting this claim. Also, since the period of any oscillations is likely to
be $\approx 2t_{i}$, a much greater duration of experimental data is required
for confirmation. The dissolutive case (black solid curve) in figure 8 clearly
demonstrates oscillations of a similar period, amplitude and duration to the
simulation results presented here as well as a contact angle hysteresis (not
shown). It should be noted that oscillatory spreading also occurs in other
systems such as water droplets on glass Schiaffino and Sonin (1997).
### V.3 Non-equilibrium interface energy analysis
The driving force for spreading on a planar substrate is often characterized
by the spreading coefficient given by,
$S^{\text{equ}}\left(t\right)=\gamma_{sv}^{\text{equ}}-\left(\gamma_{sl}^{\text{equ}}+\gamma_{lv}^{\text{equ}}\cos{\theta}\left(t\right)\right)$
(16)
where the $\gamma^{\text{equ}}$ are equilibrium values of the interface
energies and $\theta$ is the observed contact angle. The utility of Eq. (16)
is clearly limited to circumstances where the interface energies remain close
to their equilibrium values during spreading. A number of authors AKSAY et al.
(1974); Eustathopoulos ; Frenznick et al. (2008); Yin et al. (2009) have
suggested that this limitation may be overcome by replacing the equilibrium
interface energies with their instantaneous values in Eq. (16). This yields a
new spreading coefficient
$\tilde{S}\left(t\right)=\tilde{\gamma}_{sv}\left(t\right)-\left(\tilde{\gamma}_{sl}\left(t\right)+\tilde{\gamma}_{lv}\left(t\right)\cos{\theta}\left(t\right)\right)$
(17)
where the $\tilde{\gamma}$ are instantaneous interface energies. In principal,
the use of $\tilde{\gamma}$ rather than $\gamma^{\text{equ}}$ provides a more
accurate description of the driving force for spreading, particularly in the
case where the timescale for spreading, $t_{i}$, is much faster than the
interface equilibration timescale, $t_{\text{diff}}$. Since the solid-fluid
interface remains planar over the time scales of interest in the simulations,
using a horizontal force balance alone and ignoring the vertical imbalance
when deriving Eq. (17) can be viewed as a reasonable assumption. An
alternative expression to Eq. (17) can be derived if the solid-fluid interface
is non-planar using a more general Neumann’s triangle horizontal and vertical
force balance. In the following discussion, the expression used to calculate
the $\tilde{\gamma}$ is described and then $S$ is used to analyze the
influence of $\xi$ on the spreading dynamics.
It is a substantial advantage of our approach that we are able to develop an
explicit expression for the instantaneous interface energies, allowing us to
test the utility of $\tilde{S}$ as a metric for spreading. In order to
calculate $\tilde{S}$ using the results of the present calculations, we begin
with two equivalent expressions for the equilibrium energy of a planar
interface:
$\begin{array}[]{ll}\gamma&=\int_{-\infty}^{\infty}\left[\epsilon_{1}T|\nabla\rho_{1}|^{2}+\epsilon_{2}T|\nabla\rho_{2}|^{2}+\epsilon_{\phi}T|\nabla\phi|^{2}\right]dl\\\
&=2\int_{-\infty}^{\infty}\left[f-f^{\infty}-\mu_{1}^{\infty}\left(\rho_{1}-\rho_{1}^{\infty}\right)-\mu_{2}^{\infty}\left(\rho_{2}-\rho_{2}^{\infty}\right)\right]dl\end{array}$
(18)
where the $\infty$ superscript represents the value in the far field, and all
the fields have equilibrium profiles. The equivalence of the expressions in
Eq. (18) can be demonstrated by first writing down the Euler-Lagrange equation
derived from the free energy functional in Eq. (24) with additional Lagrange
multiplier terms for the conservation of both species and then integrating
once. We now assert that a plausible measure of the instantaneous interface
energy is
$\tilde{\gamma}\left(t\right)=\int_{l}\left[\epsilon_{1}T|\nabla\rho_{1}|^{2}+\epsilon_{2}T|\nabla\rho_{2}|^{2}+\epsilon_{\phi}T|\nabla\phi|^{2}\right]dl$
(19)
where $l$ is a line segment that both intersects and is normal to the
interface being measured with $\int_{l}dl>\delta$. All fields in Eq. (19) are
measured at time $t$. In general, the quantity $\tilde{\gamma}$ is a useful
heuristic when the gradients are confined to the interface region. The
numerical integration of Eq. (19) is conducted at a distance of $2\delta$ from
the triple-line location perpendicular to each local interface over a distance
of $1.5\delta$. The integration points on the respective interfaces are chosen
to be as near to the triple-line location as possible while avoiding the large
variations in the value of $\tilde{\gamma}$ that occur close to the triple-
line location Villanueva et al. (2009). Clearly, we could have defined another
instantaneous interface energy as,
$\gamma^{*}(t)=2\int\left[f-f^{\infty}-\mu_{1}^{\infty}\left(\rho_{1}-\rho_{1}^{\infty}\right)-\mu_{2}^{\infty}\left(\rho_{2}-\rho_{2}^{\infty}\right)\right]dl$
(20)
As one approaches equilibrium $\gamma^{*}\rightarrow\tilde{\gamma}$, but
dynamically the quantities are different. It would appear that $\gamma^{*}$ is
less useful than $\tilde{\gamma}$, as $\gamma^{*}$ requires the fields to be
near the far field (equilibrium) values at the integration limits extremes for
the value to “make sense” as an interface excess quantity. It is instructive
to observe the $\tilde{\gamma}$ behavior over time (see figure 9). The values
of $\tilde{\gamma}$ differ substantially from their equilibrium values for
most of the simulation. The $\tilde{\gamma}_{sv}$ appear independent of $\xi$,
which is a reasonable expectation, as $\xi$ sets the liquid concentration.
Increasing $\xi$ results in an increase in both $\tilde{\gamma}_{lv}$
and$\tilde{\gamma}_{sl}$. In figure 9 large oscillations can be observed in
the solid-liquid interface energy (red curve). These oscillations are due to
the spatially varying values of $\tilde{\gamma}_{sl}$ along the solid-liquid
interface in conjunction with the oscillations in the $\tilde{\gamma}_{sl}$
integration line location moving in unison with the triple-line location
during the oscillatory phase of motion.
Using our definition of $\tilde{\gamma}$ and the apparent contact angle,
$\theta$, we can now calculate dynamic values of both $\tilde{S}$ and
$S^{\text{equ}}$, which are presented in figure 10. The curves decrease
rapidly from their maximum value and become negative at about $t=t_{i}$ and
then oscillate in conjunction with the triple-line radial position
oscillations. Eventually, the values of $\tilde{S}$ become quite small
($<10\%$ of its original value for $\xi=0$) although the drop is still
spreading. Assuming $\tilde{S}$ quantifies the driving force for spreading,
then the differences in $\tilde{S}$ that occur for different values of $\xi$
at early times may explain both the deviations observed in the spreading
extent during the inertial regime ($t<t_{i}$) and the deviations in the
oscillation amplitudes in figure 3. The small values of $\tilde{S}$ when
compared with $S^{\text{equ}}$ at late times suggest that the spreading has
become quasi-static in nature and is bound to the evolving values of the
$\tilde{\gamma}$. The evolution of the $\tilde{\gamma}_{sl}$ occurs on a time
scale associated with $t_{\text{diff}}$ while the hydrodynamic adjustment of
the contact angle occurs on a time scale associated with $t_{i}$. Thus, the
contact angle can adjust rapidly to balance the horizontal forces and suggests
that the spreading is limited by interface equilibration at late times.
### V.4 Dissipation analysis
Much of the literature surrounding droplet spreading is concerned with
characterizing dissipation mechanisms from the point of view of an
irreversible thermodynamic process Saiz and Tomsia (2004); de Gennes (1985);
Brochard-Wyart and de Gennes (1992). In this spirit, this section provides an
analysis of the entropy production, yielding the magnitudes of the various
dissipation mechanisms in our model, which should, in turn, provide guidance
on the formulation of simplified models. The expression used here for the
total entropy production rate is given by SEKERKA and BI (2002),
$\dot{S}_{\text{PROD}}=\frac{M}{T^{2}}|\partial_{j}\left(\mu_{1}^{\text{NC}}-\mu_{2}^{\text{NC}}\right)|^{2}+\frac{M_{\phi}}{T^{2}}\left(\frac{\partial
f}{\partial\phi}-\epsilon_{\phi}T\partial_{j}^{2}\phi\right)^{2}+\frac{\nu}{2T}\left(\partial_{i}u_{k}+\partial_{k}u_{i}\right)\partial_{i}u_{k}$
(21)
where each term in the sum is a distinct dissipation mechanism (diffusion,
solid interface relaxation, and viscous flow).
The comprehensive overview of wetting by de Gennes de Gennes (1985) identified
three main mechanisms for dissipation in spreading droplets: a viscous
dissipation concerned with the “rolling motion” of the fluid within
$100\text{\,}\mu\mathrm{m}$ of the triple line, a viscous dissipation in the
precursor film and a highly localized dissipation at the triple line
associated with “triple-line friction”. In the present work, the precursor
film is absent, however, both viscous dissipation in the bulk fluid and local
triple-line dissipation are present, but are conflated within the viscous
dissipation term in Eq. (21). In most models of droplets spreading, the chosen
model for slip relaxation at the triple line influences the underlying
dissipation mechanism for the spreading droplet. For example, a molecular
kinetics model of slip generally implies a local triple-line dissipation,
while a hydrodynamic model of slip, such as Cox’s model Cox (1986) or Tanner’s
law TANNER (1979), both examples of de Gennes’ “rolling motion”, implies non-
localized dissipation Saiz and Tomsia (2004); Brochard-Wyart and de Gennes
(1992). We are reminded that this model employs diffuse interfaces, and thus
no explicit slip condition is postulated, but such slip is a direct
consequence of the model.
Figure 11 presents color contour plots of the entropy production rates at
various times. The plots show the magnitude, location and mechanism of entropy
production for the non-dissolutive case (the dissolutive cases are only
slightly different). The color mapping is rescaled in figure 11 based on the
$\max\left(\dot{S}_{\text{PROD}}\right)$ value for each image. For example,
the total entropy production rate in figure 11 (d) is only 0.4% of the value
in figure 11 (a). If we were considering a non-isothermal system, there would
be a further term in expression 21 containing temperature gradients, an effect
not considered in this work.
At very early times (not shown), the entropy production is highly localized at
the solid-fluid interface region as $\phi$ locally equilibrates. Subsequently
(not shown), pressure waves are observed as the liquid-vapor interface
equilibrates, and viscosity is the dominant mode of dissipation. By
$t=0.1t_{i}$, the pressure waves have mostly subsided and the spreading is
well under way. At this stage, the dominant dissipation mechanism remains
viscous but is now highly localized at the triple-line. As the inertial time
scale is approached in figure 11 (b), the dominant mechanism alternates
between diffusive and viscous as the droplet oscillates during the
$t_{i}<t<10t_{i}$ stage. The viscous dissipation remains highly localized at
the triple line, while the diffusive dissipation mostly occurs in the solid-
liquid interface with some occurring along the solid-vapor interface. This
correlates with figure 9, which shows that the solid-liquid interface is far
from local equilibrium until much later times. At later times (figure 11 (c)),
dissipation is mainly due to local interface equilibration along the solid-
liquid and solid-vapor interface regions. The proportion of the numerically
integrated value of $\int\dot{S}_{\text{PROD}}dV$ for each term in Eq. (21)
(diffusive, phase field, viscous) is (a) (0.51, 0.06, 0.43), (b) (0.73, 0.04,
0.23), (c) (0.84, 0.02, 0.14) and (d) (0.85, 0.07, 0.08) for each subplot in
figure 11. These proportions demonstrate the growing influence of diffusive
dissipation and the reduction in viscous dissipation as the system transitions
from the inertial regime to the diffusive regime.
### V.5 Remarks
The temporal adjustment to the equilibrium interface profiles is extremely
complex and intimately related to the interface width and the interpolated
values of the dynamic coefficients ($\nu$ and $\mathrm{bar}{M}$), see Eqs. (5)
and (6). The choice for the interpolation parameter $a$ in Eqs. (5) and (6)
biases the coefficients to have values close to the bulk fluid values in the
interface region facilitating the fastest interface dynamics possible within
the bounds set by the bulk values. The parameter $a$ is tuned to a value of 4,
as larger values do not increase the interface equilibration rate while
smaller values considerably reduce the equilibration rate.
The equilibration of the density and phase field interface profiles is fast
compared to that of the concentration field. The interface profile of the
density field, $\rho$, is adjusted rapidly by hydrodynamics alone, while the
interface profile of the concentration field, $\rho_{1}/\rho$, requires inter-
diffusion between the bulk phases and the interface regions. This
compositional relaxation could, in principle, be as slow as the diffusion time
scale, $t_{\text{diff}}$ (see Table 2), although the connection is imprecise,
as this quantity is associated primarily with the motion of the interface due
to dissolution (melting) rather than the relaxation of compositional profiles
within the interface. The solid interfaces equilibrate slowly, compared to the
liquid-vapor interface, as seen in figure 9. We expect that the observed
interface relaxation time is unrealistic, when compared with experimental
studies of metallic systems, as our chosen interface width of
$\delta=$100\text{\,}\mathrm{nm}$$ is much larger than the
$\delta\approx$1\text{\,}\mathrm{nm}$$ typical of metals. This is a
shortcoming of this treatment, and results in an unphysical time scale for
local interface equilibration. Further analysis of the relationship between
$\delta$ and the equilibration rate is required, though this analysis is
beyond the scope of this work. The limitation of requiring $\delta/R\approx
0.1$ imposed by the available computation resources does not detract from the
analysis presented in this section with respect to the reduced spreading when
$\xi$ is increased, the qualitative description of the spreading regimes and
oscillations, or the quantitative comparisons with experiments.
## VI Conclusion
This paper presents results from a model of dissolutive spreading simulated in
a parameter regime where inertial effects are initially dominant. The triple-
line motion demonstrates good agreement with the $O(t^{-\nicefrac{{1}}{{2}}})$
inertial spreading rate at early times. The model also generates oscillations
characteristic of the transition from inertial to viscous or diffusive
spreading. Subsequent analysis indicates that a force balance involving the
instantaneous interface energies evaluated using the expression in Eq. (19)
can explain the variation in spreading between the hydrodynamic and
dissolutive cases. At late times, after inertial effects have ceased, the
contact angle derived from the instantaneous interface energies is within
$0.005\text{\,}\mathrm{rad}$ of the measured contact angle suggesting that the
local interface equilibration mechanism is controlling the spreading. Analysis
of the dissipation mechanism via the entropy production expression
demonstrates that dissipation occurs at the triple line during the inertial
stage, but transitions to the solid-fluid interfaces during the oscillatory
stage consistent with the instantaneous interface energy analysis. Overall,
the simulation results show good quantitative and qualitative agreement with a
number of experimental results when time is scaled with the inertial time
scale.
Modeling droplets that have both a realistic interface width and include
inertial effects is impractical with current computational resources (at least
for the model presented herein) and may require years of real time computation
on large parallel clusters. In this work, to reduce the required compute time,
the use of a realistic interface width has been sacrificed in order to
preserve the inertial effects. This has the consequence of increasing the
simulation time required for the local equilibration process across the solid-
fluid interface as discussed in section V.3. Although, this process has a
longer duration than physically appropriate in the present work, a time regime
over which the controlling mechanism for spreading is the local interface
equilibration may be entirely physical. It is noted in Protsenko et al.
Protsenko et al. (2008) that the diffusive stage may occur in two separate
parts. The first part is surmised to be the solid-liquid interface
equilibration process and takes approximately an order of magnitude longer
than the inertial time scale, which is faster than occurs here, but very
similar in nature. The second part is the melting of the substrate, which is
included in this model, but not observed as it occurs over a time scale longer
than the total duration of a typical simulation.
Further work may involve both direct comparison with molecular kinetics theory
and more detailed analysis of the impact of the interface width on the
spreading dynamics.
## VII Acknowledgements
The authors would like to acknowledge the contributions of Dr. Jonathan E.
Guyer and Dr. Walter Villanueva for their help and guidance in implementing
the numerical model and analyzing the numerical data, and Dr. Edmund B. Webb
for insightful commentary and help in setting this work in the proper context.
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## Appendix A Derivation of the Governing Equations
In this section the underlying thermodynamic and constitutive relationships
required for the derivation of Eqs. (2), (3) and (4) are presented.
As previously outlined, the fluid phases are represented by a binary, van der
Waals equation of state and the solid phase is represented by a simple linear
compressive and tensile equation of state that ignores all shear stress. The
van der Waals equation of state is given by,
$\left(P-\frac{n^{2}}{V^{2}}\left(e_{1}X_{1}+e_{2}X_{2}\right)\right)\left(V-\mathrm{bar}{v}n\right)=nRT$
(22)
where $X_{1}$ and $X_{2}$ are the concentrations of each component, $n$ is the
number of moles and $V/n=m/\rho$. All other parameters used in Eq. (22) are
defined in section II. Eq. (22) can be related to the ideal gas law, but has
modified pressure and volume terms to account for the long range attraction of
molecules and volume exclusion, respectively Kittel and Kroemer (1980);
Plischke and Bergersen (1994). The solid equation of state is given by,
$PV_{s}=2Bn\frac{V_{s}-V}{V_{s}}$ (23)
where $V_{s}/n=m/\rho_{s}^{\text{ref}}$. The Helmholtz free energies given in
Eqs. (7) and (8) are derived from (22) and (23), respectively, using the
thermodynamic identities given in Eqs. (10), (11) and (9). In order to derive
Eqs. (2), (3) and (4), it is necessary to postulate a form for the free energy
functional.
As in reference Bi and Sekerka (1998), standard non-classical diffuse
interface expressions for $\rho_{1}$, $\rho_{2}$ and $\phi$ are used, which
results in a functional of the form,
$F=\int\left[f+\frac{\epsilon_{\phi}T}{2}|\nabla\phi|^{2}+\frac{\epsilon_{1}T}{2}|\nabla\rho_{1}|^{2}+\frac{\epsilon_{2}T}{2}|\nabla\rho_{2}|^{2}\right]dV$
(24)
Using standard dissipation arguments Bi and Sekerka (1998), Eqs. (2) and (3)
are derived using,
$\frac{\partial\phi}{\partial t}+u_{j}\partial_{j}\phi=-M_{\phi}\frac{\delta
F}{\delta\phi}$
and
$\frac{\partial\rho_{1}}{\partial
t}+\partial_{j}\left(u_{j}\rho_{1}\right)=-\partial_{j}J_{1j}$
and similarly for component 2. The fluxes are given by,
$J_{1j}=-J_{2j}=-M\partial_{j}\left(\frac{\mu_{1}^{NC}-\mu_{2}^{NC}}{T}\right)$
where,
$\mu_{1}^{NC}=\frac{\delta F}{\delta\rho_{1}}$
and
$\mu_{2}^{NC}=\frac{\delta F}{\delta\rho_{2}}$
The form of the stress tensor required to derive the momentum equation is
given by,
$\sigma_{ij}=\nu\left(\partial_{j}u_{i}+\partial_{i}u_{j}\right)+t_{ij}$
using the standard assumption that the bulk viscosity, $\lambda$, is related
to the shear viscosity via $\lambda=-\frac{2}{3}\nu$. The tensor, $t_{ij}$, is
derived from a conservation law ($\partial_{j}t_{ij}=0$) based on Noether’s
theorem Anderson et al. (1998). The expression for $t_{ij}$ is given by,
$t_{ij}=g^{NC}\delta_{ij}-\partial_{j}\rho\frac{\partial
g^{NC}}{\partial\left(\partial_{i}\rho\right)}$ (25)
where
$g^{NC}=f^{NC}+\rho_{1}\lambda_{1}+\rho_{2}\lambda_{2}$ (26)
The non-classical Gibbs free energy, $g^{NC}$, is the form of the free energy
that includes Lagrange multipliers for conservation of species 1 and 2. The
Lagrange multipliers for each species are equal to $\lambda_{1}=-\mu_{1}^{NC}$
and $\lambda_{2}=-\mu_{2}^{NC}$ in equilibrium using the variational
derivative of $\int g^{NC}dV$ with respect to $\rho_{1}$ and $\rho_{2}$. Using
Eqs. (25) and (26) the form for $\partial_{i}t_{ij}$ used in Eq. (4) can be
derived,
$\partial_{j}t_{ij}=-\rho_{1}\partial_{i}\mu_{1}^{NC}-\rho_{2}\partial_{i}\mu_{2}^{NC}-\partial_{i}\phi\frac{\delta
F}{\delta\phi}$ (27)
## Appendix B Numerical Approach
In general, even for compressible systems, many conventional algorithms use
the pressure field as the independent variable rather than the density field.
This approach is thought to have more robust convergence properties Ferziger
and Perić (1996) at low Mach numbers due to the weak dependence of pressure
gradients on density, but the convergence properties deteriorate at higher
Mach numbers. In this work, due to the non-trivial nature of the pressure-
density relationship, an inversion of this relationship would be impractical
and it is more natural to solve for the density field rather than the pressure
field. Due to the mesh collocation of the density and velocity fields, an
interpolation scheme, known as Rhie-Chow interpolation Rhie and Chow (1983),
is employed to ensure adequate velocity-pressure coupling.
The calculation of triple-line velocities is necessarily noisy, with
fluctuations on a timescale of $\Delta x/U$, where $\Delta x$ is the fixed
grid spacing. In figure 5, the curves are constructed using a 20 point boxcar
(equally-weighted) averaging scheme collected at every 10 time steps during
the simulation. We note that the sign changes in the blue curve
($t_{i}<t<10t_{i}$) in figure 5 correspond to the triple-line oscillations,
and are not due to the averaging scheme. The velocity fluctuations will be
small when $U$ is large. Indeed, at early times, when $t/t_{i}<1$, $U$ is
relatively large and the results are smooth. At later times, when
$t/t_{i}>10$, the averaging scheme does not smooth out the noise, as the
spreading rate is greatly reduced. This can be seen in the noisy behavior at
long times for the $\operatorname{Oh}=5.7\times 10^{-3}$ curve (blue) in
figure 5. The noise in the low velocity regime of figure 6 also reflects this
behavior.
The measurements for $\theta$ are calculated using the tangent to the liquid-
vapor interface at a distance of 1.3 $\delta$ from the triple-line location.
In general, this distance results in a reasonable approximation to the
apparent contact angle.
### B.1 Parasitic Currents
Parasitic currents are a common source of numerical errors when computing
flows with interface energy driving forces that have large
$\operatorname{Ca}$. Typically, for the systems of interest in this paper,
$\operatorname{Ca}\approx 10^{-2}$, but parasitic velocities were still found
to be a source of numerical error, particularly when trying to evaluate
equilibrium solutions. Parasitic currents are characterized by quasi-steady
flow fields that do not dissipate over time despite the system reaching
equilibrium in all other respects. This can result in equilibrium errors in
both the density and concentration fields. Jamet et al. Jamet et al. (2002) as
well as other researchers have demonstrated that parasitic currents can be
eliminated by recasting the momentum equation in a form that only conserves
momentum to the truncation error of the discretization rather than machine
precision. The form of the momentum equation that eliminates parasitic
currents is written in terms of the chemical potentials and is given by,
$\frac{\partial\left(\rho u_{i}\right)}{\partial t}+\partial_{j}\left(\rho
u_{i}u_{j}\right)=\partial_{j}\left(\nu\left[\partial_{j}u_{i}+\partial_{i}u_{j}\right]\right)-\rho_{1}\partial_{i}\mu_{1}^{NC}-\rho_{2}\partial_{i}\mu_{2}^{NC}$
(28)
for binary liquid-vapor system. The discretized form of Eq. (28) is known as
an energy conserving discretization in contrast to the momentum conserving
discretization, which results when the momentum equation is written in terms
of the pressure (see Eq. (13)).
### B.2 Convergence
Some simulations in this paper are tested for convergence with grid sizes of
180$\times$125, 360$\times$250 and 720$\times$500 using the triple-line and
drop height positions against time as the metrics for convergence. Production
runs for the results presented use 360$\times$250 grids. Details of these
convergence tests can be found in rea . Convergence at the $n^{\text{th}}$
time-step is achieved when the $k^{\text{th}}$ iteration within the time step
satisfies the residual condition $\beta_{n}^{k}/\beta_{n}^{0}<1\times 10^{-1}$
for each of the equations where $\beta_{n}^{k}$ is the $L_{2}$-norm of the
residual at the the $k^{\text{th}}$ iteration of the $n^{\text{th}}$ time
step. Further decreases in the residual make little difference to the dynamic
positions of the drop height and triple-line. Numerical calculations indicate
that, in the course of a simulation, $\operatorname{Ma}$ ranges from values
that require compressible flow solvers (density based with
$\operatorname{Ma}>2\times 10^{-1}$) to values for which compressible flow
solvers have trouble with accuracy and convergence for traditional segregated
solvers ($\operatorname{Ma}<2\times 10^{-1}$). The shift to low
$\operatorname{Ma}$ generally occurs when the system is quite close to
equilibrium and is not believed to affect the dynamic aspects of the
simulation, which are of most interest in this paper. In general, for low Mach
number flows, preconditioners are used to improve the convergence properties
of segregated solvers. In this work, it was found that using a coupled solver
along with a suitable preconditioner greatly improved the convergence
properties. The preconditioners are available as part of the Trilinos software
suite Heroux et al. (2003). The coupled convergence properties can be further
improved by employing physics based preconditioners that change the nature of
the equations based on the value of $\operatorname{Ma}$ Keshtiban et al.
(2004), but are not used in this work.
Parameter | Value | Unit
---|---|---
$\nu_{f}$ | 2. | 0$\times$10-3 | k g /( s ⋅ m )
$\nu_{s}$ | 2. | 0$\times$104 | k g /( s ⋅ m )
$\epsilon_{1}$ | 2. | 0$\times$10-16 | m 7/( K ⋅ k g ⋅^ 2 s )
$\epsilon_{2}$ | 2. | 0$\times$10-16 | m 7/( K ⋅ k g ⋅^ 2 s )
$T$ | 6. | 5$\times$102 | K
$m$ | 1. | 18$\times$10-1 | k g / mol
$R$ | 8. | 31 | J /( K ⋅ mol )
$v_{a}$ | 1. | 0 |
$e_{1}$ | -4. | 56$\times$10-1 | J ⋅^ 3 m /^ 2 mol
$e_{2}$ | -4. | 56$\times$10-1 | J ⋅^ 3 m /^ 2 mol
$\mathrm{bar}{v}$ | 1. | 3$\times$10-5 | ^ 3 m / mol
$A_{1}$ | 2. | 83$\times$104 | J / mol
$A_{2}$ | 5. | 64$\times$104 | J / mol
$\rho_{s}^{\text{ref}}$ | 7. | 84$\times$10-5 | k g /^ 3 m
$B$ | 2. | 02$\times$105 | J / mol
$W$ | 1. | 27$\times$105 | N /^ 2 m
$\epsilon_{\phi}$ | 1. | 0$\times$10-9 | N / K
$M_{\phi}$ | 1. | 0$\times$104 | K ⋅^ 2 m /( N ⋅ s )
$\mathrm{bar}{M}_{f}$ | 1. | 0$\times$10-7 | k g ⋅ s ⋅ K /^ 3 m
$\mathrm{bar}{M}_{s}$ | 1. | 0$\times$10-11 | k g ⋅ s ⋅ K /^ 3 m
$R_{0}$ | 1. | 0$\times$10-6 | m
$\delta$ | 1. | 0$\times$10-7 | m
$\rho_{l}^{\text{equ}}$ | 7. | 35$\times$103 | k g /^ 3 m
Table 1: Various parameter values. Time scale | Symbol | Expression | Value (s)
---|---|---|---
capillary | $t_{c}$ | $\nu_{f}R_{0}/\gamma_{lv}$ | 1. | 05$\times 10^{-10}$
phase field | $t_{\phi}$ | $\delta^{2}/\epsilon_{\phi}M_{\phi}$ | 1. | 0$\times 10^{-9}$
inertial | $t_{i}$ | $\sqrt{\rho_{l}^{\text{equ}}R_{0}/\gamma_{lv}}$ | 1. | 97$\times 10^{-8}$
convection | $t_{a}$ | $R_{0}/U$ | 1. | 97$\times 10^{-8}$
viscous | $t_{\nu}$ | $\rho_{l}^{\text{equ}}R_{0}^{2}/\nu_{l}$ | 3. | 68$\times 10^{-6}$
interface diffusion | $t_{\text{diff}}$ | $\delta^{2}/4K^{2}D_{f}$ | 7. | 69$\times 10^{-4}$
bulk diffusion | $t_{\text{d}}$ | $R_{0}^{2}/D_{f}$ | 1. | 04$\times 10^{-3}$
solid deformation | $t_{s}$ | $\delta\nu_{s}/\gamma_{lv}$ | 1. | 05$\times 10^{-2}$
instantaneous convection | $t_{a}^{*}$ | $R_{0}/U^{*}$ | . |
Table 2: Complete list of time scales referred to in this paper. Parameter | Symbol | Expression | Value
---|---|---|---
Peclet number | $\operatorname{Pe}$ | $UR_{0}/D_{f}=t_{d}/t_{a}$ | 5. | $31\times 10^{4}$
Reynolds number | $\operatorname{Re}$ | $UR_{0}\rho_{l}^{\text{equ}}/\nu_{f}=t_{\nu}/t_{a}$ | 1. | $87\times 10^{2}$
Weber number | $\operatorname{We}$ | $\operatorname{Re}\,\operatorname{Ca}=t_{\nu}t_{c}/t_{a}^{2}$ | 1. | 0
Mach number | $\operatorname{Ma}$ | $U/c$ | 5. | $72\times 10^{-2}$
Unnamed | $\operatorname{Q}$ | $m\gamma_{lv}/RT\rho_{l}^{\text{equ}}R_{0}$ | 5. | $64\times 10^{-2}$
effective dimensionless slip length | $\lambda$ | $\delta/R_{0}$ | 5. | $0\times 10^{-2}$
Capillary number | $\operatorname{Ca}$ | $U\nu_{f}/\gamma_{lv}=t_{c}/t_{a}$ | 5. | $35\times 10^{-3}$
Ohnesorge number | $\operatorname{Oh}$ | $\sqrt{\operatorname{Ca}/\operatorname{Re}}=t_{c}/t_{i}$ | 5. | $35\times 10^{-3}$
instantaneous Reynolds number | $\operatorname{Re}^{*}$ | $U^{*}R_{0}\rho_{l}^{\text{equ}}/\nu_{f}=t_{\nu}/t_{a}^{*}$ | . |
instantaneous Capillary number | $\operatorname{Ca}^{*}$ | $U^{*}\nu_{f}/\gamma_{lv}=t_{c}/t_{a}^{*}$ | . |
Table 3: Relevant dimensionless numbers.
Figure 1: The phase diagram for the system of parameters presented in Table 1.
Each region represents a possible equilibrium state for a mixture of solid
(S), liquid (L) and vapor (V) phases. The red dots represent the initial
conditions for the $\xi=0.1$ simulation discussed in section IV. The black dot
marks the liquid equilibrium condition. The liquid and vapor phases are thick
in component 1 while the solid phase is thick in component 2.
Figure 2: Sequential configurations of the liquid-vapor and solid-fluid
interfaces for $\xi=0$ and $\operatorname{Oh}=5.7\times 10^{-3}$ with darker
tones indicating later times. The curves demonstrate the extreme inertial
effects on the droplet. The droplet starts as a sphere in tangent contact with
the substrate. The drop height then rises considerably as the capillary wave
initiated from the triple line arrives at the top of the droplet. Although
large amplitude ($\approx R_{0}/5$) oscillations occur in the triple-line
position, the largest contact angle oscillation is only
$\approx$0.03\pi\text{\,}\mathrm{rad}$$.
Figure 3: The spreading radius versus time for various values of $\xi$ with
$\operatorname{Oh}=5.7\times 10^{-3}$. As $\xi$ increases, the spreading rate
and extent of spreading is slightly reduced.
Figure 4: The spreading radius, $r_{tl}$, versus time for various values of
$\operatorname{Oh}$ with $\xi=0$. The oscillations are eliminated for the
largest value of $\operatorname{Oh}$.
Figure 5: The dimensionless spreading rate against the dimensionless time with
varying $\operatorname{Oh}$ and $\xi=0$. The spreading occurs in three
distinct intervals.. The sign changes in the blue curve correspond to the
triple-line oscillations during the transition from the inertial to the
diffusive regime.
Figure 6: The observed contact angle against $\operatorname{Ca}^{*}$ for
$\xi=0$ and $\operatorname{Oh}=$5.7e-3$$.
Figure 7: The radial position of the triple line scaled against the final
radial position, $R_{f}$, against time (scaled with $t_{i}$) for $\xi=0$, Au-
Ni experimental results and Cu-Mo experimental results. The experimental
results are digitized from Saiz et al. Saiz and Tomsia (2004); Saiz et al.
(2007). The inertial time scale, $t_{i}$ for the Au-Ni system is calculated
using $\rho=$1.1\times 10^{4}\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$$,
$\gamma=$1.0\text{\,}\mathrm{J}\text{\,}{\mathrm{m}}^{-2}$$, $R_{0}=$1\times
10^{-3}\text{\,}\mathrm{m}$$. The inertial time scale for the Cu-Mo system is
calculated using $\rho=$8.9\times
10^{3}\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$$,
$\gamma=$1.3\text{\,}\mathrm{J}\text{\,}{\mathrm{m}}^{-2}$$ and
$R_{0}=$1\times 10^{-3}\text{\,}\mathrm{m}$$. The value of $t_{i}$ is
$1.9\times 10^{-8}\text{\,}\mathrm{s}$ for this work, $3.4\times
10^{-3}\text{\,}\mathrm{s}$ for the Au-Ni system and $2.6\times
10^{-3}\text{\,}\mathrm{s}$ for the Cu-Mo system. This figure shows the
reasonable agreement between the simulation and experimental data when scaled
by the inertial time scale and the agreement with the
$(t/4t_{i})^{\nicefrac{{1}}{{2}}}$ spreading rate.
Figure 8: The spreading radius versus time for $\xi=0$ and $\xi=0.9$. The
black curves are Cu-Si experiments digitized from Protosenko et al.. The
inertial time scale, $t_{i}$, for the Cu-Si system is calculated using
$\rho=$8.9\times 10^{3}\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$$,
$\gamma=$1.3\text{\,}\mathrm{J}\text{\,}{\mathrm{m}}^{-2}$$ and
$R_{0}=$8.2\times 10^{-4}\text{\,}\mathrm{m}$$. The value of $t_{i}$ is
$1.9\times 10^{-8}\text{\,}\mathrm{s}$ for this work and $2.0\times
10^{-3}\text{\,}\mathrm{s}$ for the Cu-Si system
Figure 9: The instantaneous interface energies $\tilde{\gamma}$ plotted
against time for $\xi=0$ and $\xi=0.9$. Both $\tilde{\gamma}_{lv}$ and
$\tilde{\gamma}_{sl}$ are larger for the $\xi=0.9$ curve.
Figure 10: The scaled spreading coefficient versus scaled time for various
values of $\xi$.
Figure 11: Contour plots of the entropy production rate at (a) $t=0.1t_{i}$,
(b) $t=t_{i}$, (c) $t=10t_{i}$ and (d) $t=20t_{i}$. The color intensity
represents the magnitude of either $\sqrt[8]{\dot{S}_{\text{PROD}}}$ (less
focused) on the left panel or $\sqrt{\dot{S}_{\text{PROD}}}$ (more focused) on
the right panel. The colors represent the specific entropy production
mechanism given by the terms in Eq., (21) (diffusive, phase field, viscous),
with red, green and blue representing the first (diffusion), second (solid
interface relaxation) and third (viscous flow) terms, respectively.
|
arxiv-papers
| 2010-06-24T21:15:51 |
2024-09-04T02:49:11.182462
|
{
"license": "Public Domain",
"authors": "Daniel Wheeler, James A. Warren and William J. Boettinger",
"submitter": "Daniel Wheeler",
"url": "https://arxiv.org/abs/1006.4881"
}
|
1006.4901
|
T. Hagihara et al.X-Ray Spectroscopy of Galactic Hot Gas along the PKS
2155–304 Sight Line 2009/12/162010/3/29
Galaxy: disk - Galaxy: halo - X-rays: diffuse background - X-rays: ISM
# X-ray Spectroscopy of Galactic Hot Gas along the PKS 2155-304 Sight Line
Toshishige Hagihara11affiliation: Institute of Space and Astronautical
Science, Japan Aerospace Exploration Agency, 3-1-1, Yoshinodai, Chuo,
Sagamihara, 252-5210 Yangsen Yao22affiliation: University of Colorado, CASA,
389 UCB, Boulder, CO 80309, USA Noriko Y. Yamasaki11affiliationmark: Kazuhisa
Mitsuda11affiliationmark:
Q. Daniel Wang33affiliation: Department of Astronomy, University of
Massachusetts, Amherst, MA 01003, USA Yoh Takei11affiliationmark: Tomotaka
Yoshino11affiliationmark: and Dan McCammon44affiliation: Department of
Physics, University of Wisconsin, Madison, 1150 University Avenue, Madison, WI
53706, USA Present Address is NEC corporation, Nisshin-cho 1-10, Fuchu, Tokyo
183- 8551 hagihara@astro.isas.jaxa.jp, yamasaki@astro.isas.jaxa.jp
hagihara@astro.isas.jaxa.jp, yamasaki@astro.isas.jaxa.jp
###### Abstract
We present a detailed spectroscopic study of the hot gas in the Galactic halo
toward the direction of a blazer PKS 2155-304 ($z=$0.117). The OVII and OVIII
absorption lines are measured with the Low and High Energy Transmission
Grating Spectrographs aboard Chandra, and the OVII, OVIII, and NeIX emission
lines produced in the adjacent field of the PKS 2155-304 direction are
observed with the X-ray Imaging Spectrometer aboard Suzaku. Assuming
vertically exponential distributions of the gas temperature and the density,
we perform a combined analysis of the absorption and emission data. The gas
temperature and density at the Galactic plane are determined to be
$2.5(+0.6,-0.3)\times 10^{6}$ K and $1.4(+0.5,-0.4)\times 10^{-3}$ cm-3 and
the scale heights of the gas temperature and density are $5.6(+7.4,-4.2)$ kpc
and $2.3(+0.9,-0.8)$ kpc, respectively. These values are consistent with those
obtained in the LMC X-3 direction.
## 1 Introduction
X-ray observations of edge-on spiral galaxies revealed the existence of hot
gas at temperatures of $\sim$ 106 K extending a few kpc beyond the disk (e.g.
[Wang et al. (2001), Wang et al. (2003), Strickland et al. (2004), Li et al.
(2008), Yamasaki et al. (2009)]). The origin of energy and material in such a
hot halo has not been clarified. Feedback from supernovae (SNe) as galactic
wind or fountain and heated primordial gas are possible candidates (Norman &
Ikeuchi, 1989). In any cases, halo gas plays important roles in galactic
evolution through chemical circulation and interaction between galaxies and
the intergalactic medium.
The hot gaseous halo in and around the Milky-Way has been investigated for a
long time. For instance, ROSAT All Sky Survey (RASS) quantitatively mapped the
spatial distribution of the Soft X-ray Background emission (SXB; Snowden et
al. (1997)). The Cosmic X-ray Background (CXB) component extrapolated from the
discrete hard X-ray sources could explain only about half of the SXB, leaving
the soft X-ray emission below 1 keV being of a diffuse origin. With the high
resolution X-ray microcalorimeter flying on a sounding rocket, McCammon et al.
(2002) detected emission lines of hydrogen- and helium-like oxygen, neon, and
iron ions from about 1 steradian of the sky, which suggests that the emitting
gas is of a thermal nature and at temperatures of T$\sim 10^{6}$ K. The
existence of the hot gas in and around the Milky-Way is consistent with the
Chandra observations of nearby edge-on spiral galaxies. However, because these
emission data carry very little distance information, the properties of the
global hot gas, like its density, temperature, and their distributions, are
still poorly understood.
A combined analysis of high resolution absorption and emission data provides
us with a powerful diagnostic of properties of the absorbing/emitting plasma.
Absorption lines measure the column density of the absorbing material, which
is an integration of the density of the absorbing ions along a sight line. In
contrast, emission line intensity is sensitive to the emission measure, which
is proportional to the density square of the emitting plasma. Thus, a
combination of the emission and absorption data naturally yields the density
and the size of the corresponding absorbing/emitting gas.
With significantly improved spectral resolution of current X-ray instruments,
we are now able to observe the needed high resolution absorption and emission
lines produced in the hot plasma. For instance, the X-ray absorption lines at
$z=0$, in particular the helium- and hydrogen-like OVII and OVIII lines, are
detected in spectra of many galactic and extragalactic sources (e.g. Futamoto
et al. (2004); Yao & Wang (2005); Williams et al. (2007)). Recently, Fang et
al. (2006) and Bregman & Lloyd-Davies (2007) find that the OVII absorption
line can always be detected in an AGN spectrum as long as the spectrum is of
high signal-to-noise ratio. On the other hand, the X-ray Imaging Spectrometer
(XIS) aboard Suzaku can also resolve emission lines produced in a diffuse
emitting plasma at temperatures of T$\sim 10^{6}$ K. And indeed, the OVII and
OVIII lines have been detected in nearly all directions (e.g., Smith et al.
(2007); Shelton et al. (2007)). Recently, a systematical study of emission
lines of the hot gas in and around the Galaxy has been conducted by Yoshino et
al. (2009), who report the OVII and OVIII lines in 14 blank sky observations
with the XIS and conclude that the line-of-sight mean temperatures of the
emitting gas has a narrow distribution around $2.3\times 10^{6}$ K. Since the
ion fractions of OVII and OVIII and their K-transition emissivities are very
sensitive to gas temperature at $\sim 10^{6}$ K , a combined analysis of these
emission and absorption lines will also constrain the gas temperature and its
distribution without the complexity of relative chemical abundances of metal
elements.
Although this combined analysis method has long been applied in the
ultraviolet wavelength band (Shull & Slavin, 1994), its application in the
X-ray band just began. Complementing the high resolution absorption data
observed with Chandra with the broadband emission data obtained with RASS, Yao
& Wang (2007) firstly attempted to conduct the combined analysis in the X-ray
band to infer the hot gas properties in our Galaxy. They also proposed a model
for the Galactic disk assuming the temperature and density of the hot gas
fading off exponentially along the vertical direction. They concluded that the
OVII and OVIII absorption lines observed along the Mrk 421 sight line are
consistent with the Galactic disk origin. Yao et al. (2009) further
constrained this disk model by jointly analyzing the high resolution
absorption data obtained with Chandra along the LMC X-3 sight line and
emission data observed with Suzaku in the vicinity of the sight line. They
estimated gas temperature and density at the Galactic plane and their scale
heights as 3.6 (+0.8, $-$0.7) $\times 10^{6}$ K and 1.4 (+2.0, $-$1.0) $\times
10^{-3}$ cm-3 and 1.4 (+3.8, $-$1.2) kpc and 2.8 (+3.6, $-$1.8) kpc,
respectively. These results are consistent with the early findings by Yao &
Wang (2007), i.e., the SXB can be explained by a kpc-scale halo around our
Galaxy.
In this paper, we present the second case study of the combined analysis of
high resolution absorption and emission lines. The absorption lines are
observed with Chandra along a blazer, PKS 2155–304 sight line and the emission
lines are obtained with Suzaku observations of the vicinity of the sight line.
In Section 2, we describe our observations and data reduction process. We
perform our data analysis in Section 3 and discuss our results in Section 4.
## 2 Observations and Data Reduction
Table 1: Suzaku Observation Log | Sz1 | Sz2
---|---|---
($\alpha$, $\delta$) in J2000 (degrees) | (329.2236,$-$30.5193) | (330.1731,$-$29.9560)
($\ell$,$b$) in Galactic coordinate (degrees) | (17.1809,$-$51.8544) | (18.2418,$-$52.6081)
Observation ID | 503082010 | 503083010
Observation start times (UT) | 18:32:39, 2008 Apr 29 | 08:31:41, 2008 May 2
Observation end times (UT) | 08:30:08, 2008 May 2 | 17:30:19, 2008 May 4
Exposure time | 90ks | 87ks
Exposure after the data reduction | 51.1ks | 56.3ks
### 2.1 Chandra Observations and Data Reduction
Chandra observed PKS 2155–304 many times. There are two grating spectrographs
(the low and high energy transmission grating spectrographs; LETG and HETG)
and two sets of detectors ( the advanced CCD imaging spectrometer; ACIS and
the high resolution camera; HRC) aboard Chandra 111please refer to the Chandra
Observatory Guide for more information:
http://cxc.harvard.edu/proposer/POG/html/index.html. In this work, we used all
observations available to the date of 2009 March, except for some observations
made with non-standard configuration of ACIS (i.e., putting source outside the
CCD-S3 chip) to avoid spectral resolution degradation. The data used in this
work include 46 observations with an accumulated exposure time of 1.07 Ms.
We followed the standard scripts to calibrate the observations 222Please refer
to the CIAO script for more information: http://cxc.harvard.edu/ciao/guides/.
When extracting the grating spectra and calculating the instrumental response
files, we used the same energy grid for all observations with different
grating instruments and/or with different detectors for ease of the adding
process described in the following. For those HETG observations, we only use
the first order grating spectra of the medium energy grating (MEG) to utilize
its large effective area at lower ($<1$ keV) energy. For those observations
taken with the HRC, we further followed the procedure presented in Yao et al.
(2009) to extract the first order spectra of the LETG. We then added the first
grating order spectra of all observations to obtain a single stacked spectrum
and a corresponding instrumental response file.
### 2.2 Suzaku Observations and Data Reduction
(80mm, 50mm)figure1.eps
Figure 1: RASS 0.1 - 2.4 keV band X-ray map in the vicinity of PKS 2155-304
(the bright source at the center) and the XIS field of view of the two
presented observations.
(80mm,50mm)figure2.eps
Figure 2: (a) XIS light curve in 0.3-2.0 keV and (b) solar wind proton flux
calculated using the data of ACE SWEPAM in Sz1 (top) and Sz2 (bottom)
observation periods. The time is plotted from the beginning of each
observation with Suzaku. The time bin of proton data are shifted 5000 seconds
to correct for the travel time of the solar wind from the ACE satellite to the
Earth. The dashed lines in the bottom panel indicate the threshold of the
proton flux as $4\times 10^{8}$ cm-2 s-1.
We observed the emission of the hot diffuse gas toward two off-fields of the
PKS 2155-304 sight line during the AO2 program (Table 1). To minimize
confusion by stray lights from the PKS 2155-304 and to average out the
possible spatial gradient of the diffuse emission intensity, the two fields
were chosen to be 30′ away from the PKS 2155–304 and in nearly opposite
directions (Fig. 1). With this configuration and the roll angle of the XIS
field of view, we estimate that stray lights from PKS 2155-304 contribute no
more than 10% to the observed X-ray emission in 0.3–1.0 keV energy range. Our
observation pointings are away from the southern edge of Radio Loop I
(Berkhuijsen et al., 1971). Thus we consider that there is no contamination of
the emission from the Loop I in our observations. This is supported by the
observational results that there is no EUV enhancement in this direction
(Sembach et al., 1997).
Our observations were taken with the CCD camera X-ray Imaging Spectrometer
(XIS; Koyama et al. (2007)) on board Suzaku (Mitsuda et al., 2007). The XIS
was set to the normal clocking mode and the data format was either $3\times 3$
or $5\times 5$, and the spaced-raw charge injection (SCI) was applied to the
data during the observations. We used processed data version 2.2.7.18 for the
two observations. In this work, we only used the spectra obtained with XIS1.
Compared to the other two front side-illuminated CCDs, XIS0 and XIS3, XIS1 is
a backside-illuminated CCD chip and is of high sensitivity at photon energies
below 1 keV. We found no point sources in the FOV, thus we used the full CCD
field of view in further analysis to increase the photon counts because X-ray
from the calibration sources do not affect the soft X-ray spectrum below 5
keV.
We adopted the standard data selection criteria to obtain the good time
intervals (GTIs), i.e. excluding exposures when the line of sight of Suzaku is
elevated above the Earth rim by less than 20∘ and exposures with the “cut-off
rigidity” less than 8 GV. We checked the column density of the neutral oxygen
in the Sun-lit atmosphere in the line of sight during the selected GTIs, and
excluded the exposures when the column density is larger than $1.0\times
10^{15}$ cm-2 to avoid significant neutral oxygen emission from Earth’s
atmosphere (Smith et al., 2007). We created X-ray images in 0.4–1.0 keV energy
range for the two observations, and found no obvious discrete X-ray sources in
the fields.
In the last step, we excluded those events severely contaminated by the X-ray
emission induced by the solar-wind charge exchange (SWCX) from geocorona
(Fujimoto et al. 2007), meeting either of the following two criteria by
Yoshino et al. (2009). The first one is the solar wind flux (Fig.2). We used
the solar wind data obtained with the Solar Wind Electron Proton and Alpha
Monitor (SWEPAM) aboard the Advanced Composition Explorer (ACE ) and removed
the time intervals when the proton flux in the solar wind exceeds $4\times
10^{8}$ cm-2 s-1 (Masui et al., 2009). ACE is in L1 of the Solar-Earth system,
1.5$\times 10^{6}$ km away from the Earth and assuming average solar wind
velocity as 300 km s-1, we corrected traveling time of the solar wind from L1
to the Earth. The second criteria is the Earth-to-magnetopause (ETM) distance
in the line sight of Suzaku (Fujimoto et al., 2007), which is required to be
$>5R_{E}$. We found that about 20% and 5% of the exposure time of our 1st and
2nd observations meets the first criteria and no time meets the second
criteria. Thus we exclude that 20% and 5% from 1st and 2nd observations and
used the remaining time in further analysis. We also checked the light curve
of XIS 1 in the energy range of 0.3 to 2.0 keV in the observation periods and
found no evidence for variation (Fig. 2).
We constructed instrumental response files (rmfs) and effective area files
(arf) by running the scripts xisrmfgen and xissimarfgen (Ishisaki et al.,
2007). To take into account the diffuse stray light effects, we used a 20′′
radius flat field as the input emission in calculating the arf. We also
included in the arf file the degradation of low energy efficiency due to the
contamination on the XIS optical blocking filter. The versions of calibration
files used here were ae_xi1_quanteff_20080504.fits,
ae_xi1_rmfparam_20080901.fits, ae_xi1_makepi_20080825.fits and
ae_xi1_contami_20071224.fits. We estimated the non-X-ray-background from the
night Earth database using the method described in Tawa et al. (2008).
We grouped the spectra to have a minimum number of counts in each channel
$\geq$ 50 and used energy range of 0.4–5.0 keV in our analysis. This range is
broad enough for constraining the continuum and also covers the H- and He-like
emission lines of N, O, Ne, Mg, and the L transition of Fe. The OVII, OVIII,
and NeIX lines are clearly visible in the spectra (Section 3.2).
## 3 Spectral Analysis and Results
We carried out our data analysis with the Xspec software package, adopting the
solar abundances as given in Anders & Grevesse (1989). (Hereafter, use of
italic type indicates Xspec models and their parameters.) Errors quoted
throughout this paper are single parameter errors given at the 90 % confidence
level, unless specified otherwise. Sections 3.1 and 3.2 give a discussion of
our separate analyses of the absorption and emission data, while sections 3.3
and 3.4 give a discussion of the jointly–analyzed data under the uniform and
exponential disk models.
### 3.1 Chandra X-ray Absorption Spectrum
We first measured the equivalent widths (EWs) of the absorption lines of the
highly ionized oxygen ions. Becuase measurement of these narrow absorption
lines is relevant only to the local continuum, we fit the final PKS 2155-304
spectrum between 0.55 and 0.7 keV as shown in figure 3 using a power-law model
modified with absorption by the neutral ISM(wabs). The column density of
neutral hydrogen was fixed to 1.47$\times 10^{20}$ cm-2, which is the value
determined by the LAB Survey of Galactic HI in this direction (Kalberla et
al., 2005). Three Gaussian functions were used to model the OVII Kα, OVIII Kα,
and OVII Kβ absorption lines (model A1). The measured EWs were found to be
consistent with those reported by Williams et al. (2007). The results are
summarized in Table 2
Once the equivalent widths were determined, we applied an absorption line
model, absem, to replace the gaussian functions in order to probe the
properties of the absorbing gas. Assuming the temperature and density
distributions of the hot plasma, the absem model, which is a revision of the
absline model of Yao & Wang (2005), can be used to jointly fit the emission
and absorption spectra. (See Yao & Wang (2007) and Yao et al. (2009) for a
detailed description.) For a gas with a uniform density and a single
temperature, the diagnostic procedure is summarized as follows: (1) A joint
analysis of OVII Kα and OVII Kβ directly constrains the OVII column density
and the Doppler dispersion velocity ($v_{\rm b}$). With the constrained
$v_{\rm b}$, adding the OVIII Kα line in the analysis also yields the column
density of OVIII (model A2). (2) Because the column density ratio of OVII and
OVIII is sensitive to the gas temperature, a joint analysis of the OVII and
OVIII lines will naturally constrain the gas temperature (model A3). (3)
Assuming the solar abundance for oxygen and given the constrained gas
temperature, the OVII (or OVIII ) column density can be converted to the
corresponding hot phase hydrogen column density (model A4). Table 3 gives the
results of our fits. The constrained OVII column density, 5.9 (+1.2, $-$0.9)
$\times 10^{15}$ cm-2 is comparable to typical values $\sim 10^{16}$ cm-2
obtained from AGN observations given in two systematic studies (Fang et al.
(2006) and Bregman & Lloyd-Davies (2007)).
(80mm, 50mm)figure3.eps
Figure 3: Chandra spectrum of PKS 2155-304 between 0.55 and 0.7 keV. Fitted model is A4. Table 2: Spectral fitting results of absorption data with model A1 Model | | OVII Kα | OVIII Kα | OVII Kβ
---|---|---|---|---
A1 | Centroid (eV) | $573.8^{+0.1}_{-0.2}$ | $653.1^{+0.4}_{-0.4}$ | $665.8^{+0.1}_{-0.4}$
| Sigma (eV) | $0.32^{+0.25}_{-0.32}$ | $1.01^{+0.62}_{-0.54}$ | $0.01^{+0.97}_{-0.01}$
| Equivalent Width (eV) | $0.354^{+0.075}_{-0.071}$ | $0.377^{+0.116}_{-0.102}$ | $0.119^{+0.058}_{-0.058}$
Model A1:wabs(power-law+$3\times$Gaussian) |
Table 3: Spectral fitting results of absorption data with model A2-A4 Model | $v_{b}$ | $\log$ [Column Density] | $\log T$ | $\chi^{2}$/dof
---|---|---|---|---
| (km s-1) | (cm-2) | (K) |
| | $N_{\rm O\emissiontype{VII}}$ | $N_{\rm O\emissiontype{VIII}}$ | $N_{\rm H_{Hot}}$ | |
A2 | $294^{+149}_{-220}$ | $15.76^{+0.07}_{-0.08}$ | $15.56^{+0.09}_{-0.12}$ | $\cdots$ | $\cdots$ | 489.82/474
A3 | $375^{+124}_{-158}$ | $15.77^{+0.08}_{-0.07}$ | $\cdots$ | $\cdots$ | $6.27^{+0.02}_{-0.03}$ | 498.01/474
A4 | $290^{+152}_{-220}$ | $\cdots$ | $\cdots$ | $19.08^{+0.06}_{-0.07}$ | $6.28^{+0.02}_{-0.02}$ | 489.84/474
Model A2,A3,A4:wabs(power-law)$\times$absem$\times$absem$\times$absem |
### 3.2 Suzaku X-ray Emission Spectra
The Suzaku data were modeled in order to constrain the emission measure and
the temperature of the halo. For this purpose, we first modeled the SXB using
a multiple component model, since the SXB emission is a superposition of such
components. We detail this model below.
#### 3.2.1 Foreground and Background Emission
We assumed that the SXB consists of four dominant components: (1) the Local
Hot Bubble (LHB), (2) Solar Wind Charge eXchange in the heliosphere (SWCX),
(3) a hot gaseous Galactic halo, (4) the cosmic X-ray background emission
(CXB; mainly from unresolved extragalactic sources such as AGNs). Because the
contribution from unresolved Galactic sources is expected to be negligible at
high galactic latitudes ($|b|>30^{\circ}$), we did not consider such a
contribution. The CXB spectrum is well described by a power-law.
In a study of 14 Suzaku blank sky observations, Yoshino et al. (2009) found
that there are at least 2 LU (photons ${\rm s^{-1}cm^{-2}str^{-1}}$) of OVII
line emission even in those directions where the attenuation length for the
line is less than 300 pc. This emission is considered to come from the SWCX
and LHB, though these contributions are difficult to separate with the current
CCD energy resolution. After Smith et al. (2007) and Henley et al. (2007),
Yoshino et al. (2009) found that it can be well represented by a model
consisting of unabsorbed, optically thin thermal emission from a
collisionally-ionized plasma. The best-fit temperature of this model is log
$T$ = 6.06. We therefore use a $\sim 10^{6}$ K plasma of 2 LU OVII surface
brightness as the SWCX+LHB component. The uncertainty of this estimate is
discussed in section 4.1.
Except for the SWCX+LHB component, the observed emission has been absorbed by
the foreground ISM. In the following analysis, we also fix the neutral
hydrogen column density to be $1.47\times 10^{20}~{}{\rm cm^{-2}}$ (Kalberla
et al. (2005)).
(80mm,50mm)figure4.eps
Figure 4: Suzaku spectra between 0.4 and 5.0 keV of Sz1 (top) and Sz2 (bottom)
are plotted. Fitted model is E2 (wabs(power-lawCXB \+ vmekalhalo) +
vmekalLHB+SWCX \+ 3$\times$gaussians). The O and Ne abundance of the
vmekalhalo (green, dash-dotted) and vmekalLHB+SWCX (blue, dotted) are set to
be zero and three gaussians (magenta, solid) represent OVII Kα, (OVII Kβ \+
OVIII Kα) and NeIX Kα emission lines.
(80mm,50mm)figure5.eps
Figure 5: Relation between OVII and OVIII surface brightnesses for the 14 (Yoshino et al. (2009))+2 (this work) sky fields observed with Suzaku. The horizontal and vertical bars of data points show the 1 $\sigma$ errors of the estimate. The contribution of OVII Kβ emission is corrected for OVIII Kα. The diagonal lines show the relation between OVIIand OVIII, assuming an offset OVII emission of 2.1 LU and emission from a hot plasma of the temperature and the absorption column density are shown. The Galactic absorption column density of the observation fields are indicated by the maker size of the data points. Table 4: Spectral fitting results for emission data with the model E1 Model | Data | CXB | LHB+SWCX | halo | $\chi^{2}$/dof
---|---|---|---|---|---
| | Norm a | $\log T$ | Normb | $\log T$ | Normb | N/O | Ne/O | Fe/O |
| | | (K) | | (K) | | | | |
E1 | Sz1 | $8.40^{+0.38}_{-0.40}$ | 6.06(fixed) | 4.3(fixed) | $6.26^{+0.06}_{-0.04}$ | $3.3^{+1.0}_{-0.8}$ | $6.0^{+2.5}_{-1.9}$ | $6.5^{+3.7}_{-2.5}$ | $7.4^{+13.8}_{-4.8}$ | 148.61/136
E1 | Sz2 | $6.45^{+0.36}_{-0.43}$ | 6.06(fixed) | 4.3(fixed) | $6.35^{+0.03}_{-0.03}$ | $3.2^{+0.5}_{-0.4}$ | $4.7^{+2.0}_{-1.6}$ | $2.4^{+1.2}_{-0.9}$ | $1.0^{+0.8}_{-0.5}$ | 147.33/141
E1 | Sz1+Sz2(Sz1) | $8.30^{+0.35}_{-0.39}$ | 6.06(fixed) | 4.3(fixed) | $6.33^{+0.02}_{-0.02}$ | $3.0^{+0.3}_{-0.3}$ | $5.8^{+1.6}_{-1.3}$ | $3.3^{+1.2}_{-0.9}$ | $1.7^{+1.2}_{-0.7}$ | 306.82/282
| Sz1+Sz2(Sz2) | $6.50^{+0.36}_{-0.39}$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ |
E1 † | Sz1+Sz2(Sz1) | $8.38^{+0.35}_{-0.36}$ | 6.06(fixed) | 0.0(fixed) | $6.25^{+0.03}_{-0.02}$ | $4.9^{+0.7}_{-0.6}$ | $4.2^{+1.0}_{-0.8}$ | $4.5^{+1.4}_{-1.2}$ | $4.7^{+3.0}_{-1.6}$ | 313.48/282
| Sz1+Sz2(Sz2) | $6.59^{+0.34}_{-0.39}$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ |
E1 ‡ | Sz1+Sz2(Sz1) | $8.25^{+0.38}_{-0.37}$ | 6.06(fixed) | 7.5(fixed) | $6.37^{+0.03}_{-0.03}$ | $2.3^{+0.3}_{-0.3}$ | $6.9^{+2.4}_{-1.9}$ | $3.2^{+1.3}_{-1.0}$ | $1.5^{+0.9}_{-0.5}$ | 299.45/282
| Sz1+Sz2(Sz2) | $6.47^{+0.36}_{-0.38}$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ |
$\uparrow$ indicates linked parameters
Sz1+Sz2: simultaneous fitting of the data Sz1 and Sz2
Model E1:wabs(power-lawCXB \+ vmekalhalo) + mekalLHB+SWCX
Emission measure of mekal LHB+SWCX is fixed to 0.0043cm-6 which corresponds to
2.0 LU of OVII Kα emission
† Emission measure of mekalLHB+SWCX is set to 0 as the lower limit
‡ Emission measure of mekalLHB+SWCX is set to the upper limit which
corresponds to 3.5 LU of OVII Kα emission
a in unit of photons cm-2 s-1str-1 eV-1 @1keV
b Emission measure 10-3 $\int n_{e}n_{p}dl$: in unit of cm-6 pc
Table 5: Surface brightness of OVII, OVIII and NeIX Model | Data | CXB | halo | OVII Kαc | OVII Kβ+ | OVIII Kαc | NeIX Kαc | $\chi^{2}$/dof
---|---|---|---|---|---|---|---|---
| | Norm a | Normb | N | Fe | | OVIII Kαc | | |
E2 | Sz1 | $8.21^{+0.62}_{-0.27}$ | $4.2^{+0.3}_{-0.8}$ | $6.0$ (fixed) | 7.4 (fixed) | $5.00^{+0.69}_{-0.80}$ | $1.45^{+0.33}_{-0.51}$ | $1.10^{+0.39}_{-0.56}$ | $0.65^{+0.12}_{-0.26}$ | 136.39/132
E2 | Sz2 | $6.37^{+0.53}_{-0.26}$ | $4.5^{+0.7}_{-0.6}$ | 4.7 (fixed) | 1.0 (fixed) | $5.15^{+0.66}_{-0.86}$ | $1.98^{+0.53}_{-0.37}$ | $1.62^{+0.59}_{-0.42}$ | $0.58^{+0.10}_{-0.29}$ | 150.59/137
model E2: wabs(power-lawCXB \+ vmekalhalo) + vmekalLHB+SWCX \+
3$\times$gaussians, where O and Ne abundances of two vmekal are set to 0
a in unit of photons cm-2 s-1str-1 eV-1 @1keV
b Emission Measure 10-3 $\int n_{e}n_{p}dl$: in unit of cm-6 pc
c in unit of LU = photons s-1 cm-2 str-1
#### 3.2.2 Spectral Fitting
To probe the halo gas properties, we used the following model to fit our
spectra (model E1):
${\it wabs(power-law_{CXB}+vmekal_{halo})+mekal_{LHB+SWCX}}$, with the photon
index of the CXB fixed at 1.4 and with the normalization as a free parameter.
The temperature and the corresponding emission measure (and thus the
normalization) of the mekalLHB+SWCX component were set to $1.2\times 10^{6}$ K
and $0.0043$ pc cm-6, respectively, correspondind to 2 LU of OVII Kα line
emission. In the halo component, we fixed the abundance ratio of oxygen to
hydrogen to the solar value, and allowed the abundances of nitrogen, neon, and
iron vary.
This model fit the spectra from both pointings consistently, except for an
apparently higher neon and iron abundance in Sz1 (Table 4) which would be
caused by a lower temperature of the Sz1 halo component. It is important to
clarify whether this is caused by statistical effects or by a true difference
in plasma temperature. The surface brightness of each line is a better
indicator for this purpose.
We next evaluated the surface brightness of OVII and OVIII lines by modifying
model E1 (this is model E2). We set the O and Ne abundance of the halo and
LHB+SWCX to zero and used three Gaussian emission lines to represent OVII Kα,
(OVII Kβ \+ OVIII Kα) and NeIX Kα emission (Fig. 4). Since the XIS resolution
is not high enough to enable us to distinguish the OVII Kβ (656 eV) and OVIII
Kα (653 eV) lines, they were modeled as a single line. This model fitted both
spectra with a $\chi^{2}$/dof of 135.52/132 and 150.59/137 respectively.
Assuming the ratio between OVII Kβ, and OVII K${}_{\alpha}(=\mu)$ intensities
is 0.07 (see footnote 3) 33footnotetext: $\mu$ is a slow function of the
plasma temperature for thermal emission and here the value is 0.056. If the
emission is due to SWCX, $\mu=0.083$ (Kharchenko et al., 2003). We averaged
these two values and used $\mu=0.07$ here. See Yoshino et al. (2009) section
3.1 for details., we calculated the OVII, OVIII and NeIX surface brightnesses
as listed in Table 5. Intensities of these lines between the two fields are
consistent to within the 90% confidence level, and we assume that the
temperature difference is not essential. We plotted the OVII and OVIII surface
brightness over the Yoshino et al. (2009) results (Fig. 5, with 1 $\sigma$
error) and found that the OVII and OVIII surface brightness of the PKS
2155-304 direction matches the trend of the other 14 fields.
We next fitted both data sets simultaneously with model E1 by linking
parameters of the halo component in both observations. The results are shown
in Table 5. (Fig. 6). The emission measure for the model is 3.0 (+0.3, $-$0.3)
$\times 10^{-3}$ cm-6 pc and the temperature is 2.1 (+0.1, $-$0.1) $\times
10^{6}$ K. McCammon et al. (2002) reported the emission measure and
temperature of the absorbed thermal component (=halo) as 3.7 $\times 10^{-3}$
cm-6 pc and 2.6 $\times 10^{6}$ K which are comparable to our values.
(80mm,50mm)figure6.eps
Figure 6: Suzaku spectra between 0.4 and 2.0 keV. Sz1 (top) and Sz2 (bottom)
observations are plotted. Fitted model is E1 (wabs(power-law+vmekal halo)
+mekalLHB+SWCX) and parameters of the halo components are linked in both
spectra.
### 3.3 Combined Analysis
Up to now, we have analyzed the absorption and emission data separately and
confirmed that the models including the halo component fit both data with a
temperature of 1.91(+0.09, $-$0.09) $\times 10^{6}$ K for the absorption and
2.14 (+0.15, $-$0.14) $\times 10^{6}$ K for the emission spectra.
Assuming that both plasmas are common and uniform, the plasma length and
density can be calculated using the emission measure and the column density.
The length and density are found to be 4.0 (+1.9, $-$1.4) kpc and 7.7 (+2.3,
$-$1.7) $\times 10^{-4}$ cm-3, respectively. The errors of the calculated
values are overestimated, since these errors are not independent. Moreover,
important plasma parameters such as temperature and velocity dispersion were
not considered in this simple calculation.
In this section, using the combined analysis, we will try to determine the
physical conditions of the halo plasma, including the density, the temperature
and their distribution.
#### 3.3.1 Uniform Disk Model
The first step in our combined analysis was to try the simplest model: an
isothermal plasma with uniform density extending up to $h$ kpc above the disk
(model C1).
To perform this combined analysis the emission measure and column density have
to be linked with a common parameter. We chose the equivalent hydrogen column
density ($N_{\rm H_{Hot}}$) and scale height ($h$) as the control parameters
and calculated the emission measure. The relation of the density $n$, scale
height $h$, column density $N_{\rm H_{Hot}}$ and galactic latitude $b$ is
described as $N_{\rm H_{Hot}}=nh/{\rm sin}b$. Thus we can use the A4 model for
absorption data directly, and revise the E1 model to use the vabmkl instead of
the mekal model. The vabmkl model, an extension of the mekal model, was
constructed for the combined fit and used the column density and plasma length
as the fit parameters. (see Yao et al. (2009) for a detailed model
description). For the halo components of the emission spectra, we fixed the
abundance ratio of oxygen to hydrogen to the solar value and allowed the
abundances of nitrogen, neon, and iron to vary again. All parameters except
for the normalization of the CXB components are linked over the two sets of
emission data. We put lower and upper limits (70-440 km s-1) to the velocity
dispersion ($v_{b}$) which represent the 90 % error range of the values
obtained by the absorption analysis.
The model C1 fits both data sets ($\chi^{2}$/dof=802.78/754) and the results
are given in Table 6. The column density and temperature are consistent with
the A4 model (Table 3), while the temperature and abundance of Ne and Fe are
not consistent with the E1 model (Table 4). This is because the temperature is
mostly constrained by the absorption data and the lower temperature for the
emission spectra preferred the higher abundance to describe the Ne and Fe
lines. The plasma length is 4.2 (+1.5, $-$1.2) kpc and suggests that under the
isothermal assumption the halo expands beyond the Galactic disk ($\sim$1 kpc).
Table 6: Combined spectral fitting results with the uniform disk model Model | Data | CXB | halo | $\chi^{2}$/dof
---|---|---|---|---
| | Norma | $\log N_{\rm H_{Hot}}$ | $h$ | $\log T$ | $v_{b}$⋆ | N/O | Ne/O | Fe/O |
| | | (cm-2) | (kpc) | (K) | (km s-1) | | | |
C1 | Emission:Sz1 | $8.38^{+0.39}_{-0.38}$ | $19.08^{+0.06}_{-0.07}$ | $4.2^{+1.5}_{-1.2}$ | $6.27^{+0.02}_{-0.02}$ | $\cdots$ | $4.9^{+1.4}_{-1.0}$ | $5.2^{+1.4}_{-1.5}$ | $5.0^{+1.6}_{-1.7}$ | 802.78/754
| Emission:Sz2 | $6.57^{+0.39}_{-0.38}$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\cdots$ | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| Absorption | $\cdots$ | $\uparrow$ | $\cdots$ | $\uparrow$ | $286^{+154}_{-206}$ | $\cdots$ | $\cdots$ | $\cdots$ |
$\uparrow$ indicates linked parameters
model C1: wabs(power-law+vmekal)+mekal for the emission,
wabs(power)$\times$(absem)3 for the absorption
⋆Parameter range is limited to 70-440 km s-1
ain unit of photons cm-2 s-1 str-1 eV-1 @1keV
#### 3.3.2 Exponential Disk Model
Observations of edge-on galaxies(ex. Wang et al. (2003), Li et al. (2008),
Yamasaki et al. (2009)) have revealed that the intensities of X-ray emission
from extended hot gas decreases exponentially as a function of height from the
galactic plane. As a next step in our analysis, we employed another simple
model to fit the data: an exponential distribution model (Yao et al. (2009)).
In this model, the density $n$ and temperature $T$ of the hot gas are
distributed according to the following equation,
$n=n_{0}e^{-Z/h_{n}\xi},\hskip 10.00002ptT=T_{0}e^{-Z/h_{T}\xi},\hskip
10.00002pt\gamma=h_{T}/h_{n}$ (1)
where $Z$ is the vertical distance from the Galactic plane, $n_{0}$ and
$T_{0}$ are the density and temperature at the plane, and $h_{n}$ and $h_{T}$
are the scale heights of the density and temperature, respectively, and $\xi$
is the filling factor, which is assumed to be 1 in this paper. Thus the
equivalent hydrogen column density of the hot gas ($N_{\rm H_{Hot}}$) is
calculated as $N_{\rm H_{Hot}}=\int_{0}^{\infty}ndl=\int_{0}^{\infty}n_{0}{\rm
exp}(-Z/h_{n})dZ/{\rm sin}b=n_{0}h_{n}/{\rm sin}b$.
The models vabmkl and absem can also be used in an exponential disk model
using the additional parameter $\gamma$ (see Yao et al. (2009) for detailed
description). We therefore used the same model as used in the uniform model
here (model C2). For fit parameters, for convenience we used the column
density $N_{\rm H_{Hot}}$ instead of $n_{0}$.
We jointly fitted the emission and absorption data using this exponential disk
model. The parameters obtained are summarized in Table 7. We first fixed the
velocity dispersion ($v_{b}$) at 290 km s-1. We next examined the robustness
of the temperature ($T_{0}$), column density ($N_{\rm H_{HOT}}$), and scale
height ($h_{n}$), as a function of $\gamma$, $v_{b}$, and the intensity of
foreground SWCX intensity. We found that all parameters are consistent to
within 90% statistical errors. When we fitted with $v_{b}$ allowed to vary
freely, the best-fit value of $v_{b}$ became 54${}^{+19}_{-13}$ km s-1. Though
this is above the thermal velocity ($\sim$ 30 km s-1), it is a smaller value
than that obtained from the absorption spectrum which determined the ratio
between the OVII Kα and Kβ lines. In the exponential disk model, low ($3\times
10^{5}{\rm K}<T<10^{6}$ K) temperature plasma can exist in the outer regions,
which contribute only to the OVII absorption line. This might cause the
smaller $v_{b}$ value. The cooling time of such low temperature plasmas is
very short, and the actual situation will not follow such a simple exponential
model in this temperature range. We therefore fixed $v_{b}$ at 290 km s -1, as
the best-fit value from the absorption analysis.
Confidence contours of $h_{n}$, $T_{0}$ and $N_{\rm H_{Hot}}$ versus gamma are
plotted in figure 7, over-laid on those of the LMC X-3 direction (Yao et al.,
2009). We then obtained the scale height for the temperature gradient as
$h_{t}=5.6^{+7.4}_{-4.2}$ kpc and the gas density at the galactic plane as
$n_{0}=(1.4^{+0.5}_{-0.4})\times 10^{-3}$ cm-3 (Figure 8). This values is
typical for the mid-plane plasma density (Cox, 2005). As the high temperature
plasma close to the Galactic plane can emit Fe and Ne lines efficiently, the
spectrum can be fitted without an abundance of heavy element higher than the
solar value. The emission weighted temperature calculated with best fitted
parameters using the intensity ratio of OVIII to OVII becomes $2.2(+0.1,-0.1)$
$\times 10^{6}$ K.
Table 7: Combined spectral fitting results with the exponential disk model Model | Data | CXB | halo | $\chi^{2}$/dof
---|---|---|---|---
| | Norma | $\log N_{\rm H_{Hot}}$ | $h_{n}$ | $\log T_{0}$ | $v_{b}$⋆ | $\gamma$ | N/O | Ne/O | Fe/O |
| | | (cm-2) | (kpc) | (K) | (km s-1) | | | | |
C2 | Emission:Sz1 | $8.26^{+0.36}_{-0.37}$ | $19.10^{+0.08}_{-0.07}$ | $2.3^{+0.9}_{-0.8}$ | $6.40^{+0.09}_{-0.05}$ | $\cdots$ | $2.44^{+1.11}_{-1.41}$ | $5.8^{+1.6}_{-1.3}$ | $3.1^{+1.6}_{-1.2}$ | $1.5^{+1.0}_{-0.7}$ |
| Emission:Sz2 | $6.46^{+0.36}_{-0.36}$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\cdots$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| Absorption | $\cdots$ | $\uparrow$ | $\cdots$ | $\uparrow$ | 290 (fixed) | $\uparrow$ | $\cdots$ | $\cdots$ | $\cdots$ | 792.76/757
C2 | Emission:Sz1 | $8.20^{+0.39}_{-0.42}$ | $19.13^{+0.07}_{-0.07}$ | $2.2^{+0.5}_{-0.7}$ | $6.48^{+0.04}_{-0.04}$ | $\cdots$ | 1.0(fixed) | $6.1^{+1.8}_{-1.4}$ | $2.4^{+0.9}_{-0.9}$ | $1.0^{+0.6}_{-0.4}$ |
| Emission:Sz2 | $6.40^{+0.38}_{-0.41}$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\cdots$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| Absorption | $\cdots$ | $\uparrow$ | $\cdots$ | $\uparrow$ | 290 (fixed) | $\uparrow$ | $\cdots$ | $\cdots$ | $\cdots$ | 795.64/758
C2 | Emission:Sz1 | $8.25^{+0.33}_{-0.38}$ | $19.10^{+0.07}_{-0.07}$ | $2.4^{+0.9}_{-0.7}$ | $6.38^{+0.02}_{-0.03}$ | $\cdots$ | 3.5(fixed) | $5.6^{+1.1}_{-1.3}$ | $3.3^{+1.2}_{-0.8}$ | $1.7^{+0.3}_{-0.5}$ |
| Emission:Sz2 | $6.45^{+0.33}_{-0.37}$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\cdots$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| Absorption | $\cdots$ | $\uparrow$ | $\cdots$ | $\uparrow$ | 290 (fixed) | $\uparrow$ | $\cdots$ | $\cdots$ | $\cdots$ | 793.64/758
C2 | Emission:Sz1 | $8.17^{+0.37}_{-0.38}$ | $19.41^{+0.19}_{-0.16}$ | $5.1^{+3.9}_{-4.8}$ | $6.51^{+0.16}_{-0.10}$ | $\cdots$ | $0.43^{+1.16}_{-0.23}$ | $5.7^{+1.5}_{-1.3}$ | $2.3^{+1.0}_{-1.0}$ | $1.0^{+0.4}_{-0.5}$ |
| Emission:Sz2 | $6.37^{+0.37}_{-0.38}$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\cdots$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| Absorption | $\cdots$ | $\uparrow$ | $\cdots$ | $\uparrow$ | 70 (fixed) | $\uparrow$ | $\cdots$ | $\cdots$ | $\cdots$ | 789.75/757
C2 | Emission:Sz1 | $8.27^{+0.38}_{-0.33}$ | $19.13^{+0.06}_{-0.07}$ | $1.9^{+0.5}_{-0.3}$ | $6.40^{+0.04}_{-0.05}$ | $\cdots$ | $2.84^{+1.34}_{-1.64}$ | $5.8^{+1.7}_{-1.3}$ | $3.2^{+1.1}_{-1.0}$ | $1.6^{+0.5}_{-0.8}$ |
| Emission:Sz2 | $6.47^{+0.38}_{-0.32}$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\cdots$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| Absorption | $\cdots$ | $\uparrow$ | $\cdots$ | $\uparrow$ | 440 (fixed) | $\uparrow$ | $\cdots$ | $\cdots$ | $\cdots$ | 795.44/757
C2† | Emission:Sz1 | $8.31^{+0.33}_{-0.41}$ | $19.08^{+0.05}_{-0.07}$ | $1.5^{+0.6}_{-0.5}$ | $6.36^{+0.03}_{-0.07}$ | $\cdots$ | $3.39^{+2.40}_{-1.97}$ | $4.9^{+1.1}_{-0.9}$ | $3.2^{+0.7}_{-0.5}$ | $2.0^{+0.9}_{-0.8}$ |
| Emission:Sz2 | $6.52^{+0.33}_{-0.41}$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\cdots$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| Absorption | $\cdots$ | $\uparrow$ | $\cdots$ | $\uparrow$ | $290$(fixed) | $\uparrow$ | $\cdots$ | $\cdots$ | $\cdots$ | 803.71/757
C2‡ | Emission:Sz1 | $8.16^{+0.38}_{-0.39}$ | $19.16^{+0.09}_{-0.08}$ | $3.7^{+1.8}_{-1.1}$ | $6.51^{+0.10}_{-0.07}$ | $\cdots$ | $1.47^{+0.52}_{-1.05}$ | $7.2^{+2.4}_{-1.8}$ | $2.2^{+1.3}_{-1.0}$ | $0.9^{+0.6}_{-0.4}$ |
| Emission:Sz2 | $6.36^{+0.38}_{-0.38}$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\cdots$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| Absorption | $\cdots$ | $\uparrow$ | $\cdots$ | $\uparrow$ | $290$(fixed) | $\uparrow$ | $\cdots$ | $\cdots$ | $\cdots$ | 783.37/757
$\uparrow$ indicates linked parameters
model C2: wabs(power-law+vabmkl)+mekal fot the emission,
wabs(power)$\times$(absem)3 for the absorption
†Emission measure of mekalLHB+SWCX is set to 0 as the lower limit.
‡Emission measure of mekalLHB+SWCX is set to upper limit which corresponds to
3.5 LU OVII Kα emission
ain unit of photons cm-2 s-1 str-1 eV-1 @1keV
(80mm, 150mm)figure7.eps
Figure 7: 68%, 90%, and 99% confidence contours of $h_{n}$, $T_{0}$, and
$N_{\rm H_{Hot}}$ vs. $\gamma$, obtained from the combined fits to the X-ray
absorption and emission data. Colored thick lines are for the PKS 2155-304
sight line, while the black thin lines are for the LMC X-3 sight lines (Yao et
al. (2009)). In the panel (a) the scale height of the temperature ($h_{T}$) is
constant along the dashed lines.
## 4 Discussion
### 4.1 Uncertainty due to of LHB and SWCX
Because our knowledge about the temporal and spatial variability of the SWCX
and the LHB is limited, there are uncertainties due to the assumption of their
intensity. These uncertainties could result in large uncertainties in our
results.
To assess this uncertainty, we estimated the lower and upper values of the LHB
and SWCX contributions and evaluated the parameters of the halo components
again. The lower limit of the contribution is zero. As for the upper limit, we
adopt 3.5 LU for the OVII emission, as obtained by the MBM 12 shadowing
observation (Smith et al. (2007)). As the heliospheric SWCX is caused by the
collision between the Solar wind and the neutral ISM, the estimated emissivity
has a peak around the ecliptic plane (Koutroumpa et al. (2007) , Lallement et
al. (2004)). MBM 12 is located at ($\lambda,\beta$)=(47.4, 2.6) in ecliptic
coordinates, while PKS2155-304 is at ($\lambda,\beta$)=(321.2, $-$16.8). Thus
we assume that the heliospheric SWCX contribution in the PKS 2155-304
direction could not be larger than that for MBM12.
The results using these lower and upper limits are shown in Table 4 and Table
7. Though the best fit values are slightly changed, they are consistent with
the previous analysis.
(80mm,50mm)figure8.eps
Figure 8: 68%, 90%, and 99% confidence contours of $T_{0}$ and $N_{\rm
H_{Hot}}$ vs. scale height $h_{n}$ obtained in the joint fit to the X-ray
absorption and emission data. In the upper panel the density at the plane
$n_{0}$ is constant along the solid and dashed lines.
### 4.2 Comparison with the Results for LMC X-3
We compared our results with those of the LMC X-3 direction, as is summarized
in Table 8. The directions of the LMC X-3 and PKS 2155-304 are (l,b) =
(273.6,$-$32.1) and (17.7,$-$52.2). The fact that we obtained similar values
for the two directions indicates that the hot halo is common in the big
picture and can be explained with the exponential model of the column density,
scale height and temperature as $\sim 2\times 10^{19}$ cm-2, a few kpc and
$\sim 2\times 10^{6}$ K. As the distances to the targets are 50 kpc for LMC
X-3 and 480 Mpc for PKS 2155-304, the consistency of the parameters of the
exponential disk suggests that there is little contribution from beyond LMC
X-3, or from a very extended halo of a 100 kpc scale.
Table 8: Disk model parameters for two sight lines Direction | log $N_{\rm H_{Hot}}$ | $h_{n}$ | $logT_{0}$ | $\gamma$ | Ne | Fe |
---|---|---|---|---|---|---|---
| (cm-2) | (kpc) | (K) | | | |
PKS 2155-304 | $19.10^{+0.08}_{-0.07}$ | $2.3^{+0.9}_{-0.8}$ | $6.40^{+0.09}_{-0.05}$ | $2.44^{+1.11}_{-1.41}$ | $3.1^{+1.6}_{-1.2}$ | $1.5^{+1.0}_{-0.7}$ |
LMC X-3† | $19.36^{+0.22}_{-0.21}$ | $2.8^{+3.6}_{-1.8}$ | $6.56^{+0.11}_{-0.10}$ | $0.5^{+1.2}_{-0.4}$ | $1.7^{+0.6}_{-0.4}$ | $0.9^{+0.2}_{-0.2}$ |
† from Yao et al. (2009) |
### 4.3 Distribution of the OVII and OVIII Emitting/ Absorbing Gas and Its
Origin
We calculated the distribution of OVII and OVIII ions and their emissivities
assuming the best fit parameters at $\gamma=2.44$ and at $\gamma=1.0$ and 3.5
(Fig. 9).
We then estimated the total radiative energy loss from the thick disk
distributed exponentially. Assuming solar abundances, best fit parameters and
ionization fraction and emissivity as taken from
SPEX444http://www.sron.nl/index.php?option=com_content
&task=view&id=125&Itemid=279, we obtained the energy loss rate as a function
of the distance from the Galactic plane $Z$ (Fig. 10). We then integrated the
energy loss rate until the temperature of the exponential disk become lower
than $10^{5.5}$ K. Because our results are based on X-ray observations, it is
difficult to detect plasma of T $<10^{5.5}$ K. We obtained a total radiative
energy loss rate of $7.2\times 10^{36}$ erg s-1 kpc-2 in 0.001–40 keV and
$1.8\times 10^{35}$ erg s-1 kpc-2 in 0.3–8.0 keV. These values are consistent
with the X-ray luminosity of other spiral galaxies (Strickland et al., 2004).
We next compared the energy loss rate with the energy input rate from SNe.
According to Ferrière (1998), the SN rate near the sun is 19 Myr-1 kpc-2 for
type II SNe and 2.6 Myr-1 kpc-2 for type Ia SNe, respectively. Assuming each
SN explosion releases 1 $\times 10^{51}$ ergs, the total input energy is then
7 $\times 10^{38}$ ergs s-1 kpc-2. If 1 % of the SN explosion energy is input
to the hot halo, the total energy loss can be compensated.
(80mm,50mm)figure9.eps
Figure 9: The density of OVII and OVIII ion(top) and the emissivity of OVII
and OVIII lines (bottom) as a function of the height from the galactic plane
under the best fit parameter of $\gamma$=2.44 (solid line), $\gamma$=1.0
(dashed line) and $\gamma$=3.5 (dash-dotted line).
(80mm,50mm)figure10.eps
Figure 10: Radiative energy loss rate (red, solid) and cooling time (blue,
dashed) as a function of the distance from the Galactic plane. The temperature
is indicated by the solid black line. The emissivity is calculated from the
mekal model, using a script made by Sutherland.
(http://proteus.pha.jhu.edu/$\sim$dks/Code/
Coolcurve_create/index.html)
### 4.4 Consistency with OVI Absorbing Gas
It is not clear that our model is consistent beyond $\sim 5$ kpc where the
temperature of the gas is below $\sim 10^{6.0}$ K and OVI ion becomes
dominant.
Williams et al. (2007) found two local OVI absorption lines in the FUSE PKS
2155-304 spectrum and reported column densities of 1.10$\pm 0.1\times 10^{14}$
and 8.7$\pm 0.4\times 10^{13}$ cm-2. Our exponential disk model expects OVI
column densities of $3.8\times 10^{13}$, $1.4\times 10^{14}$, and $2.1\times
10^{13}$ cm-2 with the best fit parameters when $\gamma$=2.44, 1.0, and 3.5
respectively.
Howevera plasma emitting OVI lines cools very rapidly and it would be
difficult to maintain such plasma existing high above the Galactic plane.
Radiative cooling is accelerated by the density fluctuations. Thus OVI
absorbing gas can be a patchy or blob-like condensation. To discuss this
problem, energy and matter flow models are needed, which is beyond the focus
of this paper.
## 5 Summary
We have analyzed high resolution X-ray absorption/emission data observed by
Chandra and Suzaku to determine the physical state of the global hot gas along
the PKS 2155-304 direction.
1. 1.
Suzaku clearly detected OVII Kα, OVIII Kα and OVII Kβ lines. The surface
brightnesses of OVII and OVIII in this direction can be understood in the same
scheme as obtained by other 14 observations (Yoshino et al. (2009)).
2. 2.
By the absorption analysis, column density is measured as 3.9 ($+0.6,-0.6$)
cm-3 pc and temperature is measured as 1.91 ($+0.09,-0.09$) $\times 10^{6}$ K.
By the emission analysis, emission measure is measured as 3.0 ($+0.3,-0.3$)
$\times 10^{-3}$ cm-6 pc and temperature is measured as 2.14 ($+0.15,-0.14$)
$\times 10^{6}$ K.
3. 3.
Combined analysis using the exponential disk model gives a good fit with
$\chi^{2}$/dof of 789.65/756 to both emission and absorption spectra. The gas
temperature and density at the Galactic plane are determined to be
$2.5(+0.6,-0.3)\times 10^{6}$ K and $1.4(+0.5,-0.4)\times 10^{-3}$ cm-3 and
the scale heights of the gas temperature and density $5.6(+7.4,-4.2)$ kpc and
$2.3(+0.9,-0.8)$ kpc, respectively.
4. 4.
The results obtained by the combined analysis are consistent with those for
the LMC X-3 direction. This suggest that the global hot gas surrounding our
Galaxy has common structure.
Part of this work was financially supported by Grant-in-Aid for Scientific
Research (Kakenhi) by MEXT, No. 20340041, 20340068, and 20840051. TH
appreciates the support from the JSPS research fellowship and the Global COE
Program ”the Physical Sciences Frontier”, MEXT, Japan
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|
arxiv-papers
| 2010-06-25T02:29:12 |
2024-09-04T02:49:11.195105
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Toshishige Hagihara, Yangsen Yao, Noriko Y. Yamasaki, Kazuhisa\n Mitsuda, Q. Daniel Wang, Yoh Takei, Tomotaka Yoshino, Dan McCammon",
"submitter": "Noriko Yamasaki",
"url": "https://arxiv.org/abs/1006.4901"
}
|
1006.4925
|
11institutetext: State Key Laboratory of Virtual Reality Technology and
Systems, Beihang University, Beijing 100191, China
22institutetext: Tetherless World Constellation, Rensselaer Polytechic
Institute, Troy, NY 12180 USA
# Simulating information creation in social Semantic Web applications
Xixi Luo 11 Xiaowu Chen 11 Qingping Zhao 11 Joshua Shinavier 22
###### Abstract
Appropriate ranking algorithms and incentive mechanisms are essential to the
creation of high-quality information by users of a social network. However,
evaluating such mechanisms in a quantifiable way is a difficult problem.
Studies of live social networks of limited utility, due to the subjective
nature of ranking and the lack of experimental control. Simulation provides a
valuable alternative: insofar as the simulation resembles the live social
network, fielding a new algorithm within a simulated network can predict the
effect it will have on the live network. In this paper, we propose a
simulation model based on the actor-concept-instance model of semantic social
networks, then we evaluate the model against a number of common ranking
algorithms. We observe their effects on information creation in such a
network, and we extend our results to the evaluation of generic ranking
algorithms and incentive mechanisms.
## 1 Introduction
The Social Semantic Web [1], [3] is a fairly new development that combines
technologies, strategies and methodologies from the Semantic Web and social
networks. It organizes its information by means of semi-formal ontologies,
taxonomies or folksonomies, and it places a great deal of importance on
community-driven semantics. In the Social Semantic Web, the islands of the
Social Web can be interconnected with semantic technologies, and Semantic Web
applications are enhanced with the wealth of knowledge inherent in user-
generated content [5].
Since most of the information in a social network is contributed by online
users, guiding users to create high-quality information is an important
research topic. This paper proposes a model to simulate information creation
in Social Semantic Web applications. “A simulation is an imitation of the
operation of a real world process or system over time” [2]. Simulating
information creation in semantic social networks can be used for the purpose
of:
* •
predicting changes in the application. For example, the number of users in the
application can be simulated. The likely effect of various courses of action
can then be observed in the behavior of the model.
* •
exploring the dynamics of the application. Changing simulation settings and
observing the result can provide valuable insight into the most important
factors driving the evolution of the social network.
* •
exploring new policies or mechanisms without disrupting ongoing operation of
the real system. New policies or mechanisms can be tested without committing
to a change in the actual social network.
* •
studying Social Semantic Web applications in general.
Within the Semantic Web domain, simulations have been used for research into
incentive mechanisms such as content trust[6]. However, these simulations are
intended to validate specific incentive mechanisms. In contrast, the
simulation framework proposed here can be used to evaluate general purpose
incentive mechanisms. This paper is a first step in exploring the simulation
of information creation in social semantic web applications. The simulation
model presented here is composed of actors who carry out actions in the
application, and drivers, or factors which affect information creation.
Drivers can be classified as cost drivers or reward drivers, in that they
determine the cost to an actor or the reward for an actor of carrying out a
particular action. To simulate human being’s instinctive reaction, only the
reward of an action exceed the cost of the action will the actor carry it out
in this simulation.
There are some researches about cost estimation model[12][4], but so far we
haven’t find any reward estimation model which is combined together with the
cost to simulate the execution of actions.
To demonstrate the simulation model, four ranking systems – in-degree,
PageRank, HITS and random ranking – are tested in an experimental simulation.
We will first introduce the simulation model in greater detail, followed by
four incentive mechanisms. We will then use the proposed simulation model to
simulate the four ranking systems, and discuss the experimental results.
## 2 Simulation Model
Before we formally define the simulation model, we need to introduce some key
concepts.
* •
A system is “a collection of entities (e.g. actors, concepts, and instances in
this paper) that interact together over time to accomplish one or more goals”
[2].
* •
A model is “an abstract representation of a system, usually containing
structural, logical, or mathematical relationships that describe a system in
terms of state, entities and their attributes, sets, processes, events,
activities, and delay.” [2]
Building upon the above definitions, we will introduce the model’s entities,
attributes, activities, processes and states in turn.
### 2.1 Entities
An entity is any object or component in the system that requires explicit
representation in the model. The entities of this model are drawn from the
actor-concept-instance model of ontologies [10], which contains the basic atom
entities of an social semantic web application, the actor’s participation of
constructing concept and instance make the social semantic web social, and the
concepts make the social semantic web distinct from other web applications
from the semantic point of view. Without either one, we can’t say it is a
social semantic web application. Entities are the subject of real-valued
attributes which are subject to various distributions.
In this model, a concept (also known as a schema) is any tag, class, taxonomy
or ontology which can be used to annotate or describe various data. The
granularity of what we understand as a concept may vary widely, even within a
single application. For example, in Freebase111www.freebase.com, we can
consider both “types” and “domains” as concepts, depending on the requirements
of the evaluation.
Furthermore, we associate with each concept a quality attribute to indicate
its rightness, completeness, ease of comprehension, and so on. The quality of
a concept ranges from 0.0 (the lowest quality) to 1.0 (the highest quality).
Note that in real systems, there is no such attributes like quality to be
known, as it is not possible to ask each users for a quality for each concept.
We propose the quality to estimate the average trust level of the concept from
users.
Since this model is a open model, we can always personalize the model
according to the system requirements by using different attributes and even
more than one attributes. For example, we can set rightness or completeness
and more as attributes instead of using an single attribute quality.
An instance is the main carrier of information in this model. An instance can
be a Web page, a photograph, an audio or video file, or any other object
identifiable with a URI. Instances may be annotated with arbitrary concepts by
actors, or users. As with concepts, we use a “quality” attribute for instances
which ranges from 0.0 to 1.0.
An annotation is an actor-concept-instance tuple indicating that a particular
actor has associated a particular instance with a particular concept. In the
following, if $C$ is a set of concepts, $I$ is a set of instances, and $U$ is
a set of actors (users), then let $A\subset U\times C\times I$ be the set of
all possible annotations, with which actors in $U$ associate the instances of
$I$ with concepts in $C$.
### 2.2 The simulation process
As illustrated in Figure 1, the simulation process as a whole involves:
* •
the choice of an activity to carry out and actor to carry it out
* •
the calculation of cost and reward of the action, or activity
* •
the execution of the activity, if chosen, with corresponding effects on the
simulation environment
* •
the incremental ranking of entities
* •
the optional recording of system state
Prior to simulation, candidate entities and activities for use in the
simulation are generated. A stop condition determines the end of the
simulation, which is followed by simulation analysis. We will introduce each
of these events in turn.
Figure 1: the simulation process
#### 2.2.1 candidate generation
The simulation starts with a preparatory phase in which entity and activity
candidates are generated. The candidates’ attributes follow specific
distributions according to the requirements of the simulation. For example, in
our experiment, actor candidates are associated with an “expertise” value
which follows a normal distribution with a mean of 0.5 and a standard
deviation of 0.5, the value above 1.0 or below 0.0 will be rounded up or down.
Concept and instance candidates are associated with a “quality” value which
follows a normal distribution with a mean of 0.5 and a standard deviation of
0.5. Activity candidates are distributed evenly among instance creation,
concept creation, and semantic annotation.
#### 2.2.2 choice of actor and activity
The body of the simulation consists of multiple iterations, each of which
begins with the choice of an activity and an actor to carry it out. The actor
is chosen randomly from among the actor candidates, and likewise, an activity
is chosen randomly from among the activity candidates. Then the estimated cost
and reward of the activity is calculated: if the cost is smaller than the
reward, then the actor will carry out the activity. Otherwise, the execution
of the activity fails, and the simulation proceeds to the beginning of the
next iteration. The calculation of the cost and reward of activities will be
described in section 2.4.
#### 2.2.3 activity execution
If the chosen activity is found to be worthwhile (i.e. if the estimated reward
exceeds the estimated cost), then it is carried out. The effects of the
execution of the activity on the simulation environment vary according to the
different activities. For more details, see 2.3.
#### 2.2.4 incremental ranking
After an activity is carried out, the ranking of entities may need to be
updated. This ranking is an important factor in the choice of the next
activity, and it also influences the estimation of cost and reward. In
general, the reward of an activity is higher if it involves a high-ranking
entity, which reflects the greater visibility of the entity, in an application
which features a recommendation system, and the greater inclination of users
to choose it over less highly ranked entities.
Depending on the requirements of the simulation, the ranking can be updated
after every iteration, or only occasionally. To avoid excessive computational
overhead, it should be possible to update the ranking incrementally, taking
into account the changes which have occurred since it was last updated. We
will discuss the details of the ranking system in the section of 3.
#### 2.2.5 recording of system state
To track the progress of the simulation, we need to record system state
throughout the simulation process. In this experiment, we have chosen to
record the extent of concept reuse, the quality of the most highly-ranked
entities, and the rate of execution of potential activities. We will discuss
the details of recording of system state in section of 2.5.
#### 2.2.6 stop condition
The stop condition is when the time of successfully executed semantic
annotation activities arrived a predefined number. For example, in this paper,
the stop condition is 1000 times.
### 2.3 activities
In this paper, we will describe seven types of activities: user registration,
publishing of a concept, publishing of an instance, semantic annotation,
linking of actors, linking of concepts, and linking of instances.
* •
user registration is required in most social web applications, so that user
activities can be tracked.
* •
publishing a concept or instance is analogous to publishing resources on the
ordinary Web. We are able to distinguish between two distinct types of
resources – concepts and instances – and we consider the publishing of a
concept and the publishing of an instance to be distinct activities.
Publishing an instance is the act of creating a web page, or uploading a
photograph or an audio or video file to share with the community. Publishing a
concept, on the other hand, involves creating a tag, class, taxonomy or
ontology which may be used to annotate instances. Since concepts may exist at
different levels of granularity, even within the same application, we may
consider more than one type of concept. In Freebase, for example, both domains
and types are concepts, where a type is part of an domain: it is a finer-
grained concept.
* •
semantic annotation distinguishes social semantic networks from most other Web
applications. Actors associate instances with concepts to express a meaningful
relationship. The concept’s semantic value is used to organize or classify
instances.
* •
linking actors makes a semantic social network social, in that the
relationships between actors in a semantic social network comprise a social
network. Linking actors is the activity of establishing a basic relationship
between the actors, such as a “friend” relationship, or a “knows”
relationship.
* •
linking concepts establishes semantic relationships among concepts. For
example, simple hierarchical relationships are analogous to the sub- and
superclass relationships of ontology languages such as OWL. Such linkage among
concepts adds semantic value to instances annotated with those concepts.
* •
linking instances is similar to linking concepts. Examples include linking to
a Web page from another Web page, linking a photograph to a set of
photographs, and so on.
### 2.4 Activity customization
The detail of the execution of actions can be customized. After customization,
the reward and cost driver will be chosen and customized, which is the basis
to calculate the estimated cost and reward during the simulation process. In
this paper, the customization of the activities will be shown in the
following:
#### 2.4.1 publishing concepts
To publish a concept, the simulation randomly chooses a concept from the
concept candidates, then the estimated cost and reward are calculated. If the
reward exceeds the cost, then the concept is published. The estimated cost and
reward are affected by the cost/reward drivers. The cost drivers of the
“publish concept” activity are:
* •
CQ (concept quality), which is defined when the concept candidates are
generated, whose values range from 0.0 to 1.0.
* •
AE (the actor’s expertise), which is defined when the actor candidates are
generated, whose values range from 0.0 to 1.0.
* •
CS (the concept’s size), which is defined when the concept is generated, whose
values range from 0.0 to 1.0.
* •
AE_PC (the actor’s expertise in publishing concepts). This is a cost driver
which has a value of 1.0 when the actor has never published a concept, and
0.75 when the actor has only published one. In general, it is 1.0 divided by
the number of concepts the actor has published.
* •
UE_PC (user effort for publishing a concept) is a cost driver which has a
default value of 1.0. The higher the level is, the more it will cost to
publish a concept.
We calculate the expected cost using the following formula, $\alpha$ is a
prefixed parameter:
$\displaystyle cost=CS^{\alpha}\times CDs$ (1) $\displaystyle
CDs=(CQ+AE+AE\\_PC)/3\times UE\\_PC$ (2)
The reward drivers (whose values likewise range from 0.0 to 1.0) of publishing
a concept action are:
* •
CQ (concept quality)
* •
TCQ (the top concept’s quality). The top concept is the one with the highest
reputation according to the recommendation system (see Section 3). The reason
we’ve chosen this as a reward driver is that the reward for an actor to
publish a concept is lower if there already exists a highly ranked concept.
* •
TCP (the top concept’s popularity) is related to the total number of concepts.
If the total number of published concepts is less than 10, for instance, or if
the total number of published instances is less than 10, then the actor may
still think he or she has a significant chance to create a very popular
concept. In this case, the value of TCP should be high. The greater the
proportion of instances, among all instances, annotated by the best concept,
the more “dominant” the concept is. In this case, the expected reward is low:
its value is is 1.0 minus the proportion just mentioned.
We calculate the expected reward by the following formula,$\beta$ is a
prefixed parameter:
$\displaystyle reward=CQ^{\beta}\times RDs$ (3) $\displaystyle
RDs=(TCQ+TCP)/2$ (4)
#### 2.4.2 publishing instances
Instances are chosen from the instance candidates, which are generated at the
beginning of the simulation. The cost drivers for publishing instances include
the instance type, instance size, and AE_PI, the actor’s expertise at
publishing instances. Calculation of AE_PI is similar to that of AE_PC.
Therefore, the formula used to calculate the expected cost is as follows
(where UE_PI is the user effort for publishing instances):
$cost=AE\\_PI\times UE\\_PI$ (5)
To make things simpler, the reward drivers can be summarized as IQ (instance
quality).
$reward=IQ$ (6)
#### 2.4.3 semantic annotation
In order to simulate the process of semantic annotation, one concept and one
instance should first be chosen from the candidates, such that the chosen
concept will be used to annotated the instance. In reality, a user would tend
to choose the concepts that he is familiar with or that are easy to get. In
our simulation, we let the actor randomly choose a concept from his or her own
concepts in addition to the top 10 concepts according to the recommendation
system. If there is no recommendation system, then the actor will randomly
choose one concept from his or her own concepts in addition to 10 random
concepts. The procedure of choosing an instance is as follows: firstly, the
actor will annotate his or her own instances until all instances have been
annotated, at which point random instances are chosen.
The cost drivers of semantic annotation include:
* •
AE_SA (the actor’s expertise at semantic annotation), which is similar to
AE_PC and E_PI
* •
CC (the cost of choosing a concept), which is 0.0 if the concept is created by
the actor but increases with ranking. For example, the cost of a concept in
the top 10, is 0.1, while between the top 10 and top 20 it is 0.2 and so on.
If the concept is not in the top 100, then the cost is 1.0.
* •
CI (the cost of choosing an instance),which is calculated similarly to CC
The formula is as follows, where the UE_SA is the level of user effort of
semantic annotation:
$cost=(AE\\_SA+CC+CI)/3\times UE\\_SA$ (7)
The reward drivers of semantic annotation include:
* •
CV (concept visibility). If the concept is the top 1 then the reward is very
high: 1.0. If it is only in the top 10, the reward is also high: 0.75. In
general, the reward decreasies with the rank: 1.0/(rank/10).
* •
IV (instance visibility), which is calculated similarly to CV
* •
CQ (the concept’s quality)
* •
IQ (the instance’s quality)
The formula is as follows:
$cost=(CV+IV+CQ+IQ)/4$ (8)
### 2.5 System state
System state is a collection of variables that contain all the information
necessary to describe the system at any time. The bellowing are the states we
propose to record during the simulation.
#### 2.5.1 degree of concept reuse
We use an entropy-based method to measure the degree of reuse of concepts. If
we were to simply use entropy to measure the uncertainty of concepts, the
formula would be:
$H(X)=-\sum_{i}^{n}p(c_{i})\log p(c_{i})$ (9)
where $p(c_{i})=Pr(X=c_{i})=\frac{|A_{c_{i}}|}{|A|}$ with $|A_{c_{i}}|$ the
number of annotations using concept $c_{i}$, and $|A|$ the total number of
annotations. For convenience, equation 9 may be expressed in the following as
$H(X)=H(p(c_{1}),\dots,p(c_{n}))$ .
For example, if there is only one concept in an application, and all instances
are associated with this concept, then $H(X)=-1\times\log 1=0$. For the
example in Figure2,
$H(X)=-((\frac{3}{5}\times\log\frac{3}{5})+(\frac{2}{5}\times\log\frac{2}{5}))=0.67$.
Figure 2: multiple concepts without un-annotated instances
However, this simple metric falls short when applied to applications with
instances which are not annotated. Consider the examples in Figure 3 and 3.
There are two un-annotated instances in Figure 3. According to Equation 9, the
value of $H$ for both examples should be 0, which is to say that all instances
are annotated by the same concept. However, this is unintuitive for $i_{4}$
and $i_{5}$, which are not annotated at all.
Figure 3: Single concept
Our solution to this problem is to import a virtual concept $c_{v}$ to the
concept set $C$ to form a new set $C^{*}$, and then to annotate each of the
un-annotated instances “evenly” by each concept. For example, if there are 99
concepts and one un-annotated instance, we will add one virtual concept for a
total of 100 concepts, then for each concept, add $1/100^{th}$ of an
annotation between the concept and instance.
See Figure 4. After adding a virtual concept and distributing $i_{4}$ and
$i_{5}$ to $c_{1}$ and $c_{v}$ respectively, the value of $H$ becomes $0.50$.
For details about measuring degree of concept reuse, see [9].
Figure 4: single schema with un-annotated documents
#### 2.5.2 the quality of the top concepts
We record the quality of the top entities to see how the overall quality of
the top entities is affected by different ranking algorithms. Normally, the
top entities are the most popular entities or the entities with top reputation
provided by a recommendation system which ranks entities according to their
reputation, using a specific algorithm.
#### 2.5.3 rate of execution of activities
The rate of execution of activities is another attribute which we should
track. We consider every iteration as a unit of time. The total number of
iterations can be considered as the total time of the simulation before it
satisfies the stop requirement. The rate of execution of the semantic
annotation activity is the number of successfully executed semantic
annotations divided by the total number of semantic annotation activities that
are chosen in the simulation. Since the stop condition is defined in terms of
successfully executed semantic annotation activities, the execution rate of
the semantic social network is an indication of how long the simulation takes.
For example, if the execution rate of semantic annotation is $50\%$ and the
stop condition is $1,000$ successful semantic annotation activities, then the
total number of semantic annotations is $2,000$. Moreover, since the different
activities are chosen randomly, the total semantic annotation is proportional
to the total activities. Therefore, the execution rate of semantic annotation
activities can be used to indicate how long the simulation takes. The greater
the execution rate is, the less time the simulation takes, and vice versa.
## 3 Ranking Algorithm
In this section, we will introduce four ranking algorithms which we are going
to evaluate in this paper. These ranking algorithms will be applied to
recommendation system, whose purpose is to guide users in their choice of one
entity or another from the pool of published entities. For example, users
choose between ontologies with which to annotate their instances, other users
to add as friends a social network, or related instances to those they have
created. The following ranking Algorithms are applied in order to rank
ontologies, instances, and actors, providing ranked results to assist users in
their decisions. We assume the ranking algorithm is incremental, so at the end
of each iteration of the simulation process, the ranking will be updated.
### 3.1 Random ranking
A random ranking is the baseline case. By “random” we mean that there is no
recommendation mechanism at all: users choose objects at will. For
convenience, they will tend to choose the entities they are already familiar
with, such as the entity they themselves have published. If the user hasn’t
published any resources, then they will randomly choose any entity from the
pool of candidates.
### 3.2 Indegree
The indegree technique simply ranks nodes in a weighted, directed graph
according to the total weight of edges directed at each node. This is a very
simple, and often effective, technique.
### 3.3 Hits
HITS[7], or Hyperlink-Induced Topic Search, is a link analysis algorithm based
on the notions of “hubs” and “authorities”, which are defined in a mutual
recursion. Highly-ranked hubs are those nodes which link to highly-ranked
authorities. Highly-ranked authorities, in turn, are those nodes to which
highly-ranked hubs link. HITS typically operates over a defined subset of the
overall network.
### 3.4 PageRank
The PageRank algorithm[11], like HITS, ranks nodes recursively according to
the link structure of the network. Unlike HITS, PageRank produces a global
ranking of all nodes in the network. It is very often used in search engines.
The MultiRank[8] algorithm described in our previous work applies PageRank to
the intermediate weighted graph described above.
## 4 simulation results
The simulation environment is set up as follows: In the pool of objects, 100
actors, 1000 ontologies and instances, and 20,000 actions are generated. The
100 actors comprise a unchanging group of users, which is to say that we
ignore the user registration process to make the simulation simpler to
explain, since we want to focus the most important aspects of the model. The
stop condition is that the semantic annotation action is successfully executed
1,000 times. At each time step, an actor and an action are randomly chosen,
then the estimated cost and reward are calculated. If the reward exceeds the
cost, then the actor proceeds to execute the action. Otherwise, the actor does
nothing. Every time an annotation action is successfully executed, we will
record the entropy of the ontology in order to monitor ontology reuse, and
also to keep track of the top popular otologies, in terms of quality. After
the stop event occurs, we compute the execution rate of the semantic
annotation actions for the purpose of estimating the execution time.
In this experiment, we only consider the publishing of concepts, the
publishing of instances, and semantic annotation, and for each of these
actions, we also consider the level of user effort during the calculation of
estimated cost.
### 4.1 user effort
The level of user effort indicates the degree of effort required of the user
of a particular application in order to complete a task. User effort varies by
application. The value of the user effort level not limited to 0 and 1, since
user effort is used to calculate the cost. Low values for user effort, such as
0.1, mean that it cost less to execute the action, whereas higher values, such
as 2.0, indicate higher cost. 1.0 is a default value of user effort, indicate
the average level of user effort.
Here, we only consider the difference in user effort of semantic annotation.
For the first group of experiments, user effort of semantic annotation has a
value of 1.0. For the second group, user effort of semantic annotation has a
value of 2.0. We don’t consider changing the level of user effort for
publishing otologies or instances in this experiment, since in practice, the
variance among application is not very high. Therefore, we consider user
effort of these actions to have a fixed value of 1.0.
The reason we set the level of semantic annotation user effort at 2.0 is that
a lot of applications don’t support semantic annotation very well. We would
like to see whether this will have a strong effect.
### 4.2 analysis of results
In this section, we will compare four simulation results, which include
ontology entropy, the quality of the most-used otologies, and execution rate
of semantic annotation. The baseline is the random mechanism. The first group
simulation setting is with semantic annotation support level 1.0, the second
group simulation setting is with semantic annotation support level of 2.0. In
this paragraph, we first discuss the four different results among different
mechanism, and then compare the result with different semantic annotation
support level.
After the execution session is the recording session, during which some
statistical information is collected. The statistical information can be
collected on every iteration, or only when some specific condition is
satisfied. For example, the tracking of ontology reuse is conditional on a
change in semantic annotations: we only calculate concept reuse when a
semantic annotation activity occurs. What follows is a summary of the
information we are going to collect at the end of each iteration.
#### 4.2.1 ontology entropy
Figure 5: entropy of concept reuse with SPL = 1.0 and 2.0
The figure5 and 5 show that the random mechanism results in the highest
entropy, followed by those based on the Hits, Indegree and PageRank
algorithms. This means that, in terms of promoting ontology reuse, PageRank is
the most effective, in the case where the user has higher cost to semantic
annotation (e.g. when the semantic annotation level is 2.0).
When the semantic annotation support level is 1.0, the best incentive
mechanisms with respect to ontology reuse are PageRank and Indegree, followed
by Hits, then Random. In semantic annotation, lower user effort is better for
ontology reuse than higher effort. For higher semantic annotation effort, the
ontology entropy for PageRank and Indgree are between 0.7 and 0.8 (stable
phase), while between 0.6 and 0.7 (stable phase) for lower effort. The
ontology entropy for Hits also increases from between 0.7 and 0.8 to between
0.8 and 0.9 when the user effort of semantic annotation increases from 1.0 to
2.0. For the random mechanism, user effort makes little difference.
This can be explained by the fact that with low user effort for semantic
annotation, users are encouraged to create annotations, increasing the
likelihood of ontology reuse.
#### 4.2.2 quality of top ontologies
Figure 6: top 1 and top 10 quality of concept with SPL = 1.0 and 2.0
The top ontologies are the ones which have been reused the most. Reuse
encompasses the activities of importing an existing ontology, and of using an
ontology to annotate instances. In this experiment, we have only considered
annotation, so the top ontology is the one which annotates the most instances.
After ranking ontologies accordingly, we find the quality of the top 1
ontology and the average quality of the top 10 ontologies.
Figures 6666 plot the quality of the top 1 and top 10 ontologies. The column
on the left (6 and 6) plots the quality resulting from a user effort level of
1.0, while the column on the right (66) plots the quality resulting from a
user effort level of 2.0. We can see that higher user effort results in higher
quality. For the top 1 ontology, when the user effort level increases from 1.0
to 2.0, the quality increases correspondingly from a value between 0.7 and 0.8
to a value between 0.8 and 1.0. In the case of the top 10 ontologies, when the
user effort level increases from 1.0 to 2.0, the quality increases from a
value between 0.6 and 0.7 to a value between 0.7 and 0.8, again confirming the
positive effect of user effort on quality.
The top row (6 and 6) plots the quality of the top 1 ontology, while the
bottom row (66) plots the quality of the top 10 ontologies, respectively, with
user effort level of 1.0 and 2.0. We found significant differences between the
top 1 and top 10 cases, in terms of quality. For example, the indegree
mechanism results in the highest quality in the top 1 case, but the lowest
quality in the top 10 case, which means that the top 1 ontology has a much
higher quality than that of the other ontologies in the top 10. That is to
say, the use of Indegree causes top ontologies to dominate. This is valuable
from the point of view of promoting ontology reuse.
When the user effort of annotation is 2.0 (meaning that relatively high effort
is required to annotate instances), the quality of the top 1 and top 10
ontologies is as depicted in the figure 6and 6. For the top 1 ontology,
Indegree yields the highest average quality, followed by MultiRank, Random and
Hits. The quality produced by Indegree, MultiRank, and Random starts between
0.6 and 0.7 (which is the average quality of all the ontologies), and then
increases to above 0.9 after 200 iterations, while Hits decrease to below 0.9
(still above 0.8) after around 400 iterations. Here, the Random mechanism
seems to do better than Hits, which seems counterintuitive. However, it does
make sense that the greater the user effort required for semantic annotation,
the less important the ranking mechanism is. As a result it becomes
increasingly unlikely that the user will execute the action. Does this mean
that less supportive environments actually facilitate ontology reuse? We will
see in the following section that this is not the case.
#### 4.2.3 rate of semantic annotation
The rate of creation of semantic annotations is an important factor for
estimating the length of the rest of the simulation. The stop condition of the
simulation is a specific number of successfully executed semantic annotation
actions, therefore the execution rate of semantic annotation can be used to
estimate how long will it take to arrive at this predefined number of actions.
The lower the semantic annotation rate, the longer the simulation takes, which
corresponds to the reality that if users don’t have a high rate of semantic
annotation, then the system takes a long time to accumulate a specific amount
of annotation information.
Figure 7: Semantic Annotation Rate with SPL = 1.0 and 2.0
The 7 is the semantic annotation rate when SPL is 1.0, and the 7 is the
semantic annotation rate when SPL is 2.0. When it cost less (i.e. when SPL is
1.0) to perform semantic annotation, there is no difference between the four
mechanisms. When it cost more (i.e. when SPL is 2.0), MultiRank takes the
least time, followed by Indegree and Random, and then Hits, which is to say
that the if the system uses the MultiRank ranking system, it will take less
time to arrive at the predefined number of semantic annotations.
## 5 Conclusion and Future Work
In this paper, we have illustrated the use of a simulation to evaluate ranking
algorithms and incentive mechanisms in a quantifiable way. Using this
technique, different mechanisms can be tested in advance of their deployment
in a live social network environment. We have made two assumptions:
* •
it is possible to mimic the behavior of a real social network by means of a
simulation
* •
the simulation model described above is appropriate for modeling information
creation in semantic social networks
In the past, ranking algorithms have tended to be evaluated only subjectively:
if the algorithm produces subjectively accurate results, then it is an
appropriate algorithm. The technique presented in this paper provides a
measurable alternative. However, the question naturally arises of whether this
technique itself is appropriate. How do we evaluate it? Until such time as
there are formal techniques for evaluating social network simulations, we must
rely on subjective evaluation, choosing the simulation model as sensibly as
possible and allowing the results to speak for themselves. We have thoroughly
described our simulation model and justified these choices wherever possible:
our object model is based on the actor-concept-instance model of social-
semantic networks, where change in the network is driven by an iterative
process of user actions. Users are guided by metrics of cost, reward in
conjunction with predefined metrics of quality as well as the ranking
algorithms under investigation. We have presented the results of applying our
technique to several common ranking algorithms, and we have shown that these
results are reasonable.
In future, we intend to test our simulation model against multi-relational
ranking algorithms which take advantage of the “semantics” of social-semantic
networks. We plan to release our implementation software under an open-source
license, so that it can be freely reused by developers of social networking
applications.
## 6 Acknowledgements
We would like to thank Jim Hendler and Deborah McGuinness for their valuable
feedback on various drafts of this paper.
## References
* [1] Sören Auer, Chris Bizer, Claudia Müller, and Anna V. Zhdanova (eds.), _Proceedings of the SABRE conference on Social Semantic Web_ , Leipzig, Germany, September 2007.
* [2] Jerry Banks, John Carson, Barry L. Nelson, and David Nicol, _Discrete-event system simulation (4th edition)_ , 4 ed., Prentice Hall, December 2004.
* [3] Andreas Blumauer and Tassilo Pellegrini (eds.), _Social Semantic Web: Die Konvergenz von Social Software, Web 2.0 und Semantic Web_ , X.media.press, 2008.
* [4] Barry W. Boehm, Ellis Horowitz, Ray Madachy, Donald Reifer, Bradford K. Clark, Bert Steece, Winsor A. Brown, Sunita Chulani, and Chris Abts, _Software cost estimation with cocomo ii_ , Prentice Hall PTR, January 2000.
* [5] John G. Breslin, Alexandre Passant, and Stefan Decker, _The social semantic web_ , 1 ed., Springer, October 2009.
* [6] Yolanda Gil and Donovan Artz, _Towards content trust of web resources_ , J. Web Sem. 5 (2007), no. 4, 227–239.
* [7] Jon M. Kleinberg, _Authoritative sources in a hyperlinked environment_ , Journal of the ACM 46 (1999), 668–677.
* [8] X. Luo and J. Shinaver, _MultiRank: Reputation Ranking for Generic Semantic Social Networks_ , Proceedings of the WWW 2009 Workshop on Web Incentives (WEBCENTIVES 09), Madrid, 2009.
* [9] Xixi Luo and Joshua Shinavier, _Entropy-based metrics for evaluating schema reuse_ , ASWC (Asunción Gómez-Pérez, Yong Yu, and Ying Ding, eds.), Lecture Notes in Computer Science, vol. 5926, Springer, 2009, pp. 321–331.
* [10] Peter Mika, _Ontologies Are Us: A unified model of social networks and semantics_ , Web Semantics: Science, Services and Agents on the World Wide Web 5 (2007), no. 1, 5–15.
* [11] Lawrence Page, Sergey Brin, Rajeev Motwani, and Terry Winograd, _The pagerank citation ranking: Bringing order to the web_ , Tech. report, Stanford Digital Library Technologies Project, 1998.
* [12] Elena Simperl, Igor Popov, and Tobias Bürger, _Ontocom revisited: Towards accurate cost predictions for ontology development projects_ , 2009, pp. 248–262.
|
arxiv-papers
| 2010-06-25T07:57:26 |
2024-09-04T02:49:11.203734
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xixi Luo, Xiaowu Chen, Qingping Zhao, and Joshua Shinavier",
"submitter": "Xixi Luo",
"url": "https://arxiv.org/abs/1006.4925"
}
|
1006.5027
|
On Spherically Symmetric Non-Static Space-Times Admitting Homothetic Motions
Ragab M. Gad111Email Address: ragab2gad@hotmail.com
Mathematics Department, Faculty of Science,
Minia University, 61915 El-Minia, EGYPT.
###### Abstract
Spherically symmetric solutions admitting a homothetic Killing vector field
(HKVF) either orthogonal , $\eta_{\bot}$, or parallel, $\eta_{||}$, to the
4-velocity vector field, $u^{a}$, are studied. New self-similar solution of
Einstein’s field equation is found in the case when HKVF is in a general form.
Some physical properties of the obtained solution are examined.
PACS: 04.20.-q-Classical general relativity.
PACS: 04.20.-Jb- Exact solutions.
## 1 Introduction
Recently, symmetries in general relativity have attracted much attention, not
only because of their classical physical significance, but also because they
simplify Einstein field equations. Many survey articles are given to discuss
the concept of these symmetries from the mathematical and physical viewpoints
(see for example [1])
One of the most important symmetries is the self-similarity. Self-similar
solutions of the Einstein field equations are of great interest in physics
because they are often found to play an important role in describing the
asymptotic properties of more general solutions [2]. These solutions have
relevance in astrophysics and critical phenomena in gravitational collapse
(see for example [3] \- [7] and references therein).
In a recent paper [8], Gad and Hassan studied a non-static spherically
symmetric solutions. They assumed that these space-times admit a homothetic
vector field orthogonal to the 4-velocity vector, $u^{a}$, and obtained an
exact solution. This solution has non-vanishing expansion, acceleration and
shear. They derived another solution by assuming, in additional to space-like
homothetic motion, the matter in this fluid is represented by perfect fluid.
This solution has zero expansion.
Many exact solutions has been derived by imposing the condition of the
existence of a conformal Killing vector orthogonal to the 4-velocity (see for
example [10], [9]).
In the present paper we study the cases when space-time admitting HKVF either
orthogonal or parallel to 4-velocity vector. Several authors have studied the
solutions admitting the first symmetry. Most of them have restricted their
intention to the solutions discovered by Gutman-Bosal’ke [16], which are given
in another form by Wesson [20]. These solutions are denoted by (GBW). Collins
and Land [14] have studied these solutions as well as a stiff equation of
state. Sussman [19] investigated the properties of them, and obtained
interesting results.
The second aim of this paper is to obtain an exact self-similar solution and
explore some of its physical properties.
The paper has been organized as follows: In the next section, we shall comment
on the singularities inherent the solutions obtained in [8] and we examine
when these singularities could be possible naked. We find the form of HKVF
when it is either orthogonal or parallel to $u^{a}$. We shall derive an exact
new self-similar solution. In section 3, we shall discuss the physical
properties of the obtained solution. Finally, in section 4, we shall conclude
the results.
## 2 Homothetic Motion
A global vector field $\eta$ on a space-time $M$ is called homothetic if
either one of the following conditions holds on a local chart:
$\pounds_{\eta}g_{ab}=\eta_{a;b}+\eta_{b;a}=2\Phi g_{ab},\qquad H_{a;b}=\Phi
g_{ab}+F_{ab},$ (2.1)
where $\Phi$ is a constant on $M$, $\pounds$ stands for the lie derivative
operator, a semi-colon denotes a covariant derivative with respect to the
metric connection, and $F_{ab}=-F_{ba}$ is the so-called homothetic bivector.
If $\Phi\neq 0$, $\eta$ is called proper homothetic and if $\Phi=0$, $\eta$ is
called Killing vector field on $M$.
For a geometrical interpretation of (2.1) we refer the reader to [15], [16],
and for a physical properties we refer for example to [3].
For the study of non-static spherically symmetric motion, we used the model
given by [21]
$ds^{2}=\alpha d\nu^{2}+2\beta d\nu dr-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (2.2)
where $\alpha$ and $\beta$ are positive function of $\nu$ and $r$.
Gad and Hassan [8] assumed an additional symmetry to the spherically
symmetric, space-like homothetic motion, and they obtained
$\alpha=r^{2}h(\nu)\qquad\beta=rf(\nu),$ (2.3)
where $h(\nu)$ and $f(\nu)$ are arbitrary positive functions. In additional to
the above symmetry, they assumed that the matter is represented by a perfect
fluid and found the relation between $f(\nu)$ and $h(\nu)$ as follows
$h(\nu)=\frac{1}{2}f^{2}(\nu).$ (2.4)
This solution is scalar-polynomial singular along $r=0$ [8].
In the following we examine when this singularity could be possible naked. To
do this, we consider the transverse radial null geodesics. The equations
governing these geodesics are
$f(\nu)\ddot{\nu}+(f^{\prime}(\nu)-h(\nu))\dot{\nu}=0,$ (2.5)
$r^{2}h(\nu)\dot{\nu}^{2}+2rf(\nu)\dot{\nu}\dot{r}=0.$ (2.6)
It is clear from equation (2.6) that the ingoing null geodesics are the line
$\nu$ = constant. The outgoing geodesics obey the equation
$\frac{dr}{d\nu}=-\frac{rh(\nu)}{2f(\nu)}.$ (2.7)
By integrating this equation, we get
$r=c_{1}\exp\big{(}-\int{\frac{h(\nu)}{2f(\nu)}}d\nu\big{)},\qquad c_{1}\neq
0,$ (2.8)
we can see the structure of the space-time by examining equation (2.8). For
example, if there exists a solution of equation (2.8) which starts from the
singularity and ends at the future null infinity, the singularity is globally
naked. Unfortunately, we cannot solve equation (2.8) unless the special
choices of the functions $f(\nu)$ and $h(\nu)$ are given. Now, we have two
cases are depending on the value of integrand
$\int{\frac{h(\nu)}{f(\nu)}}d\nu$, inside equation (2.8).
1. 1.
If the integrand has negative values, then the geodesics will never meet the
singular point $r=0$.
2. 2.
If the integrand has positive values, then the geodesics are meeting the
singular point $r=0$
###### Proposition 2.1
All non-static spherically symmetric solutions described by metric (2.2) admit
a homothetic vector field orthogonal to the 4-velocity vector in the form
$\eta_{\bot}=\Phi r\partial_{r}$.
proof:
Consider the homothetic Killing equation (2.1) and $\eta$ is a homothetic
Killing vector field having the general form
$\eta=A(\nu,r)\partial_{\nu}+\Gamma(\nu,r)\partial_{r}.$ (2.9)
For the metric (2.2), we have
$u^{a}=\frac{1}{\sqrt{\alpha(\nu,r)}}.$
If $\eta_{\bot}$ is everywhere orthogonal to $u^{a}$, then
$\eta^{a}_{\bot}=\Gamma(\nu,r)\delta^{a}_{r}$
By straightforward calculations, using (2.3) and the Christoffel symbols of
second kind (see Appendix), we get that this vector satisfies the condition
(2.1) if $\Gamma(\nu,r)=\Phi r$.
According to the above proposition and using (2.4), the following result has
been established
###### Proposition 2.2
All perfect fluid solutions described by the metric (2.2) admit a homothetic
vector field orthogonal to the 4-velocity vector in the form $\eta_{\bot}=\Phi
r\partial_{r}$.
Now, we study the case when the homothetic vector field is parallel to the
four-velocity vector field.
###### Proposition 2.3
All non-static spherically symmetric solutions described by metric (2.2) admit
a homothetic vector field parallel to the 4-velocity vector in the form
$\eta_{||}=\Phi\nu\partial_{\nu}$.
proof:
Consider the general form of HVF (2.9) and using the relation,since HVF is
parallel to $u^{a}$,
$\eta^{a}_{||}=const.u^{a},$
then
$\eta^{a}_{||}=A(\nu,r)\delta^{a}_{\nu}$
By straightforward calculations, using (2.3) and the Christoffel symbols of
second kind (see Appendix), we get that this vector satisfies the condition
(2.1) if $A(\nu,r)=\Phi\nu$.
By the same manner, see proposition (2.2), we can prove that if the fluid is a
perfect fluid, then it admits HVF parallel to $u^{a}$ in the form
$\eta_{||}=\Phi\nu\partial_{\nu}$.
According to the above propositions, the HVF $\eta$ takes the following form
$\eta=\phi r\partial_{r}+\phi\nu\partial_{\nu}$ (2.10)
This vector satisfies the conditions (2.1).
Now we assume that the line element (2.2) admits HVF (2.10), then the (non-
trivial) equations arising from (2.1), are
$\nu\beta_{\nu}+r\beta_{r}=0,$ $r\alpha_{r}+\nu\alpha_{\nu}=0.$
Using equations (2.3) and (2.4), the solutions of the above equations are
$\alpha=\frac{1}{2}(\frac{r}{\nu})^{2},$ $\beta=(\frac{r}{\nu}).$
According to the above results, the line element (2.2) can be written in the
following form
$ds^{2}=\frac{1}{2}(\frac{r}{\nu})^{2}d\nu^{2}+2(\frac{r}{\nu})d\nu
dr-r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})$ (2.11)
In the following section, we shall discuss some of the physical properties of
the obtained solution given by (2.11).
## 3 Physical Properties
### 3.1 Kinematic of the Velocity Field
For a given space-time the kinematics properties (acceleration, expansion
scalar, rotation, shear and scalar shear) are respectively defined as below
[16]:
The acceleration is defined by
$\dot{u}_{a}=u_{a;b}u^{b}.$
The expansion scalar, which determines the volume behavior of the fluid, is
defined by
$\Theta=u^{a}_{;a}.$
The rotation is given by
$\omega_{ab}=u_{[a;b]}+\dot{u}_{[a}u_{b]}.$
The shear tensor, which provides the distortion arising in a fluid flow
leaving the volume invariant, can be found by
$\sigma_{ab}=u_{(a;b)}+\dot{u}_{(a}u_{b)}-\frac{1}{3}\Theta h_{ab},$
where $h_{ab}=g_{ab}+u_{a}u_{b}$.
The shear invariant is given by
$\sigma^{2}=\frac{1}{2}\sigma_{ab}\sigma^{ab}.$
For the solution given by (2.11)
The acceleration is
$\dot{u}_{a}=-\frac{1}{2r}\delta^{1}_{a}$
For the expansion scalar, we find
$\Theta=0.$
The only non-vanishing components of rotation is given by
$\omega_{41}=\frac{1}{\sqrt{2}\nu}.$
The only non-zero component of the shear tensor is
$\sigma_{11}=-\frac{2\sqrt{2}}{r},$
and the shear scalar, is given by
$\sigma^{2}=\frac{1}{r^{2}}$
### 3.2 Pressure and Density
In addition to self-similarity, we assume that the matter is represented by a
perfect fluid, that is, the Einstein field equations, $G_{ab}=-\kappa T_{ab}$,
are satisfied with the energy momentum tensor
$T_{ab}=(\rho+p)u_{a}u_{b}-pg_{ab}.$
For the line element (2.11), the Einstein field equations reduce to the
following equations
$\frac{1}{r^{2}}=\kappa(\rho+p),$ $\frac{1}{2r^{2}}=\kappa p.$
From the above equations, we obtain the expression for the pressure and
density in the form
$p=\rho=\frac{1}{2\kappa r^{2}}.$
### 3.3 Tidal Forces
The components of the Riemann curvature tensor $R^{a}_{bcd}$, which describe
tidal forces (relative acceleration) between two particles in free fall, are
the components $R^{i}_{0j0}$, ($i,j=1,2,3$), [17].
For the line element (2.11), we obtain
$R^{1}_{010}=0,$
and the only non-vanishing relevant components are
$R^{2}_{020}=R^{3}_{030}=\frac{1}{4\nu^{2}}.$
Then the equations of geodesic deviation (Jacobi equations), which connected
the behavior of nearby particles and curvature, are reduce to the following
equations
$\frac{D^{2}\zeta^{r}}{d\tau^{2}}=0,$ (3.12)
$\frac{D^{2}\zeta^{\theta}}{d\tau^{2}}=-\frac{1}{2r^{2}}\zeta^{\theta},$
(3.13) $\frac{D^{2}\zeta^{\phi}}{d\tau^{2}}=-\frac{1}{2r^{2}}\zeta^{\phi},$
(3.14)
where $\zeta^{r},\ \zeta^{\theta},\ \zeta^{\phi}$ are the components of Jacobi
vector field.
Hence, equation (3.12) indicates tidal forces in radial direction will not
stretch an observer falling in this fluid. The equations (3.13)and (3.14) are
indicate a pressure or compression in the transverse directions, that is, the
tidal forces will not squeeze the observer in the transverse directions.
## 4 Conclusion
In the theory of general relativity, there are different types of self-
similarity. To distinguish between them we refer the reader to the topical
review by Carr and Coley [3]. In this paper we have restricted our intention
to the first type of self-similarity, which characterized by the existence of
a homothetic Killing vector field. We have obtained the form of homothetic
Killing vector field when it is either orthogonal or parallel to the
4-velocity vector field. In the case when HKVF takes a general form, we have
derived self-similar solution. This solution has zero expansion, non-vanishing
acceleration and non-vanishing shear and satisfies the equation of state
$\rho=p$. Furthermore, we have shown that the tidal forces in radial direction
will not stretch an observer falling in this fluid and they not squeeze him in
transverse directions.
## Appendix
We use $(x^{0},x^{1},x^{2},x^{3})=(\nu,r,\theta,\phi)$ so that the non-
vanishing Christoffel symbols of the second kind of the line element (2.2) are
$\displaystyle\Gamma^{1}_{11}$ $\displaystyle=\frac{\beta_{r}}{\beta},$
$\displaystyle\Gamma^{2}_{12}$ $\displaystyle=\frac{1}{r},$
$\displaystyle\Gamma^{1}_{22}$ $\displaystyle=-\frac{r\alpha}{\beta^{2}},$
$\displaystyle\Gamma^{3}_{13}$ $\displaystyle=\frac{1}{r},$
$\displaystyle\Gamma^{1}_{01}$ $\displaystyle=\frac{\alpha_{r}}{2\beta},$
$\displaystyle\Gamma^{2}_{33}$ $\displaystyle=-\sin\theta\cos\theta,$
$\displaystyle\Gamma^{1}_{00}$
$\displaystyle=-\frac{\alpha(\beta_{\nu}-\frac{1}{2}\alpha_{r})}{\beta^{2}}+\frac{\alpha_{\nu}}{2\beta},$
$\displaystyle\Gamma^{3}_{23}$ $\displaystyle=\cot\theta,$
$\displaystyle\Gamma^{0}_{00}$
$\displaystyle=\frac{\beta_{\nu}-\frac{1}{2}\alpha_{r}}{\beta}.$
## References
* [1] Hall G. S., Grav. Cosmol., 2, 270 (1996); Gen. Rel. Grav., 30, 1099 (1998).
* [2] Hsu L. and Wainwright J. Class. Quantum Grav., 3, 1105 (1986).
* [3] Carr B. J. and Coley A. A., Class. Quantum Grav., 16, R31 (1999).
* [4] Carr B. J. and Gundlach C., Phys. Rev. D 67, 024035 (2003).
* [5] Choptuik M. W., Phys. Rev. Lett., 70, 9 (1993).
* [6] Gundlach C., Phys. Rep., 376, 339 (2003).
* [7] Carr B. J. and Coley A. A., "The Similarity Hypothesis in General Relativity", gr-qc/0508039.
* [8] Gad R. M. and Hassan M. M., Il Nuovo Cimento B 118, 759 (2003).
* [9] Gad R. M., Il Nuovo Cimento B 117, 533 (2002).
* [10] Kitamura S., Class. Quantum Grav., 11, 195 (1994).
* [11] Bicknell G. and Henriksen R. N., Astrophys. J., 219, (1978),1043.
* [12] Bondi H., Van der Burg M. G. J. and Metzner A. W. K., Proc. R. Soc. London A, 269, (1962), 21.
* [13] Cahill M. E. and Taub A. H., (1971). Commun. Math. Phys., 21, (1971), 1.
* [14] Collins M. E. and Lang J. M., Class. Quantum Gravit., 4, (1987), 61.
* [15] Eardley D. M., Commun. Math. Phys., 37, (1974), 287.
* [16] Kramer D., Stephani H., MacCallum M. A. H. and Herlt E., "Exact Solution of Einstein’s Field Equations" (Cambridge University Press, Cambridge, 1980).
* [17] Misner C., Thorne K. and Wheeler J., "Gravitation", Freeman, San Francisco (1973).
* [18] Ori A. and Piran T., Phys. Rev. D, 42, (1990), 1068.
* [19] Sussman R. A., J. Math. Phys., 32, (1991), 223.
* [20] Wesson P. S., J. Math. Phys., 19, (1978), 2283.
* [21] Zhao Zheng, Scince in China A, 26, (1993), 178.
|
arxiv-papers
| 2010-05-12T12:23:06 |
2024-09-04T02:49:11.213503
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ragab M. Gad",
"submitter": "Ragab Gad",
"url": "https://arxiv.org/abs/1006.5027"
}
|
1006.5065
|
Further author information: (Send correspondence to R.F.E)
R.F.E.: E-mail: ron.elsner@nasa.gov, Telephone: 256 961 7765
S.L.O.: E-mail: steve.o’dell@nasa.gov, Telephone: 256 961 7776
B.D.R: E-mail: brian.ramsey@nasa.gov, Telephone: 256 961 7784
M.C.W.: Email: martin@smoker.msfc.nasa.gov, Telephone: 256 961 7798
# Methods of optimizing X-ray optical prescriptions for wide-field
applications
Ronald F. Elsnera Stephen L. O’Della Brian D. Ramseya and Martin C.
Weisskopfa aNASA Marshall Space Flight Center Space Science Office VP62
Huntsville AL 35812
###### Abstract
We are working on the development of a method for optimizing wide-field X-ray
telescope mirror prescriptions, including polynomial coefficients, mirror
shell relative displacements, and (assuming 4 focal plane detectors) detector
placement along the optical axis and detector tilt. With our methods, we hope
to reduce number of Monte-Carlo ray traces required to search the multi-
dimensional design parameter space, and to lessen the complexity of finding
the optimum design parameters in that space. Regarding higher order polynomial
terms as small perturbations of an underlying Wolter I optic design, we begin
by using the results of Monte-Carlo ray traces to devise trial analytic
functions, for an individual Wolter I mirror shell, that can be used to
represent the spatial resolution on an arbitrary focal surface. We then
introduce a notation and tools for Monte-Carlo ray tracing of a polynomial
mirror shell prescription which permits the polynomial coefficients to remain
symbolic. In principle, given a set of parameters defining the underlying
Wolter I optics, a single set of Monte-Carlo ray traces are then sufficient to
determine the polymonial coefficients through the solution of a large set of
linear equations in the symbolic coefficients. We describe the present status
of this development effort.
###### keywords:
X-ray astronomy, X-ray optics, ray trace, wide field-of-view optimization
## 1 Introduction
In 1992, Burrows, Burg, and Giacconi[1] showed how by adding higher order
polynomial terms to Wolter I prescriptions, and hence giving up some on-axis
spatial resolution, one can obtain prescriptions for reflecting surfaces that
provide improved average spatial resolution over a wide field-of-view (say
$\sim 30$ arcmin). Such so-called polynomial optics would be particularly
useful for moderately deep to deep surveys, to be carried out by observatories
such as for the proposed Wide-Field X-ray Telescope (WFXT) mission[2], and for
solar X-ray observations. Procedures for optimizing the design of wide-field
X-ray telescopes utilize Monte-Carlo methods for determining the design
parameters, including specification of the polynomial coefficients[1, 3, 4, 5,
6, 7]. Monte-Carlo ray traces are performed over a range of design parameters,
and the final design determined according to some optimization criterion and
methods. Since the number of mirror shells per module is typically large
($\sim$ 50—100), these procedures are presently complicated and computer
intensive.
The present paper is a report on the current status of an on-going study[8] of
the properties of Wolter I and polynomial optical prescriptions with the
ultimate goal of simplifying the procedures for optimizing their designs.
Since a polynomial prescription can typically be viewed as a small pertubation
to an underlying Wolter I design, we begin with Monte-Carlo studies of the
properties of Wolter I mirror shells relevant to wide-field designs,
attempting to deduce analytic formulae for representing the geometric area and
spatial resolution as functions of source position on the sky relative to the
pointing axis, focal length, mirror shell segment length and shell
intersection radius. We then outline a method, valid when the polynomial
coefficients are sufficiently small, for ray tracing polynomial optics keeping
the polynomial coefficients in symbolic form.
A merit function providing a measure of spatial resolution averaged over the
field-of-view (FOV) is defined in §2, while the parameter space over which we
have carried out Monte-Carlo ray traces is described in §3. We note in §4 that
the spatial resolution, when averaged over the FOV as in the merit function,
as a function of source position relative to the optical axis, is a simple sum
of terms up to second order in (1) the mirror shell displacement relative to
the nominal on-axis focus, and (2) the tilt angle for the CCD detector array.
In §5 we arrive at the important conclusion that the spatial resolution on an
arbitrary focal surface for a set of nested mirror shells may be written as
the sum of two terms. The first is a sum over the spatial resolution of the
individual shells on that surface, weighted by their effective area. The
second is a sum over a kind of weighted variance of the mean ray positions for
the individual telescopes on that surface. In §6, we introduce a compact
notation for representing ray trace variables such as position or direction
vectors, including polynomial coefficients in symbolic form. In §7, we discuss
the outer product of two vectors, a concept from linear algebra necessary for
the development of the methods introduced in this paper. In §8, we specify the
basic operations of a polynomial optic algebra, which are addition,
subtraction, multiplication, division, and the taking of square roots. Given a
direction vector, $\vec{k_{1}}$, and initial position
$\vec{x_{1}}=(x_{1},y_{1},z_{1})$, §9 shows how to propagate a ray from
$\vec{x_{1}}$ to axial position $z_{2}$ and determine the other coordinates,
$x_{2}$ and $y_{2}$, thus determining the final position $\vec{x_{2}}$. We
define our coordinate system and the mirror surface prescriptions for
polynomial X-ray optics in §10. In §11, we list the tasks required to trace
rays through X-ray optics. In the future, we plan to show how the tools
presented in this paper are used to accomplish these tasks while keeping the
polymonial coefficients in symbolic form, and to provide concrete examples.
Some closing remarks are provided in §12.
## 2 Merit function
For X-ray survey applications, such as the proposed Wide-Field X-ray Telescope
(WFXT) mission[2], one desires a large effective collecting area over a broad
energy range combined with good spatial resolution over a wide FOV. The
geometric area available is essentially pre-determined by the diameter of the
launch vehicle faring, the number of desired telescope modules (which are
constrained by the desired FOV and, in the absence of extendable optical
benches, the focal lengths permitted by the launch vehicle faring), and the
number of mirror shells per module allowed by mass and manufacturing
constraints. In our work, we have therefore concentrated on optimizing the
spatial resolution average over the FOV, by minimizing the merit function:
$M\ \equiv\ \frac{\int_{\phi=0}^{2\pi}\ d\phi\
\int_{\theta=0}^{\theta_{FOV}}\theta\ d\theta\ w(\theta,\phi)\
\sigma^{2}(\theta,\phi)}{\int_{\phi=0}^{2\pi}\ d\phi\
\int_{\theta=0}^{\theta_{FOV}}\theta\ d\theta\ w(\theta,\phi)},$ (1)
where $\theta$ is the polar off-axis angle for the incident X-rays, $\phi$ is
the azimuthal angle for the incident X-rays, and $w(\theta,\phi)$ is a
weighting factor. By symmetry, the average in Eq. (1) may be restricted to
$\phi\in[0,\pi/4]$ for a typical detector setup consisting of four tilted
CCDs, each occupying a single quadrant. This statement neglects any
repositioning of the detectors to place the on-axis aim point on one of them.
The quantity $\sigma^{2}(\theta,\phi)$ is the variance in the position of rays
reaching the focal surface. This focal surface may be curved or tilted with
respect to the flat plane perpendicular to the optical axis and passing
through the nominal on-axis best focus. The variance,
$\sigma^{2}(\theta,\phi)$, is given by
$\sigma^{2}(\theta,\phi)\ =\ [\ (\ <x^{2}>\ -\ <x>^{2}\ )\ +\ (\ <y^{2}>\ -\
<y>^{2}\ )\ +\ (\ <z^{2}>\ -\ <z>^{2}\ )\ ]$ (2)
where $x$, $y$ and $z$ are the positions of the rays on the chosen focal
surface, and $<q>$ denotes an average of the quantity $q$. All rays incident
on the detector are included in the averages in Eq. (2), independent of the
mirror shell from which they exited.
We have found that the coefficients of the polynomial terms modifying Wolter I
optics for wide-field applications may be regarded as small for our purposes.
Therefore, we treat them as small perturbations to the underlying Wolter I
design. It is for this reason that we have sought analytical fitting functions
for the contributions to $\sigma^{2}(\theta,\phi)$ for Wolter I optics. This
is also the justification for the procedure we describe later in this paper
for ray tracing polynomial X-ray optics keeping the polynomial coefficients as
symbolic and unevaluated until optimized.
## 3 Monte-Carlo ray traces
In order to explore the dependences of $\sigma^{2}(\theta,\phi)$ on mirror
shell parameters, we carried out an extensive series of Monte-Carlo ray traces
of single shell Wolter I optics for nominal focal lengths, $f$, of 5.5 m,
mirror shell segment lengths (2 segments per shell), $\ell_{s}$, of 10, 15, 20
and 40 cm, and shell intersection radii, $r_{0,s}$, of 15, 30, 45 and 60 cm.
Here the subscript $s$ denotes a shell number. Figure (1) plots the locations
of these ray traces in the $\ell_{s}$ vs. $r_{0,s}$ plane. In most cases, the
number of rays incident on the shell aperture was 50,000; for the point in
Figure (1) marked with the biggest dot the number of incident rays was
100,000. The mid-size dots show locations of ray traces for focal lengths of
4.5, 5.0 and 6.0 m. We used the results for $\sigma^{2}$ from these ray traces
to devise trial analytic functions for representing
$\sigma_{s}^{2}(\theta,\phi)$ as a function of the angles $\theta$, $\phi$,
and of $f$, $\ell_{s}$ and $r_{0,s}$.
---
Figure 1: Mirror segment length $\ell_{s}$ vs. intersection radius $r_{0,s}$,
with points showing locations in the $(r_{),s},\ell_{s})$ plane of Monte-Carlo
ray traces with 50,000 incident rays for focal lengths of 5.5 m. The largest
dot shows the location of additional ray traces with 100,000 incident rays at
a focal length of 5.5 m. The largest and mid-size dots show locations of
additional ray traces with 50,000 incident rays for focal lengths of 4.5, 5.0
and 6.0 m. Note there are 2 segments per shell. The solid line represents our
reconstruction of the wide-field telescope design described in Ref. 7 using
their design constraints. The vertical dashed lines show constant values for
nominal graze angles of 25, 50 and 90 arcmin at the intersection plane for
chosen values of $r_{0}$. The curved dashed lines show constant values of 5
and 15 arcmin for $\theta_{coma}$ [see §5, Eq. (17)], in the
$(\ell_{s},r_{0,s})$ plane.
The solid curve in Figure (1) shows our reconstruction of the relationship for
$\ell_{s}$ vs. $r_{0,s}$ for the 3 telescope module, 82 mirror shell per
module wide-field design described and discussed in Ref. Conconi10. We carried
out this reconstruction using the design constraints provided in their Table
(2). While certain of our assumptions may vary from theirs, in general we
expect our reconstruction to be close to their actual design.
## 4 Single mirror shell
For Monte-Carlo ray traces of a single mirror shell $s$, we define the
geometric area, $A_{geom,s}$, as
$A_{geom,s}(\theta)\ \equiv\ A_{inc,s}\ n_{s}(\theta)\ /\ n_{inc,s}(\theta),$
(3)
where $A_{inc,s}$ is the entrance aperture for shell $s$, $n_{inc,s}$ the
number of rays incident on that aperture, and $n_{s}$ the number of doubly
reflected rays exiting the mirror shell.
On a detector tilted by an angle $\theta_{tilt}$ with one corner at the
$(x,y)$ origin, but displaced along the optical axis by an amount $\delta
z_{s}$ from the flat plane perpendicular to the optical axis at the nominal
on-axis focus, the variance, or square of the RMS dispersion may be written in
the form
$\sigma_{s}^{2}(\theta,\phi,\delta z_{s},\theta_{tilt})\ =\ a_{s}\ +\\\ 2\
b_{s}\ \delta z_{s}\ +\ c_{s}\ \delta z_{s}^{2}\ +\ 2\ d_{s}\
\tan{\theta_{tilt}}\ +\ 2\ e_{s}\ \delta z_{s}\ \tan{\theta_{tilt}}\ +\ f_{s}\
\tan^{2}{\theta_{tilt}}.$ (4)
Evaluating the merit function [Eq. (1)] for this shell leads to
$\displaystyle M(\delta z_{s},\theta_{tilt})$ $\displaystyle=$ $\displaystyle
a_{s,M}\ +\ 2\ b_{s,M}\ \delta z_{s,M}\ +\ c_{s,M}\ \delta z_{s,M}^{2}$ (5)
$\displaystyle+\ 2\ d_{s,M}\ \tan{\theta_{tilt}}\ +\ 2\ e_{s,M}\ \delta z_{s}\
\tan{\theta_{tilt}}\ +\ f_{s,M}\ \tan^{2}{\theta_{tilt}},$
where the subscript $M$ denotes an average over the FOV like that in Eq. (1).
In order to carry out the integrals over the FOV, it is advantageous to have
analytic forms for the coefficients $a_{s}$, $b_{s}$, $c_{s}$, $d_{s}$,
$e_{s}$ and $f_{s}$. Minimizing Eq. (4) with respect to $\delta z_{s}$ and
$\tan{\theta_{tilt}}$, we find
$\tan{\theta_{tilt}}\ =\ \left(\frac{b_{s,M}\ e_{s,M}\ -\ c_{s,M}\
d_{s,M}}{c_{s,M}\ f_{s,M}\ -\ e_{s,M}^{2}}\right)$ (6)
$\delta z_{s}\ =\ \left(\frac{d_{s,M}\ e_{s,M}\ -\ b_{s,M}\ f_{s,M}}{c_{s,M}\
f_{s,M}\ -\ e_{s,M}^{2}}\right)$ (7)
Expressions (4)—(7) are general and applicable to any surface prescription for
grazing incidence X-ray optics.
Below we use the notation:
$\sum_{x,y}\ <(x,y)\left(\frac{k_{(x,y)}}{k_{z}}\right)>\ =\
<x\left(\frac{k_{x}}{k_{z}}\right)>\ +\ <y\left(\frac{k_{y}}{k_{z}}\right)>,$
(8)
and similarly for other combinations of terms. Angle brackets around a
quantity, $<q>$, denote an average over that quantity on an the focal surface.
Angle brackets with the subscript 0, $<q>_{0}$, denote an average in the flat
plane perpendicular to the optical axis at the nominal on-axis focal position.
We find that the coefficients $a_{s}$, $b_{s}$, $c_{s}$, $d_{s}$, $e_{s}$ and
$f_{s}$ can be expressed in terms of averages in that flat plane. Assuming a
detector in the first quadrant (both $x$ and $y$ positive), here we provide
some examples of coefficient definitions along with the trial fitting
functions that we find useful for representing the results of our Monte-Carlo
ray traces:
$\sigma_{s}^{2}(0,0)\ \equiv\ a_{s}\ \equiv\ \sum_{(x,y)}\ (\
<(x,y)^{2}>_{0,s}\ -\ <(x,y)>_{0,s}^{2}\ ),$ (9)
The behavior of $a_{s}$ as a function of $\theta$ is complicated by effects
due to coma. At present, we are working with the three trial fitting
functions, $a_{0}(\theta)$, $a_{1}(\theta)$ and $a_{2}(\theta)$, for Wolter I
optics. We define a useful function, $g$, and then $a_{0}(\theta)$,
$a_{1}(\theta)$ and $a_{2}(\theta)$:
$g(\theta,\ \zeta,\ \xi)\ =\ 1\ +\ \zeta\ \tan{\theta}\ +\ \xi\
\tan^{2}{\theta}\\\ $
$\displaystyle a_{fit}(\theta)\ =\ a_{coma}(\theta)\ +\ a_{m}(\theta),\ (m=0,\
1,\ 2)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\displaystyle a_{coma}(\theta)\ =\ (\tan{4\alpha_{0}}\ /\ 2\ )^{4}\
\tan^{2}{\theta}\ \ \ \ \ \ \ \ \ \ a_{0}(\theta)\ =\ (\ 2\ \mu_{a,0}\ \ell\
/\ \tan{4\alpha_{0}}\ )^{2}\ \tan^{4}{\theta}\ g(\theta,\ \zeta_{a},\
\xi_{a})$ (10) $\displaystyle a_{1}(\theta)\ =\ a_{coma}(\theta)\ +\
a_{0}(\theta)\ \ \ \ \ \ \ \ \ \ a_{2}(\theta)\ =\
\left(\sqrt{a_{coma}(\theta)}\ +\ \sqrt{a_{0}(\theta)}\right)^{2},\ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ $
We note that $a_{1}$ defined here is equivalent to Eq. (2) in Ref. LVS72.
Additional examples of coefficient definitions and trial fitting functions
applicable to Wolter I optics are:
$\displaystyle b_{s}$ $\displaystyle\equiv$ $\displaystyle\sum_{(x,y)}\
\left[<(x,y)\left(\frac{k_{(x,y)}}{k_{z}}\right)>_{0,s}\ -\
<(x,y)>_{0,s}<\left(\frac{k_{(x,y)}}{k_{z}}\right)>_{0,s}\right],$
$\displaystyle b_{fit}(\theta)$ $\displaystyle=$ $\displaystyle 2\ \mu_{b}\
\ell\ \tan^{2}{\theta}\ g(\theta,\ \zeta_{b},\ \xi_{b}),$
$\displaystyle c_{s}$ $\displaystyle\equiv$ $\displaystyle\sum_{(x,y)}\
\left[<\left(\frac{k_{(x,y)}}{k_{z}}\right)^{2}>_{0,s}\ -\
<\left(\frac{k_{(x,y)}}{k_{z}}\right)>_{0,s}^{2}\right],$ $\displaystyle
c_{fit}(\theta)$ $\displaystyle=$ $\displaystyle[\ \mu_{c}\ \tan{4\alpha_{0}}\
g(\theta,\ \zeta_{c},\ \xi_{c})\ ]^{2}.$
Definitions and trial fitting functions for $d_{s}$, $e_{s}$, and $f_{s}$ are
lengthy, so for reasons of readability we provide them in Appendix A.
## 5 Nested mirror shells
Consider a set of $S$ nested telescopes. We assume uniform illumination of the
entrance aperture for the full array of telescopes. The number of rays through
the $s$-th telescope is $n_{s}$, and the total number of rays through the
array of nested telescopes is
$N\ \equiv\ \sum_{s=1}^{S}n_{s}.$ (13)
We designate the $(x,y,z)$ position coordinates on an arbitrary focal surface
for the full array of the $k$-th ray through the $s$-th telescope by $(\
x_{s,k},\ y_{s,k},\ z_{s,k}\ )$. We find that the total variance, or square of
the RMS disperion, for the full set of nested shells may be written in the
form
$\sigma^{2}\ =\ \sigma_{1}^{2}\ +\ \sigma_{2}^{2},$ (14)
where
$\sigma_{1}^{2}\ =\ \sum_{s=1}^{S}\ \left(\frac{n_{s}-1}{N-1}\right)\
\sigma_{s}^{2}(\delta z_{s},\theta_{tilt}),\\\ $
with $\sigma_{s}^{2}(\delta z_{s},\theta_{tilt})$ given by Eq. (4). The second
term on the right-hand-side of Eq. (14) is given by
$\sigma_{2}^{2}\ =\ \left(\frac{N}{N-1}\right)\ \sum_{(x,y,z)}\
\left[\sum_{s=1}^{S}\frac{n_{s}}{N}<(x,y,z)_{s}>^{2}\ -\
\left(\sum_{s=1}^{S}\frac{n_{s}}{N}<(x,y,z)_{s}>\right)^{2}\right],$ (15)
Eqs. (14)—(15) show that the variance, $\sigma^{2}$, for the full set of
nested shells has two contributions. The first, Eq. (5), is a sum over the
variances for the individual telescopes, on the chosen focal surface, weighted
essentially by their relative effective geometric areas $[(n_{s}/N)\simeq
A_{geom,s}(\theta)/\sum_{s}A_{geom,s}(\theta)]$. The second, Eq. (15), is a
sum over a kind of weighted variance of the means, $<(x,y,z)_{s}>$, for the
individual telescopes on that focal surface. This second contribution can be
viewed as arising from the differences in the best focal surfaces for the
individual mirror shells from that for the full set of nested shells (see Ref.
[Conconi10]). Expressions (14)—(15) are general and applicable to any surface
prescription for grazing incidence X-ray optics.
For best performance, Eqs. (14)—(15) mean that the minimization of
$\sigma_{M}^{2}$, and thus the optimization of the parameters $\theta_{tilt}$
and $(\delta z_{s},s=1,2,3,...S)$, must be done simultaneously, rather than
following Eqs. (6) and (7) for the individual shells. In principle this can be
done using matrix methods, although the number of linear equations involved is
large for current wide field designs which approach 100 nested mirror shells
(see Ref. [Conconi10]).
We have also shown a need for expressions for $<(x,y,z)_{s}>$, and terms,
$<(x,y,z)_{s}>^{2}$ and cross terms $<(x,y,z)><(x,y,z)>$, that can then be
derived to the appropriate order, for the individual shells. We find to the
appropriate order
$\displaystyle<(x,y)>_{s}$ $\displaystyle=$ $\displaystyle
a^{\prime}_{(x,y),s}\ +\ b^{\prime}_{(x,y),s}\ \delta z_{s}\ +\
d^{\prime}_{(x,y),s}\ \tan{\theta_{tilt}}\ +\ e^{\prime}_{(x,y),s}\delta
z_{s}\ \tan{\theta_{tilt}}\ +\ f^{\prime}_{(x,y),s}\ \tan^{2}{\theta_{tilt}}$
$\displaystyle<z>_{s}$ $\displaystyle=$ $\displaystyle(\ a^{\prime}_{x,s}\ +\
a^{\prime}_{y,s}\ )\tan_{\theta{tilt}}\ +\ b^{\prime}_{x,s}\ +\ (\
b^{\prime}_{y,s}\ )\ \delta z_{s}\ \tan{\theta_{tilt}}\ +\ (\
d^{\prime}_{x,s}\ +\ d^{\prime}_{y,s}\ )\ \tan^{2}{\theta_{tilt}}.$
We provide the definitions of $a^{\prime}_{(x,y),s}$, $b^{\prime}_{(x,y),s}$,
$d^{\prime}_{(x,y),s}$, $e^{\prime}_{(x,y),s}$ and $f^{\prime}_{(x,y),s}$ in
Appendix A. For use below, we define an angle, $\theta_{coma}$, at which
$a_{coma}(\theta)$ and $a_{0}(\theta)$ [see Eq. (4)], with $\zeta_{a}=0$ and
$\xi_{a}=0$, are equal:
$\tan{\theta_{coma}}\ =\ \left(\frac{1}{8}\right)\left(\frac{f}{\ell}\right)\
\tan^{3}{4\alpha_{0}}.$ (17)
We have not yet finished devising analytic expressions for
$e_{(x,y),s}^{\prime}$ and for $f_{(x,y),s}^{\prime}$, but here provide trial
fitting functions for for $a_{(x,y),s}^{\prime}$, $b_{(x,y),s}^{\prime}$ and
$d_{(x,y),s}^{\prime}$:
$a^{\prime}_{(x,y),fit}\ =\ f\ (\ 1\ +\ \delta f_{(x,y),a}\ )\ \tan{\theta}\
\left[1\ +\ \left(\frac{3}{4}\right)\ \tan^{2}{\theta}\right]\ g(\theta,\
\zeta_{(x,y),a},\ \xi_{(x,y),a}),$ (18)
$\displaystyle b^{\prime}_{(x,y),fit,0}(\theta,\phi)\ =\ -\ \mu_{(x,y),b}\
\tan{\theta}\ (\cos{\phi},\ \sin{\phi})\ g(\theta,\ \zeta_{(x,y),b},\
\xi_{(x,y),b})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $\displaystyle
k_{(x,y),coma}(\theta)\ =\ p_{(x,y),coma}\ \sin{[\ \pi\ \zeta_{(x,y),coma}\
(\theta\ /\theta_{coma})\ ]}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ $ (19) $\displaystyle k_{(x,y),damp}(\theta)\ =\ \exp{[\ -\
\xi_{(x,y),coma}\ (\theta/\theta_{coma})^{2}\ ]}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $\displaystyle
b^{\prime}_{(x,y),fit}(\theta,\phi)\ =\ k_{(x,y),damp}(\theta)\
k_{(x,y),coma}(\theta)\ +\ (1\ -\ k_{(x,y),damp}(\theta))\ [\ q_{(x,y),coma}\
+\ b^{\prime}_{(x,y),fit,0}(\theta,\phi)\ ],$
$\displaystyle d^{\prime}_{(x,y),fit}(\theta,\phi)$ $\displaystyle=$
$\displaystyle-\ f\ (1\ +\ \delta f_{(x,y),d})(1\ +\ 2\
\tan^{2}{4\alpha_{0}})\ \tan^{2}{\theta}$ $\displaystyle\times\ (\cos{\phi},\
\sin{\phi})\ (\cos{\phi}\ +\ \sin{\phi})\ g(\theta,\ \zeta_{(x,y),d},\
\xi_{(x,y),d}),$
In the future, we plan to provide a fuller account of our methods and results
for ray tracing Wolter I optics.
## 6 Notation for polynomial coefficients
In the past, we have written sums over rays
$\lambda\ =\ \sum_{k=1}^{n}\lambda_{k},$ (21)
where $\lambda_{k}$ is some quantity such as position along an axis or a
component of a direction vector for ray $k$, in the form of a second order
expansion in the polynomial coefficients
$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle\lambda_{0000}\ +\
u_{1}\ \lambda_{1000}\ +\ u_{2}\ \lambda_{0100}\ +\ u_{3}\ \lambda_{0010}\ +\
u_{4}\ \lambda_{0001}$ $\displaystyle+\ u_{1}^{2}\ \lambda_{2000}\ +\
u_{2}^{2}\ \lambda_{0200}\ +\ u_{3}^{2}\ \lambda_{0020}\ +\ u_{4}^{2}\
\lambda_{0002}$ $\displaystyle+\ u_{1}\ u_{2}\ \lambda_{1100}\ +\ u_{1}\
u_{3}\ \lambda_{1010}\ +\ u_{1}\ u_{4}\ \lambda_{1001}$ $\displaystyle+\
u_{2}\ u_{3}\ \lambda_{0110}\ +\ u_{2}\ u_{4}\ \lambda_{0101}\ +\ u_{3}\
u_{4}\ \lambda_{0011},$
where $u_{1}$, $u_{2}$, $u_{3}$, and $u_{4}$ are the polynomial coefficients
for the mirror shell. The form Eq. (6) assumes the coefficients are small
enough so that a second order expansion is valid. The polynomial deviations
from Wolter I optics required in the applications we have studied so far
satisfy this criterion.
We define a polynomial coefficient vector
$\vec{u}\ \equiv\ (u_{1},u_{2},u_{3},u_{4}).$ (23)
We also define the scalars
$\displaystyle\lambda_{00}\ \equiv\ \lambda_{0000}$
$\displaystyle\lambda_{01}\ \equiv\ \lambda_{1000}$
$\displaystyle\lambda_{02}\ \equiv\ \lambda_{0100}$
$\displaystyle\lambda_{03}\ \equiv\ \lambda_{0010}$
$\displaystyle\lambda_{04}\ \equiv\ \lambda_{0001}$
$\displaystyle\lambda_{11}\ \equiv\ \lambda_{2000}$
$\displaystyle\lambda_{22}\ \equiv\ \lambda_{0200}$
$\displaystyle\lambda_{33}\ \equiv\ \lambda_{0030}$
$\displaystyle\lambda_{44}\ \equiv\ \lambda_{0002}$ (24)
$\displaystyle\lambda_{12}\ =\lambda_{21}\ \equiv\ \frac{1}{2}\lambda_{1100}$
$\displaystyle\lambda_{13}\ =\lambda_{31}\ \equiv\ \frac{1}{2}\lambda_{1010}$
$\displaystyle\lambda_{14}\ =\lambda_{41}\ \equiv\ \frac{1}{2}\lambda_{1001}$
$\displaystyle\lambda_{23}\ =\lambda_{32}\ \equiv\ \frac{1}{2}\lambda_{0110}$
$\displaystyle\lambda_{24}\ =\lambda_{42}\ \equiv\ \frac{1}{2}\lambda_{0101}$
$\displaystyle\lambda_{34}\ =\lambda_{43}\ \equiv\ \frac{1}{2}\lambda_{0011},$
and the vectors
$\displaystyle\vec{\lambda_{0}}$ $\displaystyle\equiv$
$\displaystyle(\lambda_{01},\lambda_{02},\lambda_{03},\lambda_{04})$
$\displaystyle\vec{\lambda_{1}}$ $\displaystyle\equiv$
$\displaystyle(\lambda_{11},\lambda_{12},\lambda_{13},\lambda_{14})$
$\displaystyle\vec{\lambda_{2}}$ $\displaystyle\equiv$
$\displaystyle(\lambda_{21},\lambda_{22},\lambda_{23},\lambda_{24})$ (25)
$\displaystyle\vec{\lambda_{3}}$ $\displaystyle\equiv$
$\displaystyle(\lambda_{31},\lambda_{32},\lambda_{33},\lambda_{34})$
$\displaystyle\vec{\lambda_{4}}$ $\displaystyle\equiv$
$\displaystyle(\lambda_{41},\lambda_{42},\lambda_{43},\lambda_{44}).$
Finally we define a matrix with the row vectors $\vec{\lambda_{1}}$,
$\vec{\lambda_{2}}$, $\vec{\lambda_{3}}$, and $\vec{\lambda_{4}}$:
$\overline{\overline{\lambda}}\ \equiv\
\left(\begin{array}[]{c}\vec{\lambda_{1}}\\\ \vec{\lambda_{2}}\\\
\vec{\lambda_{3}}\\\ \vec{\lambda_{4}}\end{array}\right)\ =\
\left(\begin{array}[]{cccc}\lambda_{11}&\lambda_{12}&\lambda_{13}&\lambda_{14}\\\
\lambda_{21}&\lambda_{22}&\lambda_{23}&\lambda_{24}\\\
\lambda_{31}&\lambda_{32}&\lambda_{33}&\lambda_{34}\\\
\lambda_{41}&\lambda_{42}&\lambda_{43}&\lambda_{44}\end{array}\right).$ (26)
$\lambda\ =\ \lambda_{00}\ +\ \vec{u}\cdot\vec{\lambda_{0}}\ +\
\vec{u}\cdot\overline{\overline{\lambda}}\cdot\vec{u}.$ (27)
In this notation, and since the matrix $\overline{\overline{\lambda}}$ is
symmetric, derivatives of $\lambda$ with respect to the polynomial
coefficients are
$\frac{\partial\lambda}{\partial u_{j}}\ =\ \lambda_{0j}\ +\ 2\sum_{i=1}^{4}\
\lambda_{ji}\ u_{i}\ =\ \lambda_{0j}\ +\ 2\ \vec{u}\cdot\vec{\lambda_{j}}.$
(28)
For a set of nested shells, denoted by prefixes $s$ or $r$ (e.g.,
$\lambda_{s})$, we note that
$\partial\left(\sum_{r=1}^{S}\lambda_{r}\right)/\partial u_{s,j}\ =\
\frac{\partial\lambda_{s}}{\partial u_{s,j}}.$ (29)
## 7 Outer product of two vectors
Consider the two vectors
$\displaystyle\vec{a}$ $\displaystyle\equiv$
$\displaystyle(a_{1},a_{2},a_{3},a_{4})$ $\displaystyle\vec{b}$
$\displaystyle\equiv$ $\displaystyle(b_{1},b_{2},b_{3},b_{4}).$
In linear algebra, the outer product of $\vec{b}$ with $\vec{a}$ is given by
$\vec{b}\otimes\vec{a}\ \equiv\ \left(\begin{array}[]{cccc}b_{1}\ a_{1}&b_{1}\
a_{2}&b_{1}\ a_{3}&b_{1}\ a_{4}\\\ b_{2}\ a_{1}&b_{2}\ a_{2}&b_{2}\
a_{3}&b_{2}\ a_{4}\\\ b_{3}\ a_{1}&b_{3}\ a_{2}&b_{3}\ a_{3}&b_{3}\ a_{4}\\\
b_{4}\ a_{1}&b_{4}\ a_{2}&b_{4}\ a_{3}&b_{4}\ a_{4}\end{array}\right).$ (31)
The outer product is an essential tool in polynomial ray tracing algebra.
## 8 Basic operations
Consider the two polynomial objects
$\displaystyle a$ $\displaystyle=$ $\displaystyle a_{00}\ +\
\vec{u}\cdot\vec{a_{0}}\ +\ \vec{u}\cdot\overline{\overline{a}}\cdot\vec{u}$
$\displaystyle b$ $\displaystyle=$ $\displaystyle b_{00}\ +\
\vec{u}\cdot\vec{b_{0}}\ +\ \vec{u}\cdot\overline{\overline{b}}\cdot\vec{u}$
(32) $\displaystyle\vec{u}$ $\displaystyle=$
$\displaystyle(u_{1},u_{2},u_{3},u_{4}).$
Treating $\vec{u}$ as small and expanding to second order in $\vec{u}$, we now
specify how to carry out basic operations on $\vec{a}$ and $\vec{b}$. Using
these operations, it is possible to construct a Monte-Carlo ray trace code for
polynomial X-ray optics with sufficiently small but unknown coefficients
$\vec{u}$. In principle, values for the coefficients can then be derived from
a final ray bundle for any assumed merit function.
In the case of the addition operation, we note
$a\ \pm\ b\ =(a_{00}\pm b_{00})\ +\ \vec{u}\cdot(\vec{a_{0}}\pm\vec{b_{0}})\
+\
\vec{u}\cdot(\overline{\overline{a}}\pm\overline{\overline{b}})\cdot\vec{u}.$
(33)
The multiplication operation is more complicated. We want to keep terms only
to 2nd order in the polynomial coefficients $\vec{u}$. To this order we find
$a\times b\ =\ a_{00}\ b_{00}\ +\ \vec{u}\cdot(a_{00}\ \vec{b_{0}}\ +\ b_{00}\
\vec{a_{0}})\ +\ \vec{u}\cdot(a_{00}\ \overline{\overline{b}}\ +\ b_{00}\
\overline{\overline{a}}\ +\ \vec{b_{0}}\otimes\vec{a_{0}})\cdot\vec{u}.$ (34)
In particular, we note that
$(\vec{u}\cdot\vec{a_{0}})\times(\vec{u}\cdot\vec{b_{0}})\ =\
\vec{u}\cdot(\vec{b_{0}}\otimes\vec{a_{0}})\cdot\vec{u}.$ (35)
We also need the square and the cube of a polynomial object to 2nd order in
$\vec{u}$. We find
$\displaystyle a^{2}$ $\displaystyle=$ $\displaystyle a_{00}^{2}\ +\
\vec{u}\cdot(2\ a_{00}\ \vec{a_{0}})\ +\ \vec{u}\cdot(2\ a_{00}\
\overline{\overline{a}}\ +\ \vec{a_{0}}\otimes\vec{a_{0}})\cdot\vec{u}$
$\displaystyle a^{3}$ $\displaystyle=$ $\displaystyle a_{00}^{3}\ +\
\vec{u}\cdot(3\ a_{00}^{2}\ \vec{a_{0}})\ +\ \vec{u}\cdot(3\ a_{00}^{2}\
\overline{\overline{a}}\ +\ 3\ a_{00}\
\vec{a_{0}}\otimes\vec{a_{0}})\cdot\vec{u}.$
For the division operation, to second order in $\vec{u}$, we find
$\displaystyle a/b$ $\displaystyle=$
$\displaystyle\left(\frac{a_{00}}{b_{00}}\right)\ \left\\{1\ +\
\vec{u}\cdot\left[\left(\frac{\vec{a_{0}}}{a_{00}}\right)\ +\
\left(\frac{\vec{b_{0}}}{b_{00}}\right)\right]\right\\}$ $\displaystyle+\
\left(\frac{a_{00}}{b_{00}}\right)\
\left\\{\vec{u}\cdot\left[\left(\frac{\overline{\overline{a}}}{a_{00}}\right)\
-\ \left(\frac{\overline{\overline{b}}}{b_{00}}\right)\ +\
\left(\frac{\vec{b_{0}}}{b_{00}}\right)\otimes\left(\frac{\vec{b_{0}}}{b_{00}}\right)\
-\
\left(\frac{\vec{b_{0}}}{b_{00}}\right)\otimes\left(\frac{\vec{a_{0}}}{a_{00}}\right)\right]\cdot\vec{u}\right\\}.$
In the case of the square root operation, to second order in $\vec{u}$, we
find
$\displaystyle\sqrt{a}$ $\displaystyle=$ $\displaystyle\sqrt{a_{00}}\
\left\\{1\ +\ \frac{1}{2}\
\vec{u}\cdot\left(\frac{\vec{a_{0}}}{a_{00}}\right)\ +\ \frac{1}{2}\
\vec{u}\cdot\left(\frac{\overline{\overline{a}}}{a_{00}}\right)\cdot\vec{u}\
-\ \frac{1}{8}\
\vec{u}\cdot\left[\left(\frac{\vec{a_{0}}}{a_{00}}\right)\otimes\left(\frac{\vec{a_{0}}}{a_{00}}\right)\right]\cdot\vec{u}.\right\\}$
(38) $\displaystyle 1\ /\ \sqrt{a}$ $\displaystyle=$ $\displaystyle 1\ /\
\sqrt{a_{00}}\ \left\\{1\ -\ \frac{1}{2}\
\vec{u}\cdot\left(\frac{\vec{a_{0}}}{a_{00}}\right)\ -\ \frac{1}{2}\
\vec{u}\cdot\left(\frac{\overline{\overline{a}}}{a_{00}}\right)\cdot\vec{u}\
+\ \frac{3}{8}\
\vec{u}\cdot\left[\left(\frac{\vec{a_{0}}}{a_{00}}\right)\otimes\left(\frac{\vec{a_{0}}}{a_{00}}\right)\right]\cdot\vec{u}.\right\\}$
(39)
We also find
$\displaystyle a\ /\ \sqrt{b}$ $\displaystyle=$
$\displaystyle\frac{a_{00}}{\sqrt{b_{00}}}\ \left\\{1\ +\
\vec{u}\cdot\left[\left(\frac{\vec{a_{0}}}{a_{00}}\right)\ -\ \frac{1}{2}\
\left(\frac{\vec{b_{0}}}{b_{00}}\right)\right]\right\\}$ $\displaystyle+\
\frac{a_{00}}{\sqrt{b_{00}}}\
\left\\{\vec{u}\cdot\left[\left(\frac{\overline{\overline{a}}}{a_{00}}\right)\
-\ \frac{1}{2}\ \left(\frac{\overline{\overline{b}}}{b_{00}}\right)\ -\
\frac{1}{2}\
\left(\frac{\vec{b_{0}}}{b_{00}}\right)\otimes\left(\frac{\vec{a_{0}}}{a_{00}}\right)\
+\ \frac{3}{8}\
\left(\frac{\vec{b_{0}}}{b_{00}}\right)\otimes\left(\frac{\vec{b_{0}}}{b_{00}}\right)\right]\cdot\vec{u}\right\\}.$
Eq. (8) is especially useful for normalizing unit vectors such as ray
direction vectors or normal vectors to surfaces.
## 9 Propagation of rays
The new $x$ and $y$ coordinates of rays propagated from axial position $z_{1}$
to $z_{2}$ according to
$\displaystyle x_{2}$ $\displaystyle=$ $\displaystyle x_{1}\ +\ k_{x1}\ t$
$\displaystyle y_{2}$ $\displaystyle=$ $\displaystyle y_{1}\ +\ k_{y1}\ t$
(41) $\displaystyle z_{2}$ $\displaystyle=$ $\displaystyle z_{1}\ +\ k_{z1}\
t.$
Since $z_{1}$ and $k_{z1}$ are known, the parametric variable $t$ can be
expressed in terms of those quantities and the value for $z_{2}$. In
polynomial notation, we have
$\displaystyle t$ $\displaystyle=$ $\displaystyle t_{00}\ +\
\vec{u}\cdot\vec{t}\ +\ \vec{u}\cdot\overline{\overline{t}}\cdot\vec{u}\ =\
\frac{z_{2}-z_{1}}{k_{z1}}$ $\displaystyle\left[\
\left(z_{2,00}-z_{1,00}\right)+\vec{u}\cdot\left(\vec{z_{2}}-\vec{z_{1}}\right)+\vec{u}\cdot\left(\overline{\overline{z_{2}}}-\overline{\overline{z_{1}}}\right)\cdot\vec{u}\
\right]\ /\
\left(k_{z1,00}+\vec{u}\cdot\vec{k_{z1}}+\vec{u}\cdot\overline{\overline{k_{z1}}}\cdot\vec{u}\right).$
The rule for division, Eq. (8), shows how to evaluate the polynomial
components of $t$ namely ($t_{00}$, $\vec{t}$, and $\overline{\overline{t}}$)
from Eq. (9) given the known polynomial components of $z_{1}$, $z_{2}$, and
$\vec{k_{1}}$. Then, also given $x_{1}$, $y_{1}$, $k_{x1}$, and $k_{y1}$ in
the form of polynomial objects, we can apply the basic operations of addition
and multiplication defined in §8 to Eq. (9) to find $x_{2}$ and $y_{2}$ in the
form of polynomial objects also. This means that knowing the polynomial
components of the initial position and direction vector, we can compute the
polynomial components of the final position vector without knowing numerical
values for the polynomial coefficients $\vec{u}$.
## 10 Surface prescriptions for polynomial X-ray optics
We consider mirror prescriptions for the primary mirrors of the form
$\displaystyle r_{s}^{2}(z)$ $\displaystyle=$ $\displaystyle r_{0,s}^{2}\
\left[1\ +\ 2\ A_{s}\ (z/r_{0,s})\ +B_{s}\ (z/r_{0,s})^{2}\ +\ u_{a,s}\
(z/r_{0,s})^{2}\ +\ u_{b,s}\ (z/r_{0,s})^{3}\right]$ $\displaystyle=$
$\displaystyle r_{0,s}^{2}\ \left[1\ +\ 2\ A_{s}\ (z/r_{0,s})\ +B_{s}\
(z/r_{0,s})^{2}\ +\ (z/r_{0,s})^{2}\
\vec{u_{s}}\cdot\vec{\zeta_{0,s}}\right].$
For the primary (P) and secondary (S) mirror segments, we have
$\begin{array}[]{llll}P:&A_{s}=\tan{\alpha_{0,s}},&B_{s}=0,&\zeta_{0,s,P}=\left(1,\left(\frac{z}{r_{0,s}}\right),0,0\right)\\\
&&&\\\
S:&A_{s}=\tan{3\alpha_{0,s}},&B_{s}=h(\alpha_{0,s})\tan^{2}{3\alpha_{0,s}},&\zeta_{0,s,S}=\left(0,0,1,\left(\frac{z}{r_{0,s}}\right)\right).\end{array}$
(44)
Here
$\displaystyle\alpha_{0,s}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{4}\right)\
\tan^{-1}{\left(\frac{r_{0,s}}{f}\right)}$ $\displaystyle h(\alpha_{0,s})$
$\displaystyle=$ $\displaystyle 1-1/[1+2\cos{(2\alpha_{0,s})}]^{2},$
and $\vec{u}=(u_{a,s,P},u_{b,s,P},u_{a,s,S},u_{b,s,S})$. The notation and
methods introduced here are, in principle, readily extended to additional
terms in the mirror presciptions [e.g., proportional to $(z/r_{0,s})^{4}$,
$(z/r_{0,s})^{5}$, etc.].
## 11 Ray tracing polynomial X-ray optics
In order to trace rays through X-ray optics, one needs to: (1) populate the
entrances aperture with rays (both in position and direction); (2) calculate
intersections with mirror segment surfaces; (3) calculate the unit normals to
the surfaces at those intersections, including deviations due to non-ideal
surfaces; (4) determine the direction of the reflected ray; and (5) take
account of obstruction by the next innermost mirror shell. Using the tools
outlined above, all these tasks can be accomplished, to sufficent accuracy,
for polynomial X-ray optics as long as the polynomial coefficients are
sufficiently small. In the future, we plan to provide more details on this
method and its results.
## 12 Concluding remarks
The ultimate goal of our work is to reduce the complexity of design procedures
for nested grazing incidence X-ray telescopes, specifically those with Wolter
I and polynomial designs. In this paper, we have:
1. 1.
Described our use of Monte-Carlo ray traces to devise trial analytic formulae
for the coefficients of terms in the expression [Eq. (4)] for the spatial
variance of rays from a point source on an arbitrary focal surface for a
single Wolter I mirror shell. Our adopted merit function [Eq. (1)] can then be
minimized to provide the best displacement of the mirror shell along the
optical axis and the best value for the detector tilt angle.
2. 2.
Shown that for a set of nested mirror shells, the spatial variance on an
arbitrary focal surface is a sum of two terms. The first [Eq. (5)] is a sum
over the variances of the individual shells evaluated on that focal surface,
weighted by their relative effective areas. The second [Eq. (15)] is a sum
over a kind of variance for the mean positions of rays from the individual
shells on that focal surface. The existence of this second term means that it
is necessary to optimize parameters such as mirror shell displacement along
the optical axis, detector tilt angle, and polynomial coefficients
simultaneously for all mirror shells, rather than individually.
3. 3.
In §6—§11, introduced notation and mathematical tools for ray tracing
polynomial optics leaving the polynomial coefficients in symbolic form. In
principle, this simplifies the design procedure by reducing the required
number of Monte-Carlo ray traces, and permitting determination of numerical
values for the polynomial coefficients through the solution of a large number
of linear equations derived from minimization of the merit function.
Our future plans are to continue these studies, refining the trial analytic
functions for Wolter I optics, implementing a polynomial optic ray trace code
using the tools described in this paper, and hopefully providing a less
complex means for the optimization of wide-field X-ray telescope designs.
###### Acknowledgements.
We thank R. Giacconi, S. S. Murray, G. Pareschi, and all the members of the
WFXT team for many interesting and helpful discussions and ideas. We carry out
all our X-ray optics ray trace work in the symbolic mathematics system
Mathematica©[10], which makes much of our work easier, more accurate, and less
tedious.
## References
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* [2] Murray, S. S., Norman, C., Ptak, A., Giacconi, R., Weisskopf, M., Ramsey, B., Bautz, M., Vikhliniin, A., Brandt, N., Rosati, P., Weaver, H., Allen, S., and Flanagan, K., “Wide field x-ray telescope mission,” in [Space Telescopes and Instrumentation 2008: Ultraviolet to Gamma Ray ], Turner, M. J. L. and Flanagan, K. A., eds., Proc. SPIE 7011, 70111J–70111J–16 (2008).
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* [4] Roming, P. W. A., Liechty, J. C., Sohn, D. H., Shoemaker, J. R., Burrows, D. N., and Garmire, G. P., “Markov chain monte carlo algorithms for optimizing grazing incidence optics for wide-field x-ray survey imaging,” in [X-Ray Optics for Astronomy: Telescopes, Multilayers, Spectrometers, and Missions ], Gorenstein, P. and Hoover, R. B., eds., Proc. SPIE 4496, 146–153 (2002).
* [5] Conconi, P., Pareschi, G., Campana, S., Chincarini, G., and Tagliaferri, G., “Wide-field x-ray imaging for future missions, including xeus,” in [Optics for EUV, X-ray, and Gamma-Ray Astronomy ], Citterio, O. and O’Dell, S. L., eds., Proc. SPIE 5168, 334–345 (2004).
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## Appendix A Additional coefficient expressions
We assume a detector in the first quadrant (both $x$ and $y$ positive). First
we provide definitions for the coefficients $d_{s}$, $e_{s}$ and $f_{s}$ (see
§4):
$d_{s}\ \equiv\ \sum_{(x,y)}\
\left[<(x,y)\left(\frac{k_{(x,y)}}{k_{z}}\right)(x+y)>_{0,s}\ -\
<(x,y)>_{0,s}<\left(\frac{k_{(x,y)}}{k_{z}}\right)(x+y)>_{0,s}\right],$ (46)
$\displaystyle e_{s}$ $\displaystyle\equiv$ $\displaystyle\sum_{(x,y)}\
\left[<\left(\frac{k_{(x,y)}}{k_{z}}\right)^{2}(x+y)>_{0,s}\ -\
<\left(\frac{k_{(x,y)}}{k_{z}}\right)>_{0,s}<\left(\frac{k_{(x,y)}}{k_{z}}\right)(x+y)>_{0,s}\right]$
$\displaystyle+\ \sum_{(x,y)}\
\left[<(x,y)\left(\frac{k_{(x,y)}}{k_{z}}\right)\left(\frac{k_{x}+k_{y}}{k_{z}}\right)>_{0,s}\
-\
<(x,y)>_{0,s}<\left(\frac{k_{(x,y)}}{k_{z}}\right)\left(\frac{k_{x}+k_{y}}{k_{z}}\right)>_{0,s}\right],$
$f_{s}\ =\ f_{xy,s}\ +\ f_{z,s},$ (48)
$\displaystyle f_{xy,s}$ $\displaystyle\equiv$ $\displaystyle\sum_{(x,y)}\
\left[<(x+y)^{2}\left(\frac{k_{(x,y)}}{k_{z}}\right)^{2}>_{0,s}\ -\
<(x+y)\left(\frac{k_{(x,y)}}{k_{z}}\right)>_{0,s}^{2}\right]$ $\displaystyle+\
2\ \sum_{(x,y)}\
\left[<(x,y)(x+y)\left(\frac{k_{(x,y)}}{k_{z}}\right)\left(\frac{k_{x}+k_{y}}{k_{z}}\right)>_{0,s}\
-\
<(x,y)>_{0,s}<(x+y)\left(\frac{k_{(x,y)}}{k_{z}}\right)\left(\frac{k_{x}+k_{y}}{k_{z}}\right)>_{0,s}\right],$
$f_{z,s}\ \equiv\ <(x+y)^{2}>_{0}\ -\ <x+y>_{0}^{2}.$ (50)
We typically find $f_{z,s}\ \ll\ f_{xy,s}$. We define
$\displaystyle g(\theta,\ \zeta,\ \xi)$ $\displaystyle\equiv$ $\displaystyle
1\ +\ \zeta\ \tan{\theta}\ +\ \xi\ \tan^{2}{\theta}$ $\displaystyle
h_{0}(\phi,\ \eta)$ $\displaystyle\equiv$ $\displaystyle 1\ +\ \eta\
\sin{2\phi}$ (51) $\displaystyle h(\theta,\phi,\ \zeta,\ \xi,\ \eta_{0},\
\eta_{\zeta},\ \eta_{\xi})$ $\displaystyle\equiv$ $\displaystyle h_{0}(\phi,\
\eta_{0})\ +\ \zeta\ \tan{\theta}\ h_{0}(\phi,\ \eta_{\zeta})\ +\ \xi\
\tan^{2}{\theta}\ h_{0}(\phi,\ \eta_{\xi}).$
Note that $h(\theta,\phi,\ \zeta,\ \xi,\ 0,\ 0,\ 0)\ =\ g(\theta,\phi,\ \xi)$.
Now we provide trial fitting functions for the coefficients $d_{s}$, $e_{s}$
and $f_{s}$:
$d_{fit}(\theta,\phi)\ =\ 2\ \mu_{d}\ f\ \ell\ \tan^{3}{\theta}\
h(\theta,\phi,\ \zeta_{d},\ \xi_{d},\ \eta_{0,d},\ \eta_{\zeta,d},\
\eta_{\xi,d}),$ (52)
$e_{fit}(\theta,\phi)\ =\ \mu_{e}\ f\ \tan^{2}{4\alpha_{0}}\ \tan{\theta}\
h(\theta,\phi,\ \zeta_{e},\ \xi_{e},\ \eta_{0,e},\ \eta_{\zeta,e},\
\eta_{\xi,e}),$ (53)
$f_{xy,fit}(\theta,\phi)\ =\ 2\ \mu_{f,xy}\ (\ f\ \tan{4\alpha_{0}}\
\tan{\theta}\ )^{2}\ h(\theta,\phi,\ \zeta_{f,xy},\ \xi_{f,xy},\
\eta_{0,f,xy},\ \eta_{\zeta,f,xy},\ \eta_{\xi,f,xy}),$ (54)
$f_{z,fit}(\theta,\phi)\ =\ \mu_{f,z}\ \tan^{3}{\theta}\ h(\theta,\phi,\
\zeta_{f,z},\ \xi_{e},\ \eta_{0,f,z},\ \eta_{\zeta,f,z},\ \eta_{\xi,f,z}).$
(55)
We now provide the definitions of $a^{\prime}_{(x,y),s}$,
$b^{\prime}_{(x,y),s}$, $d^{\prime}_{(x,y),s}$, $e^{\prime}_{(x,y),s}$ and
$f^{\prime}_{(x,y),s}$ (see §5):
$\displaystyle a^{\prime}_{(x,y),s}\ =\ <(x,y)>_{0,s}$ $\displaystyle
b^{\prime}_{(x,y),s}\ =\ <\left(\frac{k_{(x,y)}}{k_{z}}\right)>_{0,s}$
$\displaystyle d^{\prime}_{(x,y),s}\ =\
<\left(\frac{k_{(x,y)}}{k_{z}}\right)(x+y)>_{0,s},$ (56) $\displaystyle
e^{\prime}_{(x,y),s}\ =\
<\left(\frac{k_{(x,y)}}{k_{z}}\right)\left(\frac{k_{x}+k_{y}}{k_{z}}\right)>_{0,s},$
$\displaystyle\ \ \ \ \ f^{\prime}_{(x,y),s}\ =\
<\left(\frac{k_{(x,y)}}{k_{z}}\right)(x+y)\left(\frac{k_{x}+k_{y}}{k_{z}}\right)>_{0,s}.$
|
arxiv-papers
| 2010-06-25T20:38:12 |
2024-09-04T02:49:11.221175
|
{
"license": "Public Domain",
"authors": "Ronald F. Elsner, Stephen L. O'Dell, Brian D. Ramsey, and Martin C.\n Weisskopf",
"submitter": "Ronald Elsner",
"url": "https://arxiv.org/abs/1006.5065"
}
|
1006.5087
|
# Gaussian Z-Interference Channel with a Relay Link: Achievability Region and
Asymptotic Sum Capacity ††thanks: Manuscript submitted to the IEEE
Transactions on Information Theory on Sept 3, 2008, resubmitted on June 10,
2010 and revised on June 8, 2011. The material in this paper has been
presented in part at the IEEE International Symposium on Information Theory
and its Applications (ISITA), Auckland, New Zealand, December 2008, and in
part at the IEEE Information Theory and Applications (ITA) Workshop, San
Diego, CA, February 2009. The authors are with the Electrical and Computer
Engineering Department, University of Toronto, 10 King’s College Road,
Toronto, Ontario M5S 3G4, Canada (email: zhoulei@comm.utoronto.ca;
weiyu@comm.utoronto.ca). This work was supported in part by the Natural
Science and Engineering Research Council (NSERC) of Canada under the Canada
Research Chairs program, and in part by the Ontario Early Researcher Awards
program. Kindly address correspondence to Lei Zhou (zhoulei@comm.utoronto.ca).
Lei Zhou, Student Member, IEEE and Wei Yu, Senior Member, IEEE
###### Abstract
This paper studies a Gaussian Z-interference channel with a rate-limited
digital relay link from one receiver to another. Achievable rate regions are
derived based on a combination of Han-Kobayashi common-private power splitting
technique and either a compress-and-forward relay strategy or a decode-and-
forward strategy for interference subtraction at the other end. For the
Gaussian Z-interference channel with a digital link from the interference-free
receiver to the interfered receiver, the capacity region is established in the
strong interference regime; an achievable rate region is established in the
weak interference regime. In the weak interference regime, the decode-and-
forward strategy is shown to be asymptotically sum-capacity achieving in the
high signal-to-noise ratio and high interference-to-noise ratio limit. In this
case, each relay bit asymptotically improves the sum capacity by one bit. For
the Gaussian Z-interference channel with a digital link from the interfered
receiver to the interference-free receiver, the capacity region is established
in the strong interference regime; achievable rate regions are established in
the moderately strong and weak interference regimes. In addition, the
asymptotic sum capacity is established in the limit of large relay link rate.
In this case, the sum capacity improvement due to the digital link is bounded
by half a bit when the interference link is weaker than a certain threshold,
but the sum capacity improvement becomes unbounded when the interference link
is strong.
###### Index Terms:
multicell processing, relay channel, receiver cooperation, Wyner-Ziv coding,
Z-interference channel.
## I Introduction
The classic interference channel models a communication situation in which two
transmitters communicate with their respective intended receivers while
mutually interfering with each other. The interference channel is of
fundamental importance for communication system design, because many practical
systems are designed to operate in the interference-limited regime. The
largest known achievability region for the interference channel is due to Han
and Kobayashi [1], where a common-private power splitting technique is used to
partially decode and subtract the interfering signal. The Han-Kobayashi scheme
has been shown to be capacity achieving in a very weak interference regime [2,
3, 4] and to be within one bit of the capacity region in general [5].
This paper considers a communication model in which the classic interference
channel is augmented by a noiseless relay link between the two receivers. We
are motivated to study such a relay-interference channel because in practical
wireless cellular systems, the uplink receivers at the base-stations are
connected via backhaul links and the downlink receivers may also be capable of
establishing an independent communication link for the purpose of interference
mitigation.
Figure 1: Gaussian Z-interference channel with a relay link: (a) Type I; (b)
Type II.
This paper explores the use of relay techniques for interference mitigation.
We focus on the simplest interference channel model, the Gaussian
Z-interference channel (also known as the one-sided interference channel), in
which one of the receivers gets an interference-free signal, the other
receiver gets a combination of the intended and the interfering signals, and
the channel is equipped with a noiseless link of fixed capacity from one
receiver to the other. The Z-interference channel is of practical interest
because it models a two-cell cellular network with one user located at the
cell edge and another user at the cell center. (The cell-edge user is
sometimes referred to as in a soft-handoff mode [6].) Depending on the
direction of the noiseless link, the proposed model is named the Type I or the
Type II Gaussian Z-relay-interference channel in this paper as shown in Fig.
1.
The Type I Gaussian Z-relay-interference channel has a digital relay link of
finite capacity from the interference-free receiver to the interfered
receiver. Our main coding strategy for the Type I channel is a decode-and-
forward strategy, in which the relay link forwards part of the interference to
the interfered receiver using a binning technique for interference
subtraction. This paper shows that decode-and-forward is capacity achieving
for the Type I channel in the strong interference regime, and is
asymptotically sum-capacity achieving in the weak interference regime. In
addition, in the weak interference regime, every bit of relay link rate
increases the sum rate by one bit in the high signal-to-noise ratio (SNR) and
high interference-to-noise ratio (INR) limit.
The Type II Gaussian Z-relay-interference channel differs from the Type I
channel in that the direction of the digital link goes from the interfered
receiver to the interference-free receiver. Our main coding strategy for the
Type II channel is based on a combination of two relaying strategies: decode-
and-forward and compress-and-forward. In the proposed scheme, the interfered
receiver, which decodes the common message and observes a noisy version of the
neighbor’s private message, describes the common message with a bin index and
describes the neighbor’s private message using a quantization scheme. It is
shown that, in the strong interference regime, a special form of the proposed
relaying scheme, which uses decode-and-forward only, is capacity achieving. In
the weak interference regime, the proposed scheme reduces to pure compress-
and-forward. Further, when the interference link is weaker than a certain
threshold, the sum-capacity gain due to the digital link for the Type II
channel is upper bounded by half a bit. This is in contrast to the Type I
channel, in which each relay bit can be worth up to one bit in sum capacity.
### I-A Related Work
The Gaussian Z-interference channel has been extensively studied in the
literature. It is one of the few examples of an interference channel (besides
the strong interference case [1, 7, 8] and the very weak interference case [2,
3, 4]) for which the sum capacity has been established. The sum capacity of
the Gaussian Z-interference channel in the weak interference regime is
achieved with both transmitters using Gaussian codebooks and with the
interfered receiver treating the interference as noise [5, 9].
The fundamental decode-and-forward and compress-and-forward strategies for the
relay channel are due to the classic work of Cover and El Gamal [10]. Our
study of the interference channel with a relay link is motivated by the more
recent capacity results for a class of deterministic relay channels
investigated by Kim [11] and a class of modulo-sum relay channels investigated
by Aleksic et al. [12], where the relay observes the noise in the direct
channel. The situation investigated in [11, 12] is similar to the Type I
Gaussian Z-relay-interference channel, where the interference-free receiver
observes a noisy version of the interference at the interfered receiver and
helps the interfered receiver by describing the interference through a
noiseless relay link.
The channel model studied in the paper is related to the work of Sahin et al.
[13, 14, 15], Marić et al. [16], Dabora et al. [17], and Tian and Yener [18],
where the achievable rate regions and the relay strategies are studied for an
interference channel with an additional relay node, and where the relay
observes the transmitted signals from the inputs and contributes to the
outputs of both channels. In particular, [16], [17] propose an interference-
forwarding strategy which is similar to the one used for the Type I channel in
this paper. In a similar setup, the works of Ng et al. [19] and Høst-Madsen
[20] study the interference channel with analog relay links at the receiver,
and use the compress-and-forward relay strategy to obtain capacity bounds and
asymptotic results.
This paper is closely related to the work of Wang and Tse [21], Prabhakaran
and Viswanath [22], and Simeone et al. [23], where the interference channel
with limited receiver cooperation is studied. In [23], the achievable rates of
a Wyner-type cellular model with either uni- or bidirectional finite-capacity
backhaul links are characterized. In [21], a more general channel model in
which a two-user Gaussian interference channel is augmented with bidirectional
digital relay links is considered, and a conferencing protocol based on the
quantize-map-and-forward strategy of [24] is proposed.
The present paper considers a special case of the channel model in [21], i.e.,
a simplified Gaussian Z-interference channel model with a unidirectional
digital relay link. By focusing on this special case, we are able to derive
concrete achievability results and upper bounds and obtain insights on the
rate improvement due to the relay link. For example, while [21] adopts a
universal power splitting ratio of [5] at the transmitter to achieve the
capacity region to within 2 bits, this paper adapts the power splitting ratio
to channel parameters, and shows that in the weak interference regime a relay
link from the interference-free receiver to the interfered receiver is much
more beneficial than a relay link in the opposite direction for a
Z-interference channel.
### I-B Outline of the Paper
The rest of this paper is organized as follows. Section II presents
achievability results for the Type I Gaussian Z-relay-interference channel
using the decode-and-forward strategy. Capacity results are established for
the strong interference regime; asymptotic sum-capacity result is established
for the weak interference regime in the high SNR/INR limit. Section III
presents achievability results for the Type II Gaussian Z-relay-interference
channel using a combination of the decode-and-forward scheme and the compress-
and-forward scheme. Capacity results are derived in the strong interference
regimes; asymptotic sum-capacity result is established for all channel
parameters in the limit of large relay link rate. Section IV contains
concluding remarks.
## II Gaussian Z-Interference Channel with
a Relay Link: Type I
### II-A Channel Model and Notations
The Gaussian Z-interference channel is modeled as follows (see Fig. 1(a)):
$\left\\{\begin{array}[]{l}Y_{1}=h_{11}X_{1}+h_{21}X_{2}+Z_{1}\\\
Y_{2}=h_{22}X_{2}+Z_{2}\end{array}\right.$ (1)
where $X_{1}$ and $X_{2}$ are the transmit signals with power constraints
$P_{1}$ and $P_{2}$ respectively, $h_{ij}$ represents the real-valued channel
gain from transmitter $i$ to receiver $j$, and $Z_{1}$, $Z_{2}$ are the
independent additive white Gaussian noises (AWGN) with power $N$. In addition,
the Type I Gaussian Z-relay-interference channel is equipped with a digital
noiseless link of fixed capacity $R_{0}$ from receiver 2 to receiver $1$.
Each transmitter $i$ independently encodes a message $m_{i}$ into a codeword
$X_{i}^{n}(m_{i})$ using a codebook $\mathcal{C}_{i}^{n}$ of $2^{nR_{i}}$
length-$n$ codewords satisfying an average power constraint $P_{i}$. Let
$V^{n}$ be the output of the digital link from receiver $2$ to receiver $1$
taken from a relay codebook $\mathcal{C}_{R}^{n}$, where
$|\mathcal{C}_{R}^{n}|\leq 2^{nR_{0}}$. Receiver $1$ uses a decoding function
$\hat{m}_{1}=f_{1}^{n}(Y_{1}^{n},V^{n})$. Receiver $2$ uses a decoding
function $\hat{m}_{2}=f_{2}^{n}(Y_{2}^{n})$. The average probability of error
for user $i$ is defined as
$P_{e,i}^{n}=\mathbb{E}\left[\textrm{Pr}(\hat{m}_{i}\neq m_{i})\right]$. A
rate pair $(R_{1},R_{2})$ is said to be achievable if for every $\epsilon>0$
and for all sufficiently large $n$, there exists a family of codebooks
$(\mathcal{C}_{i}^{n},\mathcal{C}_{R}^{n})$, and decoding functions
$f_{i}^{n}$, $i=1,2$, such that $\max_{i}\\{P_{e,i}^{n}\\}<\epsilon$. The
capacity region is defined as the set of all achievable rate pairs.
To simplify the notation, the following definitions are used throughout this
paper:
$\displaystyle\mathsf{SNR_{1}}=\frac{|h_{11}|^{2}P_{1}}{N}$
$\displaystyle\mathsf{SNR_{2}}=\frac{|h_{22}|^{2}P_{2}}{N}$
$\displaystyle\mathsf{INR_{2}}=\frac{|h_{21}|^{2}P_{2}}{N}$
$\displaystyle\gamma(x)=\frac{1}{2}\log(1+x)$
where $\log(\cdot)$ is base 2. In addition, denote $\overline{\beta}=1-\beta$,
and let $(x)^{+}=\max\\{x,0\\}$.
### II-B Achievable Rate Region
This paper uses a combination of the Han-Kobayashi common-private power
splitting technique and a decode-and-forward strategy for the Gaussian
Z-relay-interference channel, in which a common information stream is decoded
at receiver $2$, then binned and forwarded to receiver $1$ for subtraction.
The main result of this section is the following achievability theorem.
###### Theorem 1
For the Type I Gaussian Z-interference channel with a digital relay link of
limited rate $R_{0}$ from the interference-free receiver to the interfered
receiver as shown in Fig. 1(a), in the weak interference regime defined by
$0\leq\mathsf{INR_{2}}<\min\\{\mathsf{SNR_{2},INR_{2}^{*}}\\}$, the following
rate region is achievable:
$\bigcup_{0\leq\beta\leq
1}\left\\{(R_{1},R_{2})\left|R_{1}\leq\gamma\left(\frac{\mathsf{SNR}_{1}}{1+\beta\mathsf{INR}_{2}}\right),\right.\right.\\\
R_{2}\leq\min\left\\{\gamma(\mathsf{SNR}_{2}),\gamma(\beta\mathsf{SNR}_{2})+\right.\\\
\left.\left.\gamma\left(\frac{\overline{\beta}\mathsf{INR}_{2}}{1+\mathsf{SNR}_{1}+\beta\mathsf{INR}_{2}}\right)+R_{0}\right\\}\right\\},$
(2)
where
$\mathsf{INR_{2}^{*}}=\left((1+\mathsf{SNR}_{1})(2^{-2R_{0}}\mathsf{(1+SNR_{2})}-1)\right)^{+}.$
(3)
In the strong interference regime defined by
$\min\\{\mathsf{SNR_{2},INR_{2}^{*}}\\}\leq\mathsf{INR_{2}}<\mathsf{INR}_{2}^{*}$,
the capacity region is given by
$\left\\{(R_{1},R_{2})\left|\begin{array}[]{rll}R_{1}&\leq&\gamma(\mathsf{SNR_{1}})\\\
R_{2}&\leq&\gamma(\mathsf{SNR_{2}})\\\
R_{1}+R_{2}&\leq&\gamma(\mathsf{SNR_{1}+INR_{2}})+R_{0}\end{array}\right.\right\\}.$
(4)
In the very strong interference regime defined by $\mathsf{INR_{2}\geq
INR_{2}^{*}}$, the capacity region is given by
$\left\\{(R_{1},R_{2})\left|\begin{array}[]{l}R_{1}\leq\gamma(\mathsf{SNR_{1}})\\\
R_{2}\leq\gamma(\mathsf{SNR_{2}})\\\ \end{array}\right.\right\\}.$ (5)
Figure 2: Common-private power splitting for Type I channel.
###### Proof:
We use the Han-Kobayashi [1] common-private power splitting scheme with
Gaussian inputs to prove the achievability of the rate regions (2), (4) and
(5). As depicted in Fig. 2, user $1$’s signal $X_{1}$ is intended for decoding
at $Y_{1}$ only. User $2$’s signal $X_{2}$ is the superposition of the private
message $U_{2}$ and the common message $W_{2}$, i.e., $X_{2}=U_{2}+W_{2}$. The
private message can only be decoded by the intended receiver $Y_{2}$, while
the common message can be decoded by both receivers. Independent Gaussian
codebooks of sizes $2^{nS_{1}}$, $2^{nS_{2}}$ and $2^{nT_{2}}$ are generated
according to i.i.d. Gaussian distributions $X_{1}\sim\mathcal{N}(0,P_{1})$,
$U_{2}\sim\mathcal{N}(0,\beta P_{2})$, and
$W_{2}\sim\mathcal{N}(0,\overline{\beta}P_{2})$, respectively, where
$0\leq\beta\leq 1$. The encoded sequences $X_{1}^{n}$ and
$X_{2}^{n}=U_{2}^{n}+W_{2}^{n}$ are then transmitted over a block of $n$ time
instances.
Decoding takes place in two steps. First, $(W_{2}^{n},U_{2}^{n})$ are decoded
at receiver $2$. The set of achievable rates $(T_{2},S_{2})$ is the capacity
region of a Gaussian multiple-access channel, denoted here by
$\mathcal{C}_{2}$, where
$\left\\{\begin{array}[]{rll}T_{2}&\leq&\gamma(\overline{\beta}\mathsf{SNR_{2}})\\\
S_{2}&\leq&\gamma(\beta\mathsf{SNR_{2}})\\\
S_{2}+T_{2}&\leq&\gamma(\mathsf{SNR_{2}}).\end{array}\right.$ (6)
After $(W_{2}^{n},U_{2}^{n})$ are decoded at receiver $2$,
$(X_{1}^{n},W_{2}^{n})$ are then decoded at receiver $1$ with $U_{2}^{n}$
treated as noise, but with the help of the relay link. This is a multiple-
access channel with a rate-limited relay $Y_{2}^{n}$, who has complete
knowledge of $W_{2}^{n}$. This channel is a special case of the multiple-
access relay channel studied in [25] and [26]. It is straightforward to show
that a decode-and-forward relay strategy is capacity achieving in this special
case and its capacity region $\mathcal{C}_{1}$ is the set of $(S_{1},T_{2})$
for which
$\left\\{\begin{array}[]{rll}S_{1}&\leq&\gamma\left(\displaystyle\frac{\mathsf{SNR_{1}}}{1+\beta\mathsf{INR_{2}}}\right)\\\
T_{2}&\leq&\gamma\left(\displaystyle\frac{\overline{\beta}\mathsf{INR_{2}}}{1+\beta\mathsf{INR_{2}}}\right)+R_{0}\\\
S_{1}+T_{2}&\leq&\gamma\left(\displaystyle\frac{\mathsf{SNR_{1}}+\overline{\beta}\mathsf{INR_{2}}}{1+\beta\mathsf{INR_{2}}}\right)+R_{0}.\end{array}\right.$
(7)
An achievable rate region of the Gaussian Z-interference channel with a relay
link is then the set of all $(R_{1},R_{2})$ such that $R_{1}=S_{1}$ and
$R_{2}=S_{2}+T_{2}$ for some $(S_{1},T_{2})\in\mathcal{C}_{1}$ and
$(S_{2},T_{2})\in\mathcal{C}_{2}$. Further, since $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$ depend on the common-private power splitting ratio $\beta$,
the convex hull of the union of all such $(R_{1},R_{2})$ sets over all choices
of $\beta$ is achievable.
A Fourier-Motzkin elimination method (see e.g. [27]) can be used to show that
for each fixed $\beta$, the achievable $(R_{1},R_{2})$’s form a pentagon
region characterized by
$\mathcal{R}_{\beta}=\left\\{(R_{1},R_{2})\left|\begin{array}[]{l}\displaystyle
R_{1}\leq\gamma\left(\mathsf{\frac{SNR_{1}}{1+\beta INR_{2}}}\right)\\\
\displaystyle R_{2}\leq\min\\{\gamma(\mathsf{SNR_{2}}),\gamma(\mathsf{\beta
SNR_{2}})+\\\
\displaystyle\qquad\qquad\quad\gamma\left(\mathsf{\frac{\overline{\beta}INR_{2}}{1+\beta
INR_{2}}}\right)+R_{0}\\}\\\ \displaystyle R_{1}+R_{2}\leq\gamma(\mathsf{\beta
SNR_{2}})+\\\
\qquad\qquad\quad\gamma\displaystyle\left(\mathsf{\frac{SNR_{1}+\overline{\beta}INR_{2}}{1+\beta
INR_{2}}}\right)+R_{0}\\\ \end{array}\right.\right\\}.$ (8)
The convex hull of the union of these pentagons over $\beta$ gives the
complete achievability region. It happens that the union of the pentagons,
i.e. $\bigcup_{0\leq\beta\leq 1}\mathcal{R}_{\beta}$, is already convex.
Therefore, convex hull is not needed. In the following, we give an explicit
expression for $\bigcup_{0\leq\beta\leq 1}\mathcal{R}_{\beta}$.
Figure 3: The union of rate region pentagons when $\mathsf{INR_{2}\leq
SNR_{2}}$.
Consider first the regime where $\mathsf{INR_{2}}\leq\mathsf{SNR_{2}}$. Ignore
for now the constraint $R_{2}\leq\gamma(\mathsf{SNR_{2}})$ and focus on an
expanded pentagon defined by $\\{(R_{1},R_{2})\left|R_{1}\leq
f_{1}(\beta),R_{2}\leq f_{2}(\beta),R_{1}+R_{2}\leq f_{3}(\beta)\right.\\}$,
where $f_{1}(\beta)$ is the $R_{1}$ constraint in (8), $f_{2}(\beta)$ is the
second term of the min expression in the $R_{2}$ constraint in (8), and
$f_{3}(\beta)$ is the $R_{1}+R_{2}$ constraint in (8).
It is easy to verify that when $\beta=1$, the expanded pentagon reduces to a
rectangular region, as shown in Fig. 3. Further, as $\beta$ decreases from 1
to 0, $f_{1}(\beta)$ monotonically increases and both $f_{2}(\beta)$ and
$f_{3}(\beta)$ monotonically decrease, while $f_{2}(\beta)-f_{3}(\beta)$
remains a constant in the regime where $\mathsf{INR_{2}\leq SNR_{2}}$. Since
$f_{1}(\beta)$, $f_{2}(\beta)$ and $f_{3}(\beta)$ are all continuous functions
of $\beta$, as $\beta$ decreases from $1$ to $0$, the upper-right corner point
of the expanded pentagon moves vertically downward in the $R_{2}-R_{1}$ plane,
while the lower-right corner point moves downward and to the right in a
continuous fashion. Consequently, the union of these expanded pentagons is
defined by $R_{1}\leq\gamma(\mathsf{SNR_{1}})$,
$R_{2}\leq\gamma(\mathsf{SNR_{2}})+R_{0}$, and lower-right corner points of
the pentagons $(R_{1},R_{2})$ with
$\left\\{\begin{array}[]{lll}R_{1}&=&\gamma\left(\displaystyle\frac{\mathsf{SNR}_{1}}{1+\beta\mathsf{INR}_{2}}\right)\\\
R_{2}&=&\gamma(\beta\mathsf{SNR}_{2})+\gamma\left(\displaystyle\frac{\overline{\beta}\mathsf{INR}_{2}}{1+\mathsf{SNR}_{1}+\beta\mathsf{INR}_{2}}\right)+R_{0}\end{array}\right.$
(9)
where $0\leq\beta\leq 1$. We prove in Appendix -A that such a region is convex
when $\mathsf{INR_{2}\leq SNR_{2}}$. Thus, convex hull is not needed. Finally,
incorporating the constraint $R_{2}\leq\gamma(\mathsf{SNR_{2}})$ gives the
achievable region (2).
Figure 4: The union of rate region pentagons when $\mathsf{INR_{2}\geq
SNR_{2}}$.
Now, consider the regime where $\mathsf{INR_{2}\geq SNR_{2}}$. In this regime,
$f_{1}(\beta)$, $f_{2}(\beta)$ and $f_{3}(\beta)$ are all increasing functions
as $\beta$ goes from 1 to 0\. Consequently, $\bigcup_{0\leq\beta\leq
1}\mathcal{R}_{\beta}=\mathcal{R}_{0}$, as illustrated in Fig. 4. Therefore,
convex hull is not needed. Thus, the achievable rate region simplifies to
$\left\\{(R_{1},R_{2})\left|\begin{array}[]{l}R_{1}\leq\gamma(\mathsf{SNR_{1}})\\\
R_{2}\leq\min\\{\gamma(\mathsf{SNR_{2}}),\mathsf{\gamma(INR_{2})}+R_{0}\\}\\\
R_{1}+R_{2}\leq\gamma(\mathsf{SNR_{1}+INR_{2}})+R_{0}\end{array}\right.\right\\},$
(10)
which is equivalent to (4) by noting that
$\mathsf{\gamma(INR_{2})}+R_{0}\geq\gamma(\mathsf{SNR_{2}})$ (11)
when $\mathsf{INR_{2}\geq SNR_{2}}$.
We have so far obtained the achievable rate regions for the regimes
$\mathsf{INR_{2}\leq SNR_{2}}$ and $\mathsf{INR_{2}\geq SNR_{2}}$ as in (2)
and (4) respectively. Both expressions can be further simplified in some
specific cases. Inspecting Figs. 3 and 4, it is easy to see that when
$\mathsf{INR_{2}\geq INR_{2}^{*}}$, where $\mathsf{INR_{2}^{*}}$ is as defined
in (3), the horizontal line $R_{2}=\gamma(\mathsf{SNR_{2}})$ is below the
lower-right corner point corresponding to $\beta=0$, i.e.,
$\gamma(\mathsf{SNR_{2}})\leq\gamma\displaystyle\left(\mathsf{\frac{INR_{2}}{1+SNR_{1}}}\right)+R_{0}.$
(12)
Therefore, in both the $\mathsf{INR_{2}\leq SNR_{2}}$ (Fig. 3) and the
$\mathsf{INR_{2}\geq SNR_{2}}$ (Fig. 4) regimes, whenever
$\mathsf{INR_{2}}\geq\mathsf{INR_{2}^{*}}$, the achievable rate region reduces
to a rectangle as in (5). This is the very strong interference regime.
Noting the fact that $\mathsf{INR_{2}^{*}}$ can be greater or less than
$\mathsf{SNR_{2}}$ depending on $R_{0}$, we see that the achievability result
for the Type I channel is divided into the weak, strong, and very strong
interference regimes as in (2), (4) and (5) respectively.
Finally, it is possible to prove a converse in the strong and very strong
interference regimes. The converse proof is presented in Appendix -B. ∎
It is important to note that the achievable region of Theorem 1 is derived
assuming fixed powers $P_{1}$ and $P_{2}$ at the transmitters. It is possible
that time-sharing among different transmit powers may enlarge the achievable
rate region. For simplicity in the presentation of closed-form expressions for
achievable rates, time-sharing is not explicitly incorporated in the
achievability theorems in this paper.
### II-C Numerical Examples
It is instructive to numerically compare the achievable regions of the
Gaussian Z-interference channel with and without the relay link. First,
observe that when $R_{0}=0$, the achievable rate region (2) and the capacity
region results (4) (5) reduce to previous results obtained in [1] and [8].
In the strong and very strong interference regimes, the capacity region of a
Type I Gaussian Z-relay-interference channel is achieved by transmitting
common information only at $X_{2}$. In the very strong interference regime,
the relay link does not increase capacity, because the interference is already
completely decoded and subtracted, even without the help of the relay. In the
strong interference regime, the relay link increases the capacity by helping
the common information decoding at $Y_{1}$. In fact, a relay link of rate
$R_{0}$ increases the sum capacity by exactly $R_{0}$ bits. As a numerical
example, Fig. 5 shows the capacity region of a Gaussian Z-interference channel
in the strong interference regime with and without the relay link. The channel
parameters are set to be $\mathsf{SNR}_{1}=\mathsf{SNR}_{2}=25$dB,
$\mathsf{INR}_{2}=30$dB. The capacity region without the relay is the dash-
dotted pentagon. With $R_{0}=2$ bits, the capacity region expands to the
dashed pentagon region, which represents an increase in sum rate of exactly 2
bits. As $R_{0}$ increases to $4$ bits, the channel falls into the very strong
interference regime. The capacity region becomes the solid rectangular region.
Figure 5: Capacity region of the Gaussian Z-interference channel in the strong
interference regime with and without a digital relay link of Type I. Figure 6:
Achievable rate region of the Gaussian Z-interference channel in the weak
interference regime with and without a digital relay link of Type I.
In the weak interference regime, the achievable rate region in Theorem 1 is
obtained by a Han-Kobayashi common-private power splitting scheme. By
inspection, the effect of a relay link is to shift the rate region curve
upward by $R_{0}$ bits while limiting $R_{2}$ by its single-user bound
$\gamma(\mathsf{SNR}_{2})$. Interestingly, although the relay link of rate
$R_{0}$ is provided from receiver $2$ to receiver $1$, it can help $R_{2}$ by
exactly $R_{0}$ bits, while it can only help $R_{1}$ by strictly less than
$R_{0}$ bits. As a numerical example, Fig. 6 shows the achievable rate region
of a Gaussian Z-interference channel with
$\mathsf{SNR}_{1}=\mathsf{SNR}_{2}=25$dB and $\mathsf{INR}_{2}=20$dB. The
solid curve represents the rate region achieved without the relay link. The
dashed rate region is with a relay of rate $R_{0}=1$ bit. For most part of the
curve, $R_{0}$ provides a 1-bit increase in $R_{2}$, but a less than 1-bit
increase in $R_{1}$.
It is illustrative to identify the correspondence between the various points
in the rate region and the different common-private splittings in the weak
interference regime. Point $A$ corresponds to $\beta=1$. This is where the
entire $X_{2}$ is private message. In this case, it is easy to verify that the
first term of $R_{2}$ in (2) is less than the second term:
$\gamma(\mathsf{SNR}_{2})<\gamma(\beta\mathsf{SNR}_{2})+\gamma\left(\frac{\overline{\beta}\mathsf{INR}_{2}}{1+\mathsf{SNR}_{1}+\beta\mathsf{INR}_{2}}\right)+R_{0}$
(13)
As $\beta$ decreases, more private message is converted into common message,
which means that less interference is seen at receiver $1$. As a result,
$R_{1}$ increases, $R_{2}$ is kept at a constant (since (13) continues to
hold). Graphically, as $\beta$ decreases from $1$, the achievable rate pair
moves horizontally from point $A$ to the right until it reaches point $B$,
corresponding to some $\beta^{*}$, after which the second term of $R_{2}$ in
(2) becomes less than the first term $\gamma(\mathsf{SNR}_{2})$. The value of
$\beta^{*}$ can be computed as
$\beta^{*}=\frac{(1+\mathsf{SNR}_{1})(1+\mathsf{SNR}_{2})-2^{2R_{0}}(1+\mathsf{SNR}_{1}+\mathsf{INR}_{2})}{2^{2R_{0}}\mathsf{SNR}_{2}(1+\mathsf{SNR}_{1}+\mathsf{INR}_{2})-\mathsf{INR}_{2}(1+\mathsf{SNR}_{2})}.$
(14)
As $\beta$ decreases further from $\beta^{*}$, more private message is
converted into common message, which makes $R_{1}$ even larger. However, when
$\beta<\beta^{*}$, the amount of common message can be transmitted is
restricted by the interference link $h_{21}$ and the digital link rather than
the direct link $h_{22}$. Therefore, user $2$’s data rate cannot be kept as a
constant; $R_{2}$ goes down as user $1$’s rate goes up. As shown in Fig. 6,
the achievable rate pair moves from point $B$ to point $C$ as $\beta$
decreases from $\beta^{*}$ to $0$. Point $C$ corresponds to where the entire
$X_{2}$ is common message.
### II-D Asymptotic Sum Capacity
Practical communication systems often operate in the interference-limited
regime, where both the signal and the interference are much stronger than
noise. In this section, we investigate the asymptotic sum capacity of the Type
I Gaussian Z-relay-interference channel in the weak interference regime where
noise power $N\rightarrow 0$, while power constraints $P_{1}$, $P_{2}$,
channel gains $h_{ij}$, and the digital relay link rate $R_{0}$ are kept
fixed. In other words,
$\mathsf{SNR}_{1},\mathsf{SNR}_{2},\mathsf{INR_{2}}\rightarrow\infty$, while
their ratios are kept constant.
Denote the sum capacity of a Type I Gaussian Z-interference channel with a
relay link of rate $R_{0}$ by $C_{sum}(R_{0})$. Without the digital relay
link, or equivalently $R_{0}=0$, the sum capacity of the classic Gaussian
Z-interference channel in the weak interference regime (i.e.
$\mathsf{INR_{2}\leq SNR_{2}}$) is given by [9, 5]:
$C_{sum}(0)=\gamma(\mathsf{SNR}_{2})+\gamma\left(\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{2}}\right),$
(15)
which is achieved by independent Gaussian codebooks and treating the
interference as noise at the receiver. In the high SNR/INR limit, the above
sum capacity becomes
$C_{sum}(0)\approx\frac{1}{2}\log\left(\frac{\mathsf{SNR_{2}}(\mathsf{SNR_{1}+INR_{2}})}{\mathsf{INR}_{2}}\right),$
(16)
where the notation $f(x)\approx g(x)$ is used to denote $\lim f(x)-g(x)=0$. In
the above expression, the limit is taken as $N\rightarrow 0$.
Intuitively, with a digital relay link of finite capacity $R_{0}$, the sum-
rate increase due to the relay must be bounded by $R_{0}$. The following
theorem shows that in the high SNR/INR limit, the asymptotic sum-capacity
increase is in fact $R_{0}$ in the weak-interference regime.
###### Theorem 2
For the Type I Gaussian Z-interference channel with a digital relay link of
limited rate $R_{0}$ from the interference-free receiver to the interfered
receiver as shown in Fig. 1(a), when
$\mathsf{INR_{2}}\leq\mathsf{\min\\{SNR_{2},INR_{2}^{*}\\}}$, the asymptotic
sum capacity is given by
$C_{sum}(R_{0})\approx C_{sum}(0)+R_{0}.$ (17)
###### Proof:
We first prove the achievability. As illustrated in Fig. 3 the sum rate of the
Type I Gaussian Z-relay-interference channel is achieved with
$\beta=\beta^{*}$, where $\beta^{*}$ is as derived in (14). In the high
SNR/INR limit, we have
$\lim_{N\rightarrow
0}\beta^{*}=\frac{2^{-2R_{0}}}{1+(1-2^{-2R_{0}})\frac{\mathsf{INR_{2}}}{\mathsf{SNR_{1}}}}.$
(18)
Substituting this $\beta^{*}$ into the achievable rate pair in (2), we obtain
the asymptotic rate pair as
$\left\\{\begin{array}[]{l}\displaystyle
R_{1}\approx\frac{1}{2}\log\left(1+\frac{\mathsf{SNR_{1}}}{\mathsf{INR_{2}}}\right)+R_{0}\\\
\displaystyle R_{2}\approx\frac{1}{2}\log(\mathsf{SNR_{2}})\end{array}\right.$
(19)
which gives the following asymptotic sum rate:
$\displaystyle R_{sum}$ $\displaystyle\approx$
$\displaystyle\frac{1}{2}\log\left(\frac{\mathsf{SNR_{2}}(\mathsf{SNR_{1}+INR_{2}})}{\mathsf{INR}_{2}}\right)+R_{0}$
(20) $\displaystyle\approx$ $\displaystyle C_{sum}(0)+R_{0}.$
The converse proof starts with Fano’s inequality. Denote the output of the
digital relay link over the $n$-block by $V^{n}$. Since the digital link has a
capacity limit $R_{0}$, $V^{n}$ is a discrete random variable with
$H(V^{n})\leq nR_{0}$. For a codebook of block length $n$, we have
$\displaystyle n(R_{1}+R_{2})$ (21) $\displaystyle\leq$ $\displaystyle
I(X_{1}^{n};Y_{1}^{n},V^{n})+I(X_{2}^{n};Y_{2}^{n})+n\epsilon_{n}$
$\displaystyle=$ $\displaystyle
I(X_{1}^{n};Y_{1}^{n})+I(X_{1}^{n};V^{n}|Y_{1}^{n})+I(X_{2}^{n};Y_{2}^{n})+n\epsilon_{n}$
$\displaystyle{\leq}$ $\displaystyle
I(X_{1}^{n};Y_{1}^{n})+H(V^{n}|Y_{1}^{n})+I(X_{2}^{n};Y_{2}^{n})+n\epsilon_{n}$
$\displaystyle{\leq}$ $\displaystyle
I(X_{1}^{n};Y_{1}^{n})+I(X_{2}^{n};Y_{2}^{n})+nR_{0}+n\epsilon_{n}$
$\displaystyle{\leq}$ $\displaystyle nC_{sum}(0)+nR_{0}+n\epsilon_{n},$
where $\epsilon_{n}\rightarrow 0$ as $n$ goes to infinity. Note that this
upper bound holds for all ranges of $\mathsf{SNR_{1}}$, $\mathsf{SNR_{2}}$,
and $\mathsf{INR_{2}}$. This, when combined with the asymptotic achievability
result proved earlier, gives the asymptotic sum capacity
$C_{sum}(R_{0})\approx C_{sum}(0)+R_{0}$. ∎
The above proof focuses on the sum-capacity achieving power splitting ratio
$\beta^{*}$. As $\beta\leq\beta^{*}$, the achievable rate pair goes from point
$B$ to point $C$ along the dashed curve as shown in Fig. 6. It turns out that
for any fixed $0<\beta\leq\beta^{*}$, the sum rate also asymptotically
approaches the upper bound, thus providing an alternative proof for Theorem 2.
To see this, fix some arbitrary $0<\beta\leq\beta^{*}$, the sum rate
corresponding to this $\beta$ is given in Theorem 1 as
$\displaystyle R_{sum}$ $\displaystyle=$
$\displaystyle\gamma\left(\frac{\mathsf{SNR}_{1}}{1+\beta\mathsf{INR}_{2}}\right)+\gamma(\beta\mathsf{SNR}_{2})+$
(22)
$\displaystyle\qquad\gamma\left(\frac{\overline{\beta}\mathsf{INR}_{2}}{1+\mathsf{SNR}_{1}+\beta\mathsf{INR}_{2}}\right)+R_{0}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\log\left(\mathsf{\frac{1+\beta
SNR_{2}}{1+\beta INR_{2}}}\right)+\gamma(\mathsf{SNR_{1}+INR_{2}})+R_{0}$
$\displaystyle\approx$
$\displaystyle\frac{1}{2}\log\left(\frac{\mathsf{SNR_{2}}(\mathsf{SNR_{1}+INR_{2}})}{\mathsf{INR}_{2}}\right)+R_{0},$
which is the asymptotic sum capacity. This calculation implies that in the
high SNR/INR regime, the dashed curve in Fig. 6 has an initial slope of -1 as
$\beta$ goes from $\beta^{*}$ to $0$.
Interestingly, decode-and-forward is not the only way to asymptotically
achieve the sum capacity of the Type I channel. The following shows that a
compress-and-forward relaying scheme, although strictly suboptimal in finite
SNR/INR, becomes asymptotically sum-capacity achieving in the high SNR/INR
limit in the weak interference regime, thus giving yet another proof of
Theorem 2.
In the compress-and-forward scheme, no common-private power splitting is
performed. Each receiver only decodes the message intended for it.
Specifically, receiver $2$ compresses its received signal $Y_{2}$ into
$\hat{Y}_{2}$, then forwards it to receiver $1$ through the digital link
$R_{0}$.
Clearly, the rate of user $2$ is given by
$R_{2}=\max_{p(x_{2})}I(X_{2};Y_{2}).$ (23)
Using the Wyner-Ziv coding strategy [28, 10], for a fixed $p(x_{2})$, the
following rate for user $1$ is achievable:
$R_{1}=\max_{p(x_{1})p(\hat{y}_{2}|y_{2})}I(X_{1};Y_{1},\hat{Y}_{2})$ (24)
under the constraint
$I(Y_{2};\hat{Y}_{2}|Y_{1})\leq R_{0}.$ (25)
The optimization in (24) is in general hard. Here, we adopt independent
Gaussian codebooks with $X_{1}\sim\mathcal{N}(0,P_{1})$ and
$X_{2}\sim\mathcal{N}(0,P_{2})$, and a Gaussian quantization scheme for the
compression of $Y_{2}$:
$\hat{Y}_{2}=Y_{2}+e$ (26)
where $e$ is a Gaussian random variable independent of $Y_{2}$, with a
distribution $\mathcal{N}(0,\sigma^{2})$. We show in Appendix -C that this
choice of $p(x_{1})p(x_{2})p(\hat{y}_{2}|y_{2})$ gives the following
achievable rate pair:
$\left\\{\begin{array}[]{l}\displaystyle
R_{1}=\gamma\left(\frac{\mathsf{SNR_{1}}}{1+\mathsf{INR_{2}}}\right)+R_{0}-\delta_{0}(R_{0})\\\
\displaystyle R_{2}=\gamma(\mathsf{SNR_{2}})\end{array}\right.$ (27)
where
$\displaystyle\delta_{0}(R_{0})=$
$\displaystyle\gamma\left(\frac{(2^{2R_{0}}-1)(1+\mathsf{SNR_{2}}+\mathsf{INR_{2}})(1+\mathsf{SNR_{1}}+\mathsf{INR_{2}})}{(\mathsf{1+INR_{2}})(\mathsf{(1+SNR_{1})(1+SNR_{2})+INR_{2}})}\right).$
Let $N\rightarrow 0$, the above rate pair asymptotically goes to
$\left\\{\begin{array}[]{l}\displaystyle
R_{1}\approx\frac{1}{2}\log\left(1+\frac{\mathsf{SNR_{1}}}{\mathsf{INR_{2}}}\right)+R_{0}\\\
R_{2}\approx\frac{1}{2}\log(\mathsf{SNR_{2}})\end{array}\right.$ (28)
which again achieves the asymptotic sum capacity (17). We remark that this is
akin to the capacity result for a class deterministic relay channel [11],
where both decode-and-forward and compress-and-forward are shown to be
capacity achieving.
Although we have demonstrated the asymptotic sum-rate optimality of the point
$B$ and all points between $B$ and $C$ in the weak interference regime as
$N\rightarrow 0$ (while the ratios of SNRs and INRs are kept fixed), we remark
that the achievable region (2) may not be asymptotically optimal in other
regimes. For example, in the regime where
$\mathsf{SNR}_{2}\gg\mathsf{INR}_{2}$, both the $R_{1}+R_{2}$ and
$2R_{1}+R_{2}$ values at point $C$ ($\beta=0$) are unbounded away from their
corresponding upper bounds as shown by Wang and Tse [21, Lemma 5.1] (Eq. (22)
and Eq. (26)). To close this gap, one can use Wang and Tse’s quantize-map-and-
forward approach [21], which in fact achieves the capacity region of the
general Gaussian interference channel with bidirectional links to within a
constant number of bits.
## III Gaussian Z-Interference Channel with
a Relay Link: Type II
### III-A Achievable Rate Region
As a counterpart of the Type I channel considered in the previous section,
this section studies the Type II channel, where the relay link goes from the
interfered receiver to the interference-free receiver as shown in Fig. 1(b).
Intuitively, when the interference link is weak, the digital link would not be
as efficient as in the Type I channel, because receiver $1$’s knowledge of
$X_{2}$ is inferior to that of the receiver $2$. However, when the
interference link is very strong, receiver $1$ becomes a better receiver for
$X_{2}$ than receiver $2$, in which case the digital link is capable of
increasing user $2$’s rate by as much as $R_{0}$.
The main difference between the Type I and the Type II channels is that in the
Type I channel, the relay ($Y_{2}$) observes a noisy version of the
interference at the relay destination ($Y_{1}$). In addition, the interference
consists of messages intended for $Y_{2}$. Thus, the decoding and the
forwarding of the interference is a natural strategy. In the Type II channel,
the relay ($Y_{1}$) observes a noisy version of the intended signal at the
relay destination ($Y_{2}$). Thus, decode-and-forward and compress-and-forward
can both be used. The following achievability theorem is based on a
combination of the Han-Kobayashi scheme (with $\beta$ being the common-private
splitting ratio) and two relay strategies, where the relay decodes then
forwards the common information using a rate $R_{a}$ and compresses then
forwards the private information using a rate $R_{b}$, with
$R_{a}+R_{b}=R_{0}$, as shown in Fig. 7. In addition, the presence of common
information gives rise to the possibility of compressing a combination of
private and common messages. A parameter $\alpha$ accounts for the combination
of private and common message compression.
Unlike the Type I channel, the achievable rate region for the Type II Gaussian
Z-relay-interference channel has a more complicated structure. In addition to
the weak, strong and very strong interference regimes, there is a new
moderately strong regime, where a combination of the decode-and-forward and
the compress-and-forward strategies is proposed. The proposed scheme reduces
to pure compress-and-forward in the weak interference regime, and pure decode-
and-forward in the strong interference regime.
Figure 7: Common-private power splitting for Type II channel with
$R_{0}=R_{a}+R_{b}$, where $R_{a}$ is used to decode-and-forward $W_{2}$, and
$R_{b}$ is used to compress-and-forward $U_{2}$.
###### Theorem 3
For the Type II Gaussian Z-interference channel with a digital relay link of
limited rate $R_{0}$ from the interfered receiver to the interference-free
receiver as shown in Fig. 1(b), in the weak interference regime defined by
$\mathsf{INR_{2}\leq SNR_{2}}$, the following rate region is achievable:
$\bigcup_{0\leq\beta\leq
1}\left\\{(R_{1},R_{2})\left|R_{1}\leq\gamma\left(\frac{\mathsf{SNR}_{1}}{1+\beta\mathsf{INR}_{2}}\right),\right.\right.\\\
R_{2}\leq\gamma(\mathsf{\beta
SNR_{2}})+\left.\gamma\left(\mathsf{\frac{\overline{\beta}INR_{2}}{1+SNR_{1}+\beta
INR_{2}}}\right)+\delta(\beta,R_{0})\right\\},$ (29)
where
$\delta(\beta,R_{0})=\gamma\left(\frac{\beta(2^{2R_{0}}-1)\mathsf{INR_{2}}}{2^{2R_{0}}(\mathsf{1+\beta
SNR_{2}})+\mathsf{\beta INR_{2}}}\right).$ (30)
In the moderately strong interference regime, defined by
$\mathsf{SNR_{2}\leq INR_{2}}\leq
2^{2R_{0}}(1+\mathsf{SNR_{2}})-1\stackrel{{\scriptstyle\bigtriangleup}}{{=}}\mathsf{INR}_{2}^{\dagger},$
(31)
the following rate region is achievable:
$\mathrm{co}\left\\{\bigcup_{\alpha\in\mathbb{R},0\leq\beta\leq
1,\;R_{a}+R_{b}\leq R_{0}}\mathcal{R}_{\alpha,\beta}(R_{a},R_{b})\right\\},$
(32)
where “co” denotes convex hull and $\mathcal{R}_{\alpha,\beta}(R_{a},R_{b})$
is a pentagon region given by
$\left\\{(R_{1},R_{2})\left|\begin{array}[]{l}R_{1}\leq\gamma(\mathsf{\frac{SNR_{1}}{1+\beta
INR_{2}}})\\\
R_{2}\leq\min\left\\{\gamma(\mathsf{SNR_{2}})+R_{b}+\eta(\alpha,\beta,R_{a}),\right.\\\
\qquad\qquad\quad\gamma(\mathsf{\beta
SNR_{2}})+\gamma\left(\mathsf{\frac{\overline{\beta}INR_{2}}{1+\beta
INR_{2}}}\right)\\\
\qquad\qquad\quad\left.+\zeta(\alpha,\beta,R_{a})\right\\}\\\
R_{1}+R_{2}\leq\gamma(\beta\mathsf{SNR_{2}})+\gamma\left(\mathsf{\frac{SNR_{1}+\overline{\beta}INR_{2}}{1+\beta
INR_{2}}}\right)\\\
\qquad\qquad\quad+\zeta(\alpha,\beta,R_{a})\end{array}\right.\right\\},$ (33)
where
$\zeta(\alpha,\beta,R_{a})=\gamma\left(\frac{\beta\mathsf{INR_{2}}}{(1+\beta\mathsf{SNR_{2}})(1+\frac{\sigma^{2}}{N})}\right),$
(34)
and
$\eta(\alpha,\beta,R_{a})=\\\
\gamma\left(\frac{(1+2\alpha\overline{\beta}+\alpha^{2}\overline{\beta})\mathsf{INR_{2}}+\beta\overline{\beta}\alpha^{2}\mathsf{INR_{2}}\mathsf{SNR_{2}}}{(1+\mathsf{SNR_{2}})(1+\frac{\sigma^{2}}{N})}\right)$
(35)
with
$\frac{\sigma^{2}}{N}=\frac{1+\mathsf{SNR_{2}}+(1+2\alpha\overline{\beta}+\alpha^{2}\overline{\beta})\mathsf{INR_{2}}+\beta\overline{\beta}\alpha^{2}\mathsf{INR_{2}}\mathsf{SNR_{2}}}{(2^{2R_{a}}-1)(1+\mathsf{SNR_{2}})}.$
(36)
In the strong interference regime defined by
$\mathsf{INR}_{2}^{\dagger}\leq\mathsf{INR_{2}}\leq\mathsf{(1+SNR_{1})}\mathsf{INR}_{2}^{\dagger}\stackrel{{\scriptstyle\bigtriangleup}}{{=}}\mathsf{INR}_{2}^{{\ddagger}},$
(37)
the capacity region is given by
$\left\\{(R_{1},R_{2})\left|\begin{array}[]{rll}R_{1}&\leq&\gamma(\mathsf{SNR_{1}})\\\
R_{2}&\leq&\gamma(\mathsf{SNR_{2}})+R_{0}\\\
R_{1}+R_{2}&\leq&\gamma(\mathsf{SNR_{1}+INR_{2}})\end{array}\right.\right\\}.$
(38)
In the very strong interference regime defined by
$\mathsf{INR_{2}}\geq\mathsf{INR}_{2}^{{\ddagger}},$ (39)
the capacity region is given by
$\left\\{(R_{1},R_{2})\left|\begin{array}[]{l}R_{1}\leq\gamma(\mathsf{SNR_{1}})\\\
R_{2}\leq\gamma(\mathsf{SNR_{2}})+R_{0}\end{array}\right.\right\\}.$ (40)
###### Proof:
See Appendix -D. ∎
### III-B Numerical Examples
Figure 8: Achievable region of the Gaussian Z-interference channel in the
strong interference regime with and without a digital relay link of Type II.
In the strong and very strong interference regimes, the entire $X_{2}$ is
common information. The relay expands the capacity region by decoding
$X_{2}^{n}$ at receiver $1$ and forwarding its bin index to receiver $2$. The
boundaries of the strong and very strong regimes depend on the relay link
rate. As a numerical example, Fig. 8 shows how the capacity region of a Type
II channel is expanded by the relay link in the strong and very strong
interference regimes. Here, $\mathsf{SNR_{1}=SNR_{2}=20}$ dB and
$\mathsf{INR_{2}=55}$ dB. Without the digital link, this is a Gaussian
Z-interference channel in the very strong interference regime [8], where
$\mathsf{INR_{2}\geq SNR_{2}(1+SNR_{1})}$ and the capacity region is a
rectangle as depicted by the dash-dotted region in Fig. 8. With a 2-bit
digital link, $R_{2}$ is expanded by exactly 2 bits. The Z-interference
channel remains in the very strong interference regime, where the capacity
region is given by (40) and depicted by the dashed rectangular region in Fig.
8. When $R_{0}=4$ bits, the Z-interference channel now falls into the strong
interference regime. The capacity region as given by (38) now becomes a solid
pentagon region. Further increase in the rate of the digital link can increase
the maximum $R_{2}$ but not the sum rate.
In the weak interference regime where $\mathsf{INR_{2}\leq SNR_{2}}$, Theorem
3 shows that a pure compress-and-forward for the private message should be
used for relaying. Intuitively, this is because when the interference link is
weak the common message rate is limited by the interference link, which cannot
be helped by relaying. Thus, the digital link needs to focus on helping the
decoding of private message at $Y_{2}$ by compress-and-forward. As a numerical
example, Fig. 9 shows the achievable rate region of a Gaussian Z-interference
channel with $\mathsf{SNR_{1}=SNR_{2}=20}$ dB and $\mathsf{INR_{2}=15}$ dB
with and without the relay link. The dashed region denoted by points
$A^{\prime}$ and $B$ represents the rate region achieved without the digital
link. The solid rate region denoted by points $A$ and $B$ is with a 2-bit
digital link. From the rate pair expression (29), the effect of the digital
link is to shift the rate region of the channel without the relay upward by
$\delta(\beta,R_{0})$ bits. Since $\delta(\beta,R_{0})$ is monotonically
decreasing as $\beta$ decreases from 1 to 0, for fixed $R_{1}$, the largest
increase in $R_{2}$ corresponds to $\delta(1,R_{0})$, i.e. the increase from
point $A^{\prime}$ to $A$. Note that $A$ and $A^{\prime}$ are the maximum sum-
rate points of the Type II Gaussian Z-interference channel with and without
the relay respectively. They correspond to all-private message transmission,
which is in contrast to the Type I case where the maximum sum rate is achieved
with some $\beta^{*}\neq 1$. Finally, we note that the relay does not affect
point $B$, which corresponds to $\beta=0$, because $\delta(0,R_{0})=0$.
\begin{overpic}[scale={0.6}]{./figures/Z_weak_qtz_typeII.eps}
\centering\put(35.0,63.0){\small$A$} \put(33.0,55.0){\small$A^{\prime}$}
\put(87.0,11.0){\small$B$} \@add@centering\end{overpic} Figure 9: Capacity
region of the Gaussian Z-interference channel in the weak interference regime
with and without a digital relay link of Type II.
### III-C Sum-Capacity Upper Bound
By Theorem 3, an achievable sum rate of the Type II Gaussian Z-interference
channel with a relay link of rate $R_{0}$ in the weak interference regime is
$R_{sum}=\gamma\left(\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{2}}\right)+\gamma(\mathsf{SNR_{2}})+\delta(1,R_{0}),$
(41)
which is obtained by setting $\beta=1$ in (29). Comparing with the sum
capacity of the Gaussian Z-interference channel without the relay in the weak
interference regime (15), the sum-rate increase using the relay scheme of
Theorem 3 is upper bounded by
$\displaystyle\delta(1,R_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(\frac{\mathsf{1+SNR_{2}+INR_{2}}}{\mathsf{1+SNR_{2}}+2^{-2R_{0}}\mathsf{INR_{2}}}\right)$
(42) $\displaystyle\leq$
$\displaystyle\gamma\left(\frac{\mathsf{INR_{2}}}{\mathsf{1+SNR_{2}}}\right)$
$\displaystyle\leq$ $\displaystyle\frac{1}{2},$
where $\mathsf{INR_{2}\leq SNR_{2}}$ is used in the last step. As illustrated
in the example in Fig. 9, the rate increase from point $A^{\prime}$ to point
$A$ is about 0.2 bits, which is less than 1/2 bits and is a fraction of the
2-bit relay link rate. This is in contrast to the Type I channel, where each
relay bit can increase the sum rate by up to one bit. The following theorem
provides an asymptotic sum-capacity result for the Type II channel and a proof
of the 1/2-bit upper bound when $\mathsf{INR_{2}}$ is not very strong.
###### Theorem 4
For the Type II Gaussian Z-interference channel with a digital relay link of
rate $R_{0}$ from the interfered receiver to the interference-free receiver as
shown in Fig. 1(b), when $R_{0}\rightarrow\infty$, the asymptotic sum capacity
is
$C_{sum}(\infty)=\gamma(\mathsf{SNR_{1}+INR_{2}})+\gamma\left(\mathsf{\frac{SNR_{2}}{1+INR_{2}}}\right).$
(43)
Further, when $\mathsf{INR_{2}\leq INR_{2}^{\S}}$, where
$\mathsf{INR_{2}^{\S}}$ is defined by
$\mathsf{INR_{2}^{\S}=SNR_{2}(1+SINR_{1})}$, we have
$C_{sum}(\infty)-C_{sum}(0)\leq\frac{1}{2}.$ (44)
###### Proof:
When $R_{0}=\infty$, receiver $2$ has complete knowledge of $Y_{1}^{n}$.
Starting from Fano’s inequality:
$\displaystyle n(R_{1}+R_{2})$ (45) $\displaystyle\leq$ $\displaystyle
I(X_{1}^{n};Y_{1}^{n})+I(X_{2}^{n};Y_{1}^{n},Y_{2}^{n})+n\epsilon_{n}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}$ $\displaystyle
I(X_{1}^{n};Y_{1}^{n})+I(X_{2}^{n};Y_{1}^{n},Y_{2}^{n}|X_{1}^{n})+n\epsilon_{n}$
$\displaystyle=$ $\displaystyle
I(X_{1}^{n},X_{2}^{n};Y_{1}^{n})+I(X_{2}^{n};Y_{2}^{n}|Y_{1}^{n},X_{1}^{n})+n\epsilon_{n},$
where (a) follows from the fact that $X_{1}^{n}$ is independent of
$X_{2}^{n}$. The first term in (45) is bounded by the sum capacity of the
multiple-access channel $(X_{1}^{n},X_{2}^{n},Y_{1}^{n})$:
$I(X_{1}^{n},X_{2}^{n};Y_{1}^{n})\leq n\gamma(\mathsf{SNR_{1}+INR_{2}})$ (46)
The second term in (45) is bounded by
$\displaystyle I(X_{2}^{n};Y_{2}^{n}|Y_{1}^{n},X_{1}^{n})$ (47)
$\displaystyle=$ $\displaystyle
h(Y_{2}^{n}|Y_{1}^{n},X_{1}^{n})-h(Y_{2}^{n}|Y_{1}^{n},X_{1}^{n},X_{2}^{n})$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}$
$\displaystyle\sum_{i=1}^{n}\left\\{h(Y_{2,i}|Y_{1,i},X_{1,i})-h(Z_{2,i})\right\\}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\left\\{h(h_{22}X_{2,i}+Z_{2,i}|h_{21}X_{2,i}+Z_{1,i})-h(Z_{2,i})\right\\}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}$ $\displaystyle
n\gamma\left(\mathsf{\frac{SNR_{2}}{1+INR_{2}}}\right),$
where (a) follows from the chain rule and the fact that conditioning does not
increase entropy, and (b) follows from the fact that Gaussian distribution
maximizes the conditional entropy under a covariance constraint. Combining
(46) and (47) gives the sum rate upper bound:
$C_{sum}(\infty)\leq\gamma(\mathsf{SNR_{1}+INR_{2}})+\gamma\left(\mathsf{\frac{SNR_{2}}{1+INR_{2}}}\right).$
(48)
It can be easily verified that the above sum-rate upper bound is also
asymptotically achievable. By Theorem 3, with $R_{0}=\infty$, there are only
two interference regimes: weak interference regime and moderately strong
interference regime. In the weak interference regime, a pure compress-and-
forward scheme, i.e., setting $\beta=1$ in (29) achieves (48). In the
moderately strong interference regime, setting $\beta=1$ and $R_{b}=0$ in (33)
achieves
$\gamma(\mathsf{SNR_{2}})+\gamma\left(\mathsf{\frac{SNR_{1}}{1+INR_{2}}}\right)+\gamma\left(\mathsf{\frac{INR_{2}}{1+SNR_{2}}}\right)$
(49)
which is equivalent to (48). This proves the asymptotic sum-capacity result.
Now, without the relay link, the sum capacity for the Gaussian Z-interference
channel is ([7, 8, 1, 9, 5]):
$\displaystyle C_{sum}(0)=$
$\displaystyle\left\\{\begin{array}[]{ll}\gamma(\mathsf{SNR_{2}})+\displaystyle\gamma\left(\mathsf{\frac{SNR_{1}}{1+INR_{2}}}\right)&\mathrm{if\
\ }\mathsf{INR_{2}\leq SNR_{2}}\\\
\gamma(\mathsf{SNR_{1}+INR_{2}})&\mathrm{if\ \ }\mathsf{SNR_{2}\leq
INR_{2}\leq INR_{2}^{\S}}\\\
\gamma(\mathsf{SNR_{1}})+\gamma(\mathsf{SNR_{2}})&\mathrm{if\ \
}\mathsf{INR_{2}\geq INR_{2}^{\S}}\end{array}\right.$
Comparing $C_{sum}(0)$ with the asymptotic sum capacity in the limit of large
relay rate (43), we have
$C_{sum}(\infty)-C(0)=\gamma\left(\mathsf{\frac{INR_{2}}{1+SNR_{2}}}\right)\leq\frac{1}{2}$
(51)
when $\mathsf{INR_{2}\leq SNR_{2}}$ and
$C_{sum}(\infty)-C(0)=\gamma\left(\mathsf{\frac{SNR_{2}}{1+INR_{2}}}\right)\leq\frac{1}{2}$
(52)
when $\mathsf{SNR_{2}\leq INR_{2}\leq INR_{2}^{\S}}$. Therefore, the sum-
capacity gain is upper bounded by half a bit when $\mathsf{INR_{2}\leq
INR_{2}^{\S}}$. ∎
Note that when $\mathsf{INR_{2}\geq INR_{2}^{\S}}$, the sum-capacity gain can
be larger than half a bit. In fact, in the regime where $\mathsf{INR_{2}\gg
SNR_{1},INR_{2}\gg SNR_{2}}$ and $\mathsf{SNR_{1},SNR_{2}}\gg 1$, we have
$C_{sum}(\infty)-C_{sum}(0)\approx\frac{1}{2}\log\left(\mathsf{\frac{INR_{2}}{SNR_{1}SNR_{2}}}\right),$
(53)
which can be unbounded.
The asymptotic sum capacity (43) is essentially the sum capacity of a degraded
Gaussian interference channel where the inputs are $X_{1}$ and $X_{2}$, and
outputs are $Y_{1}$ and $(Y_{1},Y_{2})$ of a Gaussian Z-interference channel.
The capacity region for the general degraded interference channel is still
open.
## IV Summary
This paper studies a Gaussian Z-interference channel with unidirectional relay
link at the receiver. When the relay link goes from the interference-free
receiver to the interfered receiver, a suitable relay strategy is to let the
interference-free receiver decode-and-forward a part of the interference for
subtraction. Interference decode-and-forward is capacity achieving in the
strong interference regime. In the weak interference regime, the asymptotic
sum capacity can be achieved with either a decode-and-forward or a compress-
and-forward strategy in the high SNR/INR limit.
When the relay link goes from the interfered receiver to the interference-free
receiver, a suitable relay strategy is a combination of decode- and compress-
and-forward of the intended message. In the strong interference regime,
decode-and-forward alone is capacity achieving. In the weak interference
regime, the combination scheme reduces to pure compress-and-forward. In the
moderately strong interference regime, a combination of both need to be used.
The direction of the relay link is crucial. In the weak interference regime, a
relay link from the interference-free receiver to the interfered receiver can
significantly increase the achievable sum rate by up to one bit for every
relay bit, while in the reversed direction, the sum rate increase is upper
bounded by half a bit regardless of the relay link rate. In contrast, in the
strong interference regime, the sum-capacity gain due to a relay from the
interference-free receiver to the interfered receiver eventually saturates,
while a relay link in the reverse direction provides unbounded sum-capacity
gain.
### -A Convexity of Achievable Rate Region (9)
This appendix shows that the region defined by
$R_{1}\leq\mathsf{\gamma(SNR_{1})}$,
$R_{2}\leq\mathsf{\gamma(SNR_{2})}+R_{0}$, and the curve
$\left\\{\begin{array}[]{lll}R_{1}&=&\gamma\left(\displaystyle\frac{\mathsf{SNR}_{1}}{1+\beta\mathsf{INR}_{2}}\right)\\\
R_{2}&=&\gamma(\beta\mathsf{SNR}_{2})+\gamma\left(\displaystyle\frac{\overline{\beta}\mathsf{INR}_{2}}{1+\mathsf{SNR}_{1}+\beta\mathsf{INR}_{2}}\right)+R_{0}\end{array}\right.$
(54)
where $0\leq\beta\leq 1$, is convex when $\mathsf{INR_{2}\leq SNR_{2}}$.
Note that, when $\beta=1$ and $\beta=0$, the curve defined by (54) meets
$R_{2}=\mathsf{\gamma(SNR_{2})}+R_{0}$ and $R_{1}=\mathsf{\gamma(SNR_{1})}$ at
points $A$ and $B$, respectively, as shown in Fig. 10. Therefore, to prove the
convexity of the region, we only need to prove that the curve (54)
parameterized by $\beta$ is concave.
Figure 10: The region defined by lines $R_{1}=\mathsf{\gamma(SNR_{1})}$,
$R_{2}=\mathsf{\gamma(SNR_{2})}+R_{0}$ and the curve (54).
First, we express $\beta$ in terms of $R_{1}$:
$\beta=\frac{1}{\mathsf{INR_{2}}}\left(\frac{\mathsf{SNR_{1}}}{2^{2R_{1}}-1}-1\right).$
(55)
Substituting this expression for $\beta$ into the expression for $R_{2}$ in
(54), we obtain $R_{2}$ as a function of $R_{1}$:
$R_{2}=\frac{1}{2}\log\left(-\nu 2^{2R_{1}}+\lambda\right)+\mu$ (56)
where $\nu=\mathsf{\frac{SNR_{2}}{INR_{2}}}-1$,
$\lambda=\mathsf{\frac{SNR_{2}}{INR_{2}}(1+SNR_{1})}-1$ and
$\mu=\gamma\left(\mathsf{\frac{1+INR_{2}}{SNR_{1}}}\right)+R_{0}$. Note that
when $\mathsf{INR_{2}\leq SNR_{2}}$, $\nu\geq 0$ and $\lambda>0$.
Observe that $R_{1}$ is a monotonic decreasing function of $\beta$. So, in the
range $0\leq\beta\leq 1$, we have
$\gamma\left(\mathsf{\frac{SNR_{1}}{1+INR_{2}}}\right)\leq
R_{1}\leq\gamma(\mathsf{SNR_{1}}).$ (57)
In this range of $R_{1}$, it is easy to verify that $-\nu
2^{2R_{1}}+\lambda>0$.
Now, taking the first and second order derivatives of $R_{2}$ with respect to
$R_{1}$ in (56), we have
$\displaystyle R_{2}^{\prime}=\frac{-\nu 2^{2R_{1}}}{-\nu
2^{2R_{1}}+\lambda},\;\;\;\;R_{2}^{\prime\prime}=\frac{-2\lambda\nu
2^{2R_{1}}}{(-\nu 2^{2R_{1}}+\lambda)^{2}}\ln 2.$ (58)
Since $\nu\geq 0$, $\lambda>0$, and $-\nu 2^{2R_{1}}+\lambda>0$, we have
$R_{2}^{\prime}\leq 0$ and $R_{2}^{\prime\prime}\leq 0$. As a result, the
curve (54) parameterized by $\beta$ is concave.
### -B Converse Proof for the Strong and Very Strong Interference Regimes in
Theorem 1
In this appendix, we prove a converse in the strong and very strong
interference regimes for the Type I channel. The converse is based on a
technique used in [1] and [8] for proving the converse for the strong
interference channel without the relay link. The idea is to show that when
$\mathsf{INR_{2}\geq\min\\{SNR_{2},INR_{2}^{*}\\}}$, if a rate pair
$(R_{1},R_{2})$ is achievable for the Gaussian Z-interference channel with a
relay link, i.e., $X_{1}^{n}$ can be reliably decoded at receiver $1$ at rate
$R_{1}$, and $X_{2}^{n}$ can be reliably decoded at receiver $2$ at rate
$R_{2}$, then $X_{2}^{n}$ must also be decodable at the receiver $1$.
First, the reliable decoding of $X_{2}^{n}$ at receiver $2$ requires
$R_{2}\leq\mathsf{\gamma(SNR_{2})}.$ (59)
To show that $X_{2}^{n}$ is also decodable at receiver $1$ when
$\mathsf{INR_{2}\geq\min\\{SNR_{2},INR_{2}^{*}\\}}$, consider the two cases
$\mathsf{SNR_{2}\leq INR_{2}^{*}}$ and $\mathsf{SNR_{2}\geq INR_{2}^{*}}$
separately.
First, when $\mathsf{SNR_{2}\leq INR_{2}^{*}}$, we have $\mathsf{INR_{2}\geq
SNR_{2}}$, or $h_{21}\geq h_{22}$. In this case, after $X_{1}^{n}$ is decoded
at receiver $1$ (possibly with the help of the relay link), receiver $1$ may
subtract $X_{1}^{n}$ from $Y_{1}^{n}$ then scale the resulting signal to
obtain
$Y_{1}^{{}^{\prime}n}=\frac{h_{22}}{h_{21}}(Y_{1}^{n}-h_{11}X_{1}^{n})=h_{22}X_{2}^{n}+\frac{h_{22}}{h_{21}}Z_{1}^{n}.$
(60)
When $h_{21}\geq h_{22}$, the Gaussian noise $\frac{h_{22}}{h_{21}}Z_{1}^{n}$
in this effective channel has a smaller variance than the noise in
$Y_{2}^{n}=h_{22}X_{2}^{n}+Z_{2}^{n}$. Since $X_{2}^{n}$ is reliably decodable
at receiver $2$, $X_{2}^{n}$ must also be reliably decodable at receiver $1$.
When $\mathsf{SNR_{2}\geq INR_{2}^{*}}$, we have $\mathsf{INR_{2}\geq
INR_{2}^{*}}$. In this case, since $X_{2}^{n}$ is reliably decoded at $Y_{2}$,
with the perfect knowledge of $X_{2}^{n}$ at receiver $2$,
$(X_{2}^{n},Y_{1}^{n},Y_{2}^{n})$ forms a deterministic relay channel [11]
with $X_{2}^{n}$ as the input, $Y_{1}^{n}$ as the output and $Y_{2}^{n}$ as
the deterministic relay. As a result, the following rate for $X_{2}^{n}$ can
be supported:
$R_{2}=\gamma\left(\mathsf{\frac{INR_{2}}{1+SNR_{1}}}\right)+R_{0}$ (61)
Since $\mathsf{INR_{2}\geq INR_{2}^{*}}$, it is easy to verify that the above
rate is always greater than the rate supported at the receiver $2$, i.e.,
$\gamma\left(\mathsf{\frac{INR_{2}}{1+SNR_{1}}}\right)+R_{0}\geq\gamma(\mathsf{SNR_{2}}),$
(62)
which implies that whenever $X_{2}^{n}$ is reliably decodable at $Y_{2}$, it
is also reliably decodable at $Y_{1}$ with the help of the relay.
Now, because both $X_{1}^{n}$ and $X_{2}^{n}$ are always decodable at receiver
$1$ in the strong interference regime, the achievable rate region of the
Gaussian Z-interference channel with a digital relay link is included in the
capacity region of the same channel in which both $X_{1}^{n}$ and $X_{2}^{n}$
are required at $Y_{1}^{n}$, and $X_{2}^{n}$ is required at $Y_{2}^{n}$.
Further, the capacity region of such a channel can only be enlarged if
$X_{2}^{n}$ is provided to $Y_{2}^{n}$ by a genie. In such a case, the channel
reduces to a Gaussian multiple-access channel with $(X_{1}^{n},X_{2}^{n})$ as
inputs, $Y_{1}^{n}$ as the output, and with the same relay link from receiver
$2$ to receiver $1$, where the relay knows $X_{2}^{n}$ perfectly. The capacity
region of such a channel is
$\left\\{(R_{1},R_{2})\left|\begin{array}[]{rll}R_{1}&\leq&\gamma(\mathsf{SNR_{1}})\\\
R_{2}&\leq&\gamma(\mathsf{INR_{2}})+R_{0}\\\
R_{1}+R_{2}&\leq&\gamma(\mathsf{SNR_{1}+INR_{2}})+R_{0}\end{array}\right.\right\\}$
(63)
Combining (63) and (59), then applying (11) gives us (4). This proves that
when $\mathsf{INR_{2}\geq\min\\{SNR_{2},INR_{2}^{*}\\}}$, the achievable rate
region of the Gaussian Z-interference channel with a relay link must be
included in (4), which, in the very strong interference regime, reduces to
(5).
### -C Evaluation of Wyner-Ziv Rate (27)
In this appendix, we show that with independent Gaussian inputs
$X_{1}\sim\mathcal{N}(0,P_{1})$ and $X_{2}\sim\mathcal{N}(0,P_{2})$, and the
Gaussian quantization scheme (26), the achievable rate described by (23), (24)
and (25) is given by (27). The technique is similar to that in [29].
With a Gaussian input $X_{2}\sim\mathcal{N}(0,P_{2})$, $R_{2}$ is given by
$\displaystyle R_{2}$ $\displaystyle=$ $\displaystyle
I(X_{2};Y_{2})=\gamma(\mathsf{SNR_{2}}).$ (64)
With the knowledge of $X_{2}$ at $Y_{2}$, $X_{1}$, $Y_{1}$ together with
$Y_{2}$ become a deterministic relay channel with a digital link. To fully
utilize the digital link, we set $\hat{Y}_{2}$ to be such that
$I(Y_{2};\hat{Y}_{2}|Y_{1})=R_{0}$. Note that $\hat{Y}_{2}=Y_{2}+e$, where
$Y_{2}$ and $e$ are independent and $e\sim\mathcal{N}(0,\sigma^{2})$. To find
$\sigma^{2}$, note that
$\displaystyle
R_{0}=h(\hat{Y}_{2}|Y_{1})-h(\hat{Y}_{2}|Y_{1},Y_{2})=\gamma\left(\frac{\sigma_{Y_{2}|Y_{1}}^{2}}{\sigma^{2}}\right)$
(65)
where $\sigma_{Y_{2}|Y_{1}}^{2}$, the conditional variance of $Y_{2}$ given
$Y_{1}$, can be calculated in a standard way. Thus, from (65), we have
$\displaystyle\sigma^{2}=\frac{N}{2^{2R_{0}}-1}\left(1+\mathsf{\frac{SNR_{2}(1+SNR_{1})}{1+SNR_{1}+INR_{2}}}\right).$
(66)
Now, we are ready to calculate $R_{1}$. First,
$\displaystyle h(\hat{Y}_{2}|Y_{1},X_{1})$ (67) $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(2\pi
e\left(\sigma^{2}+N\left(1+\mathsf{\frac{SNR_{2}}{1+INR_{2}}}\right)\right)\right)$
where $\sigma^{2}$ is given by (66). Now, the rate of user $1$ is given by
$\displaystyle R_{1}$ $\displaystyle=$ $\displaystyle
I(X_{1};Y_{1},\hat{Y}_{2})$ (68) $\displaystyle=$ $\displaystyle
I(X_{1};Y_{1})+h(\hat{Y}_{2}|Y_{1})-h(\hat{Y}_{2}|Y_{1},X_{1}).$
Clearly, with independent Gaussian inputs $X_{1}\sim\mathcal{N}(0,P_{1})$ and
$X_{2}\sim\mathcal{N}(0,P_{2})$,
$I(X_{1};Y_{1})=\gamma\left(\mathsf{\frac{SNR_{1}}{1+INR_{2}}}\right).$ (69)
Substituting (69), (67) and $h(\hat{Y}_{2}|Y_{1})$ from (65) into (68), after
some calculations, we obtain $R_{1}$ in (27).
### -D Proof of Theorem 3
We first prove the achievability of the rate region given in (32). We then
show that (32) reduces to (29) in the weak interference regime, and reduces to
(38) and (40) in the strong and very strong interference regimes,
respectively.
A two-step decoding procedure is used to prove the achievability. Consider
first the decoding of $(X_{1}^{n},W_{2}^{n})$ at $Y_{1}$. The achievable set
of $(S_{1},T_{2})$ is the capacity region of a multiple-access channel,
denoted by $\mathcal{C}_{1}$, which is just (7) with $R_{0}$ set to zero.
Next, consider the decoding of $(W_{2}^{n},U_{2}^{n})$ at receiver $2$ with
the help of a digital relay link of rate $R_{0}$. This is a multiple-access
channel with a rate-limited relay, where the relay has complete knowledge of
$W_{2}^{n}$ and a noisy observation $h_{21}U_{2}^{n}+Z_{1}^{n}$, obtained by
subtracting $X_{1}^{n}$ and $W_{2}^{n}$ from the received signal at receiver
$1$. Each of these two pieces of information is useful for decoding
$(W_{2}^{n},U_{2}^{n})$ at receiver $2$.
Now, consider a relay scheme which splits the digital link in two parts:
$R_{a}$ bits for describing $U_{2}^{n}$, and $R_{b}$ for describing $W_{2}$,
where $R_{a}+R_{b}=R_{0}$. However, since only a noisy version of $U_{2}^{n}$
is available at the relay ($Y_{1}$), a compress-and-forward strategy using
Wyner-Ziv coding ([28, 10]) may be used for describing $U_{2}^{n}$. One way to
do compress-and-forward is to quantize $h_{21}U_{2}^{n}+Z_{1}^{n}$ with
$Y_{2}^{n}$ acting as the decoder side information. However, the presence of
$W_{2}^{n}$ offers other possibilities. First, receiver $2$ may choose to
decode $W_{2}^{n}$ before decoding $U_{2}^{n}$, in which case $W_{2}^{n}$
becomes additional decoder side information for Wyner-Ziv coding. Second,
instead of quantizing $h_{21}U_{2}^{n}+Z_{1}^{n}$ with $W_{2}^{n}$ completely
subtracted from the relay’s observation, the relay may choose to subtract
$W_{2}^{n}$ partially—doing so can benefit the Wyner-Ziv rate. This second
approach is is adopted in the rest of the proof. Interestingly, the two
approaches turn out to give identical achievable rates.
Specifically, let the relay form the following fictitious signal
$\bar{Y}_{1}^{n}=h_{21}(U_{2}^{n}+W_{2}^{n})+\alpha h_{21}W_{2}^{n}+Z_{1}^{n}$
(70)
for some $\alpha\in\mathbb{R}$. The proposed relay scheme, which combines the
decode-and-forward technique and the compress-and-forward technique, is
illustrated in Fig. 11, where $W_{2}^{n}$ and $U_{2}^{n}$ are the inputs of
the multiple-access channel, $(Y_{2}^{n},\hat{Y}_{1}^{n})$ is the output, and
$\hat{Y}_{1}^{n}$ is a quantized version of $\bar{Y}_{1}^{n}$. With complete
knowledge of $W_{2}^{n}$ at the relay, the capacity of this multiple-access
relay channel, denoted by $\mathcal{C}_{2}$, is given by the set of rates
$(S_{2},T_{2})$ where
$\left\\{\begin{array}[]{rll}S_{2}&\leq&I(U_{2};Y_{2},\hat{Y}_{1}|W_{2})\\\
T_{2}&\leq&I(W_{2};Y_{2},\hat{Y}_{1}|U_{2})+R_{b}\\\
S_{2}+T_{2}&\leq&I(U_{2},W_{2};Y_{2},\hat{Y}_{1})+R_{b}\end{array}\right.$
(71)
Similar to Theorem 1, we adopt $\bar{Y}_{1}$: $\hat{Y}_{1}=\bar{Y}_{1}+e$,
where $e$ is a Gaussian random variable independent of $\bar{Y}_{1}$, with a
distribution $\mathcal{N}(0,\sigma^{2})$. With the encoder side information
$W_{2}$ at the input of the relay link and the decoder side information
$Y_{2}$ at the output of the relay link, the Wyner-Ziv coding rate for
quantizing $\bar{Y}_{1}$ into $\hat{Y}_{1}$ is given by ([30] [10])
$I(\hat{Y}_{1};W_{2},\bar{Y}_{1})-I(\hat{Y}_{1};Y_{2})\leq R_{a}$. But
$\displaystyle I(\hat{Y}_{1};W_{2},\bar{Y}_{1})-I(\hat{Y}_{1};Y_{2})$ (72)
$\displaystyle=$ $\displaystyle
I(\hat{Y}_{1};\bar{Y}_{1})+I(\hat{Y}_{1};W_{2}|\bar{Y}_{1})-I(\hat{Y}_{1};Y_{2})$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}$ $\displaystyle
I(\hat{Y}_{1};\bar{Y}_{1})-I(\hat{Y}_{1};Y_{2})$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}$ $\displaystyle
I(\hat{Y}_{1};\bar{Y}_{1}|Y_{2})$
where both $(a)$ and $(b)$ come from the fact that $\hat{Y}_{1}=\bar{Y}_{1}+e$
and $e$ is independent of $W_{2}$ or $Y_{2}$. Thus, we have
$I(\hat{Y}_{1};\bar{Y}_{1}|Y_{2})\leq R_{a}$. To fully utilize the channel, we
set $\hat{Y}_{1}$ to be such that $I(\hat{Y}_{1};\bar{Y}_{1}|Y_{2})$ is equal
to $R_{a}$. To find $\sigma^{2}$, note that
$R_{a}=h(\hat{Y}_{1}|Y_{2})-h(\hat{Y}_{1}|\bar{Y}_{1},Y_{2})=\frac{1}{2}\log\left(\frac{\sigma_{\hat{Y}_{1}|Y_{2}}^{2}}{\sigma^{2}}\right)$
(73)
where $\sigma_{\hat{Y}_{1}|Y_{2}}^{2}$ is the conditional variance of
$\hat{Y}_{1}$ given $Y_{2}$. Calculating $\sigma_{\hat{Y}_{1}|Y_{2}}^{2}$ and
substituting it into (73), we obtain (36).
Now, define
$I(U_{2};\hat{Y}_{1}|Y_{2},W_{2})\stackrel{{\scriptstyle\bigtriangleup}}{{=}}\zeta(\alpha,\beta,R_{a})$,
$I(W_{2};\hat{Y}_{1}|Y_{2},U_{2})\stackrel{{\scriptstyle\bigtriangleup}}{{=}}\xi(\alpha,\beta,R_{a})$,
and
$I(W_{2},U_{2};\hat{Y}_{1}|Y_{2})\stackrel{{\scriptstyle\bigtriangleup}}{{=}}\eta(\alpha,\beta,R_{a})$.
Applying Gaussian distributions
$W_{2}\sim\mathcal{N}(0,\overline{\beta}P_{2})$ and
$U_{2}\sim\mathcal{N}(0,\beta P_{2})$, the multiple-access relay channel
capacity region $\mathcal{C}_{2}$ in (71) becomes
$\left\\{\begin{array}[]{rll}S_{2}&\leq&\displaystyle\gamma(\beta\mathsf{SNR_{2}})+\zeta(\alpha,\beta,R_{a})\\\
T_{2}&\leq&\gamma(\overline{\beta}\mathsf{SNR_{2}})+\xi(\alpha,\beta,R_{a})+R_{b}\\\
S_{2}+T_{2}&\leq&\displaystyle\gamma(\mathsf{SNR_{2}})+\eta(\alpha,\beta,R_{a})+R_{b}.\end{array}\right.$
(74)
The computations of $\zeta(\alpha,\beta,R_{a})$, $\xi(\alpha,\beta,R_{a})$ and
$\eta(\alpha,\beta,R_{a})$ are as follows. First,
$\displaystyle\eta(\alpha,\beta,R_{a})=\frac{1}{2}\log\left(\frac{\sigma_{\hat{Y}_{1}|Y_{2}}^{2}}{N+\sigma^{2}}\right).$
(75)
Calculating $\sigma_{\hat{Y}_{1}|Y_{2}}^{2}$, we obtain (35). Likewise,
$\displaystyle\zeta(\alpha,\beta,R_{a})=\frac{1}{2}\log\left(\frac{\sigma_{\hat{Y}_{1}|Y_{2},W_{2}}^{2}}{N+\sigma^{2}}\right).$
(76)
A similar computation leads to (34). The expression of
$\xi(\alpha,\beta,R_{a})$ does not affect our final result.
Figure 11: Gaussian multiple-access channel with two digital relay links.
Finally, an achievable rate region for the Gaussian Z-relay-interference
channel is a set of $(R_{1},R_{2})$ with $R_{1}=S_{1}$ and
$R_{2}=S_{2}+T_{2}$, for which $(S_{1},T_{2})\in\mathcal{C}_{1}$ and
$(S_{2},T_{2})\in\mathcal{C}_{2}$. Combining the $\mathcal{C}_{1}$ region and
the $\mathcal{C}_{2}$ region (74) using the Fourier-Motzkin elimination
procedure, we obtain a pentagon achievable rate region
$R_{\alpha,\beta}(R_{a},R_{b})$ for each fixed $\alpha$, $0\leq\beta\leq 1$
and $R_{a}+R_{b}=R_{0}$ as shown in (33). With time-sharing, the overall
achievable rate region is given by (32). In the following, we show that (29),
(38) and (40) are all included in the above achievable rate region.
First, consider the weak interference regime, where $\mathsf{INR_{2}\leq
SNR_{2}}$. For any nonnegative $R_{b}$ and when $\mathsf{INR_{2}\leq
SNR_{2}}$, it is easy to verify that
$\gamma(\mathsf{\beta
SNR_{2}})+\gamma\left(\mathsf{\frac{\overline{\beta}INR_{2}}{1+\beta
INR_{2}}}\right)\leq\gamma(\mathsf{SNR_{2}})+R_{b}$ (77)
and $\zeta(\alpha,\beta,R_{a})\leq\eta(\alpha,\beta,R_{a})$. Thus, the second
term of the minimization in the expression of $R_{2}$ in (33) is always less
than the first term. In this case, $R_{a}$ enters the rate region expression
only through $\zeta(\alpha,\beta,R_{a})$. It is easy to verify that
$\zeta(\alpha,\beta,R_{a})$ is a monotonically increasing function of $R_{a}$.
Thus, the maximum achievability region is obtained for $R_{a}=R_{0}$ and
$R_{b}=0$. Therefore a pure quantization scheme is optimal in the weak
interference regime.
Further, $\alpha$ enters the rate region expression only through
$\zeta(\alpha,\beta,R_{0})$. Thus, we can choose $\alpha$ to maximize
$\zeta(\alpha,\beta,R_{0})$, or equivalently, to minimize $\sigma^{2}$ in
(36). Taking the derivative of (36) on $\alpha$ and setting it to zero, the
optimal $\alpha$ is
$\alpha^{*}=-\frac{1}{1+\beta\mathsf{SNR_{2}}}.$ (78)
Substituting $\alpha^{*}$ into (36), we obtain
$\frac{\sigma^{2}}{N}=\frac{1}{2^{2R_{0}}-1}\left(1+\frac{\beta\mathsf{INR_{2}}}{1+\beta\mathsf{SNR_{2}}}\right),$
(79)
which gives a derivation of (30):
$\zeta(\alpha^{*},\beta,R_{0})=\gamma\left(\frac{\beta(2^{2R_{0}}-1)\mathsf{INR_{2}}}{2^{2R_{0}}(\mathsf{1+\beta
SNR_{2}})+\mathsf{\beta
INR_{2}}}\right)\stackrel{{\scriptstyle\bigtriangleup}}{{=}}\delta(\beta,R_{0}).$
(80)
Finally, we take the union of all $\mathcal{R}_{\alpha^{*},\beta}(R_{0},0)$.
Following the same approach of the proof in Theorem 1, we can show that the
union of achievable pentagons, $\bigcup_{0\leq\beta\leq
1}\mathcal{R}_{\alpha^{*},\beta}(R_{0},0)$ is defined by
$R_{1}\leq\gamma(\mathsf{SNR_{1}})$,
$R_{2}\leq\gamma(\mathsf{SNR_{2}})+\delta(\beta,R_{0})$, and lower-right
corner points of the pentagons
$\left\\{\begin{array}[]{l}R_{1}=\gamma\left(\displaystyle\frac{\mathsf{SNR}_{1}}{1+\beta\mathsf{INR}_{2}}\right)\\\
R_{2}=\displaystyle\gamma(\beta\mathsf{SNR}_{2})+\gamma\left(\frac{\overline{\beta}\mathsf{INR}_{2}}{1+\mathsf{SNR_{1}}+\beta\mathsf{INR}_{2}}\right)+\delta(\beta,R_{0}).\end{array}\right.$
(81)
We prove in Appendix -E that this region is convex when $\mathsf{INR_{2}\leq
SNR_{2}}$. Thus, the convex hull is not needed. This establishes the region
(29) for the weak interference regime.
In the moderately strong interference regime, the achievability of (32)
follows directly from the general achievability region. In this regime, the
rate region is achieved by a mixed scheme, which includes both the decode-and-
forward and the compress-and-forward strategies.
Finally, consider the strong interference regime, where $\mathsf{INR_{2}\geq
INR_{2}^{\dagger}}$ and the very strong interference regime, where
$\mathsf{INR_{2}\geq INR_{2}^{{\ddagger}}}$. We show that (38) and (40) are
the capacity regions, respectively.
First, by setting111The value of $\alpha$ does not affect
$\mathcal{R}_{\alpha,\beta}(R_{a},R_{b})$ when $R_{a}=0$. $R_{b}=R_{0}$,
$R_{a}=0$ and $\beta=0$, the achievable rate region
$\mathcal{R}_{\alpha,\beta}(R_{a},R_{b})$ in (33) reduces to
$\left\\{(R_{1},R_{2})\left|\begin{array}[]{rll}R_{1}&\leq&\gamma(\mathsf{SNR_{1}})\\\
R_{2}&\leq&\min\left\\{\gamma(\mathsf{SNR_{2}})+R_{0},\gamma(\mathsf{INR_{2}})\right\\}\\\
R_{1}+R_{2}&\leq&\gamma(\mathsf{SNR_{1}+INR_{2}})\end{array}\right.\right\\}.$
(82)
This rate region reduces to (38) in the strong interference regime, because
$\gamma(\mathsf{SNR_{2}})+R_{0}\leq\gamma(\mathsf{INR_{2}})$ when
$\mathsf{INR_{2}\geq INR_{2}^{\dagger}}$. Thus, (38) is achievable.
Further, in the very strong interference regime, where $\mathsf{INR_{2}\geq
INR_{2}^{{\ddagger}}}$, the constraint on $R_{1}+R_{2}$ in (38) becomes
redundant. Thus, the rate region reduces to (40).
Next, we give a converse proof to show that (38) and (40) are indeed the
capacity regions in the strong and very strong interference regimes,
respectively. Following the same idea as in the converse proof of Theorem 1,
we show that if $(R_{1},R_{2})$ is in the achievable rate region for the Type
II channel, i.e., $X_{1}^{n}$ can be reliably decoded at receiver $1$ at rate
$R_{1}$, and $X_{2}^{n}$ can be reliably decoded at receiver $2$ at rate
$R_{2}$, then $X_{2}^{n}$ must also be decodable at the receiver $1$.
First, observe that by the cut-set upper bound [31], reliable decoding of
$X_{2}^{n}$ at receiver $2$ requires
$R_{2}\leq\gamma(\mathsf{SNR_{2}})+R_{0}.$ (83)
To show that $X_{2}^{n}$ must be decodable at receiver $1$, note that after
the decoding of $X_{1}^{n}$ at receiver $1$, $X_{1}^{n}$ can be subtracted
from the received signal to form
$\tilde{Y}_{1}^{n}=h_{21}X_{2}^{n}+Z_{1}^{n}.$ (84)
The capacity of this channel is $\mathsf{\gamma(INR_{2})}$. On the other hand,
$R_{2}$ is bounded by $\gamma(\mathsf{SNR_{2}})+R_{0}$, which is less than
$\mathsf{\gamma(INR_{2})}$ when $\mathsf{INR_{2}\geq INR_{2}^{\dagger}}$. So,
$X_{2}^{n}$ is always decodable based on $\tilde{Y}_{1}^{n}$.
Now, since both $X_{1}^{n}$ and $X_{2}^{n}$ are decodable at receiver $1$ in
the strong interference regime, the achievable rate region of the Gaussian
Z-relay-interference channel in the strong interference regime must be a
subset of the capacity region of a Gaussian multiple-access channel with
$X_{1}^{n}$, $X_{2}^{n}$ as inputs and $Y_{1}^{n}$ as output, which is
$\left\\{(R_{1},R_{2})\left|\begin{array}[]{rll}R_{1}&\leq&\mathsf{\gamma(SNR_{1})}\\\
R_{2}&\leq&\mathsf{\gamma(INR_{2})}\\\
R_{1}+R_{2}&\leq&\mathsf{\gamma(SNR_{1}+INR_{2})}\end{array}\right.\right\\}.$
(85)
Combining (83), (85), and observing that
$\gamma(\mathsf{SNR_{2}})+R_{0}\leq\gamma(\mathsf{INR_{2}})$ when
$\mathsf{INR_{2}\geq INR_{2}^{\dagger}}$, we proved that the achievable rate
region of the Gaussian Z-relay-interference channel must be bounded by (38)
when $\mathsf{INR_{2}\geq INR_{2}^{\dagger}}$, which reduces to (40) when
$\mathsf{INR_{2}\geq INR_{2}^{{\ddagger}}}$.
### -E Convexity of Achievable Rate Region (81)
This appendix proves that the region defined by
$R_{1}\leq\mathsf{\gamma(SNR_{1})}$,
$R_{2}\leq\mathsf{\gamma(SNR_{2})}+\delta(\beta,R_{0})$, and the curve
$\left\\{\begin{array}[]{lll}R_{1}&\leq&\gamma\left(\displaystyle\frac{\mathsf{SNR}_{1}}{1+\beta\mathsf{INR}_{2}}\right)\\\
R_{2}&\leq&\displaystyle\gamma(\beta\mathsf{SNR}_{2})+\gamma\left(\frac{\overline{\beta}\mathsf{INR}_{2}}{1+\mathsf{SNR}_{1}+\beta\mathsf{INR}_{2}}\right)+\delta(\beta,R_{0})\end{array}\right.$
(86)
where $0\leq\beta\leq 1$, is convex when $\mathsf{INR_{2}\leq SNR_{2}}$.
We follow the same idea used in Appendix -A to prove the convexity of the
above region. By Appendix -A, we can rewrite $R_{2}$ as
$R_{2}=\frac{1}{2}\log\left(-\nu
2^{2R_{1}}+\lambda\right)+\tilde{\mu}+\delta(\beta,R_{0})$ (87)
where $\tilde{\mu}=\mu-R_{0}$ is a constant, and $\nu,\lambda,\mu$ are as
defined in Appendix -A.
It is easy to verify that in the weak interference regime,
$\delta(\beta,R_{0})$ is concave in $\beta$, and $\beta(R_{1})$, as denoted in
(55), is convex in $R_{1}$. Combining this with the fact that
$\delta(\beta,R_{0})$ is a nondecreasing function of $\beta$ shows that
$\delta(\beta,R_{0})$ is a concave function of $R_{1}$. Adding
$\delta(\beta,R_{0})$ with another concave (proved in Appendix -A) term
$\frac{1}{2}\log\left(-\nu 2^{2R_{1}}+\lambda\right)+\tilde{\mu}$ gives us the
desired result that $R_{2}$ is a concave function of $R_{1}$.
Therefore, the region defined by $R_{1}\leq\mathsf{\gamma(SNR_{1})}$,
$R_{2}\leq\mathsf{\gamma(SNR_{2})}+\delta(\beta,R_{0})$ and (86) is convex.
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Lei Zhou (S’05) received the B.E. degree in electronics engineering from
Tsinghua University, Beijing, China, in 2003 and M.A.Sc. degree in electrical
and computer engineering from the University of Toronto, ON, Canada, in 2008.
During 2008-2009, he was with Nortel Networks, Ottawa, ON, Canada. He is
currently pursuing the Ph.D. degree with the Department of Electrical and
Computer Engineering, University of Toronto, Canada. His research interests
include multiterminal information theory, wireless communications, and signal
processing. He is a recipient of the Shahid U.H. Qureshi Memorial Scholarship
in 2011, and the Alexander Graham Bell Canada Graduate Scholarship for
2011-2013.
---
Wei Yu (S’97-M’02-SM’08) received the B.A.Sc. degree in Computer Engineering
and Mathematics from the University of Waterloo, Waterloo, Ontario, Canada in
1997 and M.S. and Ph.D. degrees in Electrical Engineering from Stanford
University, Stanford, CA, in 1998 and 2002, respectively. Since 2002, he has
been with the Electrical and Computer Engineering Department at the University
of Toronto, Toronto, Ontario, Canada, where he is now an Associate Professor
and holds a Canada Research Chair in Information Theory and Digital
Communications. His main research interests include multiuser information
theory, optimization, wireless communications and broadband access networks.
Prof. Wei Yu currently serves as an Associate Editor for IEEE Transactions on
Information Theory and an Editor for IEEE Transactions on Communications. He
was an Editor for IEEE Transactions on Wireless Communications from 2004 to
2007, and a Guest Editor for a number of special issues for the IEEE Journal
on Selected Areas in Communications and the EURASIP Journal on Applied Signal
Processing. He is member of the Signal Processing for Communications and
Networking Technical Committee of the IEEE Signal Processing Society. He
received the IEEE Signal Processing Society Best Paper Award in 2008, the
McCharles Prize for Early Career Research Distinction in 2008, the Early
Career Teaching Award from the Faculty of Applied Science and Engineering,
University of Toronto in 2007, and the Early Researcher Award from Ontario in
2006.
---
|
arxiv-papers
| 2010-06-26T00:49:34 |
2024-09-04T02:49:11.229811
|
{
"license": "Public Domain",
"authors": "Lei Zhou and Wei Yu",
"submitter": "Lei Zhou",
"url": "https://arxiv.org/abs/1006.5087"
}
|
1006.5158
|
# Jacob’s ladders and the nonlocal interaction of the function $Z(t)$ with the
function $\tilde{Z}^{2}(t)$ on the distance $\sim(1-c)\pi(t)$ for a collection
of disconnected sets
Jan Moser Department of Mathematical Analysis and Numerical Mathematics,
Comenius University, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA
jan.mozer@fmph.uniba.sk
###### Abstract.
It is shown in this paper that there is a fine correlation of the third order
between the values of the functions $Z[\varphi_{1}(t)]$ and $\tilde{Z}^{2}(t)$
which corresponds to two collections of disconnected sets. The corresponding
new asymptotic formula cannot be obtained within known theories of
Balasubramanian, Heath-Brown and Ivic.
###### Key words and phrases:
Riemann zeta-function
## 1\. Result
### 1.1.
Let (see [2], (3), (4))
(1.1) $\begin{split}G_{1}(x)&=G_{1}(x;T,H)=\bigcup_{T\leq t_{2\nu}\leq
T+H}\left\\{t:\ t_{2\nu}(-x)<t<t_{2\nu}(x),\ 0<x\leq\frac{\pi}{2}\right\\}\\\
G_{2}(y)&=G_{2}(y;T,H)=\\\ &=\bigcup_{T\leq t_{2\nu+1}\leq T+H}\left\\{t:\
t_{2\nu+1}(-y)<t<t_{2\nu+1}(y),\ 0<y\leq\frac{\pi}{2}\right\\}\end{split}$
(1.2) $H=T^{1/6+2\epsilon},$
and the collection of the sequences $\\{t_{\nu}(\tau)\\},\ \tau\in[-\pi,\pi],\
\nu=1,2,\dots$ be defined by the equation (see [2], (1))
(1.3) $\vartheta[t_{\nu}(\tau)]=\pi\nu+\tau;\ t_{\nu}(0)=t_{\nu}.$
### 1.2.
In this paper we obtain some new properties of the signal
$Z(t)=e^{i\vartheta(t)}\zeta\left(\frac{1}{2}+it\right)$
generated by the Riemann zeta-function. Namely, let
(1.4) $G_{1}(x)=\varphi_{1}[\mathring{G}_{1}(x)],\
G_{2}(y)=\varphi_{1}[\mathring{G}_{2}(y)]$
where $y=\varphi_{1}(T),\ T\geq T_{0}[\varphi_{1}]$ is the Jacob’s ladder. The
following theorem holds true.
###### Theorem.
(1.5)
$\begin{split}\int_{\mathring{G}_{1}(x)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t&=\frac{2}{\pi}H\sin x+\mathcal{O}(T^{1/6+\epsilon}),\\\
\int_{\mathring{G}_{2}(y)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t&=-\frac{2}{\pi}H\sin y+\mathcal{O}(T^{1/6+\epsilon}),\end{split}$
where
(1.6) $t-\varphi_{1}(t)\sim(1-c)\pi(t),\qquad t\to\infty,$
and $c$ is the Euler’s constant and $\pi(t)$ is the prime-counting function.
Let us remind another representation of the signal $Z(t)$ given by the
Riemann-Siegel formula (in its local form)
$Z(t)=2\sum_{n<P_{0}}\frac{1}{\sqrt{n}}\cos\\{\vartheta(t)-t\ln
n\\}+\mathcal{O}(T^{-1/4}),\ t\in[T,T+H],\ P_{0}=\sqrt{\frac{T}{2\pi}},$
i.e. in the form of the resultant oscillation of the system of nonlinear
oscillators
$\frac{2}{\sqrt{n}}\cos\\{\vartheta(t)-t\ln n\\};\
\vartheta(t)=\frac{t}{2}\ln\frac{t}{2\pi}-\frac{t}{2}-\frac{\pi}{8}+\mathcal{O}\left(\frac{1}{t}\right)$
(at the same time see, for example, the expression $Z[t_{\nu}(\tau)]$ by
(1.3)).
### 1.3.
Let (comp. (1.4))
(1.7) $T=\varphi_{1}(\mathring{T}),\ T+H=\varphi_{1}(\widering{T+H}).$
Since (see (1.6), (1.7))
$\mathring{T}-\varphi_{1}(\mathring{T})\sim(1-c)\frac{\mathring{T}}{\ln\mathring{T}}\
\Rightarrow\ \mathring{T}-T\sim(1-c)\frac{\mathring{T}}{\ln\mathring{T}}\
\Rightarrow\ \mathring{T}\sim T$
we have seen (see (1.2))
$\mathring{T}-(T+H)\sim(1-c)\frac{\mathring{T}}{\ln\mathring{T}}-H\sim(1-c)\frac{\mathring{T}}{\ln\mathring{T}},$
i.e. $\mathring{T}>T+H$. Then we have
(1.8) $\begin{split}&[T,T+H]\cap[\mathring{T},\widering{T+H}]=\emptyset;\
T+H<\mathring{T},\\\
&\rho\\{[T,T+H];[\mathring{T},\widering{T+H}]\\}\sim(1-c)\pi(T),\end{split}$
where $\rho$ stands for the distance of the corresponding segments (comp. [5],
(1.3), (1.6)).
###### Remark 1.
Some nonlocal interaction of the functions $Z[\varphi_{1}(t)]$ and
$\tilde{Z}^{2}(t)$ is expressed by the formula (1.5) where (see (1.7))
$t\in\mathring{G}_{1}(x)\cup\mathring{G}_{2}(y)\cap[\mathring{T},\widering{T+H}]\
\Rightarrow\ \varphi_{1}(t)\in G_{1}(x)\cup G_{2}(y)\cap[T,T+H].$
Such an interaction is connected with two collections of disconnected sets
unboundedly receding each from other (see (1.8), $\rho\to\infty$ as
$T\to\infty$) - like mutually receding galaxies (the Hubble law). Compare this
remark with the Remark 3 in [5].
###### Remark 2.
The asymptotic formulae (1.5) cannot be obtained by methods of
Balasubramanian, Heath-Brown and Ivic (comp. [1]).
This paper is a continuation of the series [3]-[15].
## 2\. The first corollaries
First of all, we obtain from (1.5)
###### Corollary 1.
(2.1)
$\int_{\mathring{G}_{1}(x)\cup\mathring{G}_{2}(y)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\left\\{\begin{array}[]{lcr}\frac{2}{\pi}(\sin x-\sin
y)H+\mathcal{O}(T^{1/6+\epsilon})&,&x\not=y\\\
\mathcal{O}(T^{1/6+\epsilon})&,&x=y,\end{array}\right.$
$\displaystyle\int_{\mathring{G}_{1}(x)}Z[\varphi_{2}(t)]\tilde{Z}^{2}(t){\rm
d}t-\int_{\mathring{G}_{2}(y)}Z[\varphi_{2}(t)]\tilde{Z}^{2}(t){\rm d}t=$
$\displaystyle\frac{2}{\pi}(\sin x+\sin y)H+\mathcal{O}(T^{1/6+\epsilon}).$
Next, in the case $x=y=\frac{\pi}{2}$ we obtain
###### Corollary 2.
$\begin{split}&\int_{\mathring{G}_{1}\left(\frac{\pi}{2}\right)\cup\mathring{G}_{2}\left(\frac{\pi}{2}\right)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\mathcal{O}(T^{1/6+\epsilon}),\\\
&\int_{\mathring{G}_{1}\left(\frac{\pi}{2}\right)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t-\int_{\mathring{G}_{2}\left(\frac{\pi}{2}\right)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\\\ &\frac{4}{\pi}H+\mathcal{O}(T^{1/6+\epsilon}),\end{split}$
where
$[\mathring{T},\widering{T+H}]\subset\mathring{G}_{1}\left(\frac{\pi}{2}\right)\cup\mathring{G}_{2}\left(\frac{\pi}{2}\right)$.
## 3\. Law of the asymptotic equality of the areas of the positive and
negative parts of the graph of the function
$Z[\varphi_{1}(t)]\tilde{Z}^{2}(t)$
Let
(3.1) $\begin{split}\mathring{G}_{1}^{+}(x)&=\left\\{t:\
t\in\mathring{G}_{1}(x),\ Z[\varphi_{1}(t)]\tilde{Z}^{2}(t)>0\right\\},\\\
\mathring{G}_{1}^{-}(x)&=\left\\{t:\ t\in\mathring{G}_{1}(x),\
Z[\varphi_{1}(t)]\tilde{Z}^{2}(t)<0\right\\},\\\
\mathring{G}_{2}^{+}(x)&=\left\\{t:\ t\in\mathring{G}_{2}(x),\
Z[\varphi_{1}(t)]\tilde{Z}^{2}(t)>0\right\\},\\\
\mathring{G}_{2}^{-}(x)&=\left\\{t:\ t\in\mathring{G}_{2}(x),\
Z[\varphi_{1}(t)]\tilde{Z}^{2}(t)<0\right\\}.\end{split}$
Then we obtain from (2.1) by (1.1), (3.1) the following
###### Corollary 3.
(3.2)
$\begin{split}&\int_{\mathring{G}_{1}^{+}(x)\cup\mathring{G}_{2}^{+}(x)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t\sim\\\
&-\int_{\mathring{G}_{1}^{-}(x)\cup\mathring{G}_{2}^{-}(x)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t,\quad T\to\infty.\end{split}$
Indeed, from (1.5) by (3.1) we have
$0<(1-\epsilon)\frac{2}{\pi}H\sin
x<\int_{\mathring{G}_{1}(x)}\leq\int_{\mathring{G}_{1}^{+}(x)}\leq\int_{\mathring{G}_{1}^{+}(x)\cup\mathring{G}_{2}^{+}(x)},$
and similarly
$0<(1-\epsilon)\frac{2}{\pi}H\sin
x<-\int_{\mathring{G}_{1}^{-}(x)\cup\mathring{G}_{2}^{-}(x)}.$
Hence, from (2.1), $x=y$, we get
$\int_{\mathring{G}_{1}^{+}(x)}+\int_{\mathring{G}_{1}^{-}(x)}+\int_{\mathring{G}_{2}^{+}(x)}+\int_{\mathring{G}_{2}^{-}(x)}=\mathcal{O}(T^{1/6+\epsilon})=o(H)$
(see (1.2), i.e. (3.2)).
###### Remark 3.
The formula (3.2) represents the law of the asymptotic equality of the areas
(measures) of the figures which correspond to the positive part and negative
part, respectively, of the graph of the function
(3.3) $Z[\varphi_{1}(t)]\tilde{Z}^{2}(t),\
t\in\mathring{G}_{1}(x)\cup\mathring{G}_{2}(x)$
with respect to the disconnected sets
$\mathring{G}_{1}^{+}(x)\cup\mathring{G}_{2}^{+}(x),\
\mathring{G}_{1}^{-}(x)\cup\mathring{G}_{2}^{-}(x)$. This is one of the laws
governing _chaotic_ behavior of the positive and negative values of the
function (3.3).
## 4\. Proof of the Theorem
### 4.1.
Let is remind that
$\tilde{Z}^{2}(t)=\frac{{\rm d}\varphi_{1}(t)}{{\rm d}t},\
\varphi_{1}(t)=\frac{1}{2}\varphi(t)$
where
$\tilde{Z}^{2}(t)=\frac{Z^{2}(t)}{2\Phi^{\prime}_{\varphi}[\varphi(t)]}=\frac{Z^{2}(t)}{\left\\{1+\mathcal{O}\left(\frac{\ln\ln
t}{\ln t}\right)\right\\}\ln t}$
(see [3], (3.9); [5], (1.3); [9], (1.1), (3.1), (3.2)). Thus, the following
lemma holds true (see [8], (2.5); [9], (3.3)).
###### Lemma.
For every integrable function (in the Lebesgue sense) $f(x),\
x\in[\varphi_{1}(T),\varphi_{1}(T+U)]$ the following is true
(4.1) $\int_{T}^{T+U}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{\varphi_{1}(T)}^{\varphi_{1}(T+U)}f(x){\rm d}x,\
U\in\left.\left(0,\frac{T}{\ln T}\right.\right],$
where $t-\varphi_{1}(t)\sim(1-c)\pi(t)$.
###### Remark 4.
The formula (4.1) remains true also in the case when the integral on the
right-hand side of eq. (4.1) is only relatively convergent improper integral
of the second kind (in the Riemann sense).
In the case (comp. (1.7)) $T=\varphi_{1}(\mathring{T}),\
T+U=\varphi_{1}(\widering{T+U})$ we obtain from (4.1)
(4.2)
$\int_{\mathring{T}}^{\widering{T+U}}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{T}^{T+U}f(x){\rm d}x.$
### 4.2.
First of all, we have from (4.2), for example,
(4.3)
$\int_{\mathring{t}_{2\nu}(-x)}^{\mathring{t}_{2\nu}(x)}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{t_{2\nu}(-x)}^{t_{2\nu}(x)}f(t){\rm d}t,$
(see (1.4). Next, in the case
$f(t)=Z[\varphi_{1}(t)]\tilde{Z}^{2}(t),\
t\in\mathring{G}_{1}(x)\cup\mathring{G}_{2}(y)$
we have the following $\tilde{Z}^{2}$-transformation
(4.4)
$\begin{split}&\int_{\mathring{G}_{1}(x)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{G_{1}(x)}Z(t){\rm d}t,\\\
&\int_{\mathring{G}_{2}(y)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{G_{2}(y)}Z(t){\rm d}t,\end{split}$
(see (1.1), (4.3)). Let us remind that we have proved the following mean-value
formulae (see [2])
(4.5) $\begin{split}&\int_{G_{1}(x)}Z(t){\rm d}t=\frac{2}{\pi}H\sin
x+\mathcal{O}(T^{1/6+\epsilon}),\\\ &\int_{G_{2}(y)}Z(t){\rm
d}t=-\frac{2}{\pi}H\sin y+\mathcal{O}(T^{1/6+\epsilon}).\end{split}$
Now, our formulae (1.5) follow from (4.4) by (4.5).
I would like to thank Michal Demetrian for helping me with the electronic
version of this work.
## References
* [1] A. Ivic, ‘The Riemann zeta-function‘, A Willey-Interscience Pub., New York, 1985.
* [2] J. Moser, ‘New consequences of the Riemann-Siegel formula‘, Acta Arith., 42, (1982), 1-10 (in russian).
* [3] J. Moser, ‘Jacob’s ladders and the almost exact asymptotic representation of the Hardy-Littlewood integral’, (2008), arXiv:0901.3973.
* [4] J. Moser, ‘Jacob’s ladders and the tangent law for short parts of the Hardy-Littlewood integral’, (2009), arXiv:0906.0659.
* [5] J. Moser, ‘Jacob’s ladders and the multiplicative asymptotic formula for short and microscopic parts of the Hardy-Littlewood integral’, (2009), arXiv:0907.0301.
* [6] J. Moser, ‘Jacob’s ladders and the quantization of the Hardy-Littlewood integral’, (2009), arXiv:0909.3928.
* [7] J. Moser, ‘Jacob’s ladders and the first asymptotic formula for the expression of the sixth order $|\zeta(1/2+i\varphi(t)/2)|^{4}|\zeta(1/2+it)|^{2}$’, (2009), arXiv:0911.1246.
* [8] J. Moser, ‘Jacob’s ladders and the first asymptotic formula for the expression of the fifth order $Z[\varphi(t)/2+\rho_{1}]Z[\varphi(t)/2+\rho_{2}]Z[\varphi(t)/2+\rho_{3}]\hat{Z}^{2}(t)$ for the collection of disconnected sets‘, (2009), arXiv:0912.0130.
* [9] J. Moser, ‘Jacob’s ladders, the iterations of Jacob’s ladder $\varphi_{1}^{k}(t)$ and asymptotic formulae for the integrals of the products $Z^{2}[\varphi^{n}_{1}(t)]Z^{2}[\varphi^{n-1}(t)]\cdots Z^{2}[\varphi^{0}_{1}(t)]$ for arbitrary fixed $n\in\mathbb{N}$‘ (2010), arXiv:1001.1632.
* [10] J. Moser, ‘Jacob’s ladders and the asymptotic formula for the integral of the eight order expression $|\zeta(1/2+i\varphi_{2}(t))|^{4}|\zeta(1/2+it)|^{4}$‘, (2010), arXiv:1001.2114.
* [11] J. Moser, ‘Jacob’s ladders and the asymptotically approximate solutions of a nonlinear diophantine equation‘, (2010), arXiv: 1001.3019.
* [12] J. Moser, ‘Jacob’s ladders and the asymptotic formula for short and microscopic parts of the Hardy-Littlewood integral of the function $|\zeta(1/2+it)|^{4}$‘, (2010), arXiv:1001.4007.
* [13] J. Moser, ‘Jacob’s ladders and the nonlocal interaction of the function $|\zeta(1/2+it)|$ with $\arg\zeta(1/2+it)$ on the distance $\sim(1-c)\pi(t)$‘, (2010), arXiv: 1004.0169.
* [14] J. Moser, ‘Jacob’s ladders and the $\tilde{Z}^{2}$ \- transformation of polynomials in $\ln\varphi_{1}(t)$‘, (2010), arXiv: 1005.2052.
* [15] J. Moser, ‘Jacob’s ladders and the oscillations of the function $|\zeta\left(\frac{1}{2}+it\right)|^{2}$ around the main part of its mean-value; law of the almost exact equality of the corresponding areas‘, (2010), arxiv:
|
arxiv-papers
| 2010-06-26T18:13:30 |
2024-09-04T02:49:11.241799
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jan Moser",
"submitter": "Michal Demetrian",
"url": "https://arxiv.org/abs/1006.5158"
}
|
1006.5190
|
# Anisotropy of graphite optical conductivity
L.A. Falkovsky L.D. Landau Institute for Theoretical Physics, Moscow 119334,
Russia Laboratoire de Physique des Solides, Univ. Paris-Sud, CNRS, UMR 8502,
F-91405 Orsay Cedex, France
###### Abstract
The graphite conductivity is evaluated for frequencies between 0.1 eV, the
energy of the order of the electron-hole overlap, and 1.5 eV, the electron
nearest hopping energy. The in-plane conductivity per single atomic sheet is
close to the universal graphene conductivity $e^{2}/4\hbar$ and, however,
contains a singularity conditioned by peculiarities of the electron
dispersion. The conductivity is less in the $c-$direction by the factor of the
order of 0.01 governed by electron hopping in this direction.
###### pacs:
78.67.-n, 81.05.Bx, 81.05.Uw
Recently, the light transmittance of graphene was found Na ; Li ; Ma in the
wide frequency region to differ from unity by the value of $\pi\alpha$, where
$\alpha$ is the fine structure constant of quantum electrodynamics. These
experimental observations are in excellent agreement with the theoretical
calculations GSC ; FV of the graphene conductance, $G=e^{2}/4\hbar$, which
does not depend on any material parameters.
This phenomenon is remarkable in two aspects. First, the fine structure
constant has been found in one measurement for the first time in solid state
physics. Second and most important, the Coulomb interaction does not disturb
the agreement between the experiment and the theory Mi ; SS . It should be
emphasize that the Coulomb interaction in graphene is poorly screened while
the carriers are absent in this gapless insulator.
In connection with this, it is interesting to study the change in the optical
conductivity going from 2d graphene to its close ”relative” , 3d graphite,
with the optical conductivity measured in Refs. TP ; KHC .
The electron properties of graphite is well described within the classical
Walles-Slonczewski-Weiss-McClure theory SW . There are many parameters in this
theory of the various order of value (see, e. g. PP ). Among them, the energy
$\gamma_{0}=3.1$ eV is largest one representing the electron in-plane hopping
between nearest neighbors at the distance $r_{0}=$1.42 $\AA$. If we are
interested in frequencies less than 3.1 eV, we can use the power
$\bf{k}-$momentum expansion of the corresponding term in the Hamiltonian,
taking only the linear approximation. Then the constant velocity $v=10^{8}$
cm/s appears. The parameter $\gamma_{1}\simeq 0.4$ eV known from optical
studies of bilayer graphene KCM ; Ba is next in the order. It describes the
interaction between the nearest layers at the distance $c_{0}$=3.35 $\AA$. The
parameters $\gamma_{3}$ and $\gamma_{4}$ give corrections of the order of 10%
to the velocity $v$. Finally, the parameters $\gamma_{2}$, $\gamma_{5}$ of the
order of 0.02 eV from the third sphere are used in order to describe the
dispersion of the conduction and valence bands in the $c-$direction. They are
usually included in order to characterize the carriers and are known from the
Shubnikov-de Haas oscillations and the cyclotron resonance. However, for the
optical transitions at relative high frequencies
$\gamma_{2},\gamma_{5}\ll\omega\ll\gamma_{0}$, we can, first, neglect the
smallest parameters $\gamma_{2},\gamma_{5}$ and, second, use the linear ${\bf
k}-$expansion with the constant velocity $v$ for in-layer directions. Our
results have the explicit analytic form.
Figure 1: Dispersion of low energy bands in graphite.
Thus, the simplified Hamiltonian of the model is given by
$H(\mathbf{k})=\left(\begin{array}[]{cccc}0&k_{+}&\gamma(z)&0\\\
k_{-}&0&0&0\\\ \gamma(z)&0&0&k_{-}\\\ 0&0&k_{+}&0\end{array}\right),$ (1)
where the velocity parameter $v$ is included in the definition of the momentum
components $k_{\pm}=v(\mp ik_{x}-k_{y})$, and the constant $\gamma_{1}$ stands
in the function $\gamma(z)=2\gamma_{1}\cos{z}$ depending on the dimensionless
$k_{z}-$component $z=k_{z}c_{0}$ along the $c$-axis, $0<z<\pi/2$.
The eigenenergies of the Hamiltonian write:
$\displaystyle\varepsilon_{1,2}=\frac{\gamma(z)}{2}\pm\sqrt{\frac{\gamma^{2}(z)}{4}+k^{2}}\,,$
(2)
$\displaystyle\varepsilon_{3,4}=-\frac{\gamma(z)}{2}\pm\sqrt{\frac{\gamma^{2}(z)}{4}+k^{2}}\,.$
The so-called ”Dirac” point of graphene, $k=0$, turns into the K-G-H line of
the graphite Brillouin zone, where the valence and conduction bands slick
together, $\varepsilon_{2,3}=0$. It should be emphasized that this
degeneration is conditioned by the lattice symmetry but is not a result of the
model.
Others two bands, $\varepsilon_{1,4}=\pm\gamma(z)$, are spaced at the distance
$\gamma(z)$ which vanishes at the H point of the Brillouin zone. This band
schema corresponds to the gapless semiconductor.
In order to calculate the optical conductivity, we use the general expression
FV
$\displaystyle\sigma^{ij}(\omega)=\frac{2ie^{2}}{(2\pi)^{3}}\int
d^{3}k\sum_{n\geq
m}\left\\{-\frac{df}{d\varepsilon_{n}}\frac{v_{nn}^{i}v_{nn}^{j}}{\omega+i\nu}\right.$
$\displaystyle+2\omega\left.\frac{v_{nm}^{i}v_{mn}^{j}[f(\varepsilon_{n})-f(\varepsilon_{m})]}{(\varepsilon_{m}-\varepsilon_{n})[(\omega+i\nu)^{2}-(\varepsilon_{n}-\varepsilon_{m})^{2}]}\right\\}\,,$
(3)
valid in the collisionless limit $\omega\gg\nu$, where $\nu$ is the collision
rate. This condition is definitely fulfilled, if the frequencies are larger
than the electron-hole overlap in graphite determined by the parameters
$\gamma_{2},\gamma_{5}$. The temperature is involved here by the Fermi-Dirac
function $f(\varepsilon)=[\exp(\frac{\varepsilon-\mu}{T})+1]^{-1}$, the
coefficient 2 takes into account the spin degeneration, and the integral is
taken over the Brillouin zone where the electron dispersions $\varepsilon_{n}$
are defined.
The first term in Eq. (Anisotropy of graphite optical conductivity) is the
intraband Drude-Boltzmann conductivity with the group velocity
$\mathbf{v}_{nn}=\partial\varepsilon_{n}/\partial{\bf k}.$
This conductivity behaves as $1/\omega$ and becomes less than the second term
for frequencies higher than the electron-hole overlap. The second term
corresponds with the electronic interband transitions accompanied by the
photon absorption. It involves the matrix elements of the velocity operator
$U^{-1}\frac{\partial H({\bf k})}{\partial{\bf k}}U,$
calculated in the representation transforming the Hamiltonian (1) to the
diagonal form with the help of the operator $U$. We find for various
transitions
$\begin{array}[]{c}v_{23}^{x}=2i(\varepsilon_{3}-\varepsilon_{2})k_{y}/N_{2}N_{3}\,,\\\
v_{12}^{x}=2(\varepsilon_{1}+\varepsilon_{2})k_{x}/N_{1}N_{2}\,,\\\
v_{14}^{x}=2i(\varepsilon_{4}-\varepsilon_{1})k_{y}/N_{1}N_{4}\,,\\\
\end{array}$
where $N_{n}^{2}=2(\varepsilon_{n}^{2}+k^{2})$ .
The calculations show that the off-diagonal components of conductivity reduce
to zero and the in-plane diagonal components are equal. For their real part,
we obtain the integral which is explicitly taken over $\varphi$ and $k$ in the
polar coordinates at the zero temperature. Thus, we meet the integral over
$k_{z}$:
$\displaystyle\text{Re}~{}\frac{\sigma}{\sigma_{0}}=\frac{1}{\pi}\int_{0}^{\pi/2}dz\left[\frac{2\gamma(z)+\omega}{\gamma(z)+\omega}\right.$
(4)
$\displaystyle\left.+\frac{2\gamma^{2}(z)}{\omega^{2}}\theta_{1}+\frac{2\gamma(z)-\omega}{\gamma(z)-\omega}\theta_{2}\right]\,,$
where $\gamma(z)=2\gamma_{1}\cos{z}$, and $\theta_{1}$, $\theta_{2}$ are the
step functions depending on $\omega-\gamma(z)$ and $\omega-2\gamma(z)$,
respectively. This integral can also be taken, but the result looks more
complicated.
Here, we introduce the conductivity $\sigma_{0}=e^{2}/4\hbar c_{0}$ which can
be named as the graphite universal conductivity. It differs from the graphene
conductivity only in the factor $1/c_{0}$ which is simply the number of the
atomic sheets in graphite per length unit in the $c-$direction. One can see,
that the graphite conductivity goes to $\sigma_{0}$ at low as well as high
frequencies compared to $\gamma(z)$ (see, also Fig. 2). However, at
$\omega=2\gamma_{1}$=0.84 eV, both the kink and the threshold are seen in the
real and imaginary parts, correspondingly. These singularities arise due to
the electron transitions between bands $2\rightarrow 1$ and $4\rightarrow 3$
(see, Fig. 1) described by the second term in Eq. (4). The position of the
singularities gives the value $\gamma_{1}$=0.42 eV, which agrees well with
optical studies of bilayer graphene.
Figure 2: Real $\sigma_{1}$ and imaginary $\sigma_{2}$ parts of the graphite
optical conductivity for the in-plane direction (per one atomic sheet in units
of $e^{2}/4\hbar$) versus frequency (in units of $2\gamma_{1}=0.84$ eV);
experimental data KHC , solid line; results of the present theory, dashed
lines.
Let us consider next the conductivity in the $c-$axis. We need now the matrix
elements $v_{nm}^{z}$. Calculations show that they are nonzero only for the
transitions $2\rightarrow 1$ and $4\rightarrow 3$:
$v_{21}^{z}=2\gamma^{\prime}(z)\varepsilon_{1}\varepsilon_{2}/N_{1}N_{2}\,,$
$v_{43}^{z}=-2\gamma^{\prime}(z)\varepsilon_{3}\varepsilon_{4}/N_{3}N_{4}\,,$
where the derivative $\gamma^{\prime}(z)=2\gamma_{1}c_{0}\sin{z}$. Using Eq.
(2), we get
$v_{21}^{z}=-v_{43}^{z}=-\gamma^{\prime}(z)k/\sqrt{\gamma^{2}(z)+4k^{2}}.$
Integrating in Eq. (Anisotropy of graphite optical conductivity) over
$\varphi$ and $k$, we obtain
$\displaystyle\text{Re}~{}\frac{\sigma^{zz}}{\sigma_{0}}=\left(\frac{\gamma_{1}c_{0}}{\hbar
v}\right)^{2}I(t)\,,$
where the integral over $k_{z}$
$I(t)=\frac{4}{\pi}\int_{0}^{\pi/2}dz\sin^{2}{z}\left(1-\frac{\cos^{2}{z}}{t^{2}}\right)\theta(t-\cos{z})$
with $t=\omega/2\gamma_{1}$. This integral can be also taken and it has in
limiting cases the very simple forms:
$I(t)=\frac{8}{3\pi}t\,,\quad t\ll 1\,,$ $I(t)=1-\frac{1}{4t^{2}},\quad
t>1\,.$
Figure 3: The real and imaginary parts of conductivity in $c-$direction; units
are the same as in Fig. 2.
The imaginary part of the conductivity in $c-$direction is given by the
$k_{z}-$integral
$\displaystyle\text{Im}~{}\frac{\sigma^{zz}}{\sigma_{0}}=\frac{4}{\pi^{2}}\left(\frac{\gamma_{1}c_{0}}{\hbar
v}\right)^{2}\int_{0}^{\pi/2}dz\sin^{2}(z)$
$\displaystyle\left[-2\frac{\gamma(z)}{\omega}+\left(1-\frac{\gamma^{2}(z)}{\omega^{2}}\right)\ln{\frac{|\gamma(z)-\omega|}{\gamma(z)+\omega}}\right]\,.$
The conductivity in the $c-$direction is shown in Fig. 3. Compared with the
in-plane conductivity, the $c-$conductivity is less by the factor
$(\gamma_{1}c_{0}/\hbar v)^{2}\sim 0.01$. This factor represents the squared
ratio of the hopping integrals for the inter- and in-layer directions
$(\gamma_{1}/\gamma_{0})^{2}\simeq\exp{(-2c_{0}/r_{0})}$.
In conclusions, for the in-plane direction, the optical conductivity of
graphite per single atomic sheet is close to the graphene universal
conductivity. However, the singularities, the kink in the real part and the
threshold in the imaginary part, appear at the frequency $\omega=2\gamma_{1}$,
where $\gamma_{1}$ is the inter-layer hopping energy for the bilayer graphene.
For the $c-$direction, the conductivity is less by the parameter representing
the ratio of the inter- and in-layer hopping energies; the real part of
conductivity increases linearly with the frequency and does not contain any
singularities.
This work was supported by the Russian Foundation for Basic Research (grant
No. 10-02-00193-a) and by the Fondation de Cooperation Scientifique Digiteo
Triangle de la Physique, 2009-069T project ”BIGRAPH”.
## References
* (1) R.R. Nair, P. Blake, A.N. Grigorenko, K.S. Novoselov, T.J. Booth, T. Stauber, N.M.R. Peres, A.K. Geim, Science 320, 5881 (2008).
* (2) Z.Q. Li, E.A. Henriksen, Z. Jiang, Z. Hao, M.C. Martin, P. Kim, H.L. Stormer, D.N. Basov, Nature Physics 4, 532 (2008).
* (3) K.F. Mak, M.Y. Sfeir, Y. Wu, C.H. Lui, J.A. Misewich, and Tony F. Heinz, Phys. Rev. Lett. 101, 196405 (2008).
* (4) V.P. Gusynin, S.G. Sharapov, and J.P. Carbotte, Phys. Rev. Lett. 96, 256802 (2006).
* (5) L.A. Falkovsky and A.A Varlamov, Eur. Phys. J. B 56, 281 (2007).
* (6) E.G. Mishchenko, Europhys. Lett. 83, 17005(2008).
* (7) D.E. Sheehy and J. Scmalian, arXiv:0906.5164vl
* (8) E.A. Taft and H.R. Philipp, Phys. Rev. 138, A197 (1965).
* (9) A.B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, Phys. Rev. Lett. 100, 117401 (2008).
* (10) P.R. Wallace, Phys. Rev. 71 622 (1947); J.W. McClure, Phys. Rev. 108, 612 (1957); J.C. Slonczewski and P.R. Weiss, Phys. Rev. 109, 272 (1958);
* (11) B. Partoens and F.M. Peeters, Phys. Rev. B 74, 075404 (2006).
* (12) A.B. Kuzmenko, I. Crassee,, D. van der Marel, P. Blake, and K.S. Novoselov, Phys. Rev. 80, 165406 (2009).
* (13) Z.Q. Li, E.A. Henriksen, Z. Jiang, Z. Hao, M.C. Martin, P. Kim, H.L. Stormer, and D.N. Basov, Phys. Rev. Lett. 102, 037403 (2009).
|
arxiv-papers
| 2010-06-27T09:26:04 |
2024-09-04T02:49:11.246589
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L.A. Falkovsky",
"submitter": "L. A. Falkovsky",
"url": "https://arxiv.org/abs/1006.5190"
}
|
1006.5212
|
# Generalized Projective Representations for sl(n+1)111Research supported by
NSFC Grant 10701002
$\mbox{Yufeng Zhao}^{1}$ and $\mbox{Xiaoping Xu}^{2}$
1 LMAM, School of Mathematical Sciences
Peking University, Beijing, 100871, P. R. China
2 Hua Loo-Keng Key Mathematical Laboratory
Institute of Mathematics, Academy of Mathematics and Systems Sciences
Chinese Academy of Sciences, Beijing, 100190, P. R. China
###### Abstract
It is well known that $n$-dimensional projective group gives rise to a non-
homogenous representation of the Lie algebra $sl(n+1)$ on the polynomial
functions of the projective space. Using Shen’s mixed product for Witt
algebras (also known as Larsson functor), we generalize the above
representation of $sl(n+1)$ to a non-homogenous representation on the tensor
space of any finite-dimensional irreducible $gl(n)$-module with the polynomial
space. Moreover, the structure of such a representation is completely
determined by employing projection operator techniques and well-known
Kostant’s characteristic identities for certain matrices with entries in the
universal enveloping algebra. In particular, we obtain a new one parameter
family of infinite-dimensional irreducible $sl(n+1)$-modules, which are in
general not highest-weight type, for any given finite-dimensional irreducible
$sl(n)$-module. The results could also be used to study the quantum field
theory with the projective group as the symmetry.
## 1 Introduction
A projective transformation on $\mathbb{F}^{n}$ for a field $\mathbb{F}$ is
given by
$u\mapsto\frac{Au+\vec{b}}{\vec{c}\>^{t}u+d}\qquad\mbox{for}\;\;u\in\mathbb{F}^{n},$
$None$
where all the vectors in $\mathbb{F}^{n}$ are in column form and
$\left(\begin{array}[]{cc}A&\vec{b}\\\ \vec{c}\>^{t}&d\end{array}\right)\in
GL(n).$ $None$
It is well-known that a transformation of mapping straight lines to lines must
be a projective transformation. The group of projective transformations is the
fundamental symmetry of $n$-dimensional projective geometry. Physically, the
group with $n=4$ and $\mathbb{F}=\mathbb{R}$ consists of all the
transformations of keeping free particles including light signals moving with
constant velocities along straight lines (e.g., cf. [GWZ1-2]). Based on the
embeddings of the poincaré group and De Sitter group into the projective group
with $n=4$ and $\mathbb{F}=\mathbb{R}$, Guo, Wu and Zhou [GWZ1-2] proposed
three kinds of special relativity.
In this paper, we give a representation-theoretic exploration on the impact of
projective transformations. Note that the Lie algebra of $n$-dimensional
projective group is spanned by the following differential operations
$\\{\partial_{x_{j}},x_{i}\partial_{x_{j}},x_{i}\sum_{r=1}^{n}x_{r}\partial_{x_{r}}\mid
i,j=1,2,...,n\\},$ $None$
which is isomorphic to the special linear Lie algebra $sl(n+1)$. Through the
above operators, we obtain a representation of $sl(n+1)$ on the polynomial
algebra $\mathbb{F}[x_{1},...,x_{n}]$. The non-homogeneity of (1.3) motivates
us to generalize the above representation of $sl(n+1)$ to a non-homogenous
representation on the tensor space of any finite-dimensional irreducible
$gl(n)$-module with $\mathbb{F}[x_{1},...,x_{n}]$ via Shen’s mixed product for
Witt algebras (cf. [Sg1-3]) (also known as Larsson functor (cf. [La])). It
turns out that the structure of such generalized projective representations
can be completely determined by employing projection operator techniques (cf.
[Gm1]) and well-known Kostant’s characteristic identities for certain matrices
with entries in the universal enveloping algebra (cf. [K]). In particular, we
obtain a new one parameter family of infinite-dimensional irreducible
$sl(n+1)$-modules for any given finite-dimensional irreducible $sl(n)$-module.
Denote by $\mathbb{Z}$ the ring of integers and by $\mathbb{N}$ the additive
semigroup of nonnegative integers. For any two integers $m$ and $n$, we denote
$\overline{m,n}=\left\\{\begin{array}[]{ll}\\{m,m+1,\cdots,n\\}&\mbox{if}\;m\leq
n,\\\ \emptyset&\mbox{otherwise}.\end{array}\right.$ $None$
Let ${\cal A}$ be a commutative associative algebra over $\mathbb{F}$. If
$\\{D_{i}\ |\ i\in\overline{1,n}\\}$ is a set of commuting derivations of
${\cal A}$, the set of derivations ${\cal
W}(n)=\\{\sum\limits_{i=1}^{n}a_{i}D_{i}|\ a_{i}\in{\cal A}\\}$ forms a Lie
algebra via the following Lie brackets:
$[\sum\limits_{i=1}^{n}a_{i}D_{i},\sum\limits_{i=1}^{n}b_{i}D_{i}]=\sum\limits_{i,j=1}^{n}(a_{j}D_{j}(b_{i})-b_{j}D_{j}(a_{i}))D_{i}.$
$None$
Let $E_{r,s}$ be the square matrix with 1 as its $(r,s)$-entry and 0 as the
others. The general linear Lie algebra $gl(n)$ is the Lie algebra of $n\times
n$ matrices over ${\mathbb{F}}$ with a vector space basis $\\{E_{i,j}\mid
i,j\in\overline{1,n}\\}$. For any $gl(n)$-module $V$, we define an action
$\pi$ of the Lie algebra ${\cal W}(n)$ on ${\cal A}\otimes_{\mathbb{F}}V$ by
$\pi(\sum\limits_{i=1}^{n}a_{i}D_{i})=\sum_{i,j=1}^{n}D_{i}(a_{j})\otimes
E_{i,j}+\sum\limits_{i=1}^{n}a_{i}D_{i}\otimes\mbox{Id}_{V}.$ $None$
Then $\pi$ gives a representation of ${\cal W}(n)$ (cf. [Sg1-3]) and the
functor from $gl(n)$-modules to ${\cal W}(n)$-modules was also later known as
Larsson functor (cf. [L]). The structure of the module ${\cal
A}\otimes_{\mathbb{F}}V$ was determined by Rao [R] when ${\cal
A}=\mathbb{F}[x_{1}^{\pm 1},...,x_{n}^{\pm 1}]$, $D_{i}=\partial_{x_{i}}$ and
$V$ is a finite-dimensional irreducible $gl(n)$-module. Lin and Tan [LT] did
the similar thing when ${\cal A}$ is the algebra of quantum torus. The first
author of this paper [Z] determined the ${\cal W}(n)$-module structure of
${\cal A}\otimes_{\mathbb{F}}V$ in the case that ${\cal A}$ is a certain semi-
group algebra and $D_{i}$ are locally-finite derivations in [X]. Throughout
this paper, we always assume $\mbox{char}\>\mathbb{F}=0$.
Take ${\cal H}=\sum\limits_{i=1}^{n}\mathbb{F}E_{i,i}$ as a Cartan subalgebra
of $gl(n)$. Assume ${\cal A}=\mathbb{F}[x_{1},...,x_{n}]$ and let
$D_{i}=\partial_{x_{i}}$. Embed $sl(n+1)$ into ${\cal W}(n)$ via (1.3). Then
the space ${\cal A}\otimes_{\mathbb{F}}V$ forms an $sl(n+1)$-module with the
representation $\pi|_{sl(n+1)}$, which we call a generalized projective
representation.
Characteristic identities have a long history. The first person to exploit
them was Dirac [D], who wrote down what amounts to the characteristic identity
for the Lie algebra $so(1,3)$. This particular example is intimately connected
with the problem of describing the structure of relativistically invariant
wave equations. Such identities have been shown to be powerful tools for the
analysis of finite dimensional representations of Lie groups (cf. [BB], [BG],
[F]). It has been shown by Kostant [K] (also cf. [Gm4]) that the
characteristic identities for semi-simple Lie algebras also hold for infinite
dimensional representations. Moreover, one may construct projection operators
analogous to the projection operators of Green [G], and Bracken and Green [BG]
in finite dimensions. We refer [Gh], [Gm1-Gm3], [Ha], [L], [LG], [M], [OCC]
and [O] for the other works on the identities. Using projection operator
techniques and Kostant’s characteristic identities, we prove:
Main Theorem. Let $V$ be a finite-dimensional irreducible $gl(n)$-module with
highest weight $\mu$. We have the following conclusions:
(i) The space ${\cal A}\otimes_{\mathbb{F}}V$ is an irreducible
$sl(n+1)$-module if and only if
$\mu(E_{1,1})+\sum\limits_{j=1}^{n}\mu(E_{j,j})\not\in-\mathbb{N}\bigcup\overline{2,1+\mu(E_{1,1}-E_{2,2})}$
$None$
and
$\mu(E_{i,i})+\sum\limits_{j=1}^{n}\mu(E_{j,j})-i\not\in\overline{1,\mu(E_{i,i}-E_{i+1,i+1})}\qquad\mbox{for}\;i\in\overline{2,n-1}.$
$None$
(ii) If one of the conditions in (1.7) and (1.8) fails, then both
$U(sl(n+1))(1\otimes V)$ and $({\cal
A}\otimes_{\mathbb{F}}V)/(U(sl(n+1))(1\otimes V))$ are irreducible
$sl(n+1)$-modules .
Note that all $\mu(E_{i,i}-E_{i+1,i+1})$ are nonnegative integers, which
determine the corresponding $sl(n)$-module $V$ uniquely. Moreover, the
identity matrix in $gl(n)$ are allowed to be any constant map. The above
theorem says that given a finite-dimensional irreducible $sl(n)$-module, we
can construct a new one-parameter family of explicit irreducible
$sl(n+1)$-modules via its projective representation and Shen’s mixed product.
A quantum field is an operator value function on a certain Hilbert space,
which is often a direct sum of infinite-dimensional irreducible modules of a
certain Lie algebra (group). The Lie algebra of two-dimensional conformal
group is exactly the Virasoro algebra, which is infinite-dimensional. The
minimal models of two-dimensional conformal field theory were constructed from
direct sums of certain infinite-dimensional irreducible modules of the
Virasoro algebra, where a distinguished module gives rise to a vertex operator
algebra. When $n>2$, the $n$-dimension conformal group is finite-dimensional,
whose Lie algebra is exactly isomorphic to $so(n,2)$. It is still unknown what
should a higher-dimensional conformal field theory be. Part of reason is that
we lack of enough knowledge on the infinite-dimensional irreducible
$so(n,2)$-modules that are compatible to the natural conformal representation
of $so(n,2)$. This motivates us to study explicit infinite-dimensional
irreducible modules of finite-dimensional simple Lie algebras by using non-
homogeneous polynomial representations and Shen’s mixed product for Witt
algebras. This paper is the first work in this direction.
As we mentioned earlier, projective groups are important groups in physics. In
comparison with the minimal models of two-dimensional conformal field theory,
the underlying module of the projective representation of $sl(n+1)$ should be
the distinguished module in the possible quantum field theory with the
projective group as the symmetry. The other modules would make the theory more
substantial.
The paper is organized as follows. In Section 2, we slightly generalize
Kostant’s characteristic identities and recall some facts about projection
operators based on Kostant’s work [K] and Gould’s works [Gm1, Gm4]. In Section
3, we prove (i) and (ii) in the theorem.
Acknowledgement: We would like to thank Professor Han-Ying Guo for his
interesting talk that motivates this work.
## 2 Characteristic Identities and Projection Operators
In order to keep the paper self-contained, we will first prove certain
characteristic identities for $gl(n)$, which will be used to study the
irreducibility of the generalized projective representations for $sl(n+1)$ .
Then we will recall some facts about the projection operators for $gl(n)$,
although part of the results in this section has appeared in [Gm1, Gm4], [OCC]
and [K].
### 2.1 Some Standard Facts for $gl(n)$ and $sl(n)$
Recall that $gl(n)$ is the Lie algebra of $n\times n$ matrices over
${\mathbb{F}}$ with a basis $\\{E_{i,j}\mid i,j\in\overline{1,n}\\}$ and the
Lie bracket:
$[E_{i,j},E_{k,l}]=\delta_{k,j}E_{i,l}-\delta_{i,l}E_{k,j}\qquad\mbox{for}\;\;i,j,k,l\in\overline{1,n}.$
$None$
Note that $gl(n)$ is reductive with the following decomposition of ideals
$gl(n)=sl(n)\oplus{\mathbb{F}}\mbox{I}$, where $sl(n)$ is the special linear
Lie algebra of matrices with zero trace and
$\mbox{I}=\sum\limits_{i=1}^{n}E_{i,i}$ is the identity matrix, which is
central.
Take ${\cal H}=\sum\limits_{i=1}^{n}\mathbb{F}E_{i,i}$ as a Cartan subalgebra
of $gl(n)$. If $\lambda\in{\cal{H}}^{*}$ is a weight of $gl(n)$, we identify
$\lambda$ with the $n$-tuple $\lambda=(\lambda_{1},\cdots,\lambda_{n})$, where
$\lambda_{i}=\lambda(E_{i,i})$. For $\lambda,\mu\in{\cal{H}}^{*}$, we define
$(\lambda,\mu)=\sum\limits_{i=1}^{n}\lambda_{i}\mu_{i}.$ $None$
Denote by $\varepsilon_{i}$ the weight with 1 as its $i$th coordinate and 0 as
the others, i.e.
$\varepsilon_{i}=(0,\cdots,0,\stackrel{{\scriptstyle i}}{{1}},0,\cdots,0).$
$None$
The set $\Phi^{+}=\\{\varepsilon_{i}-\varepsilon_{j}\mid 1\leq i<j\leq n\\}$
forms a set of positive roots of $gl(n)$. In this case, the half-sum of the
positive roots is given by
$\delta=\frac{1}{2}\sum\limits_{i<j}(\varepsilon_{i}-\varepsilon_{j})=\frac{1}{2}\sum\limits_{i=1}^{n}(n+1-2i)\varepsilon_{i}.$
$None$
Moreover, there exists a one-to-one correspondence between the set of finite-
dimensional irreducible $gl(n)$-modules and the set of $n$ tuples
$\lambda=(\lambda_{1},\cdots,\lambda_{n})$ such that
$\lambda_{i}-\lambda_{i+1}\in{\mathbb{N}}\ \mbox{for}\ i\in\overline{1,n-1}$.
Such an $n$ tuple $\lambda$ is called the highest weight of the corresponding
module which we denote as $V(\lambda)$.
Let $U$ denote the universal enveloping algebra of $gl(n)$ and let $Z$ be the
center of $U$. Set
$\sigma_{1}=\mbox{I},\
\sigma_{r}=\sum\limits_{i_{1},..,i_{r}=1}^{n}E_{i_{1},i_{2}}E_{i_{2},i_{3}}\cdots
E_{i_{r-1},i_{r}}E_{i_{r},i_{1}}\qquad\mbox{for}\;r\in\overline{2,n}.$ $None$
Then the center
$Z=\mathbb{F}[\sigma_{1},\cdots,\sigma_{n}].$ $None$
The subspace ${\cal H}_{0}={\cal H}\bigcap sl(n)$ is a Cartan subalgebra of
$sl(n)$ with the standard basis $\\{\alpha_{i}^{\vee}=E_{i,i}-E_{i+1,i+1}\mid
i\in\overline{1,n-1}\\}.$ Denote the dual vector space of ${\cal H}_{0}$ by
${\cal H}_{0}^{*}$ and let $\omega_{1},\omega_{2},\cdots,\omega_{n-1}$ be the
fundamental integral dominant weights in ${\cal H}_{0}^{*}$ defined by
$\omega_{i}(\alpha_{j}^{\vee})=\delta_{i,j}.$
Let $V(\psi)$ be the finite dimensional irreducible $sl(n)$-module with the
highest weight $\psi$. We can make $V(\psi)$ as a $gl(n)$-module $V(\psi,b)$
by letting the central element I act as the scalar map $b\mbox{Id}_{V(\psi)}$.
For $\vec{a}=(a_{1},...,a_{n-1})\in\mathbb{N}^{n-1}$ and $0<k\in\mathbb{Z}$,
we denote
$\displaystyle\hskip 39.83368ptI(\vec{a},k)$ $\displaystyle=$
$\displaystyle\\{(a_{1}+c_{1}-c_{2},a_{2}+c_{2}-c_{3},...,a_{n-1}+c_{n-1}-c_{n})\mid
c_{i}\in\mathbb{N}$ $\displaystyle\mbox{such
that}\;\sum_{i=1}^{n}c_{i}=k\;\mbox{and}\;c_{s+1}\leq
a_{s}\;\mbox{for}\;s\in\overline{1,n-1}\\}.\hskip 85.35826pt(2.7)$
Moreover, we set
$\omega_{\vec{a}}=\sum_{i=1}^{n-1}a_{i}\omega_{i}\qquad\mbox{for}\;\vec{a}\in\mathbb{N}^{n-1}.$
$None$
Lemma 2.1.1 (e.g., cf. Proposition 15.25 in [FH]) For any
$\vec{a}\in\mathbb{N}^{n-1}$, the tensor product of $sl(n)$-module
$V(\omega_{\vec{a}})$ with $V(k\omega_{1})$ decomposes into a direct sum:
$V(\omega_{\vec{a}})\otimes_{\mathbb{F}}V(k\omega_{1})=\bigoplus_{\vec{b}\in
I(\vec{a},k)}V(\omega_{\vec{b}}).$ $None$
Lemma 2.1.2 Let $\Pi$ be the weight set of $gl(n)$-module $V(\psi,b)$. Assume
$\psi=\sum\limits_{i=1}^{n-1}a_{i}\omega_{i}$,
$\nu=(\nu_{1},\cdots,\nu_{n})\in\Pi$ and
$(\nu_{1}-\nu_{2},\cdots,\nu_{n-1}-\nu_{n})=\psi-\sum\limits_{i=1}^{n-1}k_{i}\alpha_{i}$.
Then
$\nu_{1}=\sum\limits_{i=1}^{n-1}a_{i}+\frac{b-\sum\limits_{i=1}^{n-1}ia_{i}}{n}-k_{1},\
\nu_{n}=\frac{b-\sum\limits_{i=1}^{n-1}ia_{i}}{n}+k_{n-1},$
$\nu_{j}=\sum\limits_{i=j}^{n-1}a_{i}+\frac{b-\sum\limits_{i=1}^{n-1}ia_{i}}{n}+k_{j-1}-k_{j},\;\;j\in\overline{2,n-1}.$
$None$
Denote
$\underline{k}=(k_{1},k_{2},...,k_{n})\in\mathbb{N}^{n},\quad|\underline{k}|=\sum_{i=1}^{n}k_{i},$
$I(\mu,j)=\\{\underline{c}=(c_{1},\cdots,c_{n})\ |\ c_{i}\in\mathbb{N}\ ,\
|\underline{c}|=j,\
c_{s+1}\leq\mu_{s}-\mu_{s+1}\;\mbox{for}\;s\in\overline{1,n-1}\\}.$ $None$
It is easy to deduce the following lemma from the above two lemmas.
Lemma 2.1.3 The tensor product of $gl(n)$-module $V(\mu)$ with
$V(k\varepsilon_{1})$ decomposes into a direct sum:
$V(\mu)\otimes_{\mathbb{F}}V(k\varepsilon_{1})=\bigoplus_{\underline{c}\in
I(\mu,k)}V(\mu+\underline{c}).$ $None$
### 2.2 Characteristic Identities and Projection Operators for $gl(n)$
Now we will introduce the characteristic identities for $gl(n)$. We know that
the universal enveloping algebra $U$ of $gl(n)$ can be imbedded into $U\otimes
U$ by the associative algebra homomorphism $d:U\rightarrow U\otimes U$
determined by
$d(u)=u\otimes 1+1\otimes u\qquad\mbox{ for}\ u\in gl(n).$ $None$
Let $V(\lambda)$ be a fixed finite-dimensional $gl(n)$-module with highest
weight $\lambda$ and let $\pi_{\lambda}$ be the corresponding representation.
Kostant [K] considered the map
$\partial:U\rightarrow(\mbox{End}\>V(\lambda))\otimes_{\mathbb{F}}U;$ $\
u\mapsto 1\otimes u+\pi_{\lambda}(u)\otimes 1$ $None$
for $u\in gl(n)$ and extended $\partial$ to an associative algebra
homomorphism from $U$ to $(\mbox{End}\>V(\lambda))\otimes_{\mathbb{F}}U$. More
generally, if $d(u)=\sum\limits_{r}u_{r}\otimes v_{r}$, we have
$\partial(u)=\sum\limits_{r}\pi_{\lambda}(u_{r})\otimes v_{r}$. For $z\in Z$,
we denote
$\tilde{z}=-\frac{1}{2}[\partial(z)-\pi_{\lambda}(z)\otimes 1-1\otimes z],$
$None$
which may be viewed as an $m\times m$ ($m=\mbox{dim}V(\lambda)$) matrix with
entries in $U$.
Denote by $\chi_{\zeta}$ the central character of a highest weight
$gl(n)$-module with highest weight $\zeta$. Suppose now that $W$ is another
$gl(n)$-module admitting the central character $\chi_{\mu}$ and $\pi_{\mu}$ is
the corresponding representation. We extend $\pi_{\mu}$ to an algebra
homomorphism
$\tilde{\pi_{\mu}}:(\mbox{End}\>V(\lambda))\otimes_{\mathbb{F}}U\rightarrow(\mbox{End}\>V(\lambda))\otimes_{\mathbb{F}}(\mbox{End}\>W);$
$\ \sum\limits_{i}\rho_{i}\otimes
u_{i}\mapsto\sum\limits_{i}\rho_{i}\otimes\pi_{\mu}(u_{i}),$ $None$
where $\rho_{i}\in\mbox{End}\>V(\lambda)$ and $u_{i}\in U$. In particular, we
have
$\tilde{\pi_{\mu}}(\tilde{z})=-\frac{1}{2}[(\pi_{\lambda}\otimes\pi_{\mu})(z)-\pi_{\lambda}(z)\otimes
1-1\otimes\pi_{\mu}(z)]$ $None$
for $z\in Z$. Clearly, $\tilde{\pi_{\mu}}(\tilde{z})$ is a linear operator on
$V(\lambda)\otimes_{\mathbb{F}}W$ which may be viewed as an $m\times m$ matrix
with entries from $\mbox{End}\>W$ under a basis of $V(\lambda)$. For
$\nu\in{\cal H}^{\ast}$, we define
$f_{\nu}=-\frac{1}{2}(\chi_{\mu+\nu}-\chi_{\lambda}-\chi_{\mu}).$ $None$
Denote by $\Pi_{\lambda}$ the weight set of $V(\lambda)$.
Lemma 2.2.1 (cf. [K], [G], [OCC] ) On the space $W$, the matrix $\tilde{z}$
satisfies the following characteristic identity:
$\prod\limits_{\nu\in\Pi_{\lambda}}(\tilde{z}-f_{\nu}(z))=0\qquad\mbox{for}\;\;z\in
Z.$ $None$
By varying the module $V(\lambda)$ and the central element $z$, we obtain a
series of characteristic identities. In particular, the following
characteristic identity will be used in the proof of the main theorem:
Corollary 2.2.2 Take $V(\lambda)$ to be the dual module of $gl(n)$-module
$V(2,1,\cdots,1)$. Then the matrix $\tilde{\sigma_{2}}$ satisfies the
following characteristic identity on $W$:
$\prod\limits_{i=1}^{n}(\tilde{\sigma_{2}}-m_{i})=0,\ \
m_{i}=\frac{1}{2}(\lambda,\lambda+2\delta)-\frac{1}{2}(\lambda_{i},\lambda_{i}+2(\mu+\delta)),$
$None$
where $\lambda_{i}=(-1,\cdots,\stackrel{{\scriptstyle i}}{{-2}},\cdots,-1),\
i\in\overline{1,n}$.
Proof Note that the module $V(2,1,\cdots,1)$ has a basis
$\\{e_{1},\cdots,e_{n}\\}$ such that
$E_{i,i}(e_{k})=(1+\delta_{i,k})e_{k},\ E_{i,j}(e_{k})=\delta_{j,k}e_{i}\
(i\neq j).$ $None$
Let $\pi^{*}$ be the dual module of $gl(n,{\mathbb{F}})$-module
$V(2,1,\cdots,1)$. Then
$\pi^{*}(E_{i,i})=-(E_{i,i}+\mbox{I}),\ \pi^{*}(E_{i,j})=-E_{j,i}\ (i\neq j).$
$None$
Obviously, the representation $\pi^{*}$ is $n$-dimensional and all its weights
$\\{\lambda_{1}=(-2,-1,\cdots,-1),...,\lambda_{i}=(-1,\cdots,\stackrel{{\scriptstyle
i}}{{-2}},\cdots,-1),\cdots,\lambda_{n}=(-1,\cdots,-1,-2)\\}$ $None$
occur with multiplicity one. The matrix $\tilde{\sigma_{2}}$ in this case is
given by
$\tilde{\sigma_{2}}=-\sum\limits_{i,j}^{n}\pi^{*}(E_{j,i})E_{i,j}.$ $None$
This is the matrix
$\begin{array}[]{rcl}\tilde{\sigma_{2}}&=&\left[\begin{array}[]{cccc}\mbox{I}+E_{1,1}&E_{1,2}&\cdots&E_{1,n}\\\
E_{2,1}&\mbox{I}+E_{2,2}&\cdots&E_{2,n}\\\ \vdots&\vdots&\cdots&\vdots\\\
E_{n,1}&E_{n,2}&\cdots&\mbox{I}+E_{n,n}\end{array}\right].\end{array}$ $None$
By Lemma 2.2.1, the operator $\tilde{\sigma_{2}}$ satisfies the characteristic
identity on $W$:
$\prod\limits_{i=1}^{n}(\tilde{\sigma_{2}}-m_{i})=0,\ \
m_{i}=\frac{1}{2}(\lambda,\lambda+2\delta)-\frac{1}{2}(\lambda_{i},\lambda_{i}+2(\mu+\delta)),\
i\in\overline{1,n}.$ $None$
$\Box$
In the rest of this section, we will recall some facts about projection
operators appeared in [Gm2].
Take $gl(n)$-module $V(\lambda)=V(\varepsilon_{1})^{*}$ (resp.
$V(\varepsilon_{1})$). Then the corresponding matrices $\tilde{\sigma_{2}}$
are
$M=(E_{i,j})_{i,j=1}^{n},\ \tilde{M}=-M^{T}\in U(gl(n)),$ $None$
respectively. On the space $W$, the matrices $M$ and $\tilde{M}$ satisfy the
following characteristic identities:
$\prod\limits_{i=1}^{n}(M-d_{i})=0,\ d_{i}=\mu_{i}+n-i;$ $None$
$\prod\limits_{i=1}^{n}(\tilde{M}-\tilde{d}_{i})=0,\ \tilde{d}_{i}=n-1-d_{i},\
i\in\overline{1,n}.$ $None$
For $r\in\overline{1,n}$,
$P_{r}=\prod\limits_{r\neq
l\in\overline{1,n}}(\frac{M-d_{l}}{d_{r}-d_{l}}),\qquad\tilde{P}_{r}=\prod\limits_{r\neq
l\in\overline{1,n}}(\frac{\tilde{M}-\tilde{d}_{l}}{\tilde{d}_{r}-\tilde{d}_{l}})$
$None$
are called projection operators, which project the tensor product space
$V(\varepsilon_{1})^{*}{\otimes}_{\mathbb{F}}W$ (resp.
$V(\varepsilon_{1})\otimes_{\mathbb{F}}W$ ) onto the irreducible module
$W_{r}=P_{r}(V(\varepsilon_{1})^{*}\otimes_{\mathbb{F}}W)$ (resp.
$\tilde{W}_{r}=\tilde{P}_{r}(V(\varepsilon_{1})\otimes_{\mathbb{F}}W)$) with
central character $\chi_{\nu-\varepsilon_{r}}$ (resp.
$\chi_{\nu+\varepsilon_{r}}$).
## 3 Generalized Projective Representations
In this section, we will give the detailed construction of generalized
projective representations for the special linear Lie algebra $sl(n+1)$ and
study their irreducibility.
### 3.1 Construction of the Representations
Let ${\cal A}=\mathbb{F}[x_{1},\cdots,x_{n}]$. The operator
$p_{i}=x_{i}\sum\limits_{i=1}^{n}x_{i}\partial_{x_{i}}$ is called pseudo-
translation operator on ${\cal A}$ in physics. Note that the Lie algebra of
$n$-dimensional projective group
$L_{n+1}=\sum\limits_{i,j=1}^{n}\mathbb{F}x_{i}\partial_{x_{j}}+\sum\limits_{i=1}^{n}\mathbb{F}\partial_{x_{i}}+\sum\limits_{i=1}^{n}\mathbb{F}p_{i},$
$None$
forms a Lie subalgebra of Witt algebra
${\cal W}(n)=\\{\sum\limits_{i=1}^{n}f_{i}\partial_{x_{i}}\ |\ f_{i}\in{\cal
A}\\}.$ $None$
Moreover, we have the following Lie brackets:
$[\partial_{x_{j}},p_{i}]=\left\\{\begin{array}[]{llll}x_{i}\partial_{x_{j}},&i\neq
j,\\\
\sum\limits_{i=1}^{n}x_{i}\partial_{x_{i}}+x_{i}\partial_{x_{i}},&i=j,\end{array}\right.$
$None$ $[x_{i}\partial_{x_{j}},p_{k}]=\delta_{j,k}{p}_{i},\
[x_{i}\partial_{x_{j}},\partial_{x_{k}}]=-\delta_{i,k}\partial_{x_{j}},\
[x_{i}\partial_{x_{j}},x_{k}\partial_{x_{l}}]=\delta_{j,k}x_{i}\partial_{x_{l}}-\delta_{i,l}x_{k}\partial_{x_{j}}.$
$None$
To abbreviate, we denote
$P=\sum\limits_{i=1}^{n}\mathbb{F}p_{i},\
S=\sum\limits_{i=1}^{n}\mathbb{F}\partial_{x_{i}},\
\overline{L}_{n}=\sum\limits_{i,j=1}^{n}\mathbb{F}x_{i}\partial_{x_{j}},\
\overline{L}_{n}^{\prime}=[\overline{L}_{n},\overline{L}_{n}].$ $None$
Then
$L_{n+1}=P\oplus S\oplus\overline{L}_{n},\ [P,P]=\\{0\\},\ [S,S]=\\{0\\}.$
$None$
Moreover, $\overline{L}_{n}$ (resp. $\overline{L}_{n}^{\prime}$) is isomorphic
to $gl(n)$ (resp. $sl(n)$). It is easy to verify:
Lemma 3.1.1 The special linear Lie algebra $sl(n+1)$ is isomorphic to
$L_{n+1}$ with the following identification of Chevalley generators:
$h_{i}=x_{i}\partial_{x_{i}}-x_{i+1}\partial_{x_{i+1}},\qquad
h_{n}=\sum\limits_{i=1}^{n}x_{i}\partial_{x_{i}}+x_{n}\partial_{x_{n}},$
$None$ $e_{i}=x_{i}\partial_{x_{i+1}},\;\;f_{i}=x_{i+1}\partial_{x_{i}},\qquad
e_{n}=p_{n},\;\;f_{n}=-\partial_{x_{n}},$ $None$
for $i\in\overline{1,n-1}$.
For any finite dimensional $gl(n)$-module $V$ with highest weight
$\mu=(\mu_{1},\cdots,\mu_{n})$, we define an action $\pi$ of Witt Lie algebra
${\cal W}(n)$ on ${\cal A}\otimes_{\mathbb{F}}V$ by
$\pi(\sum\limits_{i=1}^{n}a_{i}D_{i})=\sum_{i,j=1}^{n}D_{i}(a_{j})\otimes
E_{i,j}+\sum\limits_{i=1}^{n}a_{i}D_{i}\otimes\mbox{Id}_{V}.$ $None$
Embed $sl(n+1)$ into ${\cal W}(n)$ by (3.7) and (3.8). The space ${\cal
A}\otimes_{\mathbb{F}}V$ forms an $sl(n+1)$-module with the representation
$\pi|_{sl(n+1)}$, which we call a generalized projective representation of
$sl(n+1)$.
It follows from (3.7)-(3.9) that the explicit $L_{n+1}$-module ${\cal
A}\otimes_{\mathbb{F}}V$ structure is given by:
$(x_{i}\partial_{x_{j}}).(f\otimes v)=(x_{i}\partial_{x_{j}})f\otimes
v+f\otimes E_{i,j}.v,$ $None$ $\partial_{x_{i}}.(f\otimes
v)=\partial_{x_{i}}(f)\otimes v,$ $None$ $p_{i}.(f\otimes v)=p_{i}(f)\otimes
v+x_{i}f\otimes\sum\limits_{i=1}^{n}E_{i,i}.v+\sum\limits_{j=1}^{n}x_{j}f\otimes
E_{i,j}.v$ $None$
for $i,j\in\overline{1,n}$, where $f\in{\cal A}$ and $v\in V$.
### 3.2 Irreducibility Criteria for Generalized Projective Representations
In this section, we will give two irreducibility criteria for $L_{n+1}$-module
${\cal A}\otimes_{\mathbb{F}}V$ (cf. Proposition 3.2.3 and Proposition 3.2.5).
Lemma 3.2.1 The vector space $U(P)(1\otimes_{\mathbb{F}}V)$ is an irreducible
$L_{n+1}$-submodule of ${\cal A}\otimes_{\mathbb{F}}V$, where $U(P)$ denote
the universal enveloping algebra of abelian Lie algebra $P$.
Proof Denote
$x^{\underline{c}}=x_{1}^{c_{1}}x_{2}^{c_{2}}\cdots
x_{n}^{c_{n}}\qquad\mbox{for}\
\underline{c}=(c_{1},\cdots,c_{n})\in\mathbb{N}^{n}.$ $None$
Recall that $\Pi_{\mu}$ denotes the weight set of $V$. By (3.10), we have
$(x_{i}\partial_{x_{i}}).(x^{\underline{c}}\otimes
v_{\nu})=(c_{i}+\nu_{i})x^{\underline{c}}\otimes v_{\nu},$ $None$
$(\sum\limits_{i=1}^{n}x_{i}\partial_{x_{i}}).(x^{\underline{c}}\otimes
v_{\nu})=(|\underline{c}|+\sum\limits_{i=1}^{n}\nu_{i})x^{\underline{c}}\otimes
v_{\nu},$ $None$
for any $x^{\underline{c}}\in{\cal A}$ and $\nu\in\Pi_{\mu}$.
Assume that $\\{v_{1},\cdots,v_{\ell}\\}$ is a basis of $V$. For $1\leq
k\in\mathbb{N}$, we set
$U(P)(1\otimes_{\mathbb{F}}V)_{{\langle}k\rangle}=\mbox{Span}_{\mathbb{F}}\\{p_{i_{1}}p_{i_{2}}\cdots
p_{i_{k}}(1\otimes v_{j})\ |\ 1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{k}\leq
n,j\in\overline{1,d}\\},$ $None$ $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}k\rangle}=\mbox{Span}_{\mathbb{F}}\\{x^{\underline{c}}\otimes
v_{j}\ |\ x^{\underline{c}}\in{\cal A},\ |\underline{c}|=k,\
j\in\overline{1,l}\\}.$ $None$
Then (3.14) implies that
$U(P)(1\otimes_{\mathbb{F}}V)=\bigoplus\limits_{k\in\mathbb{N}}(U(P)(1\otimes
V))_{{\langle}k\rangle}.$ $None$
Denote
$\triangle_{i,j}^{k}=\left\\{\begin{array}[]{llll}x_{j}\partial_{x_{i}}&\mbox{if}\;i\neq
j,\\\
\sum\limits_{i=1}^{n}x_{i}\partial_{x_{i}}+x_{i}\partial_{x_{i}}+k-1&\mbox{if}\;i=j,\end{array}\right.$
$None$
By induction, we can easily verify the following fomula:
$\partial_{x_{i}}.p_{i_{1}}p_{i_{2}}\cdots p_{i_{k}}(1\otimes
v_{j})=\sum\limits_{s=1}^{n}p_{i_{1}}p_{i_{2}}\cdots\hat{p}_{i_{s}}\cdots
p_{i_{k}}\triangle_{i,i_{s}}^{k}(1\otimes v_{j}),$ $None$
$\displaystyle(x_{i}\partial_{x_{l}}).p_{i_{1}}p_{i_{2}}\cdots
p_{i_{k}}(1\otimes v_{j})$ $\displaystyle=$
$\displaystyle\sum\limits_{s=1}^{n}\delta_{l,i_{s}}p_{i_{1}}p_{i_{2}}\cdots\hat{p}_{i_{s}}\cdots
p_{i_{k}}p_{i}(1\otimes
v_{j})+p_{i_{1}}p_{i_{2}}\cdots{\cal}{p}_{i_{k}}(1\otimes
E_{i,l}.v_{j}),\hskip 73.97733pt(3.21)$
${p}_{i}.{\cal}{p}_{i_{1}}{p}_{i_{2}}\cdots p_{i_{k}}(1\otimes
v_{j})={p}_{i}p_{i_{1}}p_{i_{2}}\cdots{p}_{i_{k}}(1\otimes v_{j}).$ $None$
So (3.20)-(3.22) imply that $U(P)(1\otimes_{\mathbb{F}}V)$ is an
$L_{n+1}$-submodule of ${\cal A}\otimes_{\mathbb{F}}V$. Furthermore, it is
easy to verify that
$\bigcap\limits_{i=1}^{n}\mbox{Ker}\ \partial_{x_{i}}|_{({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}k\rangle}}=\\{0\\}\ \mbox{for \ any }\
1\leq k\in\mathbb{N}.$ $None$
Thus any non-trivial submodule of $U(P)(1\otimes_{\mathbb{F}}V)$ must contain
$1\otimes V$. The irreducibility of $U(P)(1\otimes_{\mathbb{F}}V)$ follows.
$\Box$
In the rest of this section, we will investigate the condition for ${\cal
A}\otimes_{\mathbb{F}}V=U(P)(1\otimes_{\mathbb{F}}V)$.
It is obvious that ${\cal
A}\otimes_{\mathbb{F}}V=U(P)(1\otimes_{\mathbb{F}}V)$ if and only if $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}k\rangle}=(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}k\rangle}$
holds for any $k\in\mathbb{N}$.
Suppose that $B_{i}$ is an ordered basis for $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}i\rangle}$. Denote by ${P}_{i+1,i}^{j}$ the
matrix of the linear map ${p}_{j}|_{({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}i\rangle}}$ (cf. (3.12)) with respect to
the bases $B_{i}$ and $B_{i+1}$. For any $0<j\in\mathbb{Z}$, we denote
$\Gamma_{j}=\\{\hat{i}=(i_{1},i_{2},...,i_{j})\mid
i_{s}\in\overline{1,n};i_{1}\leq i_{2}\leq\cdots\leq i_{j}\\}.$ $None$
Set
$P_{\hat{i}}={P}_{j,j-1}^{i_{1}}{P}_{j-1,j-2}^{i_{2}}\cdots{P}_{1,0}^{i_{j}}.$
$None$
We order
$\Gamma_{j}=\\{\hat{k}^{1},\hat{k}^{2},...,\hat{k}^{\ell_{j}}\\}$ $None$
lexically. In particular,
$\hat{k}^{1}=(1,1,...,1),\qquad\hat{k}^{\ell_{j}}=(n,n,...,n).$ $None$
Then $(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j\rangle}$ is, as a vector
space, isomorphic to the column space of the $\ell\ell_{j}\times\ell\ell_{j}$
matrix
$M_{j}=[P_{\hat{k}^{1}},P_{\hat{k}^{2}},...,P_{\hat{k}^{\ell_{j}}}]$ $None$
(recall that $\ell=\dim V$). Denote
$I_{1}=\\{1\\}\bigcup\\{i\in\overline{2,n}\ |\ \mu_{i-1}-\mu_{i}\geq 1\\}.$
$None$
By means of the characteristic identity in Corollary 2.2.2, we get the
following necessary but not sufficient condition for the irreducibility of
$L_{n+1}$-module ${\cal A}\otimes_{\mathbb{F}}V$:
Lemma 3.2.2 For any $1\leq k\in\mathbb{N}$. If $\mu_{i}+|\mu|-i+s\neq 0$ for
any $s\in\overline{1,k},\ i\in I_{1}$, then $L_{n+1}$-module ${\cal
A}\otimes_{\mathbb{F}}V$ is irreducible.
Proof The Lemma is based on the following result:
Claim. For any $1\leq k\in\mathbb{N}$. If $\mu_{i}+|\mu|-i+s\neq 0,\ \forall\
s\in\overline{1,k},\ i\in I_{1}$, then $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}k\rangle}=(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}k\rangle}$
holds.
We will prove this claim by induction on $k$. For $k=1$,
$(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}1\rangle}$ is isomorphic as vector
space to the column space of $\ell n\times\ell n$ matrix
$\begin{array}[]{rcl}&&[{P}_{1,0}^{1},{P}_{1,0}^{2},\cdots,{P}_{1,0}^{n}]\end{array}\mbox{with}\
\begin{array}[]{rcl}{P}_{1,0}^{k}&=&\left[\begin{array}[]{c}E_{k,1}|_{V}\\\
E_{k,2}|_{V}\\\ \vdots\\\ (\mbox{I}+E_{k,k})|_{V}\\\ \vdots\\\
E_{k,n}|_{V}\end{array}\right],\end{array}$ $None$
which is exactly $\tilde{\sigma_{2}}|_{V}$ (cf. (2.25)). By Corollary 2.2.2,
the matrix $\tilde{\sigma_{2}}|_{V}$ is diagonalizable and it has full rank if
and only if all its eigenvalues are not zero, i.e.
$m_{i}=\frac{1}{2}(\lambda,\lambda+2\delta)-\frac{1}{2}(\lambda_{i},\lambda_{i}+2(\mu+\delta))=\mu_{i}+|\mu|-i+1\neq
0$ $None$
for any $i\in I_{1}$.
Now suppose that the lemma holds for $k=\iota-1$. Assume $k=\iota$. Note that
the eigenvalues of the matrix
$\left[\begin{array}[]{cccc}(\mbox{I}+E_{1,1}+s-1)|_{V}&E_{1,2}|_{V}&\cdots&E_{1,n}|_{V}\\\
E_{2,1}|_{V}&(\mbox{I}+E_{2,2}+s-1)|_{V}&\cdots&E_{2,n}|_{V}\\\
\vdots&\vdots&\cdots&\vdots\\\
E_{n,1}|_{V}&E_{n,2}|_{V}&\cdots&(\mbox{I}+E_{n,n}+s-1)|_{V}\end{array}\right]$
$None$
are $\mu_{i}+\sum\limits_{j=1}^{n}\mu_{j}-i+s\ (i\in I_{1})$ by (3.31). Thus
it is invertible if and only if $\mu_{i}+|\mu|-i+s\neq 0,\ \forall\ i\in
I_{1}$.
Observe that $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}k\rangle}=(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}k\rangle}$
if and only if the vectors
$\\{p_{i_{1}}\cdots p_{i_{k}}(1\otimes v_{j})\ |\ 1\leq i_{1}\leq\cdots\leq
i_{k}\leq n,j\in\overline{1,\ell}\\}$ $None$
are linearly independent. Equivalently, the corresponding
$\ell\ell_{k}\times\ell\ell_{k}$ system of homogeneous linear equations
$M_{k}X=0$ $None$
has only zero solution by (3.31).
Suppose
$\sum\limits_{1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{k}\leq
n,j\in\overline{1,d}}a_{i_{1},i_{2},\cdots,i_{k}}^{j}p_{i_{1}}p_{i_{2}}\cdots
p_{i_{k}}(1\otimes v_{j})=0,\ a_{i_{1},i_{2},\cdots,i_{k}}^{j}\in\mathbb{F}.$
$None$
It can be written as
$\sum\limits_{1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{k}\leq
n}p_{i_{1}}p_{i_{2}}\cdots p_{i_{k}}(1\otimes
w_{i_{1},i_{2},\cdots,i_{k}})=0,\ w_{i_{1},i_{2},\cdots,i_{k}}\in V.$ $None$
Then
$\partial_{x_{l}}.\sum\limits_{1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{k}\leq
n}p_{i_{1}}p_{i_{2}}\cdots p_{i_{k}}(1\otimes
w_{i_{1},i_{2},\cdots,i_{k}})=0,\ \forall\ l\in\overline{1,n}.$ $None$
Equivalently,
$\sum\limits_{1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{k}\leq
n}\sum\limits_{s=1}^{n}p_{i_{1}}p_{i_{2}}\cdots\hat{p}_{i_{s}}\cdots
p_{i_{k}}\triangle_{l,i_{s}}^{k}(1\otimes w_{i_{1},i_{2},\cdots,i_{k}})=0$
$None$
by (3.20). Moreover, it can be written as the form
$\sum\limits_{1\leq j_{1}\leq j_{2}\leq\cdots\leq j_{k-1}\leq
n}p_{j_{1}}p_{j_{2}}\cdots p_{j_{k-1}}(1\otimes
u_{j_{1},j_{2},\cdots,j_{k-1}})=0,$ $None$
where
$\displaystyle 1\otimes u_{j_{1},j_{2},\cdots,j_{k-1}}$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{j_{1}}\triangle_{l,i}^{k}(1\otimes
w_{i,j_{1},j_{2},\cdots,j_{k-1}})+\sum\limits_{i=j_{1}}^{j_{2}}\triangle_{l,i}^{k}(1\otimes
w_{j_{1},i,j_{2},\cdots,j_{k-1}})+\cdots$
$\displaystyle+\sum\limits_{i=j_{s-1}}^{j_{s}}\triangle_{l,i}^{k}(1\otimes
w_{j_{1},j_{2},\cdots,j_{s-1},i,j_{s},\cdots j_{k-1}})+\cdots$
$\displaystyle+\sum\limits_{i=j_{k-1}}^{n}\triangle_{l,i}^{k}(1\otimes
w_{j_{1},j_{2},\cdots,j_{k-1},i})=N_{k-1}\varpi_{n}\hskip 142.26378pt(3.40)$
with
$N_{k-1}={\small\left[\begin{array}[]{cccc}\sum\limits_{i=1}^{n}x_{i}\partial_{x_{i}}+x_{1}\partial_{x_{1}}+k-1&x_{2}\partial_{x_{1}}&\cdots&x_{n}\partial_{x_{1}}\\\
x_{1}\partial_{x_{2}}&\sum\limits_{i=1}^{n}x_{i}\partial_{x_{i}}+x_{2}\partial_{x_{2}}+k-1&\cdots&x_{n}\partial_{x_{2}}\\\
\vdots&\vdots&\cdots&\vdots\\\
x_{1}\partial_{x_{n}}&x_{2}\partial_{x_{n}}&\cdots&\sum\limits_{i=1}^{n}x_{i}\partial_{x_{i}}+x_{n}\partial_{x_{n}}+k-1\end{array}\right]}$
$None$
and
$\varpi_{n}=\left[\begin{array}[]{c}X_{1,j_{1}-1}\\\ X_{j_{1},j_{2}-1}\\\
X_{j_{2},j_{3-1}}\\\ \vdots\\\ X_{j_{k-1},n}\end{array}\right],$ $None$
in which
$X_{1,j_{1}-1}=\left[\begin{array}[]{c}1\otimes
w_{1,j_{1},j_{2},\cdots,j_{k-1}}\\\ 1\otimes
w_{2,j_{1},j_{2},\cdots,j_{k-1}}\\\ \vdots\\\ 1\otimes
w_{j_{1}-1,j_{1},j_{2},\cdots,j_{k-1}}\end{array}\right],\;\;X_{j_{1},j_{2}-1}=\left[\begin{array}[]{c}2\otimes
w_{j_{1},j_{1},j_{2},\cdots,j_{k-1}}\\\ 1\otimes
w_{j_{1},j_{1}+1,j_{2},\cdots,j_{k-1}}\\\ \vdots\\\ 1\otimes
w_{j_{1},j_{2}-1,j_{2},\cdots,j_{k-1}}\end{array}\right],$ $None$
$X_{j_{2},j_{3-1}}=\left[\begin{array}[]{c}2\otimes
w_{j_{1},j_{2},j_{2},\cdots,j_{k-1}}\\\ 1\otimes
w_{j_{1},j_{2},j_{2}+1,\cdots,j_{k-1}}\\\ \vdots\\\ 1\otimes
w_{j_{1},j_{2},j_{3}-1,\cdots,j_{k-1}}\end{array}\right],\cdots,\;\;X_{j_{k-1},n}=\left[\begin{array}[]{c}2\otimes
w_{j_{1},j_{2},\cdots,j_{k-1},j_{k-1}}\\\ 1\otimes
w_{j_{1},j_{2},\cdots,j_{k-1},j_{k-1}+1}\\\ \vdots\\\ 1\otimes
w_{j_{1},j_{2},\cdots,j_{k-1},n}\end{array}\right].$ $None$
We know that $N_{k-1}\varpi_{n}=0$ has only zero solution if and only if
$\mu_{i}+|\mu|-i+k\neq 0,\ \forall\ i\in I_{1}$. By Lemma 3.2.1 and the claim,
the Lemma is followed. $\Box$
Next, we study the structure
$(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j\rangle}$ as an
$\bar{L}_{n}$-module and finally get a necessary and sufficient condition for
the irreducibility of $L_{n+1}$-module ${\cal A}\otimes_{\mathbb{F}}V$ (cf.
Proposition 3.2.5 ).
Note that $\bar{L}_{n}\simeq gl(n)$ ( resp. $\bar{L}_{n}^{\prime}\simeq sl(n)$
) according to (3.5). Obviously, (3.12) implies that as
$\overline{L}_{n}$-module $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}$ is isomorphic to tensor product
module $V(j\epsilon_{1})\otimes_{\mathbb{F}}V$
$({\cal A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}\cong
V(j\varepsilon_{1})\otimes_{\mathbb{F}}V(\mu)=\bigoplus_{\underline{c}\in
I(\mu,j)}V(\mu+\underline{c})$ $None$
(cf. Lemma 2.1.3). Denote
$p^{\underline{c}}=p_{1}^{c_{1}}p_{2}^{c_{2}}\cdots
p_{n}^{c_{n}}\qquad\mbox{for}\
\underline{c}=(c_{1},\cdots,c_{n})\in\mathbb{N}^{n}.$ $None$
Define the linear map
$\varphi_{j}:({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}\rightarrow(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j\rangle}$
$\sum\limits_{|\underline{l}|=j,\
q\in\overline{1,l}}a_{\underline{l},q}x^{\underline{l}}\otimes
v_{q}\mapsto\sum\limits_{|\underline{l}|=j,\
q\in\overline{1,l}}a_{\underline{l},q}p^{\underline{l}}.(1\otimes v_{q}).$
$None$
It is easy to verify that $\varphi_{j}$ is an $\overline{L}_{n}$-module
homomorphism by (3.21).
For any $\underline{c}\in I(\mu,j)$ (cf. (2.11)), let $\xi_{\underline{c}}$ be
a maximal vector for highest weight module
$V(\mu+\underline{c})\subseteq({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}$. Then
$\hat{\xi}_{\underline{c}}=\varphi_{j}(\xi_{\underline{c}})$ $None$
is also a maximal vector of
$V(\mu+\underline{c})\subseteq(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j\rangle}$.
Write
$\hat{\xi}_{\underline{c}}=q_{\underline{c}}\xi_{\underline{c}},\;\ \
q_{\underline{c}}\in\mathbb{F}.$ $None$
We know that
$(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j\rangle}\bigcap
V(\mu+\underline{c})\neq\\{0\\}\Rightarrow
V(\mu+\underline{c})\subseteq(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j\rangle}.$
$None$
So
$(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j\rangle}\bigcap
V(\mu+\underline{c})\neq\\{0\\}\qquad\mbox{iff}\qquad q_{\underline{c}}\neq
0.$ $None$
In the following Lemma, we will calculate $q_{\underline{c}}$ explicitly:
Lemma 3.2.3 For any $\underline{c}\in I(\mu,j)$, we have
$q_{\underline{c}}=\prod\limits_{s=1}^{n}\prod\limits_{i=1}^{c_{s}}(\mu_{s}+|\mu|-s+i),$
$None$
where we treat $\prod\limits_{i=1}^{c_{s}}(\mu_{s}+|\mu|-s+i)=1$ if $c_{s}=0$.
Proof Let $v_{\mu}$ be a highest weight vector of the given $gl(n)$-module
$V$.
For any $\underline{c}\in I(\mu,j)$, we write a maximal vector
$\xi_{\underline{c}}$ for highest weight module $V(\mu+\underline{c})$ as
$\xi_{\underline{c}}=a_{\underline{c}}x^{{\underline{c}}}\otimes
v_{\mu}+\sum\limits_{(\nu,\underline{l})\neq(\mu,\underline{c}),{\underline{c}}+\mu={\underline{l}}+\nu,\
i\in\overline{1,m({\nu})}}a_{\nu,\underline{l}}^{i}x^{\underline{l}}\otimes
v_{\nu}^{i},\qquad 0\neq a_{\underline{c}},\
a_{\nu,\underline{l}}^{i}\in\mathbb{F},$ $None$
where $m({\nu})$ denotes the multiplicity of weight $\nu\in\Pi_{\mu}$.
Case 1. $c_{n}\neq 0$.
Note that $[x_{s}\partial_{x_{t}},\partial_{x_{n}}]=0,\ \forall\ 1\leq s<t\leq
n$. We know
$0\neq\partial_{x_{n}}.\xi_{\underline{c}}=a_{\underline{c}}c_{n}x^{{\underline{c}}-\varepsilon_{n}}\otimes
v_{\mu}+\sum\limits_{(\nu,\underline{l})\neq(\mu,\underline{c}),{\underline{c}}+\mu={\underline{l}}+\nu,\
i\in\overline{1,m({\nu})}}a_{\nu,\underline{l}}^{i}l_{n}x^{\underline{l}-\varepsilon_{n}}\otimes
v_{\nu}^{i}$ $None$
is a maximal vector for the $\overline{L}_{n}$\- highest weight module
$V(\mu+\underline{c}-\varepsilon_{n})$. Set
$\xi_{\underline{c}-\varepsilon_{n}}=\partial_{x_{n}}.\xi_{\underline{c}}.$
$None$
Obviously,
$\hat{\xi}_{\underline{c}}=a_{\underline{c}}p^{{\underline{c}}}.(1\otimes
v_{\mu})+\sum\limits_{(\nu,\underline{l})\neq(\mu,\underline{c}),{\underline{c}}+\mu={\underline{l}}+\nu,\
i\in\overline{1,m({\nu})}}a_{\nu,\underline{l}}^{i}p^{\underline{l}}.(1\otimes
v_{\nu}^{i}),$ $None$
$\hat{\xi}_{\underline{c}-\varepsilon_{n}}=\varphi_{j}(\partial_{x_{n}}.\xi_{\underline{c}})=a_{\underline{c}}c_{n}p^{{\underline{c}}-\varepsilon_{n}}.(1\otimes
v_{\mu})+\sum\limits_{(\nu,\underline{l})\neq(\mu,\underline{c}),{\underline{c}}+\mu={\underline{l}}+\nu,\
i\in\overline{1,m({\nu})}}a_{\nu,\underline{l}}^{i}l_{n}p^{\underline{l}-\varepsilon_{n}}.(1\otimes
v_{\nu}^{i}).$ $None$
Write
$\partial_{x_{n}}.\hat{\xi}_{\underline{c}}=b_{\underline{c}}(n)\hat{\xi}_{\underline{c}-\varepsilon_{n}},\qquad
b_{\underline{c}}(n)\in\mathbb{F}.$ $None$
By (3.21) and (3.54), we have
$\displaystyle\partial_{x_{n}}.\hat{\xi}_{\underline{c}}=\partial_{x_{n}}.\varphi_{j}(\xi_{\underline{c}})$
$\displaystyle=$ $\displaystyle
c_{n}a_{\underline{c}}p^{{\underline{c}}-\varepsilon_{n}}\Delta_{n,n}^{j}(1\otimes
v_{\mu})$
$\displaystyle+\sum\limits_{(\nu,\underline{l})\neq(\mu,\underline{c}),{\underline{c}}+\mu={\underline{l}}+\nu,\
i\in\overline{1,m({\nu})}}a_{\nu,\underline{l}}^{i}[\sum\limits_{s=1}^{n-1}l_{s}p^{\underline{l}-\varepsilon_{s}}(1\otimes
E_{s,n}v_{\nu}^{i})+l_{n}p^{\underline{l}-\varepsilon_{n}}\Delta_{n,n}^{j}(1\otimes
v_{\nu}^{i})].\hskip 28.45274pt(3.59)$
Denote
$E_{s,n}.v_{\mu+\varepsilon_{n}-\varepsilon_{s}}^{i}=\Im_{i}v_{\mu},\qquad\Im_{i}\in\mathbb{F}.$
$None$
Then, we have
$b_{\underline{c}}(n)=\frac{c_{n}a_{\underline{c}}(j-1+\mu_{n}+|\mu|)+\sum\limits_{s=1}^{n-1}(1+c_{s})\sum\limits_{i=1}^{m(\mu-\varepsilon_{s}+\varepsilon_{n})}a_{\mu-\varepsilon_{s}+\varepsilon_{n},\underline{c}+\varepsilon_{s}-\varepsilon_{n}}^{i}\Im_{i}}{c_{n}a_{\underline{c}}}$
$None$
by (3.58)-(3.61). For any $s\in\overline{1,n-1}$, we get
$\displaystyle 0=x_{s}\partial_{x_{n}}.\xi_{\underline{c}}$ $\displaystyle=$
$\displaystyle
c_{n}a_{\underline{c}}x^{\underline{c}+\varepsilon_{s}-\varepsilon_{n}}\otimes
v_{\mu}+\sum\limits_{(\nu,\underline{l})\neq(\mu,\underline{c}),{\underline{c}}+\mu={\underline{l}}+\nu,\
i\in\overline{1,m({\nu})}}a_{\nu,\underline{l}}^{i}[l_{n}x^{\underline{l}+\varepsilon_{s}-\varepsilon_{n}}\otimes
v_{\nu}^{i}+x^{\underline{l}}\otimes E_{s,n}.v_{\nu}^{i}].\hskip
5.69046pt(3.62)$
Therefore,
$c_{n}a_{\underline{c}}+\sum\limits_{i=1}^{m(\mu-\varepsilon_{s}+\varepsilon_{n})}a_{\mu-\varepsilon_{s}+\varepsilon_{n},\underline{c}+\varepsilon_{s}-\varepsilon_{n}}^{i}\Im_{i}=0$
$None$
Hence, we have
$b_{\underline{c}}(n)=c_{n}-n+\mu_{n}+|\mu|.$ $None$
Furthermore, by (3.55), we obtain
$\partial_{x_{n}}.\hat{\xi}_{\underline{c}}=q_{\underline{c}}\partial_{x_{n}}.\xi_{\underline{c}}=q_{\underline{c}}\xi_{\underline{c}-\varepsilon_{n}}=b_{\underline{k}}(n)\hat{\xi}_{\underline{c}-\varepsilon_{n}}=b_{\underline{k}}(n)q_{\underline{c}-\varepsilon_{n}}\xi_{\underline{c}-\varepsilon_{n}}$
$None$
which implies that
$q_{\underline{c}}=b_{\underline{c}}(n)q_{\underline{c}-\varepsilon_{n}}$
$None$
Case 2. $c_{n}=0$.
Suppose $c_{n-1}\neq 0$. We claim that $l_{n}=0$ for any $\underline{l}$ of
$a_{\nu,\underline{l}}^{i}\neq 0$ in (3.54). Indeed, we can write
$(\nu_{1}-\nu_{2},\cdots,\nu_{n-1}-\nu_{n})=\sum\limits_{i=1}^{n-1}(\mu_{i}-\mu_{i+1})\varpi_{i}-\sum\limits_{i=1}^{n-1}k_{i}\alpha_{i}.$
$None$
Since $\nu_{n}+l_{n}=\mu_{n}+c_{n}$, we have
$l_{n}=\mu_{n}-\nu_{n}=\mu_{n}-(\mu_{n}+k_{n-1})$ by (2.9). Thus we get
$l_{n}+k_{n-1}=0$. So $l_{n}=0$.
Hence,
$x_{n-1}\partial_{x_{n}}.\partial_{x_{n-1}}.\xi_{\underline{c}}=-\partial_{x_{n}}.\xi_{\underline{c}}+\partial_{x_{n-1}}.x_{n-1}\partial_{x_{n}}.\xi_{\underline{c}}=0.$
$None$
This implies that $\partial_{x_{n-1}}.\xi_{\underline{c}}$ is a maximal vector
for the highest weight $\overline{L}_{n}$-module
$V(\mu+\underline{c}-\varepsilon_{n-1})$. Repeating the process from (3.56) to
(3.64), we get
$q_{\underline{c}}=(\mu_{n-1}+|\mu|-(n-1)+c_{n-1})q_{\underline{c}-\varepsilon_{n-1}}.$
$None$
Then induction implies that
$q_{\underline{c}}=\prod\limits_{s=1}^{n}\prod\limits_{i=1}^{c_{s}}(\mu_{s}+|\mu|-s+i).$
$None$
Thus the lemma is proved. $\Box$
By Lemma 3.2.3, we know that
$(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j\rangle}=({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}$ iff $q_{\underline{c}}\neq 0$
for any $\underline{c}\in I(\mu,j),\ 0<j\in\mathbb{N}$. Thus we get the
following sufficient and necessary condition for the irreducibility of
$L_{n+1}$-module ${\cal A}\otimes_{\mathbb{F}}V$:
Proposition 3.2.4 The $L_{n+1}$-module ${\cal A}\otimes_{\mathbb{F}}V$ is
irreducible iff $\ q_{\underline{c}}\neq 0$ for any $\underline{c}\in
I(\mu,j),\ 0<j\in\mathbb{N}$.
### 3.3 Jordan-Holder Series for $L_{n+1}$-module ${\cal
A}\otimes_{\mathbb{F}}V$
In this section, we will assume that ${\cal A}\otimes_{\mathbb{F}}V\neq
U(P)(1\otimes_{\mathbb{F}}V)$ and prove the irreducibility of the quotient
module $({\cal A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)$.
First, we study the structure of $({\cal
A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)$ as an
$\overline{L}_{n}$-module (cf. Lemma 3.3.1 and Lemma 3.3.2). Recall the
$\overline{L}_{n}$-module homomorphism $\varphi_{j}$ defined by (3.47). Denote
$\mbox{Ker}(\varphi_{j})$ by ${\cal{R}}_{{\langle}j\rangle}$ and set
$I(\mu,j)^{\prime}=\\{\underline{c}\in I(\mu,j)\ |\ \ q_{\underline{c}}=0\\}.$
$None$
According to Lemma 3.2.3, we have the following result:
Lemma 3.3.1 For any $1\leq j\in\mathbb{N}$, we have
${\cal{R}}_{{\langle}j\rangle}=\bigoplus\limits_{\underline{c}\in
I(\mu,j)^{\prime}}V(\mu+\underline{c})$ and the following direct sum of
submodules for $\overline{L}_{n}$: $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}=(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j\rangle}\oplus{\cal{R}}_{{\langle}j\rangle}.$
Let $1\leq k\in\mathbb{N}$. Suppose that
$({\cal A}\otimes_{\mathbb{F}}V)_{{\langle}i\rangle}=(U(P)(1\otimes
V))_{{\langle}i\rangle}\;\;\mbox{for any }\ \;i\in\overline{0,k},$ $None$
but
$({\cal A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}\neq(U(P)(1\otimes
V))_{{\langle}j\rangle}\;\;\mbox{when}\ j\geq k+1.$ $None$
Then by Lemma 3.3.1, as $\overline{L}_{n}$-modules,
$({\cal
A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)\cong\bigoplus\limits_{j=k+1}^{\infty}{\cal{R}}_{{\langle}j\rangle}.$
$None$
Lemma 3.3.2 Let $1\leq k\in\mathbb{N}$ such that (3.72) and (3.73) hold. Then
$\bar{L}_{n}$-module ${\cal{R}}_{{\langle}k+1\rangle}$ is irreducible.
Proof Since $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}k\rangle}=(U(P)(1\otimes
V))_{{\langle}k\rangle}$, Lemma 3.2.3 implies
$q_{\underline{c}}\neq 0\qquad\forall\ \underline{c}\in I(\mu,k).$ $None$
For convenience, we denote
$q_{s}(\underline{c})=\prod\limits_{i=1}^{c_{s}}(\mu_{s}+|\mu|-s+i).$ $None$
So
$q_{\underline{c}}=\prod\limits_{s=1}^{n}q_{s}(\underline{c}).$ $None$
Assume $\underline{m}\in I(\mu,k+1)^{\prime}$, i.e. $q_{\underline{m}}=0$.
Since the $gl(n)$-modules
$V((k+1)\varepsilon_{1})\otimes_{\mathbb{F}}V\simeq({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}k+1\rangle}\subset
V(\varepsilon_{1})\otimes_{\mathbb{F}}({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}k\rangle}\simeq
V(\varepsilon_{1})\otimes_{\mathbb{F}}(V(k\varepsilon_{1})\otimes_{\mathbb{F}}V)$
$None$
in the sense of monomorphism, we know that
$\exists\quad\underline{t}\in{\mathbb{N}}^{n}\ \mbox{and}\
r\in\overline{1,n},\ \mbox{such \ that}\ \underline{t}\in I(\mu,k)\
\mbox{and}\ \underline{m}=\underline{t}+\varepsilon_{r}.$ $None$
Therefore, $m_{r}=t_{r}+1$. The fact $q_{\underline{m}}=0$ implies that
$q_{r}(\underline{m})=0$ because
$q_{s}(\underline{m})=q_{s}(\underline{t})\neq 0$ for $s\neq r$. Then
$q_{r}(\underline{t})\neq 0$ and
$q_{r}(\underline{m})=0=q_{r}(\underline{t})(\mu_{r}+|\mu|-r+t_{r}+1)$ imply
that
$\mu_{r}+|\mu|-r+t_{r}+1=0=\mu_{r}+|\mu|-r+m_{r}.$ $None$
Obviously, $1\leq m_{r}=t_{r}+1\leq|\underline{m}|=k+1$ and $\underline{m}\in
I(\mu,k+1)^{\prime}\subset I(\mu,k+1)$ imply that
$m_{r}\leq\mu_{r-1}-\mu_{r}$ $None$
by (2.11). Assume $1\leq m_{r}<k$. Then by (2.11) and (3.81), we know there
exists some $\underline{l}\in{\mathbb{N}}^{n}$ satisfying $l_{r}=m_{r}$ and
$\underline{l}\in I(\mu,k)$. So $q_{r}(\underline{l})=q_{r}(\underline{m})=0$.
Furthermore, $q_{\underline{l}}=0$, which contradicts (3.75). Therefore,
$m_{r}=k+1$, i.e. $\underline{m}=(k+1)\epsilon_{r}$.
Suppose there exists another $(k+1)\varepsilon_{s}\in I(\mu,k+1)^{\prime}$ but
$s\neq r$. Then $0=\mu_{r}+|\mu|-r+k+1=\mu_{s}+|\mu|-s+k+1$. This is
impossible, since $\mu_{s}\geq\mu_{r}$ whenever $s<r$. Thus we prove that
$|I(\mu,k+1)^{\prime}|=1$, i.e. $\bar{L}_{n}$-module
${\cal{R}}_{{\langle}k+1\rangle}$ is irreducible. $\Box$
Set
$I_{s}=\\{1\\}\bigcup\\{j\in\overline{2,n}\ |\ \mu_{j-1}-\mu_{j}\geq s\\}.$
$None$
By the above two lemmas, we can give the proof of (i) in the Main Theorem:
Proposition 3.3.3 The vector space ${\cal A}\otimes_{\mathbb{F}}V$ is an
irreducible $sl(n+1)$-module if and only if
$\forall\ 1\leq s\in\mathbb{N},\ \mu_{i}+|\mu|-i+s\neq 0\qquad\mbox{for\
any}\;\;i\in I_{s}.$ $None$
Proof Assume that there exist $1\leq s\in\mathbb{N}$ and $i\in I_{s}$
satisfying $\mu_{i}+|\mu|-i+s=0$. Then
$V(\mu+s\varepsilon_{i})\subseteq{\cal{R}}_{{\langle}s\rangle}$ by (2.11),
(3.71), (3.82) and Lemma 3.3.1. Hence, $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}s\rangle}\neq(U(P)(1\otimes
V))_{{\langle}s\rangle}$, i.e. $sl(n+1)$-module ${\cal
A}\otimes_{\mathbb{F}}V$ is reducible.
Suppose that $sl(n+1)$-module ${\cal A}\otimes_{\mathbb{F}}V$ is reducible.
Let $k$ satisfying (3.72) and (3.73). By Lemma 3.3.2, we have
$V(\mu+(k+1)\epsilon_{s})={\cal{R}}_{{\langle}k+1\rangle}$ for some $s\in
I_{k+1}$. So Lemma 3.2.3 implies $\mu_{s}+|\mu|-s+k+1=0$. $\Box$
Remark 3.3.4 The condition (3.83) is equivalent to (1.7) and (1.8) given in
the Main Theorem.
In the rest of this section, we study the relationship between any
$\bar{L}_{n}$-module ${\cal{R}}_{{\langle}j\rangle}$ and $\bar{L}_{n}$-module
${\cal{R}}_{{\langle}j+1\rangle}$ for any $j\geq k+1$ based on the
decomposition of tensor module and the projection operator techniques for
$gl(n)$ (Recall Lemma 2.1.3 and projection operators appeared in Section 2.2).
And we finally prove the irreducibility of quotient module $({\cal
A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)$ (cf. Proposition 3.3.8)
.
Lemma 3.3.5 Let $k+1\leq s\in\mathbb{N}$. For $\nu=\mu+\underline{l}$ with
$\underline{l}\in I(\mu,s+1)^{\prime}$, there exist
$\nu^{\prime}=\mu+\underline{m}$ with $\underline{m}\in I(\mu,s)^{\prime}$ and
$r\in\overline{1,n}$ such that $\nu=\nu^{\prime}+\varepsilon_{r}$ and
$\mu_{r}+|\mu|-r+m_{r}+1\neq 0$.
Proof Suppose $\nu=\mu+\underline{l}$ with $\underline{l}\in
I(\mu,s+1)^{\prime}$. Then $q_{\underline{l}}=0$.
Claim. There exists $r\in\overline{1,n}$ such that $\mu_{r}+|\mu|-r+l_{r}\neq
0$ and $l_{r}\geq 1$.
Set
$I_{\underline{l}}=\\{t\in\overline{1,n}\ |\ \mu_{t}+|\mu|-t+l_{t}=0\\}.$
$None$
It follows that $|I_{\underline{l}}|=0,1$. Otherwise,
$\mu_{t}+|\mu|-t+l_{t}=0=\mu_{q}+|\mu|-q+l_{q}$ for some $t<q$. This is
impossible because $\mu+\underline{l}$ is a highest weight implies that
$l_{t}+\mu_{t}\geq l_{q}+\mu_{q}$ whenever $t<q$. It is obvious that the claim
holds when $|I_{\underline{l}}|=0$.
Now assume $|I_{\underline{l}}|=1$ and $r_{0}\in I_{\underline{l}}$. If the
claim does not hold, then $\underline{l}=(s+1)\varepsilon_{r_{0}}$ and
$\mu_{t}+|\mu|-t\neq 0$ for any $t\neq r_{0}$. From Lemma 3.3.2, we know
${\cal{R}}_{{\langle}k+1\rangle}=V(\mu+(k+1)\varepsilon_{t})$ for some $t\in
I_{k+1}$. Thus, Lemma 3.2.3 implies
$q_{(k+1)\epsilon_{t}}=\prod\limits_{i=1}^{k+1}(\mu_{t}+|\mu|-t+i)=0$. Assume
$\mu_{t}+|\mu|-t+r=0$ for some $r\in\overline{1,k+1}$. On the other hand,
$\mu_{r_{0}}+|\mu|-r_{0}+s+1=0$ by (3.84) due to $r_{0}\in I_{\underline{l}}$.
Hence, we have $\mu_{r_{0}}-\mu_{t}=r_{0}+r-(t+s+1)$. If $r_{0}\leq t$, then
$\mu_{r_{0}}-\mu_{t}\geq 0$; which contradicts
$r_{0}+r-(t+s+1)=r_{0}-t+r-(s+1)<0$. If $r_{0}>t$, then
$\mu_{t}+l_{t}=\mu_{t}\geq\mu_{r_{0}}+l_{r_{0}}=\mu_{r_{0}}+s+1$ because
$\mu+\underline{l}=\mu+(s+1)\varepsilon_{r_{0}}$ is a highest weight.
Therefore, $\mu_{t}-\mu_{r_{0}}\geq s+1$; i.e. $t+s+1-(r_{0}+r)\geq s+1$.
Hence, $r_{0}-t+r\leq 0$. A contradiction arises. Thus the claim holds.
Suppose that $r$ satisfies the claim. Take $\nu^{\prime}=\nu-\varepsilon_{r},\
\underline{m}=\underline{l}-\varepsilon_{r}$. We claim that $\underline{m}\in
I(\mu,s)^{\prime}$. In fact, $\underline{m}\in I(\mu,s)$ by (2.11) because
$l_{r}-1\leq s$ and $l_{r}\leq\mu_{r-1}-\mu_{r}$ implies
$l_{r}-1<\mu_{r-1}-\mu_{r}$. Furthermore,
$q_{\underline{l}}=(\mu_{r}+|\mu|-r+l_{r})q_{\underline{m}}=0$ implies that
$q_{\underline{m}}=0$. Therefore, $\underline{m}\in I(\mu,s)^{\prime}$. Thus
the Lemma follows. $\Box$
Lemma 3.3.6 We have
$\partial_{l}({\cal{R}}_{{\langle}j\rangle})\not\subseteq(U(P)(1\otimes
V))_{{\langle}j-1\rangle}$ for any $l\in\overline{1,n}$ and $j>k+1.$
Proof Assume
$\partial_{l}({\cal{R}}_{{\langle}j\rangle})\subseteq(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j-1\rangle}\
\mbox{for \ some}\ l\in\overline{1,n}\;\mbox{and}\;j>k+1.$ $None$
By (3.11), we know that
$\partial_{l}(U(P)(1\otimes_{\mathbb{F}}V)_{{\langle}j\rangle})\subseteq(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j-1\rangle}.$
$None$
Then (3.85), (3.86) and Lemma 3.3.1 imply that
$\partial_{l}(({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle})\subseteq(U(P)(1\otimes
V))_{{\langle}j-1\rangle},$ $None$
that is,
$\partial_{l}(({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle})\varsubsetneq({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j-1\rangle}.$ $None$
This contradicts the fact
$\partial_{l}(({\cal A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle})=({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j-1\rangle}\;\;\ \mbox{for any}\
l\in\overline{1,n}.$ $None$
Thus the lemma follows. $\Box$
Let $\\{e_{1}^{\prime},\cdots,e_{n}^{\prime}\\}$ be a basis for
$\overline{L}_{n}$-module $V(\varepsilon_{1})$. For any $1\leq
j\in\mathbb{N}$, we define the following linear map:
$T_{j}:V(\varepsilon_{1})\otimes_{\mathbb{F}}({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}\rightarrow({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j+1\rangle}$ $T_{j}(e_{i}^{\prime}\otimes
v)=p_{i}.v,\ \forall\ v\in({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}.$ $None$
Lemma 3.3.7 The linear map $\ T_{j}\ $ is an intertwining operator from the
tensor module $V(\varepsilon_{1})\otimes_{\mathbb{F}}({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}$ to $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j+1\rangle}$ for $\overline{L}_{n}$.
Proof For any $s,t,i\in\overline{1,n}$ and $v\in({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}$, we have
$\displaystyle T_{j}(x_{s}\partial_{x_{t}}.(e_{i}^{\prime}\otimes v))$
$\displaystyle=$ $\displaystyle
T_{j}(x_{s}\partial_{x_{t}}(e_{i}^{\prime})\otimes v+e_{i}^{\prime}\otimes
x_{s}\partial_{x_{t}}.v)=T_{j}(\delta_{t,i}e_{s}^{\prime}\otimes
v+e_{i}^{\prime}\otimes x_{s}\partial_{x_{t}}.v)$ $\displaystyle=$
$\displaystyle\delta_{t,i}p_{s}.v+p_{i}.x_{s}\partial_{x_{t}}.v=x_{s}\partial_{x_{t}}.p_{i}.v=x_{s}\partial_{x_{t}}.T_{j}(e_{i}^{\prime}\otimes
v)\hskip 145.10922pt(3.91)$
by (3.21). Thus the lemma follows. $\Box$
Based on the Lemma 3.3.5, Lemma 3.3.6 and Lemma 3.3.7, we can prove (ii) of
the Main Theorem in the following:
Proposition 3.3.8 If ${\cal A}\otimes_{\mathbb{F}}V\neq
U(P)(1\otimes_{\mathbb{F}}V)$, then $\\{0\\}\subset
U(P)(1\otimes_{\mathbb{F}}V)\subset{\cal A}\otimes_{\mathbb{F}}V$ is a Jordan-
Holder Series for $L_{n+1}$-module ${\cal A}\otimes_{\mathbb{F}}V$.
Proof Suppose that $W+U(P)(1\otimes_{\mathbb{F}}V)$ is any nonzero submodule
of quotient module $({\cal
A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)$, where
$W=\bigoplus\limits_{j\geq k+1}W\bigcap{\cal{R}}_{{\langle}j\rangle}$ is a
weighted subspace of ${\cal A}\otimes_{\mathbb{F}}V$. By Lemma 3.3.6, we get
$W\bigcap{\cal{R}}_{{\langle}k+1\rangle}\neq\\{0\\}$.
By Lemma 3.3.5, we know that for any $s\geq k+1$ and $\nu=\mu+\underline{l}$
with $\underline{l}\in I(\mu,s+1)^{\prime}$, there exist
$\nu^{\prime}=\mu+\underline{m}$ with $\underline{m}\in I(\mu,s)^{\prime}$ and
$r\in\overline{1,n}$ such that $\nu=\nu^{\prime}+\varepsilon_{r}$ and
$\mu_{r}+|\mu|-r+m_{r}+1\neq 0$. Therefore, the highest weight module
$V(\nu)\;(\subseteq{\cal{R}}_{{\langle}s+1\rangle})$ of highest weight $\nu$
appears in the decomposition of $\overline{L}_{n}$-tensor module
$V(\varepsilon_{1})\otimes_{\mathbb{F}}V(\nu^{\prime})$ ($\subseteq
V(\varepsilon_{1})\otimes_{\mathbb{F}}{\cal{R}}_{{\langle}s\rangle}\subseteq
V(\varepsilon_{1})\otimes_{\mathbb{F}}({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}s\rangle}$).
Claim. There exists some maximal vector $v_{\nu}$ (resp.
$\xi_{\underline{m}+\varepsilon_{r}}$ ) of highest weight module
$V(\nu)\subseteq V(\varepsilon_{1})\otimes_{\mathbb{F}}V(\nu^{\prime})$ (resp.
$V(\nu)\subseteq{\cal{R}}_{{\langle}s+1\rangle}$ ) satisfying
$T_{s}(v_{\nu})=(\mu_{r}+|\mu|-r+m_{r}+1)\xi_{\underline{m}+\varepsilon_{r}}\neq
0.$ $None$
Assume that $w_{\nu}$ is a maximal vector of irreducible module
$V(\nu)\subseteq V(\varepsilon_{1})\otimes_{\mathbb{F}}V(\nu^{\prime})$. Since
$T_{s}:V(\varepsilon_{1})\otimes_{\mathbb{F}}({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}s\rangle}\rightarrow({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}s+1\rangle}$ is an intertwining operator
for $\overline{L}_{n}$, we know $T_{s}(w_{\nu})$ is also a maximal vector of
$\overline{L}_{n}$-module
$V(\nu)\subseteq{\cal{R}}_{{\langle}s+1\rangle}\;(\subseteq({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}s+1\rangle})$. Since the maximal vector
$w_{\nu}$ must take the following form:
$w_{\nu}=a_{r}e_{r}^{\prime}\otimes\varpi_{\nu^{\prime}}+\sum_{j=1}^{r-1}\sum_{i=1}^{m(\nu^{\prime}-\varepsilon_{j}+\varepsilon_{r})}a_{j}^{i}e_{j}^{\prime}\otimes
v_{\nu^{\prime}-\varepsilon_{j}+\varepsilon_{r}}^{i},\qquad 0\neq a_{r},\
a_{j}^{i}\in\mathbb{F},$ $None$
where $\varpi_{\nu^{\prime}}$ is a maximal vector of the highest weight module
$V(\nu^{\prime})$ and the set
$\\{v_{\nu^{\prime}-\varepsilon_{j}+\varepsilon_{r}}^{i}\mid
i\in\overline{1,m(\nu^{\prime}-\varepsilon_{j}+\varepsilon_{r}))}\\}$ is a
basis of the weight subspace
$[V(\nu^{\prime})]_{\nu^{\prime}-\varepsilon_{j}+\varepsilon_{r}}$. For
convenience, we write
$v_{\nu^{\prime}-\varepsilon_{j}+\varepsilon_{r}}^{i}=\sum\limits_{\varepsilon_{j}-\varepsilon_{r}=\sum\limits_{k=1}^{m}i_{k}(\varepsilon_{s_{k}}-\varepsilon_{t_{k}});\>s_{i}>t_{i}}a_{\underline{s},\underline{t}}^{\underline{i}}(x_{s_{1}}\partial_{t_{1}})^{i_{1}}.\cdots.(x_{s_{m}}\partial_{t_{m}})^{i_{m}}.\varpi_{\nu^{\prime}}.$
$None$
Therefore,
$T_{s}(w_{\nu})=a_{r}p_{r}.\varpi_{\nu^{\prime}}+\sum_{j=1}^{r-1}\sum_{i=1}^{m(\nu^{\prime}-\varepsilon_{j}+\varepsilon_{r})}a_{j}^{i}p_{j}.v_{\nu^{\prime}-\varepsilon_{j}+\varepsilon_{r}}^{i}.$
$None$
So (3.94) and (3.95) imply that
$T_{s}(w_{\nu})=n_{\varepsilon_{r}}.\varpi_{\nu^{\prime}}\qquad\mbox{for \
some}\ n_{\varepsilon_{r}}\in U(P\oplus\bar{L}_{n}).$ $None$
Let $\varpi_{\nu^{\prime}}={\xi}_{\underline{m}}$ (resp.
$\varpi_{\nu^{\prime}}=\hat{\xi}_{\underline{m}}$ ) in (3.96), where
$\xi_{\underline{m}}=a_{\underline{m}}x^{{\underline{m}}}\otimes
v_{\mu}+\sum\limits_{(\eta,\underline{l})\neq(\mu,\underline{m}),{\underline{m}}+\mu={\underline{l}}+\eta,\
i\in\overline{1,m({\eta})}}a_{\eta,\underline{l}}^{i}x^{\underline{l}}\otimes
v_{\eta}^{i},\qquad 0\neq a_{\underline{m}},\
a_{\eta,\underline{l}}^{i}\in\mathbb{F};$ $None$
$\hat{\xi}_{\underline{m}}=a_{\underline{m}}p^{{\underline{m}}}.(1\otimes
v_{\mu})+\sum\limits_{(\eta,\underline{l})\neq(\mu,\underline{m}),{\underline{m}}+\mu={\underline{l}}+\eta,\
i\in\overline{1,m({\eta})}}a_{\eta,\underline{l}}^{i}p^{\underline{l}}.(1\otimes
v_{\eta}^{i}),\qquad 0\neq a_{\underline{m}},\
a_{\eta,\underline{l}}^{i}\in\mathbb{F}.$ $None$
Take
$\hat{\xi}_{\underline{m}+\varepsilon_{r}}=n_{\varepsilon_{r}}.\hat{\xi}_{\underline{m}}.$
$None$
Then by Lemma 3.2.3, we have
$\hat{\xi}_{\underline{m}+\varepsilon_{r}}=q_{\underline{m}+\varepsilon_{r}}{\xi}_{\underline{m}+\varepsilon_{r}}=n_{\varepsilon_{r}}.\hat{\xi}_{\underline{m}}=q_{\underline{m}}n_{\varepsilon_{r}}.{\xi}_{\underline{m}}.$
$None$
Hence, we have
$(\mu_{r}+|\mu|-r+m_{r}+1){\xi}_{\underline{m}+\varepsilon_{r}}=n_{\varepsilon_{r}}.{\xi}_{\underline{m}}.$
$None$
Now we take
$\displaystyle v_{\nu}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{r-1}\sum_{i=1}^{m(\nu^{\prime}-\varepsilon_{j}+\varepsilon_{r})}\sum\limits_{\varepsilon_{j}-\varepsilon_{r}=\sum\limits_{k=1}^{m}i_{k}(\varepsilon_{s_{k}}-\varepsilon_{t_{k}});\>s_{i}>t_{i}}a_{j}^{i}a_{\underline{s},\underline{t}}^{\underline{i}}e_{j}^{\prime}\otimes(x_{s_{1}}\partial_{t_{1}})^{i_{1}}.\cdots.(x_{s_{m}}\partial_{t_{m}})^{i_{m}}.{\xi}_{\underline{m}}$
$\displaystyle+a_{r}e_{r}^{\prime}\otimes{\xi}_{\underline{m}}.\hskip
318.67078pt(3.102)$
Then
$T_{s}(v_{\nu})=(\mu_{r}+|\mu|-r+m_{r}+1)\xi_{\underline{m}+\varepsilon_{r}}\neq
0$ by (3.101) and (3.102). Therefore, (2.30) and (3.92) imply that
$\\{0\\}\neq(T_{s}|_{V(\varepsilon_{1})\bigotimes_{\mathbb{F}}V(\nu^{\prime})}\circ\tilde{P}_{r})(V(\varepsilon_{1})\otimes_{\mathbb{F}}V(\nu^{\prime}))=V(\nu),$
$None$
where
$\tilde{P}_{r}=\prod\limits_{l\neq
r}(\frac{\tilde{M}-\tilde{d}_{l}}{\tilde{d}_{r}-\tilde{d}_{l}}),\
\tilde{d}_{i}=i-1-m_{i}-\mu_{i},$ $None$
and $\tilde{M}$ is the matrix in (2.27). Hence,
${\cal{R}}_{{\langle}s\rangle}\subseteq W$ for any $s\geq k+1$. Thus we prove
$({\cal A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)$ is irreducible.
$\Box$
Suppose ${\cal{R}}_{{\langle}k+1\rangle}=V(\mu+(k+1)\varepsilon_{r})$ for some
$r\in I_{k+1}$. From the proof of Proposition 3.3.8, we know $({\cal
A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)$ is generated by
$\bar{\xi}_{\mu+(k+1)\varepsilon_{r}}={\xi}_{\mu+(k+1)\varepsilon_{r}}+U(P)(1\otimes_{\mathbb{F}}V)$,
i.e. $({\cal
A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)=U(L_{n+1}).\bar{\xi}_{\mu+(k+1)\varepsilon_{r}}$.
Hence, we get the following result:
Proposition 3.3.9 The $L_{n+1}$-module $({\cal
A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)$ is isomorphic to the
irreducible module $U(P)(1\otimes_{\mathbb{F}}M)$, where $M$ is a
$gl(n)$-irreducible module admitting the character
$\chi_{\mu+(k+1)\varepsilon_{r}}$.
Proof Since $\partial_{i}.{\xi}_{\mu+(k+1)\varepsilon_{r}}\in(U(P)(1\otimes
V))_{{\langle}k\rangle}=({\cal A}\otimes_{\mathbb{F}}V)_{{\langle}k\rangle}$,
we get $\partial_{i}.\bar{\xi}_{\mu+(k+1)\varepsilon_{r}}=0$ for any
$i\in\overline{1,n}$. It follows from Proposition 3.3.8 that both $({\cal
A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)$ and
$U(P)(1\otimes_{\mathbb{F}}M)$ are irreducible $L_{n+1}$-modules. So it is
easy to verify that
$\vartheta:({\cal
A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)\rightarrow
U(P)(1\otimes_{\mathbb{F}}M);x.\bar{\xi}_{\mu+(k+1)\varepsilon_{r}}\mapsto
x.(1\otimes v_{\mu+(k+1)\varepsilon_{r}})$ $None$
is an $L_{n+1}$-module isomorphism, where $x\in U(P\oplus\bar{L}_{n})$ and
$v_{\mu+(k+1)\varepsilon_{r}}$ is a maximal vector of $M$. $\Box$
From Proposition 3.3.3, Proposition 3.3.8 and Proposition 3.3.9, we get the
Main Theorem.
Remark 3.3.10 The irreducible module $U(P)(1\otimes_{\mathbb{F}}V)$ is cyclic,
i.e. it is generated by one vector. From Lemma 3.1.1 and (3.12), we know that
$U(P)(1\otimes_{\mathbb{F}}V)$ is in general not a highest weight module. We
can easily verify the following result from the Main Theorem:
Corollary 3.3.11 Assume one of the conditions in (1.7) and (1.8) fails. Denote
$i_{0}=\mbox{min}\\{i\in\overline{1,n}\ |\
\mu_{i}+|\mu|-i+1\in-\mathbb{N}\\}.$ $None$
We have:
(i) The integer $k=-\mu_{i_{0}}-|\mu|+i_{0}-1$ satisfies (3.72) and (3.73).
(ii) The irreducible module $U(P)(1\otimes_{\mathbb{F}}V)$ is finite
dimensional highest weight module with highest weight
$k\omega_{1}+\sum\limits_{i=2}^{n}m_{i-1}\omega_{i}$ iff $\ i_{0}=1$, where
$m_{i}=\mu_{i}-\mu_{i+1}$ for $i\in\overline{1,n-1}$.
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|
arxiv-papers
| 2010-06-27T14:06:34 |
2024-09-04T02:49:11.252597
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yufeng Zhao and Xiaoping Xu",
"submitter": "Xiaoping Xu",
"url": "https://arxiv.org/abs/1006.5212"
}
|
1006.5294
|
# Complete description of invariant Einstein metrics on the generalized flag
manifold $SO(2n)/U(p)\times U(n-p)$
Andreas Arvanitoyeorgos, Ioannis Chrysikos and Yusuke Sakane University of
Patras, Department of Mathematics, GR-26500 Rion, Greece
arvanito@math.upatras.gr xrysikos@master.math.upatras.gr Osaka University,
Department of Pure and Applied Mathematics, Graduate School of Information and
Technology, Osaka 560-043, Japan sakane@math.sci.osaka-u.ac.jp
###### Abstract.
We find the precise number of non-Kähler $SO(2n)$-invariant Einstein metrics
on the generalized flag manifold $M=SO(2n)/U(p)\times U(n-p)$ with $n\geq 4$
and $2\leq p\leq n-2$. We use an analysis on parametric systems of polynomial
equations and we give some insight towards the study of such systems. We also
examine the isometric problem for these Einstein metrics.
2000 Mathematics Subject Classification. Primary 53C25; Secondary 53C30,
12D05, 65H10
Keywords: homogeneous manifold, Einstein metric, generalized flag manifold,
algebraic systems of equations, resultant.
The first two authors were partially supported by the C. Carathéodory grant
#C.161 2007-10, University of Patras and the third auther was supported by
Grant-in-Aid for Scientific Research (C) 21540080
## Introduction
A Riemannian metric $g$ is called Einstein if the Ricci tensor
$\operatorname{Ric}_{g}$ satisfies the equation ${\rm Ric}_{g}=e\cdot g$, for
some $e\in\mathbb{R}$. When $M$ is compact, Einstein metrics of volume 1 can
be characterized variationally as the critical points of the scalar curvature
functional $T(g)=\int_{M}S_{g}d{\rm vol}_{g}$ on the space $\mathcal{M}_{1}$
of Riemannian metrics of volume 1. If $M=G/K$ is a compact homogeneous space,
a $G$-invariant Einstein metric is precisely a critical point of $T$
restricted to the set of $G$-invariant metrics of volume 1. As a consequence,
the Einstein equation reduces to a system of non-linear algebraic equations,
which is still very complicated but more manageable, and in some times can be
solved explicity. Thus most known examples of Einstein manifolds are
homogeneous.
In a recent work [AC] the first two authors classified all flag manifolds for
which the isotropy representation decomposes into four pairwise inequivalent
irreducible submodules, and found new invariant Einstein metrics on these
spaces. Recall that a generalized flag manifold is an adjoint orbit of a
compact semisimple Lie group $G$, or equivalently a compact homogeneous space
of the form $M=G/K=G/C(S)$, where $C(S)$ is the centralizer of a torus $S$ in
$G$.
Eventhough the problem of finding all invariant Einstein metrics on $M$ can be
facilitated by use of certain theoretical results (e.g. the work [Grv] on the
total number of $G$-invariant complex Einstein metrics), it still remains a
difficult one, especially when the number of isotropy summands increases. This
difficulty also increases when we pass from exceptional flag manifolds to
classical flag manifolds, because in the later case the Einstein equation
reduces to a parametric system. In particular, eventhough all invariant
Einstein metrics were found for every generalized flag manifold with four
isotropy summands, a partial answer was given for the spaces
$SO(2n)/U(p)\times U(n-p)$ and $Sp(n)/U(p)\times U(n-p)$.
We summarize the results obtained in [AC] about these spaces.
###### Theorem 1.
([AC]) The flag manifold $SO(2n)/U(p)\times U(n-p)$ ($n\geq 4$ and $2\leq
p\leq n-2$) admits at least six $SO(2n)$-invariant Einstein metrics. There are
two non-Kähler Einstein metrics and two pairs of isometric Kähler-Einstein
metrics.
###### Theorem 2.
([AC]) The flag manifold $Sp(n)/U(p)\times U(n-p)$ ($n\geq 2$ and $1\leq p\leq
n-1$) admits at least four $Sp(n)$-invariant Einstein metrics, which are
Kähler.
For the special case $n=2p$ the following results have been obtained:
###### Theorem 3.
([AC]) The flag manifold $SO(4n)/U(p)\times U(p)$ ($p\geq 2$) admits at least
six $SO(4n)$-invariant Einstein metrics. There are two non-isometric non-
Kähler Einstein metrics, and four isometric Kähler-Einstein metrics. In the
special case where $2\leq p\leq 6$ there are two more non-Kähler Einstein
metrics, and the total number of $SO(4n)$-invariant Einstein metrics is
exactly eight.
###### Theorem 4.
([AC]) The flag manifold $Sp(2n)/U(p)\times U(p)$ ($p\geq 1$) admits precisely
six $Sp(n)$-invariant Einstein metrics. There are four isometric Kähler-
Einstein metrics, and two non-Kähler Einstein metrics.
In the present paper we find all $SO(2n)$-invariant Einstein metrics on the
flag manifold $SO(2n)/U(p)\times U(n-p)$, by using a new approach into
manipulating the algebraic systems of equations obtained from the Einstein
equation. The coefficients of the polynomials in such systems involve
parameters, so a major difficulty appears when we try to show existence and
uniqueness of solutions. Therefore, the contribution of the present work is,
besides answering the original problem on Einstein metrics, to give some
insight towards the study of parametric systems of algebraic equations.
Our main result is the following:
Main Theorem. Let $M=SO(2n)/U(p)\times U(n-p)$ with $n\geq 4$ and $2\leq p\leq
n-2$. Then $M$ admits exactly four non-Kähler $SO(2n)$-invariant Einstein
metrics for the pairs $(n,p)=(12,6)$, $(10,5)$, $(8,4)$, $(7,4)$, $(7,3)$,
$(6,4)$, $(6,3)$, $(6,2)$, $(5,3)$, $(5,2)$, $(4,2)$, and two non-Kähler
$SO(2n)$-invariant Einstein metrics for all other cases.
The flag manifold $Sp(n)/U(p)\times U(n-p)$ will be treated in a forthcoming
paper.
## 1\. The Einstein equation on flag manifolds
Let $M=G/K=G/C(S)$ be a generalized flag manifold of a compact simple Lie
group $G$, where $K=C(S)$ is the centralizer of a torus $S$ in $G$. Let $o=eK$
be the identity coset of $G/K$. We denote by $\mathfrak{g}$ and $\mathfrak{k}$
the corresponding Lie algrebras of $G$ and $K$. Let $B$ denote the Killing
form of $\mathfrak{g}$. Since $G$ is compact and simple, $-B$ is a positive
definite inner product on $\mathfrak{g}$. With repsect to $-B$ we consider the
orthogonal decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{m}$. This
is a reductive decomposition of $\mathfrak{g}$, that is
$\operatorname{Ad}(K)\mathfrak{m}\subset\mathfrak{m}$, and as usual we
identify the tangent space $T_{o}M$ with $\mathfrak{m}$. Since $K=C(S)$, the
isotropy group $K$ is connected and the relation
$\operatorname{Ad}(K)\mathfrak{m}\subset\mathfrak{m}$ is equivalent with
$[\mathfrak{k},\mathfrak{m}]\subset\mathfrak{m}$. Thus, for a flag manifold
$M=G/K$ the notion of $\operatorname{Ad}(K)$-invariant and
$\operatorname{ad}(\mathfrak{k})$-invariant is equivalent.
Let $\chi:K\to\operatorname{Aut}(T_{o}M)$ be the isotropy representation of
$K$ on $T_{o}M$. Since $\chi$ is equivalent to the adjoint representation of
$K$ restricted on $\mathfrak{m}$, the set of all $G$-invariant symmetric
covariant 2-tensors on $G/K$ can be identified with the set of all
$\operatorname{Ad}(K)$-invariant symmetric bilinear forms on $\mathfrak{m}$.
In particular, the set of $G$-invariant metrics on $G/K$ is identified with
the set of $\operatorname{Ad}(K)$-invariant inner products on $\mathfrak{m}$.
Let $\mathfrak{m}=\mathfrak{m}_{1}\oplus\cdots\oplus\mathfrak{m}_{s}$ be a
$(-B)$-orthogonal $\operatorname{Ad}(K)$-invariant decomposition of
$\mathfrak{m}$ into pairwise inequivalent irreducible
$\operatorname{Ad}(K)$-modules $\mathfrak{m}_{i}$ $(i=1,\ldots,s)$. Such a
decomposition always exists and can be expressed in terms of
$\mathfrak{t}$-roots (cf. [AP], [AC]). Then, a $G$-invariant Riemannian metric
on $M$ (or equivalently, an $\operatorname{Ad}(K)$-invariant inner product
$\langle\ ,\ \rangle$ on $\mathfrak{m}=T_{o}M$) is given by
$g=\langle\ ,\
\rangle=x_{1}\cdot(-B)|_{\mathfrak{m}_{1}}+\cdots+x_{s}\cdot(-B)|_{\mathfrak{m}_{s}},$
(1)
where $(x_{1},\ldots,x_{s})\in\mathbb{R}^{s}_{+}$. Since
$\mathfrak{m}_{i}\neq\mathfrak{m}_{j}$ as
$\operatorname{Ad}(K)$-representation, any $G$-invariant metric on $M$ has the
above form.
Similarly, the Ricci tensor $\operatorname{Ric}_{g}$ of a $G$-invariant metric
$g$ on $M$, as a symmetric covariant 2-tensor on $G/K$ is given by
$\operatorname{Ric}_{g}=r_{1}\cdot(-B)|_{\mathfrak{m}_{1}}+\cdots+r_{s}\cdot(-B)|_{\mathfrak{m}_{s}},$
where $r_{1},\ldots,r_{s}$ are the components of the Ricci tensor on each
$\mathfrak{m}_{i}$, that is
$\operatorname{Ric}_{g}|_{\mathfrak{m}_{i}}=r_{i}\cdot(-B)|_{\mathfrak{m}_{i}}$.
These components obtain o useful description in terms of the structure
constants $[ijk]$ first introduced in [WZ]. Let $\\{X_{\alpha}\\}$ be a
$(-B)$-orthonormal basis adapted to the decomposition of $\mathfrak{m}$, that
is $X_{\alpha}\in\mathfrak{m}_{i}$ for some $i$, and $\alpha<\beta$ if $i<j$
(with $X_{\alpha}\in\mathfrak{m}_{i}$ and $X_{\beta}\in\mathfrak{m}_{j}$). Set
$A_{\alpha\beta}^{\gamma}=B([X_{\alpha},X_{\beta}],X_{\gamma})$ so that
$[X_{\alpha},X_{\beta}]_{\mathfrak{m}}=\sum_{\gamma}A_{\alpha\beta}^{\gamma}X_{\gamma}$,
and $[ijk]=\sum(A_{\alpha\beta}^{\gamma})^{2}$, where the sum is taken over
all indices $\alpha,\beta,\gamma$ with
$X_{\alpha}\in\mathfrak{m}_{i},X_{\beta}\in\mathfrak{m}_{j},X_{\gamma}\in\mathfrak{m}_{k}$
(where $[\ ,\ ]_{\mathfrak{m}}$ denotes the $\mathfrak{m}$-component). Then
$[ijk]$ is nonnegative, symmetric in all three entries, and independent of the
$(-B)$-orthonormal bases choosen for $\mathfrak{m}_{i},\mathfrak{m}_{j}$ and
$\mathfrak{m}_{k}$ (but it depends on the choise of the decomposition of
$\mathfrak{m}$).
###### Proposition 1.
([PaS]) Let $M=G/K$ be a generalized flag manifold of a compact simple Lie
group $G$ and let $\mathfrak{m}=\bigoplus_{i=1}^{s}\mathfrak{m}_{i}$ be a
decomposition of $\mathfrak{m}$ into pairwise inequivalent irreducible
$\operatorname{Ad}(K)$-submodules. Then the components $r_{1},\ldots,r_{s}$ of
the Ricci tensor of a $G$-invariant metric (1) on $M$ are given by
$r_{k}=\frac{1}{2x_{k}}+\frac{1}{4d_{k}}\sum_{i,j}\frac{x_{k}}{x_{i}x_{j}}[ijk]-\frac{1}{2d_{k}}\sum_{i,j}\frac{x_{j}}{x_{k}x_{i}}[kij],\qquad(k=1,\ldots,s).$
In wiew of Proposition 1, a $G$-invariant metric
$g=(x_{1},\ldots,x_{s})\in\mathbb{R}^{s}_{+}$ on $M$, is an Einstein metric
with Einstein constant $e$, if and only if it is a positive real solution of
the system
$\frac{1}{2x_{k}}+\frac{1}{4d_{k}}\sum_{i,j}\frac{x_{k}}{x_{i}x_{j}}[ijk]-\frac{1}{2d_{k}}\sum_{i,j}\frac{x_{j}}{x_{k}x_{i}}[kij]=e,\quad
1\leq k\leq s.$
## 2\. The generalized flag manifold $SO(2n)/U(p)\times U(n-p)$
We review some results related to the generalized flag manifold
$M=G/K=SO(2n)/U(p)\times U(n-p)$ ($n\geq 4,\ 2\leq p\leq n-2$) obtained in
[AC]. Its corresponding painted Dynkin diagram is given by
$\alpha_{1}$1$\alpha_{2}$2$\ldots$$(2\leq
p\leq\ell-2)$$\alpha_{p}$2$\ldots$2$\alpha_{\ell-1}$1$\alpha_{\ell}$1
The isotropy representation of $M$ decomposes into a direct sum
$\chi=\chi_{1}\oplus\chi_{2}\oplus\chi_{3}\oplus\chi_{4}$, which gives rise to
a decomposition
$\mathfrak{m}=\mathfrak{m}_{1}\oplus\mathfrak{m}_{2}\oplus\mathfrak{m}_{3}\oplus\mathfrak{m}_{4}$
of $\mathfrak{m}=T_{o}M$ into four irreducible inequivalent
$\operatorname{ad}(\mathfrak{k})$-submodules. The dimensions
$d_{i}=\dim\mathfrak{m}_{i}\ (i=1,2,3,4)$ of these submodules can be obtained
by use of Weyl’s formula [AC, pp. 204-205, p. 210] and are given by
$d_{1}=2p(n-p),\ d_{2}=(n-p)(n-p-1),\ d_{3}=2p(n-p),\ d_{4}=p(p-1).$
According to (1), a $G$-invariant metric on $M=G/K$ is given by
$\left\langle\ ,\
\right\rangle=x_{1}\cdot(-B)|_{\mathfrak{m}_{1}}+x_{2}\cdot(-B)|_{\mathfrak{m}_{2}}+x_{3}\cdot(-B)|_{\mathfrak{m}_{3}}+x_{4}\cdot(-B)|_{\mathfrak{m}_{4}},$
(2)
for positive real numbers $x_{1},x_{2},x_{3},x_{4}$. We will denote such
metrics by $g=(x_{1},x_{2},x_{3},x_{4})$.
It is known ([Nis]) that if $n\neq 2p$ then $M$ admits two non-equivalent
$G$-invariant complex structures $J_{1},J_{2}$, and thus two non-isometric
Kähler-Einstein metrics which are given (up to scale) by (see also [AC,
Theorem 3])
$g_{1}=(n/2,\ n+p-1,\ n/2+p-1,\ p-1)$
---
$g_{2}=(n/2,\ n-p-1,\ 3n/2-p-1,\ 2n-p-1)$.
(3)
If $n=2p$ then $M$ admits a unique $G$-invariant complex structure with
corresponding Kähler-Einstein metric (up to scale) given by $g=(p,\ p-1,\
2p-1,\ 3p-1)$ (cf. also [AC, Theorem 8] where all isometric Kähler-Einstein
metrics are listed).
The Ricci tensor of $M$ has been computed in [AC] and is given as follows:
###### Proposition 2.
The components $r_{i}$ of the Ricci tensor for a $G$-invariant Riemannian
metric on $M$ determined by (2) are given as follows:
$\ \
\left.\begin{tabular}[]{l}$r_{1}=\displaystyle\frac{1}{2x_{1}}+\frac{c_{12}^{3}}{2d_{1}}\Big{(}\frac{x_{1}}{x_{2}x_{3}}-\frac{x_{2}}{x_{1}x_{3}}-\frac{x_{3}}{x_{1}x_{2}}\Big{)}+\frac{c_{13}^{4}}{2d_{1}}\Big{(}\frac{x_{1}}{x_{3}x_{4}}-\frac{x_{4}}{x_{1}x_{3}}-\frac{x_{3}}{x_{1}x_{4}}\Big{)}$\\\
$r_{2}=\displaystyle\frac{1}{2x_{2}}+\frac{c_{12}^{3}}{2d_{2}}\Big{(}\frac{x_{2}}{x_{1}x_{3}}-\frac{x_{1}}{x_{2}x_{3}}-\frac{x_{3}}{x_{1}x_{2}}\Big{)}$\\\
$r_{3}=\displaystyle\frac{1}{2x_{3}}+\frac{c_{12}^{3}}{2d_{3}}\Big{(}\frac{x_{3}}{x_{1}x_{2}}-\frac{x_{2}}{x_{1}x_{3}}-\frac{x_{1}}{x_{2}x_{3}}\Big{)}+\frac{c_{13}^{4}}{2d_{3}}\Big{(}\frac{x_{3}}{x_{1}x_{4}}-\frac{x_{4}}{x_{1}x_{3}}-\frac{x_{1}}{x_{3}x_{4}}\Big{)}$\\\
$r_{4}=\displaystyle\frac{1}{2x_{4}}+\frac{c_{13}^{4}}{2d_{4}}\Big{(}\frac{x_{4}}{x_{1}x_{3}}-\frac{x_{3}}{x_{1}x_{4}}-\frac{x_{1}}{x_{3}x_{4}}\Big{)},$\\\
\end{tabular}\right\\}$ (4)
where $c_{12}^{3}=[123]$ and $c_{13}^{4}=[134]$.
By taking into account the explicit form of the Kähler-Einstein metrics above,
and substituting these in (4), we can find that the values of the unknown
triples $[ijk]$ are given by
$\displaystyle{c_{12}^{3}=\frac{p(n-p)(n-p-1)}{2(n-1)}}$ and
$\displaystyle{c_{13}^{4}=\frac{p(p-1)(n-p)}{2(n-1)}}$.
A $G$-invariant metric $g=(x_{1},x_{2},x_{3},x_{4})$ on $M=G/K$ is Einstein if
and only if, there is a positive constant $e$ such that
$r_{1}=r_{2}=r_{3}=r_{4}=e$, or equivalently
$r_{1}-r_{3}=0,\quad r_{1}-r_{2}=0,\quad r_{3}-r_{4}=0.$ (5)
By substituting the values of $d_{i}\ (i=1,2,3,4)$ and $c_{12}^{3},c_{13}^{4}$
into the components of the Ricci tensor, System (5) is equivalent to the
following equations:
$\left.\begin{tabular}[]{r}$(x_{1}-x_{3})(-x_{1}x_{2}+px_{1}x_{2}-x_{2}x_{3}+px_{2}x_{3}-x_{1}x_{4}+nx_{1}x_{4}$\\\
$-px_{1}x_{4}+2x_{2}x_{4}-2nx_{2}x_{4}-x_{3}x_{4}+nx_{3}x_{4}-px_{3}x_{4})=0$\\\
$4(n-1)x_{3}x_{4}(x_{2}-x_{1})+(n+p-1)x_{4}(x_{1}^{2}-x_{2}^{2})-(n-3p-1)x_{3}^{2}x_{4}$\\\
$+(p-1)x_{2}(x_{1}^{2}-x_{3}^{2}-x_{4}^{2})=0$\\\
$4(n-1)x_{1}x_{2}(x_{4}-x_{3})+(2n-p-1)x_{2}(x_{3}^{2}-x_{4}^{2})+(2n-3p+1)x_{1}^{2}x_{2}$\\\
$+(n-p-1)x_{4}(x_{3}^{2}-x_{1}^{2}-x_{2}^{2})=0$\\\ \end{tabular}\right\\}$
(6)
## 3\. Proof of the Main Theorem
We consider the equation $r_{1}-r_{3}=0$ of System (5). This is equivalent to
$\displaystyle(x_{1}-x_{3})(-x_{1}x_{2}+px_{1}x_{2}-x_{2}x_{3}+px_{2}x_{3}-x_{1}x_{4}+nx_{1}x_{4}$
$\displaystyle-
px_{1}x_{4}+2x_{2}x_{4}-2nx_{2}x_{4}-x_{3}x_{4}+nx_{3}x_{4}-px_{3}x_{4})=0.$
CASE A Let $x_{1}=x_{3}=1$. Then the system of equations $r_{1}-r_{2}=0,\
r_{3}-r_{4}=0$ becomes
$\displaystyle{x_{2}}^{2}(n+p-1)+4(n-p-1)-4(n-1){x_{2}}+(p-1){x_{2}}{x_{4}}$
$\displaystyle=$ $\displaystyle 0$ (7)
$\displaystyle{x_{2}}{x_{4}}(n-p-1)+{x_{4}}^{2}(2n-p-1)-4(n-1){x_{4}}+4(p-1)$
$\displaystyle=$ $\displaystyle 0.$ (8)
From (7) we get that
$\displaystyle
x_{4}=-\frac{({x_{2}}-2)((n+p-1){x_{2}}-2(n-p-1))}{(p-1){x_{2}}}.$ (9)
Note that $x_{4}>0$ if and only if
$\displaystyle\frac{2(n-p-1)}{n+p-1}<x_{2}<2$. By substituting equation (9)
into equation (8), we obtain the following equation:
$\displaystyle\ \
H_{n,p}(x_{2})=(n-1)n(n+p-1){x_{2}}^{4}-4(n-1)\left(2n^{2}-2n-p^{2}+p\right){x_{2}}^{3}$
$\displaystyle+2\left(12n^{3}-11n^{2}p-25n^{2}-2np^{2}+20np+14n+2p^{3}-2p^{2}-6p-2\right){x_{2}}^{2}$
$\displaystyle-8(n-1)(4n-3p-1)(n-p-1){x_{2}}+8(n-p-1)^{2}(2n-p-1)=0.$ (10)
From (8) we get that
$\displaystyle
x_{2}=-\frac{({x_{4}}-2)({x_{4}}(2n-p-1)-2(p-1))}{{x_{4}}(n-p-1)}.$ (11)
Note that $x_{2}>0$ if and only if
$\displaystyle\frac{2(p-1)}{2n-p-1}<x_{4}<2$. By substituting equation (11)
into equation (7), we obtain the following equation:
$\displaystyle\ \
G_{n,p}(x_{4})=(n-1)n(2n-p-1){x_{4}}^{4}-4(n-1)\left(n^{2}+2np-
n-p^{2}-p\right){x_{4}}^{3}$
$\displaystyle+2\left(n^{3}+9n^{2}p-7n^{2}+4np^{2}-16np+8n-2p^{3}-2p^{2}+6p-2\right){x_{4}}^{2}$
$\displaystyle-8(n-1)(p-1){x_{4}}(n+3p-1)+8(p-1)^{2}(n+p-1)=0.$ (12)
Note that the relation between $H_{n,p}(x_{2})$ and $G_{n,p}(x_{4})$ is given
by
$\displaystyle G_{n,p}(x_{4})=H_{n,n-p}(x_{4}).$ (13)
###### Proposition 3.
The equation $H_{n,p}(x_{2})=0$ has at least two solutions between
$\displaystyle x_{2}=\frac{2(n-p-1)}{n+p-1}$ and $x_{2}=2$.
###### Proof.
We consider the value $H_{n,p}(x_{2})$ at $\displaystyle
x_{2}=\frac{2(n-p-1)}{n+p-1}$ and $x_{2}=2$. We see that
$\displaystyle
H_{n,p}\left(\frac{2(n-p-1)}{n+p-1}\right)=\frac{8(p-1)^{3}(n-p-1)^{2}}{(n+p-1)^{2}}>0\quad\mbox{and
}\quad H_{n,p}\left(2\right)=8(p-1)^{3}>0.$
Now, the value $H_{n,p}(x_{2})$ at $\displaystyle x_{2}=\frac{2(n-p-1)}{n}$ is
given by
$\displaystyle
H_{n,p}\left(\frac{2(n-p-1)}{n}\right)=-\frac{16(p-1)^{2}(n-p-1)^{3}}{n^{3}}<0,$
thus the equation $H_{n,p}(x_{2})=0$ has at least two solutions between
$\displaystyle x_{2}=\frac{2(n-p-1)}{n+p-1}$ and $x_{2}=2$. ∎
We need to show that the polynomial $H_{n,p}(x_{2})$ has only one local
minimum (i.e. the two solutions obtained in Proposition 3 are unique), with
some exceptions which will also be studied.
###### Lemma 1.
For $n\geq 2p+5$ and $p\geq 4$ the equation $H_{n,p}(x_{2})=0$ has exactly two
positive solutions.
###### Proof.
We have that
$\displaystyle\ \
\frac{dH_{n,p}}{dx_{2}}=4(n-1)n(n+p-1){x_{2}}^{3}-12(n-1)\left(2n^{2}-2n-p^{2}+p\right){x_{2}}^{2}$
$\displaystyle+4\left(12n^{3}-11n^{2}p-25n^{2}-2np^{2}+20np+14n+2p^{3}-2p^{2}-6p-2\right){x_{2}}$
$\displaystyle-8(n-1)(4n-3p-1)(n-p-1),$ $\displaystyle\ \
\frac{d^{2}H_{n,p}}{d{x_{2}}^{2}}=12(n-1)n(n+p-1){x_{2}}^{2}-24(n-1)\left(2n^{2}-2n-p^{2}+p\right){x_{2}}$
$\displaystyle+4\left(12n^{3}-11n^{2}p-25n^{2}-2np^{2}+20np+14n+2p^{3}-2p^{2}-6p-2\right)$
and
$\displaystyle\ \
\frac{d^{3}H_{n,p}}{d{x_{2}}^{3}}=24(n-1)n(n+p-1){x_{2}}-24(n-1)\left(2n^{2}-2n-p^{2}+p\right).$
Note that the quadratic polynomial
$\displaystyle\frac{d^{2}H_{n,p}}{d{x_{2}}^{2}}$ attains its minimum at
$\displaystyle x_{2}=\frac{2n^{2}-2n-p^{2}+p}{n(n+p-1)}$ and we see that
$\displaystyle\ \
\frac{d^{2}H_{n,p}}{d{x_{2}}^{2}}\left(\frac{2n^{2}-2n-p^{2}+p}{n(n+p-1)}\right)=\frac{4}{n(n+p-1)}\left(n^{4}p-n^{4}-n^{3}p^{2}-6n^{3}p+3n^{3}\right.$
$\displaystyle\left.-4n^{2}p^{2}+12n^{2}p-4n^{2}-np^{4}+2np^{3}+5np^{2}-8np+2n+3p^{4}-6p^{3}+3p^{2}\right).$
We set
$\displaystyle\ \
M(n,p)=n^{4}p-n^{4}-n^{3}p^{2}-6n^{3}p+3n^{3}-4n^{2}p^{2}+12n^{2}p-4n^{2}-np^{4}+2np^{3}$
$\displaystyle+5np^{2}-8np+2n+3p^{4}-6p^{3}+3p^{2}$
and we investigate the conditions for $n,p$ such that $M(n,p)>0$ for $n\geq
2p$.
We consider the coefficients of $M(n,p)$ as a polynomial of $n-2p-5$. We can
write $M(n,p)$ as
$\displaystyle\quad
M(n,p)=(p-1)(n-2p-5)^{4}+\left(7p^{2}+6p-17\right)(n-2p-5)^{3}$
$\displaystyle+\left(18p^{3}+41p^{2}-30p-109\right)(n-2p-5)^{2}$
$\displaystyle+\left(19p^{4}+62p^{3}-26p^{2}-274p-313\right)(n-2p-5)$
$\displaystyle+6p^{5}+22p^{4}-64p^{3}-309p^{2}-491p-340.$
We put
$\begin{array}[]{lcl}a_{0}=6p^{5}+22p^{4}-64p^{3}-309p^{2}-491p-340,&&a_{1}=19p^{4}+62p^{3}-26p^{2}-274p-313,\\\
a_{2}=18p^{3}+41p^{2}-30p-109,&&a_{3}=7p^{2}+6p-17.\end{array}$
Note that
$\displaystyle
a_{0}=6(p-4)^{5}+142(p-4)^{4}+1248(p-4)^{3}+4875(p-4)^{2}+7277(p-4)+432,$
$\displaystyle a_{1}=19(p-3)^{4}+290(p-3)^{3}+1558(p-3)^{2}+3296(p-3)+1844,$
$\displaystyle a_{2}=18(p-2)^{3}+149(p-2)^{2}+350(p-2)+139,$ $\displaystyle
a_{3}=7(p-2)^{2}+34(p-2)+23.$
Thus we see that $a_{0}>0$, $a_{1}>0,a_{2}>0,a_{3}>0$ for $p\geq 4$. Therefore
we see that $\displaystyle\frac{d^{2}H_{n,p}}{d{x_{2}}^{2}}>0$ for $n\geq
2p+5$ and $p\geq 4$ and hence, $\displaystyle\frac{dH_{n,p}}{d{x_{2}}}(x_{2})$
is monotone increasing and the polynomial $H_{n,p}(x_{2})$ has only one local
minimum for $n\geq 2p+5$ and $p\geq 4$. Thus the equation $H_{n,p}(x_{2})=0$
has exactly two positive solutions. ∎
Now we examine the values $p=2$ and $p=3$.
###### Lemma 2.
(1) Let $p=2$. Then for $n\geq 7$ the equation $H_{n,2}(x_{2})=0$ has exactly
two positive solutions, and for $4\leq n\leq 6$ it has exactly four positive
solutions.
(2) Let $p=3$. Then for $n\geq 8$ the equation $H_{n,3}(x_{2})=0$ has exactly
two positive solutions, and for $6\leq n\leq 7$ it has exactly four positive
solutions.
###### Proof.
(1) For $p=2$ we have that
$\displaystyle M(n,2)=n^{4}-13n^{3}+4n^{2}+6n+12$
$\displaystyle=(n-13)^{4}+39(n-13)^{3}+511(n-13)^{2}+2307(n-13)+766.$
Thus we see that $\displaystyle\frac{d^{2}H_{n,2}}{d{x_{2}}^{2}}>0$ for $n\geq
13$, and hence, $\displaystyle\frac{dH_{n,2}}{d{x_{2}}}(x_{2})$ is monotone
increasing and the polynomial $H_{n,2}(x_{2})$ has only one local minimum for
$n\geq 13$. Thus the equation $H_{n,2}(x_{2})=0$ has exactly two positive
solutions for $n\geq 13$. For $4\leq n\leq 12$, we consider polynomials
$H_{n,2}(x_{2})$ one by one and we see that, for $7\leq n\leq 12$ the equation
$H_{n,2}(x_{2})=0$ has two positive solutions, and for $4\leq n\leq 6$ the
equation $H_{n,2}(x_{2})=0$ has four positive solutions.
$H_{12,2}(x_{2})$
$H_{11,2}(x_{2})$
$H_{10,2}(x_{2})$
$H_{9,2}(x_{2})$
$H_{8,2}(x_{2})$
$H_{7,2}(x_{2})$
$H_{6,2}(x_{2})$
$H_{5,2}(x_{2})$
(2) For $p=3$ we have that
$\displaystyle M(n,3)=2\left(n^{4}-12n^{3}-2n^{2}-2n+54\right)$
$\displaystyle=(n-13)^{4}+40(n-13)^{3}+544(n-13)^{2}+2650(n-13)+1887.$
Thus we see that $\displaystyle\frac{d^{2}H_{n,3}}{d{x_{2}}^{2}}>0$ for $n\geq
13$, and hence, $\displaystyle\frac{dH_{n,3}}{d{x_{2}}}(x_{2})$ is monotone
increasing and the polynomial $H_{n,3}(x_{2})$ has only one local minimum for
$n\geq 13$. Thus the equation $H_{n,3}(x_{2})=0$ has exactly two positive
solutions for $n\geq 13$. For $6\leq n\leq 12$, we consider polynomials
$H_{n,3}(x_{2})$ one by one and we see that, for $8\leq n\leq 12$ the equation
$H_{n,3}(x_{2})=0$ has two positive solutions, and for $6\leq n\leq 7$ the
equation $H_{n,3}(x_{2})=0$ has four positive solutions.
$H_{12,3}(x_{2})$
$H_{11,3}(x_{2})$
$H_{10,3}(x_{2})$
$H_{9,3}(x_{2})$
$H_{8,3}(x_{2})$
$H_{7,3}(x_{2})$
$H_{6,3}(x_{2})$
∎
Next, we consider the case when $2p\leq n\leq 2p+4$. We may assume that $p\geq
4$.
###### Lemma 3.
Let $n=2p$. Then the equation $H_{2p,p}(x_{2})=0$ has exactly two positive
solutions for $p\geq 7$ and four positive solutions for $4\leq p\leq 6$.
###### Proof.
We see that
$\displaystyle\ \
H_{2p,p}(x_{2})=2\left((2p-1){x_{2}}^{2}-2(2p-1){x_{2}}+2(p-1)\right)\times$
$\displaystyle\left(p(3p-1){x_{2}}^{2}-4p(2p-1){x_{2}}+2(p-1)(3p-1)\right)$
Thus, the four solutions of the equation $H_{2p,p}(x_{2})=0$ are given by
$(a)\ x_{2}=\frac{2p\pm\sqrt{2p-1}-1}{2p-1},\quad(b)\
x_{2}=\frac{2p(2p-1)\pm\sqrt{2}\sqrt{-p\left(p^{3}-7p^{2}+5p-1\right)}}{p(3p-1)}.$
(14)
Since $-p\left(p^{3}-7p^{2}+5p-1\right)$ is negative for $p\geq 7$, we see
that the equation $H_{2p,p}(x_{2})=0$ has exactly two positive solutions
$p\geq 7$ and four positive solutions for $4\leq p\leq 6$.
$H_{12,6}(x_{2})$
$H_{10,5}(x_{2})$
$H_{8,4}(x_{2})$
∎
###### Lemma 4.
Let $n=2p+1$. Then for $p\geq 4$ the equation $H_{2p+1,p}(x_{2})=0$ has
exactly two positive solutions.
###### Proof.
We see that
$\displaystyle\ \
H_{2p+1,p}(x_{2})=6p^{2}(2p+1){x_{2}}^{4}-8p^{2}(7p+5){x_{2}}^{3}+2\left(50p^{3}+36p^{2}+3p-1\right){x_{2}}^{2}$
$\displaystyle-16p^{2}(5p+3){x_{2}}+8p^{2}(3p+1)$
and
$\displaystyle\frac{d^{2}H_{2p+1,p}}{d{x_{2}}^{2}}\left(\frac{2n^{2}-2n-p^{2}+p}{n(n+p-1)}\right)=\frac{4\left(2p^{4}-18p^{3}-8p^{2}+p-1\right)}{2p+1}$
$\displaystyle=\frac{4\left(2(p-9)^{4}+54(p-9)^{3}+478(p-9)^{2}+1315(p-9)-640\right)}{2p+1}.$
Thus we see that $\displaystyle\frac{d^{2}H_{2p+1,p}}{d{x_{2}}^{2}}>0$ for
$p\geq 10$ and hence, $\displaystyle\frac{dH_{2p+1,p}}{d{x_{2}}}(x_{2})$ is
monotone increasing and the polynomial $H_{2p+1,p}(x_{2})$ has only one local
minimum for $p\geq 10$. Thus the equation $H_{2p+1,p}(x_{2})=0$ has exactly
two positive solutions.
For $4\leq p\leq 9$, we see that
$\displaystyle\frac{d^{2}H_{2p+1,p}}{d{x_{2}}^{2}}\left(\frac{2n^{2}-2n-p^{2}+p}{n(n+p-1)}\right)$
is negative and two real solutions $\alpha,\beta$ of the quadratic equation
$\displaystyle\frac{d^{2}H_{2p+1,p}}{d{x_{2}}^{2}}=0$ are given by
$\displaystyle\alpha=\frac{2p^{2}(7p+5)-\sqrt{2}\sqrt{-p^{2}\left(2p^{4}-18p^{3}-8p^{2}+p-1\right)}}{6\left(2p^{3}+p^{2}\right)},$
$\displaystyle\beta=\frac{2p^{2}(7p+5)+\sqrt{2}\sqrt{-p^{2}\left(2p^{4}-18p^{3}-8p^{2}+p-1\right)}}{6\left(2p^{3}+p^{2}\right)}.$
Since the polynomial $\displaystyle\frac{dH_{2p+1,p}}{d{x_{2}}}(x_{2})$ of
degree 3 takes a local minimum at $\displaystyle x_{2}=\beta$, we consider the
value $\displaystyle\frac{dH_{2p+1,p}}{d{x_{2}}}(\beta)$. We see that
$\displaystyle\ \ \
\frac{dH_{2p+1,p}}{d{x_{2}}}(\beta)=\frac{2}{9p^{4}(2p+1)^{2}}\left(2(p-1)^{2}\left(8p^{3}-14p^{2}-36p-15\right)p^{4}\right.$
$\displaystyle\left.+2\sqrt{2}\left(2p^{4}-18p^{3}-8p^{2}+p-1\right)\sqrt{-p^{2}\left(2p^{4}-18p^{3}-8p^{2}+p-1\right)}p^{2}\right).$
By evaluating the above expression for the integers $4\leq p\leq 9$, we see
that $\displaystyle\frac{dH_{2p+1,p}}{d{x_{2}}}(\beta)>0$ for $6\leq p\leq 9$
and $\displaystyle\frac{dH_{2p+1,p}}{d{x_{2}}}(\beta)<0$ for $4\leq p\leq 5$.
Thus the polynomial $H_{2p+1,p}(x_{2})$ has only one local minimum for $6\leq
p\leq 9$, and $H_{2p+1,p}(x_{2})$ has two local minima and one local maximum
for $4\leq p\leq 5$. However, we see that for $p=4,5$ the equation
$H_{2p+1,p}(x_{2})=0$ has exactly two roots between
$\displaystyle\frac{2(n-p-1)}{(n+p-1)}=\frac{2}{3}$ and $2$, and this
completes the proof.
$H_{11,5}(x_{2})$
$H_{9,4}(x_{2})$
∎
###### Lemma 5.
Let $n=2p+2$. Then for $p\geq 4$ the equation $H_{2p+2,p}(x_{2})=0$ has
exactly two positive solutions.
###### Proof.
We see that
$\displaystyle\ \
H_{2p+2,p}(x_{2})=(2p+1)(2p+2)(3p+1){x_{2}}^{4}-4(2p+1)\left(7p^{2}+13p+4\right){x_{2}}^{3}$
$\displaystyle+4(p+1)\left(25p^{2}+42p+11\right){x_{2}}^{2}-8(p+1)(2p+1)(5p+7){x_{2}}+24(p+1)^{3}$
and
$\displaystyle\frac{d^{2}H_{2p+2,p}}{d{x_{2}}^{2}}\left(\frac{2n^{2}-2n-p^{2}+p}{n(n+p-1)}\right)=\frac{2(6p^{5}-35p^{4}-88p^{3}-51p^{2}-20p-4)}{(3p+1)(p+1)}$
$\displaystyle=\frac{2\left(6(p-8)^{5}+205(p-8)^{4}+2632(p-8)^{3}+15117(p-8)^{2}+33468(p-8)+4764\right)}{(3p+1)(p+1)}.$
Thus we see that $\displaystyle\frac{d^{2}H_{2p+2,p}}{d{x_{2}}^{2}}>0$ for
$p\geq 8$ and hence, $\displaystyle\frac{dH_{2p+2,p}}{d{x_{2}}}(x_{2})$ is
monotone increasing and the polynomial $H_{2p+2,p}(x_{2})$ has only one local
minimum for $p\geq 8$. Thus the equation $H_{2p+2,p}(x_{2})=0$ has exactly two
positive solutions.
For $4\leq p\leq 7$, we see that
$\displaystyle\frac{d^{2}H_{2p+2,p}}{d{x_{2}}^{2}}\left(\frac{2n^{2}-2n-p^{2}+p}{n(n+p-1)}\right)$
is negative and the two real solutions $\alpha,\beta$ of the quadratic
equation $\displaystyle\frac{d^{2}H_{2p+2,p}}{d{x_{2}}^{2}}=0$ are given by
$\displaystyle\alpha=\frac{3(2p+1)\left(7p^{2}+13p+4\right)-\sqrt{3}\sqrt{(-2p-1)\left(6p^{5}-35p^{4}-88p^{3}-51p^{2}-20p-4\right)}}{6(p+1)(2p+1)(3p+1)},$
$\displaystyle\beta=\frac{3(2p+1)\left(7p^{2}+13p+4\right)+\sqrt{3}\sqrt{(-2p-1)\left(6p^{5}-35p^{4}-88p^{3}-51p^{2}-20p-4\right)}}{6(p+1)(2p+1)(3p+1)}.$
Since the polynomial $\displaystyle\frac{dH_{2p+2,p}}{d{x_{2}}}(x_{2})$ of
degree 3 takes local minimum at $\displaystyle x_{2}=\beta$, we consider the
value $\displaystyle\frac{dH_{2p+2,p}}{d{x_{2}}}(\beta)$. We see that
$\displaystyle\ \ \
\frac{dH_{2p+2,p}}{d{x_{2}}}(\beta)=\frac{1}{9(p+1)^{2}(2p+1)(3p+1)^{2}}\left(18(2p+1)\left(4p^{2}+7p+2\right)\right.\times$
$\displaystyle\left(p^{3}-p^{2}-6p-2\right)(p-1)^{2}+2\sqrt{3}\left(6p^{5}-35p^{4}-88p^{3}-51p^{2}-20p-4\right)\times$
$\displaystyle\left.\sqrt{(-2p-1)\left(6p^{5}-35p^{4}-88p^{3}-51p^{2}-20p-4\right)}\right).$
By substituting integer $4\leq p\leq 7$, we see that
$\displaystyle\frac{dH_{2p+2,p}}{d{x_{2}}}(\beta)>0$ for $5\leq p\leq 7$ and
$\displaystyle\frac{dH_{2p+2,p}}{d{x_{2}}}(\beta)<0$ for $p=4$. Thus the
polynomial $H_{2p+2,p}(x_{2})$ has only one local minimum for $5\leq p\leq 7$,
and $H_{2p+2,p}(x_{2})$ has two local minima and one local maximum for $p=4$.
However, we see that for $p=4$ the equation $H_{2p+2,p}(x_{2})=0$ has exactly
two roots between $\displaystyle\frac{2(n-p-1)}{(n+p-1)}=\frac{2(p+1)}{3p+1}$
and $2$.
$H_{10,4}(x_{2})$
∎
By a similar method we can prove the next two lemmas.
###### Lemma 6.
Let $n=2p+3$. Then for $p\geq 4$ the equation $H_{2p+3,p}(x_{2})=0$ has
exactly two positive solutions.
###### Lemma 7.
Let $n=2p+4$. Then for $p\geq 4$ the equation $H_{2p+4,p}(x_{2})=0$ has
exactly two positive solutions.
Therefore we have obtained the following:
###### Proposition 4.
(1) If $x_{1}=x_{3}$ and $n\geq 2p$, then $M$ admits exactly four
$SO(2n)$-invariant Einstein metrics for the pairs $(n,p)=(12,6)$, $(10,5)$,
$(8,4)$, $(7,3)$, $(6,3)$, $(6,2)$, $(5,2)$, $(4,2)$ and two
$SO(2n)$-invariant Einstein metrics for all other cases.
(2) If $x_{1}=x_{3}$ and $n\leq 2p$, then $M$ admits exactly four
$SO(2n)$-invariant Einstein metrics for the pairs $(n,p)=(12,6)$, $(10,5)$,
$(8,4)$, $(7,4)$, $(6,4)$, $(6,3)$, $(5,3)$, $(4,2)$ and two
$SO(2n)$-invariant Einstein metrics for all other cases.
###### Proof.
Part (1) is a consequence of Proposition 3 and Lemmas 1 – 7. For (2), we
consider the equation $G_{n,p}(x_{4})=0$, and the result follows from the
relation (13). ∎
CASE B Let
$\displaystyle\ \
-\,x_{1}x_{2}+px_{1}x_{2}-x_{2}x_{3}+px_{2}x_{3}-x_{1}x_{4}+nx_{1}x_{4}-px_{1}x_{4}$
$\displaystyle+\,2x_{2}x_{4}-2nx_{2}x_{4}-x_{3}x_{4}+nx_{3}x_{4}-px_{3}x_{4}=0,$
(15)
and set $x_{1}=1$. From equation (15) we obtain that
$\displaystyle
x_{3}=\frac{2(n-1){x_{2}}{x_{4}}-(n-p-1){x_{4}}-(p-1){x_{2}}}{(n-p-1){x_{4}}+(p-1){x_{2}}}.$
(16)
We need to show the following:
###### Proposition 5.
The system of equations $r_{1}-r_{2}=0$ and $r_{3}-r_{4}=0$ has no positive
solutions, except Kähler-Einstein metrics.
###### Proof.
We substitute equation (16) and $x_{1}=1$ into the equations $r_{1}-r_{2}=0$
and $r_{3}-r_{4}=0$, and we obtain the following equations :
$\displaystyle
F(x_{2},x_{4})=-(p-1){x_{2}}^{3}{x_{4}}\left(2n^{2}-4n+p^{2}+2p+1\right)-{x_{2}}^{2}{x_{4}}^{2}(3n^{3}+5n^{2}p-9n^{2}$
$\displaystyle-np^{2}-6np+7n+p^{3}-3p^{2}+3p-1)+(p-1)^{2}{x_{2}}^{4}(n+p-1)$
$\displaystyle+8(n-1)(p-1){x_{2}}^{2}{x_{4}}(n+p-1)-4(p-1)^{2}{x_{2}}^{2}(n+p-1)$
$\displaystyle+(p-1){x_{2}}{x_{4}}^{3}(n-p-1)^{2}+8(n-1){x_{2}}{x_{4}}^{2}(n-p-1)(n+p-1)$
$\displaystyle-8(p-1){x_{2}}{x_{4}}(n-p-1)(n+p-1)-4{x_{4}}^{2}(n-p-1)^{2}(n+p-1)=0,$
(17) $\displaystyle
G(x_{2},x_{4})={x_{2}}{x_{4}}^{3}(n-p-1)\left(3n^{2}-2np-2n+p^{2}-2p+1\right)$
$\displaystyle+{x_{2}}^{2}{x_{4}}^{2}(8n^{3}-6n^{2}p-18n^{2}+2np^{2}+12np+10n-p^{3}-3p^{2}-3p-1)$
$\displaystyle-(p-1)^{2}{x_{2}}^{3}{x_{4}}(n-p-1)-8(n-1)(p-1){x_{2}}^{2}{x_{4}}(2n-p-1)$
$\displaystyle+4(p-1)^{2}{x_{2}}^{2}(2n-p-1)-8(n-1){x_{2}}{x_{4}}^{2}(n-p-1)(2n-p-1)$
$\displaystyle+8(p-1){x_{2}}{x_{4}}(n-p-1)(2n-p-1)-{x_{4}}^{4}(n-p-1)^{2}(2n-p-1)$
$\displaystyle+4{x_{4}}^{2}(n-p-1)^{2}(2n-p-1)=0.$ (18)
We consider the resultant of the polynomials $F(x_{2},x_{4})$ and
$G(x_{2},x_{4})$ with respect to $x_{2}$, which is a polynomial of $x_{4}$,
say $Q(x_{4})$. We factor $Q(x_{4})$ as
$\displaystyle
Q(x_{4})=128(n-1)^{6}(p-1)^{2}{x_{4}}^{8}(n-p-1)^{4}(n{x_{4}}-2p+2)(n{x_{4}}-4n+2p+2)\times$
$\displaystyle(3n{x_{4}}-4n-2p{x_{4}}+2p-2{x_{4}}+2)(n{x_{4}}+2p{x_{4}}-2p-2{x_{4}}+2)\times$
$\displaystyle(6n^{5}{x_{4}}^{4}+8n^{5}{x_{4}}^{3}+2n^{5}{x_{4}}^{2}-3n^{4}p{x_{4}}^{4}-36n^{4}p{x_{4}}^{3}-38n^{4}p{x_{4}}^{2}-8n^{4}p{x_{4}}-17n^{4}{x_{4}}^{4}$
$\displaystyle-12n^{4}{x_{4}}^{3}+22n^{4}{x_{4}}^{2}+8n^{4}{x_{4}}+72n^{3}p^{2}{x_{4}}^{2}+56n^{3}p^{2}{x_{4}}+8n^{3}p^{2}+7n^{3}p{x_{4}}^{4}+116n^{3}p{x_{4}}^{3}$
$\displaystyle+36n^{3}p{x_{4}}^{2}-64n^{3}p{x_{4}}-16n^{3}p+15n^{3}{x_{4}}^{4}-12n^{3}{x_{4}}^{3}-60n^{3}{x_{4}}^{2}+8n^{3}{x_{4}}+8n^{3}$
$\displaystyle+8n^{2}p^{3}{x_{4}}^{3}+44n^{2}p^{3}{x_{4}}^{2}-48n^{2}p^{3}{x_{4}}-24n^{2}p^{3}-24n^{2}p^{2}{x_{4}}^{3}-260n^{2}p^{2}{x_{4}}^{2}-32n^{2}p^{2}{x_{4}}$
$\displaystyle+40n^{2}p^{2}-4n^{2}p{x_{4}}^{4}-104n^{2}p{x_{4}}^{3}+108n^{2}p{x_{4}}^{2}+112n^{2}p{x_{4}}-8n^{2}p-4n^{2}{x_{4}}^{4}+24n^{2}{x_{4}}^{3}$
$\displaystyle+44n^{2}{x_{4}}^{2}-32n^{2}{x_{4}}-8n^{2}-32np^{4}{x_{4}}^{2}-80np^{4}{x_{4}}-8np^{3}{x_{4}}^{3}-8np^{3}{x_{4}}^{2}+256np^{3}{x_{4}}$
$\displaystyle+32np^{3}+24np^{2}{x_{4}}^{3}+216np^{2}{x_{4}}^{2}-192np^{2}{x_{4}}-64np^{2}+24np{x_{4}}^{3}-136np{x_{4}}^{2}+32np$
$\displaystyle-8n{x_{4}}^{3}-8n{x_{4}}^{2}+16n{x_{4}}+32p^{5}{x_{4}}+32p^{5}+32p^{4}{x_{4}}^{2}-96p^{4}-32p^{3}{x_{4}}^{2}-128p^{3}{x_{4}}$
$\displaystyle+96p^{3}-32p^{2}{x_{4}}^{2}+128p^{2}{x_{4}}-32p^{2}+32p{x_{4}}^{2}-32p{x_{4}}).$
We first consider the cases when
$\displaystyle(n{x_{4}}-2(p-1))(n{x_{4}}-2(2n-p-1)\times$
$\displaystyle\left((3n-2(p+1)){x_{4}}-2(2n-p-1)\right)\left((n+2(p-1)){x_{4}}-2(p-1)\right)=0,$
and we claim that we only get Kähler-Einstein metrics on $SO(2n)/U(p)\times
U(n-p)$.
1) Let $\displaystyle x_{4}=\frac{2(p-1)}{n}$. Then equations (17) and (18)
reduce to
$\displaystyle\frac{(p-1)^{2}(n\,{x_{2}}-2(n+p-1))}{n^{3}}\left(n^{2}(n+p-1){x_{2}}^{3}-2(n-2)n(n-2p){x_{2}}^{2}\right.$
$\displaystyle\left.-4(n-p-1)\left(n^{2}+2np-4n-p^{2}+1\right){x_{2}}+8n(n-p-1)^{2}\right)=0,$
$\displaystyle\frac{2(p-1)^{2}(n-p-1)}{n^{4}}(n\,{x_{2}}-2(n+p-1))(n\,{x_{2}}-2(n-p+1))\times$
$\displaystyle\left(2(n-p-1)(2n-p-1)-n(p-1){x_{2}}\right)=0.$
If $n\,{x_{2}}-2(n-p+1)\neq 0$, we have
$\displaystyle\ \
\left(n^{2}(n+p-1){x_{2}}^{3}-2(n-2)n(n-2p){x_{2}}^{2}\right.$
$\displaystyle\left.-4(n-p-1)\left(n^{2}+2np-4n-p^{2}+1\right){x_{2}}+8n(n-p-1)^{2}\right)=0,$
$\displaystyle\ \
(n\,{x_{2}}-2(n-p+1))\left(2(n-p-1)(2n-p-1)-n(p-1){x_{2}}\right)=0.$
By taking the resultant of these polynomials with respect to $x_{2}$, we get
$-2048(n-1)^{2}n^{6}\left((n-p)^{2}+n-1\right)(n-p-1)^{3}(n-p),$
and we see that the resultant is non-zero for $2\leq p\leq n-2$. Thus we get
only $\displaystyle{x_{2}}=\frac{2(n+p-1)}{n}$ for a solution of equations
(17) and (18). From (16), we see $\displaystyle{x_{3}}=\frac{n+2p-2}{n}$. Thus
we obtain a Kähler-Einstein metric in this case.
Notice that this metric corresponds (up to scale) to the Kähler-Einstein
metric $g_{1}$ of (3)
2) Let $\displaystyle x_{4}=\frac{2(2n-p-1)}{n}$. Then equations (17) and (18)
reduce to
$\displaystyle\ \
-\frac{(n\,{x_{2}}-2(n-p-1))}{n^{3}}\left(-n^{2}(p-1)^{2}(n+p-1){x_{2}}^{3}\right.$
$\displaystyle+2n(p-1)\left(4n^{3}-3n^{2}p-9n^{2}+2np^{2}+10np+4n-4p^{2}-4p\right){x_{2}}^{2}$
$\displaystyle+4(2n-p-1)\left(6n^{4}+5n^{3}p-19n^{3}-13n^{2}p^{2}+21n^{2}+5np^{3}+9np^{2}\right.$
$\displaystyle\left.\left.-5np-9n-p^{4}-2p^{3}+2p+1\right){x_{2}}-8n(n-p-1)(2n-p-1)^{2}(n+p-1)\right)=0,$
$\displaystyle\ \
\frac{2(2n-p-1)(n\,{x_{2}}-2(n-p-1))}{n^{4}}\left(-n^{2}(p-1)^{2}(n-p-1){x_{2}}^{2}\right.$
$\displaystyle+4n\left(4n^{3}-5n^{2}p-7n^{2}+np^{2}+8np+3n-2p^{2}-2p\right)(2n-p-1){x_{2}}$
$\displaystyle\left.+4(n-p-1)^{2}(3n-p-1)(2n-p-1)^{2}\right)=0.$
If $n\,{x_{2}}-2(n-p-1)\neq 0$, we have
$\displaystyle\ \
\left(-n^{2}(p-1)^{2}(n+p-1){x_{2}}^{3}+2n(p-1)\left(4n^{3}-3n^{2}p-9n^{2}+2np^{2}+10np+4n\right.\right.$
$\displaystyle\left.-4p^{2}-4p\right){x_{2}}^{2}+4(2n-p-1)\left(6n^{4}+5n^{3}p-19n^{3}-13n^{2}p^{2}+21n^{2}+5np^{3}+9np^{2}\right.$
$\displaystyle\left.\left.-5np-9n-p^{4}-2p^{3}+2p+1\right){x_{2}}-8n(n-p-1)(2n-p-1)^{2}(n+p-1)\right)=0,$
$\displaystyle\ \
\left(-n^{2}(p-1)^{2}(n-p-1){x_{2}}^{2}+4n\left(4n^{3}-5n^{2}p-7n^{2}+np^{2}+8np+3n-2p^{2}-2p\right)\times\right.$
$\displaystyle\left.(2n-p-1){x_{2}}+4(n-p-1)^{2}(3n-p-1)(2n-p-1)^{2}\right)=0.$
By taking the resultant of these polynomials with respect to $x_{2}$, we get
$\displaystyle-2048(n-1)^{2}n^{6}(p-1)^{2}(n-p-1)(n-p)(2n-p-1)^{6}\left(p(n-p)+(n-1)^{2}\right)\times$
$\displaystyle\left(26n^{5}-48n^{4}p-92n^{4}+14n^{3}p^{2}+160n^{3}p+124n^{3}+12n^{2}p^{3}-64n^{2}p^{2}-180n^{2}p\right.$
$\displaystyle\left.-80n^{2}-4np^{4}-7np^{3}+63np^{2}+83np+25n+3p^{4}-2p^{3}-16p^{2}-14p-3\right).$
Now we have
$\displaystyle\ \
26n^{5}-48n^{4}p-92n^{4}+14n^{3}p^{2}+160n^{3}p+124n^{3}+12n^{2}p^{3}-64n^{2}p^{2}-180n^{2}p$
$\displaystyle-80n^{2}-4np^{4}-7np^{3}+63np^{2}+83np+25n+3p^{4}-2p^{3}-16p^{2}-14p-3$
$\displaystyle=26(n-p-1)^{5}+2(41p+19)(n-p-1)^{4}+2\left(41p^{2}+60p+8\right)(n-p-1)^{3}$
$\displaystyle+2p\left(13p^{2}+55p+30\right)(n-p-1)^{2}+\left(29p^{3}+49p^{2}+11p-1\right)(n-p-1)+8p^{2}(p+1)$
which is positive for $2\leq p\leq n-2$. Thus we see that the resultant is
non-zero and we only get $\displaystyle{x_{2}}=\frac{2(n-p-1)}{n}$ for a
solution of equations (17) and (18). From (16), we see
$\displaystyle{x_{3}}=\frac{3n-2p-2}{n}$. Thus we obtain a Kähler-Einstein
metric in this case.
Notice that this metric corresponds (up to scale) to the Kähler-Einstein
metric $g_{2}$ of (3)
3) Let $\displaystyle x_{4}=\frac{2(2n-p-1)}{3n-2(p+1)}$. By a similar method
we obtain that for $2\leq p\leq n-2$,
$\displaystyle{x_{2}}=\frac{2(n-p-1)}{3n-2p-2}$ is the only solution of
equations (17) and (18), and from (16) we see that
$\displaystyle{x_{3}}=\frac{n}{3n-2p-2}$. Thus we obtain a Kähler-Einstein
metric in this case.
4) Let $\displaystyle x_{4}=\frac{2(p-1)}{n+2(p-1)}$. By a similar method we
obtain that for $2\leq p\leq n-2$,
$\displaystyle{x_{2}}=\frac{2(n+p-1)}{n+2p-2}$ is the only positive solution
of the equations (17) and (18) for $\displaystyle\frac{n}{2}\leq p\leq n-2$,
and from (16) we see that $\displaystyle{x_{3}}=\frac{n}{n+2p-2}$.
Therefore, we obtain a Kähler-Einstein metric in all four cases.
We now denote by $T(x_{4})$ the factor of degree $4$ in the factorization of
$Q(x_{4})$. Then we can write
$\displaystyle T(x_{4})=(n-1)n^{2}(3n-4)(2n-p-1){x_{4}}^{4}$
$\displaystyle+4(n-1)n(2n-p-1)\left(n^{2}-4np-2p^{2}+8p-2\right){x_{4}}^{3}$
$\displaystyle+2(n^{5}-19n^{4}p+11n^{4}+36n^{3}p^{2}+18n^{3}p-30n^{3}+22n^{2}p^{3}-130n^{2}p^{2}+54n^{2}p$
$\displaystyle+22n^{2}-16np^{4}-4np^{3}+108np^{2}-68np-4n+16p^{4}-16p^{3}-16p^{2}+16p){x_{4}}^{2}$
$\displaystyle-8(p-1)(n-2p)(n+p-1)\left(n^{2}-6np+2n+2p^{2}+4p-2\right){x_{4}}$
$\displaystyle+8(p-1)^{2}(n-2p)^{2}(n+p-1).$
The case $n=2p$ has been studied in [AC].
We now proceed in two steps.
STEP 1. We will show that for $n\geq 4$ and $2\leq p<n/2$ the equation
$T(x_{4})=0$ has no positive solutions.
Note that $T(0)=8(p-1)^{2}(n-2p)^{2}(n+p-1)>0$ for $2\leq p<n/2$.
We have that
$\displaystyle\frac{dT}{dx_{4}}(x_{4})=4(n-1)n^{2}(3n-4)(2n-p-1){x_{4}}^{3}$
$\displaystyle+12(n-1)n(2n-p-1)\left(n^{2}-4np-2p^{2}+8p-2\right){x_{4}}^{2}$
$\displaystyle+4\left(n^{5}-19n^{4}p+11n^{4}+36n^{3}p^{2}+18n^{3}p-30n^{3}+22n^{2}p^{3}-130n^{2}p^{2}+54n^{2}p\right.$
$\displaystyle\left.+22n^{2}-16np^{4}-4np^{3}+108np^{2}-68np-4n+16p^{4}-16p^{3}-16p^{2}+16p\right){x_{4}}$
$\displaystyle-8(p-1)(n-2p)(n+p-1)\left(n^{2}-6np+2n+2p^{2}+4p-2\right).$
Note that the coefficient of ${x_{4}}^{3}$ is $4(n-1)n^{2}(3n-4)(2n-p-1)>0$.
The polynomial $T(x_{4})$ of degree $4$ attains a local minimum at
$x_{4}=u_{1}$, a local maximum at $x_{4}=u_{2}$, and a local minimum at
$x_{4}=u_{3}$.
By evaluating $\displaystyle\frac{dT}{dx_{4}}(x_{4})$ at the point
$\displaystyle\alpha=-\frac{n-2p}{2n}<0$, we have that
$\displaystyle\frac{dT}{dx_{4}}\left(-\frac{n-2p}{2n}\right)=\frac{(n-2p)}{2n}\left(2n^{5}+9n^{4}p-29n^{4}+8n^{3}p^{2}-33n^{3}p+103n^{3}-24n^{2}p^{2}\right.$
$\displaystyle\left.-16n^{2}p-112n^{2}+8np^{4}+48np^{3}-32np^{2}+92np+36n-40p^{4}+8p^{3}+8p^{2}-40p\right).$
Since we can write
$\displaystyle
2n^{5}+9n^{4}p-29n^{4}+8n^{3}p^{2}-33n^{3}p+103n^{3}-24n^{2}p^{2}-16n^{2}p-112n^{2}+8np^{4}$
$\displaystyle+48np^{3}-32np^{2}+92np+36n-40p^{4}+8p^{3}+8p^{2}-40p$
$\displaystyle=2(n-2p)^{5}+(29p-29)(n-2p)^{4}+\left(160p^{2}-265p+103\right)(n-2p)^{3}$
$\displaystyle+\left(424p^{3}-918p^{2}+602p-112\right)(n-2p)^{2}+\left(552p^{4}-1372p^{3}+1140p^{2}-356p+36\right)\times$
$\displaystyle(n-2p)+288p^{5}-768p^{4}+704p^{3}-256p^{2}+32p$
we see that $\displaystyle\frac{dT}{dx_{4}}\left(\alpha\right)>0$, thus
$u_{1}<\alpha$
Also, by evaluating $\displaystyle\frac{dT}{dx_{4}}(x_{4})$ at the point
$\displaystyle x_{4}=\beta=\frac{2(p-1)}{n}>0$, we have that
$\displaystyle\frac{dT}{dx_{4}}\left(\frac{2(p-1)}{n}\right)=-\frac{16(p-1)(n-p-1)}{n}\left(n^{3}+3n^{2}p-7n^{2}-8np^{2}+4np+8n\right.$
$\displaystyle\left.+2p^{3}+6p^{2}-6p-2\right).$
Since we can write
$\displaystyle n^{3}+3n^{2}p-7n^{2}-8np^{2}+4np+8n+2p^{3}+6p^{2}-6p-2$
$\displaystyle=$
$\displaystyle(n-2p)^{3}+(9p-7)(n-2p)^{2}+8(p-1)(2p-1)(n-2p)+2(p-1)^{2}(3p-1))>0,$
we see that $\displaystyle\frac{dT}{dx_{4}}\left(\beta\right)<0$, thus
$\beta<u_{3}$
Therefore, the three real solutions $u_{1}$, $u_{2}$, $u_{3}$ of the
polynomial $\displaystyle\frac{dT}{dx_{4}}(x_{4})$ of degree 3 satisfy
$\displaystyle u_{1}<\alpha<u_{2}<\beta<u_{3}.$
$T(x_{4})$ $\displaystyle\frac{dT}{dx_{4}}(x_{4})$
$\displaystyle\frac{dT}{dx_{4}}(x_{4})$ $T(x_{4})$
Since $T(0)>0$, in order to show that $T(x_{4})>0$ for $x_{4}>0$, we need to
prove the following:
Claim. The local minimum $T(u_{3})$ is positive.
We show our claim by dividing into two cases, namely $p=2$ and $p\geq 3$.
Case 1. $p=2$
The polynomial $T(x_{4})$ is given by
$\displaystyle
T(x_{4})=(n-1)n^{2}(2n-3)(3n-4){x_{4}}^{4}+4(n-1)n(2n-3)\left(n^{2}-8n+6\right){x_{4}}^{3}$
$\displaystyle+2\left(n^{5}-27n^{4}+150n^{3}-214n^{2}+4n+96\right){x_{4}}^{2}-8(n-4)(n+1)\left(n^{2}-10n+14\right){x_{4}}$
$\displaystyle+8(n-4)^{2}(n+1).$
Then the local minimum of $T(x_{4})$ at $x_{4}=u_{3}$ satisfies
$2/n<u_{3}<2/n+(2/n)^{2}$.
Indeed, it is
$\displaystyle\frac{dT}{dx_{4}}(x_{4})=4(n-1)n^{2}(2n-3)(3n-4){x_{4}}^{3}+12(n-1)n(2n-3)\left(n^{2}-8n+6\right){x_{4}}^{2}$
$\displaystyle+4\left(n^{5}-27n^{4}+150n^{3}-214n^{2}+4n+96\right){x_{4}}-8(n-4)(n+1)\left(n^{2}-10n+14\right).$
Then
$\displaystyle\frac{dT}{dx_{4}}(2/n)=-\frac{16(n-3)\left(n^{3}-n^{2}-16n+26\right)}{n}$
$\displaystyle=-\frac{16(n-3)\left((n-4)^{3}+11(n-4)^{2}+24(n-4)+10\right)}{n}<0$
and
$\displaystyle\frac{dT}{dx_{4}}(2/n+(2/n)^{2})=\frac{16\left(n^{6}+7n^{5}-12n^{4}-14n^{3}-152n^{2}+392n-192\right)}{n^{4}}$
$\displaystyle=\frac{16}{n^{4}}\left((n-4)^{6}+31(n-4)^{5}+368(n-4)^{4}+2194(n-4)^{3}+6848(n-4)^{2}\right.$
$\displaystyle\left.+10536(n-4)+6240\right)>0.$
Also, we have that
$\displaystyle\frac{d^{2}T}{d{x_{4}}^{2}}(x_{4})=12(n-1)n^{2}(2n-3)(3n-4){x_{4}}^{2}+24(n-1)n(2n-3)\left(n^{2}-8n+6\right){x_{4}}$
$\displaystyle+4\left(n^{5}-27n^{4}+150n^{3}-214n^{2}+4n+96\right)$
$\displaystyle=12(n-1)n^{2}(2n-3)(3n-4)\left(x_{4}+\frac{n^{2}-8n+6}{n(3n-4)}\right)^{2}+4(n^{5}-27n^{4}+150n^{3}$
$\displaystyle-214n^{2}+4n+96)-\frac{12(n-1)(2n-3)(n^{2}-8n+6)^{2}}{(3n-4)}.$
Note that
$\frac{2}{n}-(-\frac{n^{2}-8n+6}{n(3n-4)})=\frac{n^{2}-2n-2}{n(3n-4)}=\frac{(n-3)^{2}+4(n-3)+1}{n(3n-4)}>0$
and
$\displaystyle\frac{d^{2}T}{d{x_{4}}^{2}}(2/n)=4(n-3)(n-2)\left(n^{3}+2n^{2}-26n+28\right)$
$\displaystyle=4(n-3)(n-2)\left((n-4)^{3}+14(n-4)^{2}+38(n-4)+20\right)>0.$
Hence, the function $T(x_{4})$ is concave up for $x_{4}\geq 2/n$, so the local
minimum $x_{4}=u_{3}$ satisfies $2/n<u_{3}<2/n+(2/n)^{2}$.
We consider the tangent lines of the curve $T(x_{4})$ at $x_{4}=2/n$ and
$x_{4}=2/n+(2/n)^{2}$, given by the equations
$\displaystyle
z_{1}(t)=\frac{16(n-3)^{2}(3n+8)}{n^{2}}-\frac{16(n-3)\left(n^{3}-n^{2}-16n+26\right)}{n}(t-2/n)$
$\displaystyle=-\frac{16(n-3)\left(\left(n^{3}-n^{2}-16n+26\right)n\
t-2n^{3}-n^{2}+33n-28\right)}{n^{2}}$
and
$\displaystyle
z_{2}(t)=\frac{16\left(n^{6}+7n^{5}-12n^{4}-14n^{3}-152n^{2}+392n-192\right)}{n^{4}}\left(t-\frac{4}{n^{2}}-\frac{2}{n}\right)$
$\displaystyle+\frac{16\left(n^{7}+3n^{5}-28n^{4}-40n^{3}+32n^{2}+368n-192\right)}{n^{6}}$
$\displaystyle=\frac{16}{n^{6}}\left(\left(n^{6}+7n^{5}-12n^{4}-14n^{3}-152n^{2}+392n-192\right)n^{2}\
t-n^{7}-18n^{6}\right.\ \ \quad\quad\quad$
$\displaystyle\left.-n^{5}+48n^{4}+320n^{3}-144n^{2}-816n+576\right)$
respectively. These are shown in the figure.
$x_{4}=2/n$ $x_{4}=2/n+(2/n)^{2}$
Let $(x_{0},y_{0})$ be their point of intersection given by
$\displaystyle
x_{0}=\frac{2\left(n^{8}-2n^{7}-9n^{6}+64n^{5}-66n^{4}-160n^{3}+72n^{2}+408n-288\right)}{n^{2}\left(n^{7}-3n^{6}-6n^{5}+62n^{4}-92n^{3}-152n^{2}+392n-192\right)},$
$\displaystyle
y_{0}=\frac{16(n-3)}{n^{3}\left(n^{7}-3n^{6}-6n^{5}+62n^{4}-92n^{3}-152n^{2}+392n-192\right)}\times$
$\displaystyle\left(n^{10}-2n^{9}-17n^{8}+68n^{7}+94n^{6}-500n^{5}+88n^{4}-5368n^{3}+26048n^{2}-35808n+14976\right).$
Note that
$\displaystyle
n^{10}-2n^{9}-17n^{8}+68n^{7}+94n^{6}-500n^{5}+88n^{4}-5368n^{3}+26048n^{2}-35808n+14976$
$\displaystyle=(n-3)^{10}+28(n-3)^{9}+334(n-3)^{8}+2252(n-3)^{7}+9712(n-3)^{6}+29164(n-3)^{5}$
$\displaystyle+65002(n-3)^{4}+102860(n-3)^{3}+99479(n-3)^{2}+47904(n-3)+8064>0$
for $n\geq 3$. Therefore, the local minimal $T(u_{3})$ is greater than
$y_{0}$, and the claim has been proved.
Case 2. $3\leq p<n/2$.
Note that $n-p\geq p$ and
$\displaystyle
T(2(p-1)/n)=\frac{16(p-1)^{2}(n-p-1)^{2}\left(np+n+4(p-1)p\right)}{n^{2}}>0.$
Now we have
$\displaystyle\frac{dT}{dx_{4}}(x_{4})=4(n-1)n^{2}(3n-4)(2n-p-1){x_{4}}^{3}$
$\displaystyle+12(n-1)n(2n-p-1)\left(n^{2}-4np-2p^{2}+8p-2\right){x_{4}}^{2}$
$\displaystyle+4\left(n^{5}-19n^{4}p+11n^{4}+36n^{3}p^{2}+18n^{3}p-30n^{3}+22n^{2}p^{3}-130n^{2}p^{2}+54n^{2}p\right.$
$\displaystyle\left.+22n^{2}-16np^{4}-4np^{3}+108np^{2}-68np-4n+16p^{4}-16p^{3}-16p^{2}+16p\right){x_{4}}$
$\displaystyle-8(p-1)(n-2p)(n+p-1)\left(n^{2}-6np+2n+2p^{2}+4p-2\right)$
and
$\displaystyle\frac{dT}{dx_{4}}(2(p-1)/n)=-\frac{16(p-1)(n-p-1)}{n}\left(n^{3}+3n^{2}p-7n^{2}-8np^{2}+4np+8n\right.$
$\displaystyle\left.+2p^{3}+6p^{2}-6p-2\right).$
Note that
$\displaystyle n^{3}+3n^{2}p-7n^{2}-8np^{2}+4np+8n+2p^{3}+6p^{2}-6p-2$
$\displaystyle=$
$\displaystyle(n-2p)^{3}+(9p-7)(n-2p)^{2}+8(p-1)(2p-1)(n-2p)+2(p-1)^{2}(3p-1))>0,$
thus we see that $\displaystyle\frac{dT}{dx_{4}}(\beta)<0$.
Let $z_{1}(t)$ be the tangent line of the curve $T(x_{4})$ at $x_{4}=\beta$.
This is given by
$\displaystyle
z_{1}(t)=-\frac{16(p-1)(n-p-1)}{n}\left(n^{3}+3n^{2}p-7n^{2}-8np^{2}+4np+8n+2p^{3}+6p^{2}\right.$
$\displaystyle\left.-6p-2\right)\left(t-\frac{2(p-1)}{n}\right)+\frac{16(p-1)^{2}(n-p-1)^{2}\left(np+n+4(p-1)p\right)}{n^{2}}.$
We consider the point $t_{0}$ such that $z_{1}(t_{0})=0$. Then we see that
$\displaystyle
t_{0}=\frac{(p-1)\left(2n^{3}+7n^{2}p-13n^{2}-13np^{2}+2np+15n+12p^{2}-8p-4\right)}{n\left(n^{3}+3n^{2}p-7n^{2}-8np^{2}+4np+8n+2p^{3}+6p^{2}-6p-2\right)}.$
We will show that $\displaystyle\frac{dT}{dx_{4}}(t_{0})>0$ for $3\leq p\leq
n/2$. Indeed, we have
$\displaystyle\frac{dT}{dx_{4}}(t_{0})=\frac{4(p-1)(n-p-1)A(n,p)}{n\left(n^{3}+3n^{2}p-7n^{2}-8np^{2}+4np+8n+2p^{3}+6p^{2}-6p-2\right)^{3}},$
where
$\displaystyle
A(n,p)=n^{12}p-3n^{12}+15n^{11}p^{2}-68n^{11}p+85n^{11}+73n^{10}p^{3}-447n^{10}p^{2}$
$\displaystyle+1135n^{10}p-985n^{10}+68n^{9}p^{4}-730n^{9}p^{3}+3590n^{9}p^{2}-8134n^{9}p+6102n^{9}$
$\displaystyle-388n^{8}p^{5}+1743n^{8}p^{4}-3118n^{8}p^{3}-6724n^{8}p^{2}+28594n^{8}p-22347n^{8}-590n^{7}p^{6}$
$\displaystyle+3284n^{7}p^{5}-13140n^{7}p^{4}+38772n^{7}p^{3}-26930n^{7}p^{2}-48968n^{7}p+51156n^{7}$
$\displaystyle+1180n^{6}p^{7}-3852n^{6}p^{6}+17728n^{6}p^{5}-22616n^{6}p^{4}-72692n^{6}p^{3}+123252n^{6}p^{2}$
$\displaystyle+29464n^{6}p-76048n^{6}+961n^{5}p^{8}+148n^{5}p^{7}-20352n^{5}p^{6}-18940n^{5}p^{5}$
$\displaystyle+140330n^{5}p^{4}-14084n^{5}p^{3}-190872n^{5}p^{2}+29804n^{5}p+75053n^{5}-2356n^{4}p^{9}$
$\displaystyle-6225n^{4}p^{8}+24308n^{4}p^{7}+60596n^{4}p^{6}-94876n^{4}p^{5}-165630n^{4}p^{4}+164044n^{4}p^{3}$
$\displaystyle+136388n^{4}p^{2}-67312n^{4}p-49449n^{4}+1068n^{3}p^{10}+11644n^{3}p^{9}-9136n^{3}p^{8}$
$\displaystyle-59672n^{3}p^{7}-27216n^{3}p^{6}+194496n^{3}p^{5}+28056n^{3}p^{4}-177768n^{3}p^{3}-35740n^{3}p^{2}$
$\displaystyle+52804n^{3}p+21464n^{3}+32n^{2}p^{11}-6532n^{2}p^{10}-5992n^{2}p^{9}+24252n^{2}p^{8}+53120n^{2}p^{7}$
$\displaystyle-41016n^{2}p^{6}-130288n^{2}p^{5}+63304n^{2}p^{4}+77920n^{2}p^{3}-7492n^{2}p^{2}-21416n^{2}p$
$\displaystyle-5892n^{2}-64np^{12}+1024np^{11}+5664np^{10}-5120np^{9}-11616np^{8}-23168np^{7}$
$\displaystyle+44864np^{6}+27264np^{5}-37376np^{4}-12672np^{3}+5792np^{2}+4480np+928n$
$\displaystyle-768p^{11}-832p^{10}+2688p^{9}+960p^{8}+4608p^{7}-12672p^{6}+1792p^{5}+5248p^{4}$
$\displaystyle+256p^{3}-832p^{2}-384p-64.$
We shall show that $A(n,p)>0$ for $3\leq p\leq n/2$. We can write $A(n,p)$ as
a polynomial of $y=n-2p$ of the form
$A(n,p)=(p-3)y^{12}+a_{11}y^{11}+a_{10}y^{10}+a_{9}y^{9}+a_{8}y^{8}+a_{7}y^{7}+a_{6}y^{6}+a_{5}y^{5}+a_{4}y^{4}+a_{3}y^{3}+a_{2}y^{2}+a_{1}y+a_{0},$
where $a_{j}\ (j=0,\dots,11)$ can be written as follows:
$\displaystyle a_{11}=39(p-3)^{2}+94(p-3)+16$ $\displaystyle
a_{10}=667(p-3)^{3}+3268(p-3)^{2}+4604(p-3)+1424$ $\displaystyle
a_{9}=6588(p-3)^{4}+49146(p-3)^{3}+131552(p-3)^{2}+146040(p-3)+53568$
$\displaystyle
a_{8}=41696(p-3)^{5}+420063(p-3)^{4}+1666118(p-3)^{3}+3239144(p-3)^{2}$
$\displaystyle+3068032(p-3)+1121664$ $\displaystyle
a_{7}=177618(p-3)^{6}+2258984(p-3)^{5}+11898662(p-3)^{4}+33203396(p-3)^{3}$
$\displaystyle+51731888(p-3)^{2}+42627136(p-3)+14495232$ $\displaystyle
a_{6}=521336(p-3)^{7}+8010196(p-3)^{6}+52618296(p-3)^{5}+191551956(p-3)^{4}$
$\displaystyle+417348472(p-3)^{3}+544194848(p-3)^{2}+393195520(p-3)+121432064$
$\displaystyle
a_{5}=1062393(p-3)^{8}+19117036(p-3)^{7}+150329840(p-3)^{6}+674768512(p-3)^{5}$
$\displaystyle+1890947640(p-3)^{4}+3387906256(p-3)^{3}+3789854976(p-3)^{2}$
$\displaystyle+2420175872(p-3)+675510272$ $\displaystyle
a_{4}=1493910(p-3)^{9}+30767865(p-3)^{8}+281434708(p-3)^{7}+1500619596(p-3)^{6}$
$\displaystyle+5140194384(p-3)^{5}+11730160160(p-3)^{4}+17833993024(p-3)^{3}$
$\displaystyle+17419011328(p-3)^{2}+9918241792(p-3)+2508337152$ $\displaystyle
a_{3}=1416852(p-3)^{10}+32818860(p-3)^{9}+341869872(p-3)^{8}+2109020632(p-3)^{7}$
$\displaystyle+8532907744(p-3)^{6}+23658308832(p-3)^{5}+45523459968(p-3)^{4}$
$\displaystyle+60028498688(p-3)^{3}+51913028096(p-3)^{2}+26587561984(p-3)+6123782144$
$\displaystyle
a_{2}=862488(p-3)^{11}+22162788(p-3)^{10}+258695208(p-3)^{9}+1810579704(p-3)^{8}$
$\displaystyle+8442449008(p-3)^{7}+27537781712(p-3)^{6}+64116833984(p-3)^{5}$
$\displaystyle+106560650432(p-3)^{4}+123887801600(p-3)^{3}+95957073920(p-3)^{2}$
$\displaystyle+44563972096(p-3)+9401008128$ $\displaystyle
a_{1}=303264(p-3)^{12}+8551008(p-3)^{11}+110430432(p-3)^{10}+863710128(p-3)^{9}$
$\displaystyle+4556601456(p-3)^{8}+17082048928(p-3)^{7}+46660844352(p-3)^{6}$
$\displaystyle+93574409856(p-3)^{5}+136732708864(p-3)^{4}+141973649408(p-3)^{3}$
$\displaystyle+99432382464(p-3)^{2}+42173857792(p-3)+8192524288$
$\displaystyle
a_{0}=46656(p-3)^{13}+1430784(p-3)^{12}+20235744(p-3)^{11}+174764304(p-3)^{10}$
$\displaystyle+1028302272(p-3)^{9}+4352962512(p-3)^{8}+13638809216(p-3)^{7}$
$\displaystyle+32024909952(p-3)^{6}+56352955904(p-3)^{5}+73394750720(p-3)^{4}$
$\displaystyle+68769538048(p-3)^{3}+43897815040(p-3)^{2}+17110138880(p-3)+3075473408.$
We see that the coefficients $a_{j}$ ($j=0,\dots,11$) are positive for $p\geq
3$, which means that $A(n,p)>0$ for $3\leq p<n/2$. Therefore,
$\displaystyle\frac{dT}{dx_{4}}(t_{0})>0$ for $3\leq p\leq n/2$.
Now we compute $\displaystyle\frac{d^{2}T}{d{x_{4}}^{2}}(x_{4})$. We see that
$\displaystyle\frac{d^{2}T}{d{x_{4}}^{2}}(x_{4})=12(n-1)n^{2}(3n-4)(2n-p-1){x_{4}}^{2}$
$\displaystyle+24(n-1)n(2n-p-1)\left(n^{2}-4np-2p^{2}+8p-2\right){x_{4}}$
$\displaystyle+4\left(n^{5}-19n^{4}p+11n^{4}+36n^{3}p^{2}+18n^{3}p-30n^{3}+22n^{2}p^{3}-130n^{2}p^{2}+54n^{2}p+22n^{2}\right.$
$\displaystyle\left.-16np^{4}-4np^{3}+108np^{2}-68np-4n+16p^{4}-16p^{3}-16p^{2}+16p\right)$
$\displaystyle=12(n-1)n^{2}(3n-4)(2n-p-1)\left(x_{4}+\frac{n^{2}-4np-2p^{2}+8p-2}{n(3n-4)}\right)^{2}$
$\displaystyle+4\left(n^{5}-19n^{4}p+11n^{4}+36n^{3}p^{2}+18n^{3}p-30n^{3}+22n^{2}p^{3}-130n^{2}p^{2}+54n^{2}p+22n^{2}\right.$
$\displaystyle\left.-16np^{4}-4np^{3}+108np^{2}-68np-4n+16p^{4}-16p^{3}-16p^{2}+16p\right)$
$\displaystyle-12(n-1)(2n-p-1)\frac{(n^{2}-4np-2p^{2}+8p-2)^{2}}{(3n-4)}.$
Note that
$\displaystyle\beta-(-\frac{n^{2}-4np-2p^{2}+8p-2}{n(3n-4)})=\frac{n^{2}+2np-6n-2p^{2}+6}{n(3n-4)}$
$\displaystyle=\frac{(n-2p)^{2}+6(p-1)(n-2p)+6(p-1)^{2}}{n(3n-4)}>0,$
and
$\displaystyle\frac{d^{2}T}{d{x_{4}}^{2}}\left(\beta\right)=4(n-p-1)\left(n^{4}+6n^{3}p-12n^{3}+6n^{2}p^{2}-60n^{2}p+66n^{2}-8np^{3}\right.$
$\displaystyle\left.+32np^{2}+48np-80n+8p^{3}-40p^{2}+8p+24\right)$
$\displaystyle=4(n-p-1)\left((n-2p)^{4}+2(7p-6)(n-2p)^{3}+66(p-1)^{2}(n-2p)^{2}\right.$
$\displaystyle\left.+8(p-1)\left(15p^{2}-29p+10\right)(n-2p)+8(p-1)(3p-1)\left(3p^{2}-7p+3\right)\right)>0.$
Therefore, the function $T(x_{4})$ is concave up for $x_{4}\geq 2(p-1)/n$.
$x_{4}=\beta$ $x_{4}=t_{0}$
Consider the tangent line $l_{1}$ of the curve $T(x_{4})$ at $x_{4}=\beta$,
which intersects $x$-axis at a point $t_{0}$, and the tangent line $l_{2}$ of
the curve $T(x_{4})$ at $x_{4}=t_{0}$. Since
$\displaystyle\frac{dT}{dx_{4}}(\beta)<0$ and
$\displaystyle\frac{dT}{dx_{4}}(t_{0})>0$, the tangent lines $l_{1},l_{2}$
intersect at a point $(x_{0},y_{0})$ with $y_{0}>0$. Since $T(x_{4})$ is
concave up, we see that the curve $(x_{4},T(x_{4}))$ $(\beta\leq x_{4}\leq
t_{0})$ lies inside the triangle given by the three points $(\beta,T(\beta)$,
$(x_{0},y_{0})$ and $(t_{0},T(t_{0}))$.
Since the point $(u_{3},T(u_{3}))$ is inside of this triangle, it follows that
the local minimum $T(u_{3})$ is greater than $y_{0}>0$, and the claim has also
been shown in this case.
STEP 2. We consider the case that $n\geq 4$ and $n-2\geq p>n/2$.
This reduces to case STEP 1 as follows.
We consider the resultant of $F(x_{2},x_{4})$ and $G(x_{2},x_{4})$ with
respect to $x_{4}$, which is a polynomial of $x_{2}$ (instead of $x_{4}$), and
we denote this resultant by $R(x_{2})$. By factorizing $R(x_{2})$ we have that
$\displaystyle
R(x_{2})=128(n-1)^{6}(p-1)^{4}(n-p-1)^{2}{x_{2}}^{8}(n{x_{2}}-2n-2p+2)(n{x_{2}}-2n+2p+2)\times$
$\displaystyle(3n{x_{2}}-2n-2p{x_{2}}+2p-2{x_{2}}+2)(n{x_{2}}-2n+2p{x_{2}}-2p-2{x_{2}}+2)$
$\displaystyle\left(3n^{5}{x_{2}}^{4}-20n^{5}{x_{2}}^{3}+48n^{5}{x_{2}}^{2}-48n^{5}{x_{2}}+16n^{5}+3n^{4}p{x_{2}}^{4}+12n^{4}p{x_{2}}^{3}-110n^{4}p{x_{2}}^{2}\right.$
$\displaystyle+200n^{4}p{x_{2}}-104n^{4}p-10n^{4}{x_{2}}^{4}+72n^{4}{x_{2}}^{3}-178n^{4}{x_{2}}^{2}+168n^{4}{x_{2}}-40n^{4}+24n^{3}p^{2}{x_{2}}^{3}$
$\displaystyle+12n^{3}p^{2}{x_{2}}^{2}-248n^{3}p^{2}{x_{2}}+256n^{3}p^{2}-7n^{3}p{x_{2}}^{4}-44n^{3}p{x_{2}}^{3}+380n^{3}p{x_{2}}^{2}-640n^{3}p{x_{2}}$
$\displaystyle+224n^{3}p+11n^{3}{x_{2}}^{4}-92n^{3}{x_{2}}^{3}+232n^{3}{x_{2}}^{2}-200n^{3}{x_{2}}+32n^{3}-8n^{2}p^{3}{x_{2}}^{3}+84n^{2}p^{3}{x_{2}}^{2}$
$\displaystyle+48n^{2}p^{3}{x_{2}}-296n^{2}p^{3}-48n^{2}p^{2}{x_{2}}^{3}-92n^{2}p^{2}{x_{2}}^{2}+736n^{2}p^{2}{x_{2}}-440n^{2}p^{2}+4n^{2}p{x_{2}}^{4}$
$\displaystyle+56n^{2}p{x_{2}}^{3}-444n^{2}p{x_{2}}^{2}+656n^{2}p{x_{2}}-152n^{2}p-4n^{2}{x_{2}}^{4}+48n^{2}{x_{2}}^{3}-124n^{2}{x_{2}}^{2}$
$\displaystyle+96n^{2}{x_{2}}-8n^{2}-32np^{4}{x_{2}}^{2}+80np^{4}{x_{2}}+160np^{4}+8np^{3}{x_{2}}^{3}-120np^{3}{x_{2}}^{2}-256np^{3}{x_{2}}$
$\displaystyle+352np^{3}+24np^{2}{x_{2}}^{3}+120np^{2}{x_{2}}^{2}-576np^{2}{x_{2}}+224np^{2}-24np{x_{2}}^{3}+200np{x_{2}}^{2}$
$\displaystyle-256np{x_{2}}+32np-8n{x_{2}}^{3}+24n{x_{2}}^{2}-16n{x_{2}}-32p^{5}{x_{2}}-32p^{5}+32p^{4}{x_{2}}^{2}-96p^{4}$
$\displaystyle\left.+32p^{3}{x_{2}}^{2}+128p^{3}{x_{2}}-96p^{3}-32p^{2}{x_{2}}^{2}+128p^{2}{x_{2}}-32p^{2}-32p{x_{2}}^{2}+32p{x_{2}}\right).$
We denote by $S(x_{2})$ the factor of degree $4$ in the above factorization.
Then we can write
$\displaystyle S(x_{2})=(n-1)n^{2}(3n-4)(n+p-1){x_{2}}^{4}$
$\displaystyle-4(n-1)n(n+p-1)\left(5n^{2}-8np-8n+2p^{2}+8p+2\right){x_{2}}^{3}$
$\displaystyle+2\left(24n^{5}-55n^{4}p-89n^{4}+6n^{3}p^{2}+190n^{3}p+116n^{3}+42n^{2}p^{3}-46n^{2}p^{2}-222n^{2}p\right.$
$\displaystyle\left.-62n^{2}-16np^{4}-60np^{3}+60np^{2}+100np+12n+16p^{4}+16p^{3}-16p^{2}-16p\right){x_{2}}^{2}$
$\displaystyle-8(n-2p)(n-p-1)(2n-p-1)\left(3n^{2}-2np-6n-2p^{2}+4p+2\right){x_{2}}$
$\displaystyle+8(n-2p)^{2}(n-p-1)^{2}(2n-p-1).$
If we replace $p$ with $n-p$ in the polynomial $S(x_{2})$, we get exactly the
same polynomial as $T(x_{2})$, and thus we see that the equation $S(x_{2})=0$
has no positive solutions for $n-2\geq p>n/2$.
∎
The Main Theorem now follows from Propositions 4 and 5.
## 4\. The isometry problem
In this section we study the isometry problem for the new homogeneous Einstein
metrics of $M=SO(2n)/U(p)\times U(n-p)$, corresponding to the pairs $(n,p)$
which are presented in the Main Theorem. Recall that when $n=2p$, it was
proved in [AC] that the non-Kähler homogeneous Einstein metrics of the form
$g=(1,x_{2},1,x_{2})$, where $x_{2}$ is given by part $(a)$ of (14), are not
isometric. However for the special case of $2\leq p\leq 6$, the isometry
problem for the remaining two new Einstein metrics $g=(1,x_{2},1,x_{4})$,
where $x_{2}$ and $x_{4}$ are determined by part $(b)$ of (14), and
$(\ref{3})$ respectively, has not been studied yet.111Note that the first two
non-Kähler Einstein metrics on $M=SO(4p)/U(p)\times U(p)$, were obtained in
[AC, Theorem. 8] with respect to the normalization $g=(x_{1},1,x_{1},1)$. For
the special case $2\leq p\leq 6$ the new Einstein metrics are given with
respect to the normalization $g=(x_{1},1,x_{1},x_{4})$.
Let us recall the method used in [AC]. For any $G$-invariant Einstein metric
$g=(x_{1},x_{2},x_{3},x_{4})$ on $M=SO(2n)/U(p)\times U(n-p)$ (with $2\leq
p\leq n-2$) we determine a (normalized) scale invariant given by
$H_{g}=V_{g}^{1/d}S_{g}$, where $S_{g}$ is the scalar curvature of the given
metric $g$, $V_{g}=\prod_{i=1}^{4}x_{i}^{d_{i}}$ is the volume of $g$ and
$d=\sum_{i=1}^{4}d_{i}=\dim M$. In particular, the scalar curvature of $g$ is
given by
$S_{g}=\frac{1}{2}\sum_{i=1}^{4}\frac{d_{i}}{x_{i}}-\frac{[123]}{2}(\frac{x_{1}}{x_{2}x_{3}}+\frac{x_{2}}{x_{1}x_{3}}+\frac{x_{3}}{x_{1}x_{2}})-\frac{[134]}{2}(\frac{x_{1}}{x_{3}x_{4}}+\frac{x_{3}}{x_{1}x_{4}}+\frac{x_{4}}{x_{1}x_{3}})$
where $d_{i}$ and $[123]$, $[234]$ are given in Section 2. Note that $S_{g}$
is a homogeneous polynomial of degree $-1$ on the variables $x_{i}$, and the
volume $V_{g}$ is a monomial of degree $d$. Thus $H_{g}=V_{g}^{1/d}S_{g}$ is a
homogeneous polynomial of degree 0, and it is invariant under a common scaling
of the variables $x_{i}$. If two metrics are isometric then they have the same
scale invariant, so if the scale invariants $H_{g}$ and $H_{g^{\prime}}$ are
different, then the metrics $g$ and $g^{\prime}$ can not be isometric. If
$H_{g}=H^{\prime}_{g}$ we can not draw an immediate decision and conclude if
the metrics $g$ and $g^{\prime}$ are isometric or not. Finally, Kähler-
Einstein metrics which correspond to equivalent invariant complex structures
on $M$ are isometric (cf. [AC]).
In order to detect which pairs of Einstein metrics in the Main Theorem are
isometric or not, first we need to give their approximate values. Note that
the non-Kähler Einstein metrics are of the form $g=(1,x_{2},1,x_{4})$, where
$x_{2}$ is obtained by solving equation $H_{n,p}(x_{2})=0$ (see $(\ref{4})$),
if we first substitute the corresponding values of $n$ and $p$. Next, $x_{4}$
is easily obtained from (9). In the following table we present the case of
$n\neq 2p$.
Table 1 Approximate values of Einstein metrics on $M$ for pairs $(n,p)$ with
$n\neq 2p$
Pair | Einstein metrics | | |
---|---|---|---|---
$(n,p)$ | $g_{1}=(1,x_{2},1,x_{4})$ | $g_{2}=(1,x_{2},1,x_{4})$ | $g_{3}=(1,x_{2},1,x_{4})$ | $g_{4}=(1,x_{2},1,x_{4})$
$(7,4)$ | $(1,0.4661,1,0.7256)$ | $(1,0.6614,1,1.7636)$ | $(1,1.4144,1,1.3999)$ | $(1,1.5722,1,1.0631)$
$(7,3)$ | $(1,0.7256,1,0.4661)$ | $(1,1.7636,1,0.6614)$ | $(1,1.3999,1,1.4144)$ | $(1,1.0631,1,1.5722)$
$(6,4)$ | $(1,0.2680,1,0.8876)$ | $(1,0.3631,1,1.9057)$ | $(1,1.3782,1,1.5645)$ | $(1,1.5461,1,1.1658)$
$(6,2)$ | $(1,0.8876,1,0.2680)$ | $(1,1.9057,1,0.3631)$ | $(1,1.5645,1,1.3782)$ | $(1,1.1658,1,1.5461)$
$(5,3)$ | $(1,0.3241,1,0.6954)$ | $(1,0.4361,1,1.8876)$ | $(1,1.4331,1,1.5883)$ | $(1,1.6922,1,0.8952)$
$(5,2)$ | $(1,0.6954,1,0.3241)$ | $(1,1.8876,1,0.4361)$ | $(1,1.5883,1,1.4331)$ | $(1,0.8952,1,1.6922)$
Note that the values $(7,4)$ and $(7,3)$, $(6,4)$ and $(6,2)$, $(5,3)$ and
$(5,2)$, determine the quotients
$M_{1}=SO(14)/U(4)\times U(3)$, | $M^{1}=SO(14)/U(3)\times U(4)$,
---|---
$M_{2}=SO(12)/U(4)\times U(2)$, | $M^{2}=SO(12)/U(2)\times U(4)$,
$M_{3}=SO(10)/U(3)\times U(2)$, | $M^{3}=SO(10)/U(2)\times U(3)$,
respectively. In particular, as we can see from Table 1, the Einstein metrics
on $M^{i}$ are obtained from the Einstein metrics on $M_{i}$, by a permutation
of the components $x_{2},x_{4}$, for any $i=1,2,3$, and conversely.222In
general, the flag manifolds $SO(2n)/U(n-p)\times U(p)$ and $SO(2n)/U(p)\times
U(n-p)$ are isometric via an element of the Weyl group of $G$. Thus we obtain
the isometries $M_{1}\cong M^{1}$, $M_{2}\cong M^{2}$ and $M_{3}\cong M^{3}$.
This result is also obtained from Table 2, where we give the values of the
corresponding scale invariants for the Einstein metrics $g_{1},g_{2},g_{3}$
and $g_{4}$.
Also, from Table 2 we easily conclude that all non Kähler invariant Einstein
metrics on $M_{1}\cong M^{1}$, $M_{2}\cong M^{2}$ and $M_{3}\cong M^{3}$ are
not isometric, since for any case it is $H_{g_{1}}\neq H_{g_{2}}\neq
H_{g_{3}}\neq H_{g_{4}}$. This completes the examination of the case $n\neq
2p$.
Table 2 The values of the corresponding scale invariants
Scale invariants | $(7,4)$ | $(7,3)$ | $(6,4)$ | $(6,2)$ | $(5,3)$ | $(5,2)$
---|---|---|---|---|---|---
$H_{g_{1}}$ | $25.2814$ | $25.2814$ | $17.9698$ | $17.9698$ | $12.4373$ | $12.4373$
$H_{g_{2}}$ | $25.5264$ | $25.5264$ | $18.1243$ | $18.1243$ | $12.6088$ | $12.6088$
$H_{g_{3}}$ | $25.6020$ | $25.6020$ | $18.2540$ | $18.2540$ | $12.7050$ | $12.7050$
$H_{g_{4}}$ | $25.5943$ | $25.5943$ | $18.2446$ | $18.2446$ | $12.6700$ | $12.6700$
For the special case $n=2p$ with $2\leq p\leq 6$, the scale invariants
corresponding to the new non-Kähler Einstein metrics on $M=SO(4p)/U(p)\times
U(p)$ given by $g=(1,x_{2},1,x_{4})$, where $x_{2}$ and $x_{4}$ are determined
by part $(b)$ of (14), and $(\ref{3})$, respectively, are equal. However, for
$x_{2}=\frac{2p(2p-1)-\sqrt{2}\sqrt{-p\left(p^{3}-7p^{2}+5p-1\right)}}{p(3p-1)},$
$x_{4}$ is given by
$x_{4}=\frac{2p(2p-1)+\sqrt{2}\sqrt{-p\left(p^{3}-7p^{2}+5p-1\right)}}{p(3p-1)},$
and for
$x_{2}=\frac{2p(2p-1)+\sqrt{2}\sqrt{-p\left(p^{3}-7p^{2}+5p-1\right)}}{p(3p-1)},$
$x_{4}$ is given by
$x_{4}=\frac{2p(2p-1)-\sqrt{2}\sqrt{-p\left(p^{3}-7p^{2}+5p-1\right)}}{p(3p-1)}.$
Thus these two Einstein metrics on $M$ are isometric.
## References
* [AP] D. V. Alekseevsky and A. M. Perelomov: Invariant Kähler-Einstein metrics on compact homogeneous spaces, Funct. Anal. Appl. 20 (3) (1986) 171–182.
* [AC] A. Arvanitoyeorgos and I. Chrysikos: Invariant Einstein metrics on flag manifolds with four isotropy summands, Ann. Glob. Anal. Geom. 37 (4) (2010) 185–219.
* [Grv] M. M. Graev: On the number of invariant Eistein metrics on a compact homogeneous space, Newton polytopes and contraction of Lie algebras, Intern. J. Geom. Meth. Mod. Phys. 3 (5-6) (2006) 1047–1075.
* [Nis] M. Nishiyama: Classification of invariant complex structures on irreducible compact simply connected coset spaces, Osaka J. Math. 21 (1984) 39–58.
* [PaS] J-S. Park and Y. Sakane: Invariant Einstein metrics on certain homogeneous spaces, Tokyo J. Math. 20 (1) (1997) 51–61.
* [WZ] M. Wang and W. Ziller: Existence and non-excistence of homogeneous Einstein metrics, Invent. Math. 84 (1986) 177–194.
|
arxiv-papers
| 2010-06-28T09:08:59 |
2024-09-04T02:49:11.263818
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andreas Arvanitoyeorgos, Ioannis Chrysikos and Yusuke Sakane",
"submitter": "Ioannis Chrysikos",
"url": "https://arxiv.org/abs/1006.5294"
}
|
1006.5348
|
# Stable birational invariants with Galois descent and differential forms
M.Rovinsky National Research University Higher School of Economics,
Laboratory of Algebraic Geometry, 7 Vavilova Str., Moscow 117312, Russia &
Institute for Information Transmission Problems of Russian Academy of Sciences
& Independent University of Moscow marat@mccme.ru
###### Abstract.
I show that the cohomology of the generic points of algebraic complex
varieties becomes stable birational invariant, when considered ‘modulo the
cohomology of the generic points of the affine spaces’.
The author was supported by the National Science Foundation under agreement
No. DMS-0635607. During the final write-up, the author was partially supported
by RFBR grant 10-01-93113-CNRSL-a and by AG Laboratory GU-HSE, RF government
grant, ag. 11 11.G34.31.0023
These notes are concerned with certain birational invariants of smooth
algebraic varieties. All such invariants are dominant sheaves, cf. below; the
dominant sheaves are characterized in Proposition 1.7.
Two classes of invariants are of special interest: (i) stable, i.e., taking
the same values on a variety and on its direct product with an affine space,
and (ii) constant on the projective spaces. Though the latter class is a
priori wider, there are no known examples of non-stable invariants vanishing
on the projective spaces. Here an attempt of comparison is made. Namely, it is
shown that the corresponding adjoint functors coincide on the following types
of invariants: (i) of ‘level 1’, cf. Proposition 2.10 and also p.3.3, (ii)
‘related to cohomology’ (or to closed differential forms).
Differential forms play a very special rôle in the story, cf. e.g. Conjecture
1.5. Moreover, all known examples of simple invariants (as objects of an
abelian category) ‘come from’ differential forms: except for two invariants
related to the multiplicative and the additive groups
($Y\mapsto(k(Y)^{\times}/k^{\times})_{{\mathbb{Q}}}$ and $Y\mapsto k(Y)/k$,
the logarithmic and the exact differentials, cf. below), they are values of
the functor ${\mathbb{B}}^{0}$ from §1.3. For these reasons the differential
forms are studied in detail. It is shown in Corollary 2.8 that the cohomology
of the generic points of algebraic (complex) varieties becomes stable
birational invariant, when considered ‘modulo the cohomology of the generic
points of the affine spaces’.
The principal new results of §3 are Propositions 3.3 and 3.7. It is shown in
Proposition 3.3 that (i) the quotient $V^{\bullet}$ of the sheaf of algebras
of closed differential forms by the ideal generated by the exact 1-forms and
the logarithmic differentials is stable and (ii) $V^{\bullet}$ is the maximal
stable quotient of the sheaf of closed differential forms. Proposition 3.7
gives a complete description of the sheaf of closed 1-forms.
Depending on what is more convenient, we shall consider our ‘invariants’
either as dominant sheaves, or as representations, cf. §2.5. E.g., the
simplicity is more natural in the context of representations.
## 1\. Dominant presheaves and sheaves
Notations. From now on we fix an algebraically closed field $k$ of
characteristic zero, and denote by $E$ a variable coefficient field of
characteristic zero. Denote by $\text{Vec}_{E}$ the category of $E$-vector
spaces.
I am interested in birational invariants of (or ‘‘presheaves on’’)
$k$-varieties. More precisely, let ${\mathcal{S}}m_{k}^{\prime}$ be the
category, whose objects are smooth $k$-varieties and the morphisms are smooth
$k$-morphisms. Define the pretopology on ${\mathcal{S}}m_{k}^{\prime}$ by
saying that the covers are dominant morphisms. Recall, that a presheaf is a
sheaf if the following diagram is an equalizer for any covering $Y\to X$:
(1)
${\mathcal{F}}(X)\to{\mathcal{F}}(Y)\rightrightarrows{\mathcal{F}}(Y\times_{X}Y).$
The category of the sheaves of $E$-vector spaces on this site is denoted by
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}(E)$ and
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}:=\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}({\mathbb{Q}})$.
Example. For each irreducible smooth $k$-variety $X$ and integer $0\leq
q\leq\dim X$ let $\Psi_{X,q}:Y\mapsto
Z^{q}(k(X)\otimes_{k}k(Y))_{{\mathbb{Q}}}$ (${\mathbb{Q}}$-linear combinations
of irreducible subvarieties on $X\times_{k}Y$ of codimension $q$ dominant over
$X$ and $Y$.) This is a sheaf. Set $\Psi_{X}:=\Psi_{X,\dim X}$. The sheaves
$\Psi_{X}$ for all $X$ form a system of generators of
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}$.
Definition. 1\. A presheaf ${\mathcal{F}}$ is ${\mathbb{A}}^{1}$-invariant (or
stable) if
${\mathcal{F}}(X)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{F}}(X\times{\mathbb{A}}^{1})$
for all $X$.
2\. Let ${\mathcal{S}}$ be a collection of dominant morphisms in
${\mathcal{S}}m_{k}^{\prime}$ with connected fibres. Assume that
${\mathcal{S}}$ is stable under base changes of its arbitrary element by
itself: ${\rm pr}_{1}:X\times_{Y}X\to X$ belongs to ${\mathcal{S}}$ if $X\to
Y$ belongs to ${\mathcal{S}}$. A presheaf ${\mathcal{F}}$ is called an
${\mathcal{S}}$-presheaf if
${\mathcal{F}}(Y)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{F}}(X)$
for all $(X\to Y)\in{\mathcal{S}}$.
Denote by $\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}^{{\mathcal{S}}}$ the
full subcategory in $\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}$ consisting
of ${\mathcal{S}}$-sheaves. More particularly, denote by ${\mathcal{I}}_{G}$
the full subcategory in $\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}$
consisting of ${\mathbb{A}}^{1}$-invariant sheaves. Under assumptions of §1.2,
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}^{{\mathcal{S}}}\subseteq{\mathcal{I}}_{G}$.
For any dominant presheaf ${\mathcal{F}}$ denote by
$\underline{{\mathcal{F}}}$ its dominant sheafification.
For each smooth $k$-variety $Y$, we denote by $\overline{Y}$ a smooth
compactification of $Y$.
### 1.1. Examples of ${\mathbb{A}}^{1}$-invariant presheaves
In this section we consider some examples of dominant presheaves with values
in various abelian categories. They come either from algebro-geometric
constructions, or from a cohomology theory $H^{*}$ (with coefficients in a
commutative ${\mathbb{Q}}$-algebra $B$). As Example 5 suggests, those of these
examples that are ${\mathbb{A}}^{1}$-invariant sheaves, are related. This is
one of motivations for Conjecture 1.4.
An effective pure motive is a pair consisting of a smooth projective variety
and a projector in the algebra of correspondences modulo numerical
equivalence. Morphisms of co(ntra)variant pure motives are defined by
correspondences modulo numerical equivalence so that they behave as action on
the (co)homology.
Denote by ${\mathcal{M}}_{k}$ the category of covariant pure $k$-motives (and
by ${\mathcal{M}}_{k}^{\text{op}}$ its opposite, the category of contravariant
pure $k$-motives). By a well-known result of U.Jannsen, these two categories
are abelian and semisimple. A simple effective pure motive is called primitive
if it is ‘‘not divisible by the Lefschetz motive’’, the motive
$({\mathbb{P}}^{1},\pi)$, where $\pi$ induces 0 on the 0-th and the identity
on the second (co)homology.
Denote by $\overline{Y}^{\text{prim}}$ the sum in the motive of $\overline{Y}$
of all its primitive submotives; $CH^{q}$ is the (Chow) group of codimension
$q$ cycles modulo rational equivalence. We also use notations and
identifications of §2.3.
| Invariant of a connected $Y$ (dominant presheaf) | Values | stable
---|---|---|---
1 | $K_{q}(Y)_{{\mathbb{Q}}}$ for $q\geq 0$/ its sheafification | $\text{Vec}_{{\mathbb{Q}}}$ | yes/only for $q=0$
2 | $H^{q}(Y)$ for $q\geq 0$/ its sheafification $\underline{H}^{q}$ | $B$-mod | yes/only for $q=0$
3 | $\Gamma(\overline{Y},\bigotimes^{\bullet}_{{\mathcal{O}}_{\overline{Y}}}\Omega^{1}_{\overline{Y}|k})$ / its sheafification $\Gamma(\bigotimes^{\bullet}_{F}\Omega^{1}_{F|k})$, cf. Remark on p.2.5 | $\text{Vec}_{k}$ | yes/no
4 | $\Phi^{p}CH^{q}(X\times_{k}k(Y))_{{\mathbb{Q}}}$ for a smooth $X$ and a ‘‘universal’’ filtration $\Phi^{\bullet}$ on the Chow groups | |
| (e.g., $A(k(Y))_{{\mathbb{Q}}}$ for an abelian $k$-variety $A$) | $\text{Vec}_{{\mathbb{Q}}}$ | yes
5 | $\overline{Y}^{\text{prim}}=\bigoplus_{M}\overline{Y}^{\text{prim}}_{M}$ (multiplicity-one sheaf, by Proposition 1.3) | ${\mathcal{M}}_{k}^{\text{op}}$ | yes
6 | $Z^{\dim Y}(F\otimes_{k}k(Y))_{{\mathbb{Q}}}$ | $\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}^{\text{op}}$ | no
7 | $Z^{q}(Y\times_{k}F)_{{\mathbb{Q}}}$ for $q\geq 0$ / its sheafification $Z^{q}(F\otimes_{k}k(Y))_{{\mathbb{Q}}}$ | $\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}$ | only for $q=0$
| (a) composition with the evaluation functor on $X$, i.e., $\Psi_{X,q}$ | $\text{Vec}_{{\mathbb{Q}}}$ | only for $q=0$
8 | $C_{k(Y)}:={\mathcal{I}}\Psi_{X}$, cf. §1.2, (and its quotient $CH_{0}(\overline{Y}_{F})_{{\mathbb{Q}}}$) | ${\mathcal{I}}_{G}^{\text{op}}$ | yes
| (a) composition with $\mathop{\mathrm{Hom}}\nolimits_{\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}^{\text{op}}}(\underline{H}^{q}_{c},-)$: | |
| $H^{q}(\overline{Y})/N^{1}=:\underline{H}^{q}_{c}(Y)$ for any $q\geq 0$ (subsheaf of the sheaf $\underline{H}^{q}$) | $B$-mod | yes
| (a′) $k={\mathbb{C}}$: the image in $\underline{H}^{2q}(-({\mathbb{C}});{\mathbb{Q}})(Y)$ of the maximal Hodge substructure of $H^{2q}(\overline{Y}({\mathbb{C}}),{\mathbb{Q}})$ in | |
| $F^{1}$, cf. §3, p.3, (its vanishing is equivalent to the Hodge’s conjecture) | $\text{Vec}_{{\mathbb{Q}}}$ | yes
| (b) composition with $\mathop{\mathrm{Hom}}\nolimits_{\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}^{\text{op}}}(\Omega_{F|k}^{\bullet},-)$: | |
| $\Gamma(\overline{Y},\Omega^{\bullet}_{\overline{Y}|k})$ (subsheaf of the sheaf $\underline{H}^{\bullet}_{\text{dR}|k,c}$) | $\text{Vec}_{k}$ | yes
Except for
$\Gamma(\overline{Y},\bigotimes^{\bullet}_{{\mathcal{O}}_{\overline{Y}}}\Omega^{1}_{\overline{Y}|k})$,
all these invariants have Galois descent property. Except for
$Z^{q}(Y\times_{k}F)_{{\mathbb{Q}}}$ for $q>0$, $K_{q}(Y)_{{\mathbb{Q}}}$ for
$q\geq 0$ and $H^{q}(Y)$ for $q>0$, all these invariants are birational.
($N^{1}$ in example 8 (a) denotes the first term of the coniveau filtration on
$H^{*}$.)
Some of the above presheaves are defined using a compactification
$\overline{Y}$. To show that each of such presheaves is in fact well-defined
(and therefore, birationally invariant), one can use the facts that (i) any
birational map is a composition of blow-ups and blow-downs with smooth
centres, cf. [1], and (ii) the cohomology (resp., motive) of a blow-up is the
direct sum of the cohomology of the original variety and of the Gysin image
(resp., Tate twist) of the cohomology (resp., motive) of the subvariety which
is blown up. Such a presheaf is ${\mathbb{A}}^{1}$-invariant, since the
cohomology (resp., motive) of the product of a proper variety $X$ with the
projective line is the direct sum of the pull-back of the cohomology (resp.,
motive) of $X$ and of the Gysin image (resp., Tate twist) of the cohomology
(resp., motive) of $X\times\\{0\\}\cong X$.
To conclude that a birational ${\mathbb{A}}^{1}$-invariant presheaf is a
sheaf, one checks that it has the Galois descent property, so Proposition 1.7
can be applied.
###### Lemma 1.1.
For an arbitrary commutative $k$-group $A$, let ${\mathcal{H}}^{A}_{1}$ be the
presheaf $Y\mapsto\bigoplus_{y\in Y^{0}}(A(k(y))/A(k))_{{\mathbb{Q}}}$. Then
${\mathcal{H}}^{A}_{1}$ is a sheaf; it is simple (=irreducible) for simple
$A$. Let a presheaf ${\mathcal{F}}$ be the composition of the Picard functor
$Y\mapsto\mathop{\mathrm{Pic}}\nolimits^{0}(\overline{Y})$ with an additive
functor on the category of abelian $k$-varieties, e.g.
$\mathop{\mathrm{Pic}}\nolimits^{\circ}_{{\mathbb{Q}}}:Y\mapsto\mathop{\mathrm{Pic}}\nolimits^{0}(\overline{Y})_{{\mathbb{Q}}}$,
$\underline{H}^{1}_{c}:Y\mapsto H^{1}(\overline{Y})$, or
$\Omega^{1}_{|k,\text{{\rm
reg}}}:Y\mapsto\Gamma(\overline{Y},\Omega^{1}_{\overline{Y}|k})$. Then
${\mathcal{F}}$ is a sheaf and
${\mathcal{F}}=\bigoplus_{A}{\mathcal{F}}(\tilde{A})\otimes_{\mathop{\mathrm{End}}\nolimits(\tilde{A})}{\mathcal{H}}^{\tilde{A}}_{1}$,
where $A$ runs through the isogeny classes of simple abelian $k$-varieties and
$\tilde{A}$ is a representative of $A$.
Thus, such sheaves ${\mathcal{F}}$ are direct sums of copies of simple sheaves
${\mathcal{H}}^{A}_{1}$.
Proof. Suppose that ${\mathcal{A}}$ is an abelian category. Then any
semisimple object $N\in{\mathcal{A}}$ splits canonically into the direct sum
over the isomorphism classes $M$ of simple objects in ${\mathcal{A}}$ of its
$M$-isotypical parts $N_{M}$. Clearly, for any representative $\tilde{M}$ of
the isomorphism class $M$ the natural morphism
$\mathop{\mathrm{Hom}}\nolimits_{{\mathcal{A}}}(\tilde{M},N)\otimes_{\mathop{\mathrm{End}}\nolimits(\tilde{M})}\tilde{M}\to
N_{M}$ by $\varphi\otimes a\mapsto\varphi(a)$. This is an isomorphism.
Applying an additive functor
${\mathfrak{F}}:{\mathcal{A}}\to{\mathcal{B}}^{\text{op}}$ to the above
isotypical decomposition of $N$, we get a canonical isomorphism
$\prod_{M}{\mathfrak{F}}(\tilde{M})\otimes_{\mathop{\mathrm{End}}\nolimits(\tilde{M})}\mathop{\mathrm{Hom}}\nolimits_{{\mathcal{A}}}(N,\tilde{M})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathfrak{F}}(N)$,
$f\otimes l\mapsto l^{\ast}f$, where the following duality is used:
$\mathop{\mathrm{Hom}}\nolimits_{\text{mod-$\mathop{\mathrm{End}}\nolimits(\tilde{M})$}}(\mathop{\mathrm{Hom}}\nolimits_{{\mathcal{A}}}(\tilde{M},N),\mathop{\mathrm{End}}\nolimits(\tilde{M}))_{{\mathbb{Q}}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathop{\mathrm{Hom}}\nolimits_{{\mathcal{A}}}(N,\tilde{M})$.
(It is induced by the composition pairing
$\mathop{\mathrm{Hom}}\nolimits_{{\mathcal{A}}}(\tilde{M},N)\otimes\mathop{\mathrm{Hom}}\nolimits_{{\mathcal{A}}}(N,\tilde{M})\to\mathop{\mathrm{End}}\nolimits(\tilde{M})$.)
As ${\mathcal{A}}$, we take either the category of abelian $k$-varieties with
morphisms $\otimes{\mathbb{Q}}$, or the bigger category
${\mathcal{M}}_{k}^{\text{op}}$. In the case of abelian varieties, the
isomorphism classes of simple objects are the isogeny classes of simple
abelian $k$-varieties, whereas the existence of the isotypical decomposition
corresponds to the fact that for any abelian $k$-variety $B$ the natural
morphism
$\bigoplus_{A}\mathop{\mathrm{Hom}}\nolimits_{\text{ab.$k$-var}}(\tilde{A},B)\otimes_{\mathop{\mathrm{End}}\nolimits(\tilde{A})}\tilde{A}\to
B$, $\varphi\otimes a\mapsto\varphi(a)$, where $A$ runs through the isogeny
classes of simple abelian $k$-varieties and $\tilde{A}$ is a representative of
$A$, is an isogeny.
Thus, any sheaf ${\mathcal{F}}$ with semisimple values in ${\mathcal{A}}$ also
splits canonically into the direct sum over the isomorphism classes $M$ of
simple objects in ${\mathcal{A}}$ of its $M$-isotypical parts
${\mathcal{F}}_{M}$.
When ${\mathcal{A}}={\mathcal{M}}_{k}^{\text{op}}$ and ${\mathcal{F}}$ is the
dominant sheaf $Y\mapsto\overline{Y}^{\text{prim}}$, we get that
${\mathcal{F}}$ splits canonically into the direct sum of its $M$-isotypical
parts $Y\mapsto\overline{Y}^{\text{prim}}_{M}$. By Proposition 1.3, the
$M$-isotypical part $Y\mapsto\overline{Y}^{\text{prim}}_{M}$ is a simple
sheaf.
If $B={\rm Alb}(\overline{Y})$ (the Albanese variety) then
$\mathop{\mathrm{Hom}}\nolimits_{\text{ab.$k$-var}}(B,\tilde{A})=\tilde{A}(k(Y))/\tilde{A}(k)$,
and thus,
${\mathcal{F}}(Y)={\mathcal{F}}(B)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\bigoplus_{A}{\mathcal{F}}(\tilde{A})\otimes_{\mathop{\mathrm{End}}\nolimits(\tilde{A})}(\tilde{A}(k(Y))/\tilde{A}(k))$.
It is quite evident that ${\mathcal{H}}^{\tilde{A}}_{1}$ is a sheaf. By
Proposition 1.7, in the case of abelian variety $\tilde{A}$, it suffices to
check the Galois descent property, which is equivalent to the following one:
for any abelian $k$-variety $\tilde{A}$ and any finite group $H$ of its
automorphisms such that $H_{0}(H,\tilde{A})=0$ one has
${\mathcal{F}}(\tilde{A})^{H}=0$. Clearly, this property holds. The simplicity
of the sheaf ${\mathcal{H}}^{\tilde{A}}_{1}$ follows from the fact that for
any algebraically closed field extension $K|k(\tilde{A})$ and for any
subvariety $Z$ of $A$ of positive dimension there are no proper subgroups of
$\tilde{A}(K)$ containing all generic $K$-points of $Z$. (Any point of
$\tilde{A}$ is a sum of generic points of $\tilde{A}$; any sum of $\dim A$
generic $K$-points of $Z$ in sufficiently general position is a generic point
of $Z$). This argument works more naturally in the context of representations,
cf. §2. ∎
Remark. For an abelian $k$-variety $A$, the sheaf $A_{{\mathbb{Q}}}:Y\mapsto
A(k(Y))_{{\mathbb{Q}}}$ factors through the Albanese functor, but considered
as a functor to the category of torsors over abelian $k$-varieties, so
additive functors do not make sense and Lemma 1.1 is not applicable to this
sheaf. In particular, it is not semisimple.
Propositions 3.7 and 3.5 suggest that (i) the isomorphism classes of
irreducible subquotients of $\underline{H}^{\bullet}_{c}$ are the same as that
of $\Omega^{\bullet}_{|k,{\rm
reg}}:Y\mapsto\Gamma(\overline{Y},\Omega^{\bullet}_{\overline{Y}|k})$, (ii)
they can be naturally identified with the irreducible effective primitive
motives, and (iii) the isomorphism classes of irreducible subquotients of
$\underline{H}^{\bullet}$ are related to more general irreducible effective
motives, such as the Tate motive ${\mathbb{Q}}(-1)$ in the case of
$\underline{H}^{1}_{{\rm dR}/k}$.
###### Lemma 1.2.
Any dominant sheaf ${\mathcal{F}}$ with values in an abelian category with
objects of finite length (e.g., in a category of finite-dimensional vector
spaces) is ${\mathbb{A}}^{1}$-invariant.
Proof. Any smooth morphism of connected smooth $k$-varieties is covering, so
$X\times({\mathbb{A}}^{1}\times{\mathbb{A}}^{1}\smallsetminus\Delta)\stackrel{{\scriptstyle
p}}{{\longrightarrow}}X\times({\mathbb{A}}^{1}\times{\mathbb{A}}^{1}\smallsetminus\Delta)/{\mathfrak{S}}_{2}$
is a cover for any $X$. On the other hand, it is the coequalizer of
$X\times({\mathbb{A}}^{1}\times{\mathbb{A}}^{1}\smallsetminus\Delta)\stackrel{{\scriptstyle
id,(12)}}{{\rightrightarrows}}X\times({\mathbb{A}}^{1}\times{\mathbb{A}}^{1}\smallsetminus\Delta)$.
Therefore,
${\mathcal{F}}(X\times({\mathbb{A}}^{1}\times{\mathbb{A}}^{1}\smallsetminus\Delta)/{\mathfrak{S}}_{2})\stackrel{{\scriptstyle
p^{*}}}{{\longrightarrow}}{\mathcal{F}}(X\times({\mathbb{A}}^{1}\times{\mathbb{A}}^{1}\smallsetminus\Delta))$
(i) is injective, (ii) factors through the ${\mathfrak{S}}_{2}$-invariants. As
$({\mathbb{A}}^{1}\times{\mathbb{A}}^{1}\smallsetminus\Delta)/{\mathfrak{S}}_{2}\cong{\mathbb{A}}^{1}\times{\mathbb{A}}^{1}\smallsetminus\Delta$($\cong{\mathbb{A}}^{1}\times{\mathbb{G}}_{m}$),
the source and the target of $p^{*}$ are isomorphic. As they are of finite
length, the inclusion $p^{*}$ is an isomorphism. This implies that the
involution $(12)$ is identical on
${\mathcal{F}}(X\times({\mathbb{A}}^{1}\times{\mathbb{A}}^{1}\smallsetminus\Delta))$,
so in the exact sequence, defining the sheaf condition for the cover
$X\times{\mathbb{A}}^{1}\longrightarrow X$,
$0\to{\mathcal{F}}(X)\to{\mathcal{F}}(X\times{\mathbb{A}}^{1})\rightrightarrows{\mathcal{F}}(X\times{\mathbb{A}}^{1}\times{\mathbb{A}}^{1})$
the double arrow consists of equal morphisms, i.e.
${\mathcal{F}}(X)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{F}}(X\times{\mathbb{A}}^{1})$.
∎
### 1.2. Properties of
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}^{{\mathcal{S}}}$
Clearly, a subsheaf of an ${\mathcal{S}}$-sheaf is an ${\mathcal{S}}$-sheaf:
if ${\mathcal{G}}$ is a subsheaf of an ${\mathcal{S}}$-sheaf ${\mathcal{F}}$
then for any $(Y\to X)\in{\mathcal{S}}$ the parallel arrows in the upper line
in the commutative diagram
$\begin{array}[]{ccccc}{\mathcal{F}}(X)&\longrightarrow&{\mathcal{F}}(Y)&\rightrightarrows&{\mathcal{F}}(Y\times_{X}Y)\\\
\bigcup&&\bigcup&&\bigcup\\\
{\mathcal{G}}(X)&\longrightarrow&{\mathcal{G}}(Y)&\rightrightarrows&{\mathcal{G}}(Y\times_{X}Y)\end{array}$
coincide, so the parallel arrows in the lower line also coincide, i.e.
${\mathcal{G}}$ is an ${\mathcal{S}}$-sheaf.
Assume that there are generically non-finite morphisms in ${\mathcal{S}}$ with
arbitrary targets. Thus as before, ${\mathcal{I}}_{G}$ is a particular case of
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}^{{\mathcal{S}}}$. Moreover, as
restriction of any morphism $X\stackrel{{\scriptstyle f}}{{\longrightarrow}}Y$
to an open dense subset $U$ of $X$ factors through
$U\stackrel{{\scriptstyle(f,\phi)}}{{\longrightarrow}}Y\times{\mathbb{A}}^{1}\stackrel{{\scriptstyle{\rm
pr}_{Y}}}{{\longrightarrow}}Y$, one has
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}^{{\mathcal{S}}}\subseteq{\mathcal{I}}_{G}$.
1\. The categories
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}^{{\mathcal{S}}}$ and
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}$ are abelian, complete,
cocomplete and have enough injectives. (This is standard.)
2\. The section functors
$\mathop{\mathrm{Hom}}\nolimits_{\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}}(\Psi_{Y},-):{\mathcal{F}}\mapsto{\mathcal{F}}(Y)$
are exact on $\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}^{{\mathcal{S}}}$ for
all smooth $k$-varieties $Y$. As a consequence, quotients of
${\mathcal{S}}$-sheaves by their subsheaves coincide with their quotients as
presheaves: if
${\mathcal{F}}\in\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}^{{\mathcal{S}}}$
and ${\mathcal{G}}$ is a subsheaf of ${\mathcal{F}}$ then
$({\mathcal{F}}/{\mathcal{G}})(Y)={\mathcal{F}}(Y)/{\mathcal{G}}(Y)$.
3\. A sheaf is an ${\mathcal{S}}$-sheaf if and only if all its irreducible
subquotients are ${\mathcal{S}}$-sheaves.
[Proof of the ‘‘only if’’ part. As it was shown above, a subsheaf
${\mathcal{G}}$ of
${\mathcal{F}}\in\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}^{{\mathcal{S}}}$
is an ${\mathcal{S}}$-sheaf. By property 2,
$({\mathcal{F}}/{\mathcal{G}})(Y)={\mathcal{F}}(Y)/{\mathcal{G}}(Y)$, which
implies that the quotient ${\mathcal{F}}/{\mathcal{G}}$ is also an
${\mathcal{S}}$-sheaf. The ‘‘if’’ part is shown in Proposition 2.2 (in the
language of representations); cf. also Theorem 2.11.]
4\. The inclusion
${\mathcal{I}}_{G}\hookrightarrow\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}$
admits a left adjoint ${\mathcal{I}}$ and a right adjoint.
Examples of calculation of these adjoint functors are given in Propositions
3.1 and 3.3.
5\. The sheaves $C_{k(X)}:={\mathcal{I}}\Psi_{X}$ form a system of projective
generators of ${\mathcal{I}}_{G}$. [This follows from 2 and 4.]
(Remark. There are no projective objects in
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}$.)
### 1.3. Irreducible objects of ${\mathcal{I}}_{G}$
Examples. Let $M$ be a simple effective primitive pure covariant motive. Then
${\mathbb{B}}^{0}(M):Y\mapsto\mathop{\mathrm{Hom}}\nolimits_{\\{\text{{\rm
pure $k$-motives}}\\}}(\overline{Y},M)$
is a well-defined sheaf of finite-dimensional ${\mathbb{Q}}$-vector spaces
([7]).
A particular case of this example is the sheaf ${\mathcal{H}}_{1}^{A}$,
corresponding to the motive ‘‘$H_{1}(A)$’’ for any simple abelian $k$-variety
$A$.
###### Proposition 1.3 ([7]).
${\mathbb{B}}^{0}$ gives rise to a fully faithful functor
${\mathbb{B}}^{\bullet}$:
$\\{\mbox{{\rm pure $k$-motives}}\\}\longrightarrow\\{\mbox{{\rm semisimple
sheaves of finite length of finite-dimensional graded ${\mathbb{Q}}$-vector
spaces}}\\}.$
###### Conjecture 1.4 ([7]).
This is an equivalence of categories. (In other words, any irreducible sheaf
of finite-dimensional ${\mathbb{Q}}$-vector spaces is isomorphic to
${\mathbb{B}}^{0}(M)$ for a primitive irreducible effective pure motive $M$.)
This can be complemented by the following conjecture, which I consider as one
of the principal problems on ${\mathbb{A}}^{1}$-invariant sheaves.
###### Conjecture 1.5 ([8]).
Any simple ${\mathbb{A}}^{1}$-invariant sheaf can be embedded into the sheaf
$\mathop{\underline{\Omega}}^{\bullet}_{|k}:Y\mapsto\Omega^{\bullet}_{k(Y)|k}$.
This conjecture is rather strong: it implies the Bloch’s conjecture:
###### ‘‘Corollary’’ 1.6 ([8]).
Suppose that a rational map $f:Y\dasharrow X$ of smooth proper $k$-varieties
induces an injection
$\Gamma(X,\Omega^{\bullet}_{X|k})\hookrightarrow\Gamma(Y,\Omega^{\bullet}_{Y|k})$.111Example.
Let $r\geq 1$ be an integer and $X$ be a smooth proper $k$-variety with
$\Gamma(X,\Omega^{j}_{X|k})=0$ for all $r<j\leq\dim X$. Let $Y$ be a
sufficiently general $r$-dimensional plane section of a smooth projective
variety $X^{\prime}$ birational to $X$. Then, as all considered invariants are
birational, the inclusion $Y\hookrightarrow X^{\prime}$ induces an injection
$\Gamma(X,\Omega^{\bullet}_{X|k})\hookrightarrow\Gamma(Y,\Omega^{\bullet}_{Y|k})$.
Then $f$ induces a surjection $CH_{0}(Y)\to CH_{0}(X)$.
If $\Gamma(X,\Omega^{\geq 2}_{X|k})=0$ then the Albanese map induces an
isomorphism $CH_{0}(X)^{0}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\rm
Alb}(X)(k)$, where $CH_{0}(X)^{0}$ is the Chow group of 0-cycles of degree 0
and ${\rm Alb}(X)$ is the Albanese variety of $X$. (The converse, due to
Mumford, is well-known.) In that case $C_{k(X)}=CH_{0}(X_{F})_{{\mathbb{Q}}}$.
Proof. Let $C$ be the cokernel of $\alpha:CH_{0}(Y_{F})_{{\mathbb{Q}}}\to
CH_{0}(X_{F})_{{\mathbb{Q}}}$. Then the kernel of the homomorphism
$\alpha^{\ast}:\mathop{\mathrm{Hom}}\nolimits_{G}(CH_{0}(X_{F}),\Omega^{\bullet}_{F|k})\to\mathop{\mathrm{Hom}}\nolimits_{G}(CH_{0}(Y_{F}),\Omega^{\bullet}_{F|k})$
is $\mathop{\mathrm{Hom}}\nolimits_{G}(C,\Omega^{\bullet}_{F|k})$. By
Proposition 3.1, the homomorphism $\alpha^{\ast}$ coincides with the pull-back
under
$f^{\ast}:\Gamma(X,\Omega^{\bullet}_{X|k})\to\Gamma(X,\Omega^{\bullet}_{Y|k})$.
As the latter is injective, we conclude that
$\mathop{\mathrm{Hom}}\nolimits_{G}(C,\Omega^{\bullet}_{F|k})=0$. If $C\neq 0$
then it is cyclic, and thus, admits an simple quotient, and therefore, a non-
zero morphism to $\Omega^{\bullet}_{F|k}$. This contradiction implies that
$C=0$.
As the objects ${\mathbb{Q}}$ and ${\rm Alb}X(F)_{{\mathbb{Q}}}$ of
${\mathcal{I}}_{G}$ are projective ([7, §6.2]), the natural surjections
$\deg:C_{k(X)}\to{\mathbb{Q}}$ and ${\rm Alb}_{F}:\ker\deg\to{\rm
Alb}X(F)_{{\mathbb{Q}}}$ are split, so the cyclic $G$-module $C_{k(X)}$ is
isomorphic to a direct sum of type ${\mathbb{Q}}\oplus{\rm
Alb}X(F)_{{\mathbb{Q}}}\oplus\ker{\rm Alb}_{F}$. Thus,
$\mathop{\mathrm{Hom}}\nolimits_{G}(C_{k(X)},\Omega^{\bullet}_{F|k})\cong\mathop{\mathrm{Hom}}\nolimits_{G}({\mathbb{Q}}\oplus{\rm
Alb}X(F),\Omega^{\bullet}_{F|k})\oplus\mathop{\mathrm{Hom}}\nolimits_{G}(\ker{\rm
Alb}_{F},\Omega^{\bullet}_{F|k})$. By Proposition 3.1,
$\mathop{\mathrm{Hom}}\nolimits_{G}(C_{k(X)},\Omega^{\bullet}_{F|k})=\Gamma(X,\Omega^{\bullet}_{X|k})$
and $\mathop{\mathrm{Hom}}\nolimits_{G}({\mathbb{Q}}\oplus{\rm
Alb}X(F),\Omega^{\bullet}_{F|k})=\Gamma(X,\Omega^{\leq 1}_{X|k})$. If
$\Gamma(X,\Omega^{\geq 2}_{X|k})=0$ this means that
$\mathop{\mathrm{Hom}}\nolimits_{G}(C_{k(X)},\Omega^{\bullet}_{F|k})=\mathop{\mathrm{Hom}}\nolimits_{G}({\mathbb{Q}}\oplus{\rm
Alb}X(F),\Omega^{\bullet}_{F|k})$. Therefore, the $G$-module $\ker{\rm
Alb}_{F}$ should be zero, as otherwise it is cyclic, thus admits a non-zero
simple quotient, and (by Conjecture 1.5) a non-zero morphism to
$\Omega^{\bullet}_{F|k}$. It remains to take the $G$-invariants of
$\ker\deg\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}CH_{0}(X_{F})^{0}_{{\mathbb{Q}}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\rm
Alb}X(F)_{{\mathbb{Q}}}$; the torsion is controlled by Roitman’s theorem. ∎
Also this would imply that any irreducible ${\mathbb{A}}^{1}$-invariant sheaf
is a sheaf of finite-dimensional vector spaces.
Example. Let ${\mathcal{F}}$ be a simple ${\mathbb{A}}^{1}$-invariant sheaf
and suppose that it is of level 1, i.e. it is non-constant and
${\mathcal{F}}(Y)\neq 0$ for a curve $Y$, cf. also p.3.3. Then, by [7,
Corollary 6.22], ${\mathcal{F}}\cong{\mathcal{H}}^{A}_{1}$ for a simple
abelian variety $A$. Now any non-zero $\eta\in\Gamma(A,\Omega^{1}_{A|k})$
gives an embedding ${\mathcal{F}}\hookrightarrow\Omega^{1}_{|k}$ by
$[x:{\mathcal{O}}(U)\to k(Y)]\mapsto x(\eta)\in\Omega^{1}_{Y|k}(Y)$ ($U\subset
A$ is an affine open subset).
###### Proposition 1.7.
A dominant presheaf ${\mathcal{F}}$ is a sheaf if and only if the following
three conditions hold: (i) the sequence
${\mathcal{F}}(X)\to{\mathcal{F}}(X\times{\mathbb{A}}^{1})\rightrightarrows{\mathcal{F}}(X\times{\mathbb{A}}^{2})$
is exact for any smooth $k$-variety $X$,222E.g., any
${\mathbb{A}}^{1}$-invariant presheaf ${\mathcal{F}}$ satisfies the condition
(i). (ii) ${\mathcal{F}}$ is birationally invariant, (iii) it has the Galois
descent property, i.e. ${\mathcal{F}}(X)={\mathcal{F}}(Y)^{{\rm Aut}(Y|X)}$
for any Galois covering $Y\to X$.
Proof. The conditions (i)–(iii) are particular cases of the equalizer diagram
(1) for coverings by (i) projections $X\times{\mathbb{A}}^{s}\to X$, (ii) open
dense $U\subset X$, (iii) étale Galois covers $Y\to X$, respectively. Galois
descent property for any sheaf is clear, since étale morphisms with dense
images are covering and $U\times_{X}U=\coprod_{g\in{\rm Aut}(Y|X)}U_{g}$ for a
Zariski open ${\rm Aut}(Y|X)$-invariant $U\subset Y$, where $U_{g}\cong U$ is
the image of the embedding $(id_{U},g):U\hookrightarrow U\times_{X}U$.
Conversely, it is clear that any Galois-separable presheaf ${\mathcal{F}}$
satisfying (i) and (ii) is separable: if $Y\to X$ is a cover, i.e. a smooth
dominant morphism, then for any sufficiently general dominant map
$\varphi:Y\dasharrow{\mathbb{A}}^{\delta}$ (where $\delta=\dim Y-\dim X$) we
can choose a dominant étale morphism $\widetilde{Y}\to Y$ so that the
composition $\widetilde{Y}\to Y\dasharrow X\times{\mathbb{A}}^{\delta}$ is
Galois with the group denoted by $H$, and therefore, the composition
$\begin{array}[]{rccc}{\mathcal{F}}(X)\stackrel{{\scriptstyle\text{(i)}}}{{\hookrightarrow}}{\mathcal{F}}(X\times{\mathbb{A}}^{1})\stackrel{{\scriptstyle\text{(i)}}}{{\hookrightarrow}}\dots\stackrel{{\scriptstyle\text{(i)}}}{{\hookrightarrow}}&{\mathcal{F}}(X\times{\mathbb{A}}^{\delta})&\longrightarrow&{\mathcal{F}}(Y)\\\
&\downarrow\hbox to0.0pt{$\displaystyle\text{injective}$\hss}&&\downarrow\\\
&{\mathcal{F}}(\widetilde{Y})^{H}&\hookrightarrow&{\mathcal{F}}(\widetilde{Y})\end{array}$
is injective. Then in the commutative diagram
(2)
$\begin{array}[]{ccccc}{\mathcal{F}}(X)&\to&{\mathcal{F}}(\widetilde{Y})&\rightrightarrows&{\mathcal{F}}(\widetilde{Y}\times_{X}\widetilde{Y})\\\
\|&&\uparrow&&\uparrow\\\
{\mathcal{F}}(X)&\to&{\mathcal{F}}(Y)&\rightrightarrows&{\mathcal{F}}(Y\times_{X}Y)\\\
\|&&\uparrow&&\uparrow\\\
{\mathcal{F}}(X)&\to&{\mathcal{F}}(X\times{\mathbb{A}}^{\delta})&\rightrightarrows&{\mathcal{F}}(X\times{\mathbb{A}}^{\delta}\times{\mathbb{A}}^{\delta})\end{array}$
all arrows are injective, so it suffices to show the exactness of the upper
row.
Let $f$ be an element of ${\mathcal{F}}(\widetilde{Y})$. The image of $f$ in
${\mathcal{F}}(\widetilde{Y}\times_{X}\widetilde{Y})$ under the projection to
the first factor is fixed by $\\{1\\}\times H$; the image of $f$ in
${\mathcal{F}}(\widetilde{Y}\times_{X}\widetilde{Y})$ under the projection to
the second factor is fixed by $H\times\\{1\\}$. Now if $f$ is an element of
the equalizer of
${\mathcal{F}}(\widetilde{Y})\rightrightarrows{\mathcal{F}}(\widetilde{Y}\times_{X}\widetilde{Y})$
then the two images coincide, so they are fixed by the group $H\times H$. The
injectivity of both parallel arrows in the upper row of the diagram (2)
implies that $f\in{\mathcal{F}}(\widetilde{Y})^{H}$. By (iii) and the
injectivity of the vertical arrow, $f$ comes from the equalizer of the bottom
row of the diagram (2). Finally, the bottom row of the diagram (2) is exact by
Lemma 1.8, and thus, $f$ comes from ${\mathcal{F}}(X)$. ∎
###### Lemma 1.8.
Let ${\mathcal{V}}$ be a category of schemes such that for any
$X\in{\mathcal{V}}$: (i) the projection $X\times{\mathbb{A}}^{1}\to X$ is a
morphism in ${\mathcal{V}}$, (ii) any linear automorphism of any affine space
${\mathbb{A}}$ induces an automorphism of $X\times{\mathbb{A}}$ in
${\mathcal{V}}$. Let ${\mathcal{F}}$ be a presheaf on this category such that
the sequence
${\mathcal{F}}(X)\to{\mathcal{F}}(X\times{\mathbb{A}}^{1})\rightrightarrows{\mathcal{F}}(X\times{\mathbb{A}}^{2})$
is exact for any $X\in{\mathcal{V}}$. Then the sequence
${\mathcal{F}}(X)\to{\mathcal{F}}(X\times{\mathbb{A}}^{s})\rightrightarrows{\mathcal{F}}(X\times{\mathbb{A}}^{2s})$
is exact for any $X\in{\mathcal{V}}$ and any $s\geq 1$.
Proof. We proceed by induction on $s$, the case $s=1$ being trivial.
Denote by
$\mathrm{pr}_{1},\mathrm{pr}_{2}:X\times{\mathbb{A}}^{2s}\rightrightarrows
X\times{\mathbb{A}}^{s}$ the two projections.
For any $f\in{\mathcal{F}}(X\times{\mathbb{A}}^{s})$ the element
$\mathrm{pr}_{1}^{\ast}f$ is fixed by
$\Phi^{\ast}\in\mathop{\mathrm{End}}\nolimits{\mathcal{F}}(X\times{\mathbb{A}}^{s}\times{\mathbb{A}}^{s})$
for any linear automorphism $\Phi(u,v)=(u,\varphi(u,v))$ of
${\mathbb{A}}^{s}\times{\mathbb{A}}^{s}$. Similarly, $\mathrm{pr}_{2}^{\ast}f$
is fixed by
$\Psi^{\ast}\in\mathop{\mathrm{End}}\nolimits{\mathcal{F}}(X\times{\mathbb{A}}^{s}\times{\mathbb{A}}^{s})$
for any linear automorphism $\Psi(u,v)=(\psi(u,v),v)$.
Let now $f\in{\mathcal{F}}(X\times{\mathbb{A}}^{s})$ be in the equalizer of
$\mathrm{pr}_{1}^{\ast}$ and $\mathrm{pr}_{2}^{\ast}$. Then
$\mathrm{pr}_{1}^{\ast}f=\mathrm{pr}_{2}^{\ast}f$ is fixed by the group,
generated by $\Phi^{\ast}$ and $\Psi^{\ast}$ as above. Clearly, such
automorphisms $\Phi$ and $\Psi$ generate the group consisting of all linear
automorphisms $\alpha$. Then
$\mathrm{pr}_{1}^{\ast}f=\alpha^{\ast}\mathrm{pr}_{1}^{\ast}f$.
Applying the induction assumption in the case where $\alpha$ is identical on
one of the first $s$ coordinates and interchanges $i$-th and $(s+i)$-th for
other $1\leq i\leq s$, we get that $f$ belongs to the image of
${\mathcal{F}}(X\times{\mathbb{A}}^{1})\to{\mathcal{F}}(X\times{\mathbb{A}}^{s})$
under morphism induced by the projection ${\mathbb{A}}^{s}\to{\mathbb{A}}^{1}$
to one of the copies of ${\mathbb{A}}^{1}$. Then the case $s=1$ implies that
$f$ comes from ${\mathcal{F}}(X)$. ∎
Examples. 1\. A stable birationally invariant dominant presheaf with the
Galois descent is a sheaf.
2\. Example of a birationally invariant presheaf ${\mathcal{F}}$ with the
Galois descent property which is not a sheaf. Let ${\mathcal{G}}$ be a
dominant sheaf and $I\subsetneq\\{0,1,2,\dots\\}$ be a non-empty (finite or
infinite) interval. Assume that ${\mathcal{G}}(X)\neq 0$ for some $X$ with
$\dim X\not\in I$. Then the presheaf
${\mathcal{F}}:U\mapsto\left\\{\begin{array}[]{ll}{\mathcal{G}}(U)&\text{if
$\dim U\in I$}\\\ 0&\text{if $\dim U\not\in I$}\end{array}\right.$ (with the
restriction maps of ${\mathcal{G}}$, whenever possible, otherwise zero) is
birationally invariant and has the Galois descent property, but it is not a
sheaf. The sheafification of ${\mathcal{F}}$ is ${\mathcal{G}}$ if $I$ is
infinite and 0 otherwise.
Now, what are the projective generators of ${\mathcal{I}}_{G}$ from §1.2,
Property 5?
###### Conjecture 1.9.
For any smooth proper $k$-variety $X$, the sheaf $C_{k(X)}$ coincides with
$Y\mapsto CH_{0}(X_{k(Y)})_{{\mathbb{Q}}}$.
Remarks. 1\. This is known, e.g., if $X$ is a curve, cf. [7, Cor.6.21] and
Proposition 2.10 for a stronger statement. Conjecture 1.9 would imply that
${\mathcal{I}}_{G}$ is a tensor category under the operation
$({\mathcal{F}},{\mathcal{G}})\mapsto{\mathcal{I}}({\mathcal{F}}\otimes_{\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}}{\mathcal{G}})=:{\mathcal{F}}\otimes_{{\mathcal{I}}}{\mathcal{G}}$,
where $\otimes_{\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}}$ denotes the
sheafification of the tensor product presheaf. Moreover, the ‘‘Künneth
formula’’ holds:
$C_{k(X)}\otimes_{{\mathcal{I}}}C_{k(Y)}=C_{k(X\times_{k}Y)}$.
2\. It is shown in [7, Proposition 6.17] that, roughly speaking, $C_{k(X)}$ is
the quotient of generic 0-cycles on $X$ by those divisors of rational
functions on generic curves on $X$ which are generic, and thus, Conjecture 1.9
should be considered as a moving lemma.
3\. Conjecture 1.9 and the motivic conjectures imply conjectures 1.4 and 1.5.
## 2\. Alternative descriptions of ${\mathbb{A}}^{1}$-invariant sheaves
Now I want to introduce the language of representations and to use it to
explain some results and conjectures of §1, especially Conjecture 1.9.
### 2.1. Smooth representations and non-degenerate modules over algebras of
measures
For any totally disconnected Hausdorff group333cf. [4, Appendix A] $H$ an
$H$-set (group, etc.) is called smooth if the stabilizers are open.
Any smooth representation $W$ of $H$ over $E$ can be considered as a module
over the associative algebra
${\mathbb{D}}_{E}(H):=\mathop{\underleftarrow{\lim}}\limits_{U}E[H/U]$ of the
‘‘oscillating’’ measures on $H$ (for which all open subgroups and their
translates are measurable): ${\mathbb{D}}_{E}(H)\times W\to W$ is defined by
$(\alpha,w)\mapsto\beta w$ for any $\beta\in E[H]$ with the same image $E[H]$
as $\alpha$, where $w\in W^{U}$ for some open subgroup $U$ of $H$.
Passing to the inverse limit, we get the algebra structure on
${\mathbb{D}}_{E}(H)$ from ${\mathbb{D}}_{E}(H)\times E[H/U]\to E[H/U]$.
If the annihilator of $W\in\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}(E)$ in
${\mathbb{D}}_{E}(H)$ vanishes then the restriction of $W$ to any compact
subgroup $U$ contains each smooth irreducible representation of $U$.
(Otherwise, if $W$ does not contain a smooth irreducible representation $\rho$
of $U$ then the natural projector in ${\mathbb{D}}_{E}(H)$ to the
$\rho$-isotypical part would annihilate $W$.)
### 2.2. A representation theoretic setting for
(${\mathbb{A}}^{1}$-invariant) sheaves
In this section, for a group $H$ as in §2.1 and a collection $S$ of pairs of
its subgroups, we study the category
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}^{S}(E)$ of smooth
$E$-representations $W$ of $H$, satisfying $W^{U_{1}}=W^{U_{2}}$ for all
$(U_{1},U_{2})\in S$.
Theorem 2.3 explains the consistence of this notation with that of §1.
Collections $S$ and $S^{\prime}$ are called equivalent if they define the same
subcategory of
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}:=\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}^{\emptyset}$.
For any subgroup $U\subset H$ the functor $H^{0}(U,-)$ on the category of
smooth $H$-sets (or modules, etc.) coincides with
$\mathop{\underrightarrow{\lim}}\limits H^{0}(V,-)$, where the limit is taken
over the open subgroups $V$ of $H$ containing $U$. Therefore, one can assume
that the subgroups $U_{1},U_{2}$ are intersections of open ones, and in
particular, that they are closed. Further, as $W^{U_{1}}\cap
W^{U_{2}}=W^{\langle U_{1},U_{2}\rangle}$ for any $H$-module $W$ and
$U_{1},U_{2}\subseteq\langle U_{1},U_{2}\rangle$, one can assume that the
pairs $(U_{1},U_{2})\in S$ are ordered: $U_{1}\subset U_{2}$.
###### Lemma 2.1.
Assume that for any pair $(U_{1}\subset U_{2})\in S$ the functor
$H^{0}(U_{1},-)$ is exact on $\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}$.
Then the category $\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}^{S}(E)$ is
stable under passing to the subquotients in
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}(E)$, and in particular, it is
abelian. The inclusion functor
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}^{S}(E)\hookrightarrow\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}(E)$
admits a left adjoint444The diagrams
$\begin{array}[]{ccc}\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}(E)&\stackrel{{\scriptstyle{\mathcal{I}}_{S}}}{{\longrightarrow}}&\mathop{\mathcal{S}\mathrm{m}}\nolimits^{S}_{H}(E)\\\
\otimes_{E}E^{\prime}\downarrow\phantom{\otimes_{E}E^{\prime}}&&\phantom{\otimes_{E}E^{\prime}}\downarrow\otimes_{E}E^{\prime}\\\
\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}(E^{\prime})&\stackrel{{\scriptstyle{\mathcal{I}}_{S}}}{{\longrightarrow}}&\mathop{\mathcal{S}\mathrm{m}}\nolimits^{S}_{H}(E^{\prime})\end{array}\quad\mbox{and}\quad\begin{array}[]{ccc}\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}(E)&\stackrel{{\scriptstyle{\mathcal{I}}_{S}}}{{\longrightarrow}}&\mathop{\mathcal{S}\mathrm{m}}\nolimits^{S}_{H}(E)\\\
{\rm for}\uparrow\phantom{{\rm for}}&&\phantom{{\rm for}}\uparrow{\rm for}\\\
\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}(E^{\prime})&\stackrel{{\scriptstyle{\mathcal{I}}_{S}}}{{\longrightarrow}}&\mathop{\mathcal{S}\mathrm{m}}\nolimits^{S}_{H}(E^{\prime})\end{array}$
are commutative for any field extension $E^{\prime}|E$, so omitting $E$ from
the notation does not lead to a confusion. $W\longmapsto{\mathcal{I}}_{S}W$.
Proof. If a sequence $0\to W_{1}\to W\to W_{2}\to 0$ in
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}$ is exact then the sequences
$0\to W_{1}^{U_{1}}\to W^{U_{1}}\to W_{2}^{U_{1}}\to 0$ and $0\to
W_{1}^{U_{2}}\to W^{U_{2}}\to W_{2}^{U_{2}}$ are also exact. If
$W\in\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}^{S}$, i.e.
$W^{U_{1}}=W^{U_{2}}$, then $W_{1}^{U_{1}}=W_{1}\cap W^{U_{1}}=W_{1}\cap
W^{U_{2}}=W_{1}^{U_{2}}$ and $W^{U_{2}}\to W_{2}^{U_{2}}$ is surjective (since
$W^{U_{2}}=W^{U_{1}}\to W_{2}^{U_{1}}$ is surjective and factors through
$W_{2}^{U_{2}}\subseteq W_{2}^{U_{1}}$). This means that
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}^{S}$ is stable under taking
subquotients in $\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}$.
The existence of the functor ${\mathcal{I}}_{S}$ can be deduced from the
special adjoint functor theorem, cf. [5, §5.8]. However, we construct it
‘‘explicitly’’, which enables us to relate the generators of the category
${\mathcal{I}}_{G}$ to the Chow groups of 0-cycles.
Let $W^{\prime}\in\mathop{\mathcal{S}\mathrm{m}}\nolimits^{S}_{H}$. Any
$H$-homomorphism
$W\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}W^{\prime}$ factors
through the object $\alpha(W)$ of
$\mathop{\mathcal{S}\mathrm{m}}\nolimits^{S}_{H}$. We may, therefore, assume
that $\alpha$ is surjective. Let $(U_{1}\subset U_{2})\in S$. As the functor
$H^{0}(U_{1},-)$ is exact on $\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}$,
the morphism $\alpha$ induces a surjection
$W^{U_{1}}\longrightarrow(W^{\prime})^{U_{1}}$. As
$(W^{\prime})^{U_{2}}=(W^{\prime})^{U_{1}}$, the subgroup $U_{2}$ acts on
$(W^{\prime})^{U_{1}}$ trivially, and therefore, the subrepresentation
$W_{U_{1}\subset U_{2}}=\langle\sigma w-w~{}|~{}\sigma\in U_{2},~{}w\in
W^{U_{1}}\rangle_{H}$ of $H$ is contained in the kernel of $\alpha$. It
follows that $\alpha$ factors through
${\mathcal{I}}_{S}W:=W/\sum_{(U_{1}\subset U_{2})\in S}W_{U_{1}\subset
U_{2}}$.
The representation ${\mathcal{I}}_{S}W$ of $H$ is smooth, so the map
$W^{U_{1}}\longrightarrow({\mathcal{I}}_{S}W)^{U_{1}}$, induced by the
projection, is surjective, and therefore, any element
$\overline{w}\in({\mathcal{I}}_{S}W)^{U_{1}}$ can be lifted to an element
$w\in W^{U_{1}}$. Then $\sigma\overline{w}-\overline{w}$ coincides with the
projection of the element $\sigma w-w$ for any $\sigma\in U_{2}$. Notice that
$\sigma w-w\in W_{U_{1}\subset U_{2}}$, so its projection is zero, and
therefore, $\sigma\overline{w}=\overline{w}$ for any $\sigma\in U_{2}$. As
$({\mathcal{I}}_{S}W)^{U_{2}}\subseteq({\mathcal{I}}_{S}W)^{U_{1}}$, this
means that $({\mathcal{I}}_{S}W)^{U_{2}}=({\mathcal{I}}_{S}W)^{U_{1}}$, and
thus, ${\mathcal{I}}_{S}W\in\mathop{\mathcal{S}\mathrm{m}}\nolimits^{S}_{H}$.
One has
$\mathop{\mathrm{Hom}}\nolimits_{\mathop{\mathcal{S}\mathrm{m}}\nolimits^{S}_{H}}({\mathcal{I}}_{S}W,W^{\prime})=\mathop{\mathrm{Hom}}\nolimits_{\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}}(W,W^{\prime})$
for any $W\in\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}$ and
$W^{\prime}\in\mathop{\mathcal{S}\mathrm{m}}\nolimits^{S}_{H}$, i.e. the
functor ${\mathcal{I}}_{S}$ is left adjoint to the inclusion functor
$\mathop{\mathcal{S}\mathrm{m}}\nolimits^{S}_{H}\hookrightarrow\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}$.
∎
Remark. The functor ${\mathcal{I}}_{S}$ generalizes the coinvariants, since
${\mathcal{I}}_{S}=H_{0}(H,-)$ if $S=\\{(\\{1\\}\subset H)\\}$.
Examples. 1\. The functor $H^{0}(U_{1},-)$ is exact on
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}$ if, e.g., the subgroup $U_{1}$
is compact.
2\. Suppose that $H$ is the automorphism group of an algebraically closed
field extension $F|k$ of countable transcendence degree and $U_{1}$ is the
subgroup of automorphisms of $F$ over a fixed subextension of $k$ in $F$ of
infinite transcendence degree. Though $U_{1}$ need not be compact, the functor
$H^{0}(U_{1},-)$ is exact on $\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}$.
###### Proposition 2.2.
Let $H$ be a totally disconnected group and $S$ be such a collection of pairs
of its subgroups $(U_{1}\subset U_{2})$ that
1. (1)
for any pair $(U_{1}\subset U_{2})\in S$ there exists an element $\sigma\in
U_{2}$ such that (i) $(U_{1}\cap\sigma U_{1}\sigma^{-1}\subset U_{1})\in S$;
(ii) $U_{1}$ and $\sigma U_{1}\sigma^{-1}$ generate $U_{2}$, at least
topologically.
2. (2)
there exists an equivalent collection of pairs of its subgroups $(U_{1}\subset
U_{2})$, where all $U_{1}$ are compact.
Then an object of $\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}(E)$ belongs to
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}^{S}(E)$ if and only if all its
irreducible subquotients are in
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}^{S}(E)$. In particular,
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}^{S}(E)$ is a Serre subcategory of
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}(E)$.
Proof. Suppose that $W\not\in\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}^{S}$,
whereas all its irreducible subquotients are in
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}^{S}$. Then $W^{U_{1}}\neq
W^{U_{2}}$ for some pair $(U_{1}\subset U_{2})\in S$, that is there exist a
vector $v\in W^{U_{1}}\smallsetminus W^{U_{2}}$. Choose an element $\sigma\in
U_{2}$ as in condition (1) of the statement for the pair $(U_{1}\subset
U_{2})\in S$. Then $\sigma v-v=:u\neq 0$, since $U_{1}$ and $\sigma$ generate
a dense subgroup in $U_{2}$.
One may replace $W$ by its quotient by a maximal subrepresentation not
containing $u$. Then the subrepresentation $\langle u\rangle$, generated by
$u$, becomes irreducible, and thus, an object of
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}^{S}$.
By definition, $u\in W^{U_{1}}+W^{\sigma U_{1}\sigma^{-1}}\subseteq
W^{U_{1}\cap\sigma U_{1}\sigma^{-1}}$. As $\langle
u\rangle\in\mathop{\mathcal{S}\mathrm{m}}\nolimits_{H}^{S}$ and
$(U_{1}\cap\sigma U_{1}\sigma^{-1}\subset U_{1})\in S$, we conclude that $u\in
W^{U_{1}}$. This implies that $\sigma v\in W^{U_{1}}$. On the other hand,
$\sigma v\in W^{\sigma U_{1}\sigma^{-1}}$, so $\sigma v\in W^{U_{1}}\cap
W^{\sigma U_{1}\sigma^{-1}}$. The latter vector space coincides with
$W^{U_{2}}$, and thus, $v\in W^{U_{2}}$, contradicting our assumptions.
The converse follows from Lemma 2.1. ∎
### 2.3. More notations and compatibility of notations of §2.2 and §1: the
sheafification and smooth representations
From now on we fix the following notations: $F|k$ is an algebraically closed
field extension of countably infinite transcendence degree, and $G=G_{F|k}$ is
the automorphism group of the extension $F|k$.
Consider connected smooth $k$-varieties $U$ endowed with a generic $F$-point,
i.e., with a $k$-field embedding
$k(U)\stackrel{{\scriptstyle/k}}{{\hookrightarrow}}F$. For any presheaf
${\mathcal{F}}$ on ${\mathcal{S}}m_{k}^{\prime}$ we can form the direct limit
${\mathcal{F}}(F):=\mathop{\underrightarrow{\lim}}\limits{\mathcal{F}}(U)$
over such $U$. The group $G={\rm Aut}(F|k)$ acts naturally on
${\mathcal{F}}(F)$.
###### Theorem 2.3 ([4]).
* •
${\mathcal{F}}\mapsto{\mathcal{F}}(F)$ gives an equivalence of the categories
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}^{{\mathcal{S}}}(E)$ (of §1) and
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}^{S}(E)$ (of §2.2), where $S$ is
the collection of pairs $G_{F|k(X)}\subseteq G_{F|k(Y)}$ for all morphisms
$(X\to Y)\in{\mathcal{S}}$.
* •
For any presheaf ${\mathcal{F}}$, the sheaf corresponding to
${\mathcal{F}}(F)$ is the sheafification of ${\mathcal{F}}$.
### 2.4. An example: birational invariants constant on the projective spaces
Let $S$ consist of a single pair $K\subset G$ such that $K$ is a ‘maximal’
compact subgroup, i.e., any compact subgroup is conjugate to a subgroup of
$K$. Then $S$ is equivalent to the collection consisting of a single pair
$K^{\prime}\subset G$, where $K^{\prime}$ is the pointwise stabilizer of some
transcendence base of $F|k$, and also to the collection $S^{\prime}$ of pairs
$U\subset G$ such that $U$ is the pointwise stabilizer of a finite subset of a
fixed transcendence base of $F|k$. The collection $S^{\prime}$ satisfies the
assumptions of Proposition 2.2.
###### Lemma 2.4.
Let $E^{\prime}|E$ be an extension of fields, $H$ be a group and
$(\rho,W_{2})$ be an irreducible $E^{\prime}$-representation of $H$. Let
$W_{1}$ be an $E$-representation of $H$, absolutely irredicible even in
restriction to $\ker\rho$.555i.e., irredicible and with ${\rm
End}_{E[\ker\rho]}(W_{1})=E$: otherwise, if ${\rm
End}_{E[\ker\rho]}(W_{1})\neq E$ and $E^{\prime}|E$ is a non-trivial field
extension in the division $E$-algebra ${\rm End}_{E[\ker\rho]}(W_{1})$ then
the action of $E^{\prime}$ on $W_{1}$ gives a non-injective surjection of
$E^{\prime}$-representations $W_{1}\otimes_{E}E^{\prime}\longrightarrow
W_{1}$. Then the $E^{\prime}$-representation $W_{1}\otimes_{E}W_{2}$ of $H$ is
irredicible.
Proof. Let $\xi\in W_{1}\otimes_{E}W_{2}$ be a non-zero vector. It suffices to
check that the $E^{\prime}[H]$-span of $\xi$ contains $W_{1}\otimes v$ for any
$v\in W_{2}$. Any non-zero $E[\ker\rho]$-submodule in $W_{1}^{m}$ is
isomorphic to $W_{1}^{m^{\prime}}$ for some $1\leq m^{\prime}\leq m$, and
therefore, the $E[\ker\rho]$-submodule in $W_{1}\otimes_{E}W_{2}$ spanned by
$\xi$ (which is in fact a submodule in $\oplus_{i=1}^{m}W_{1}\otimes
v_{i}\cong W_{1}^{m}$ for some $m\geq 1$ and $E$-linearly independent
$v_{1},\dots,v_{m}$) contains a $E[\ker\rho]$-submodule $W_{1}^{\prime}$
isomorphic to $W_{1}$. As the endomorphisms of the $E[\ker\rho]$-module
$W_{1}$ are scalar, there exists a non-zero $m$-tuple $(a_{1},\dots,a_{m})\in
E^{n}$ such that $W_{1}^{\prime}=\\{a_{1}w\otimes v_{1}+\dots+a_{m}w\otimes
v_{m}~{}|~{}w\in W_{1}\\}$. In other words, $W_{1}^{\prime}=W_{1}\otimes
v^{\prime}$, where $v^{\prime}:=a_{1}v_{1}+\dots+a_{m}v_{m}$ is a non-zero
vector in $W_{2}$.
As any vector $v$ of $W_{2}$ is an $E^{\prime}$-linear combination of several
elements in the $H$-orbit of $v^{\prime}$, we may assume that $v=hv^{\prime}$
for some $h\in H$. Then $u\otimes v=h(h^{-1}u\otimes v^{\prime})$ for any
$u\in W_{1}$. ∎
###### Lemma 2.5.
Let $E^{\prime}|E$ be an extension of fields, $H$ be a group and
$(\rho,W_{2})$ be an irreducible $E^{\prime}$-representation of $H$. Let
$W_{1}$ be an $E$-representation of $H$ such that (i) the sum $\Sigma$ of all
proper $E$-subrepresentations of $\ker\rho$ in $W_{1}$ is proper666$\Sigma$ is
$H$-invariant: as $\ker\rho$ is a normal subgroup of $H$, the group $H$
permutes the $\ker\rho$-submodules in $W_{1}$, while $\Sigma$ is the maximal
proper $\ker\rho$-submodule in $W_{1}$. and (ii) $W_{1}/\Sigma$ is absolutely
irredicible in restriction to $\ker\rho$ and its restriction to the pointwise
stabilizer $\Xi$ of $\Sigma$ in $\ker\rho$ is non-trivial. Then any proper
$E^{\prime}$-subrepresentation of $H$ in $W_{1}\otimes_{E}W_{2}$ is contained
in $\Sigma\otimes_{E}W_{2}$.
Proof. Let $\xi\in W_{1}\otimes_{E}W_{2}$ be a vector, which is not in
$\Sigma\otimes_{E}W_{2}$. It suffices to check that the $E^{\prime}[H]$-span
$V$ of $\xi$ contains $W_{1}\otimes v$ for some non-zero $v\in W_{2}$, as then
$V$ coincides with $W_{1}\otimes_{E}W_{2}$: any vector of $W_{2}$ is an
$E^{\prime}$-linear combination of several elements in the $H$-orbit of $v$
and $W_{1}\otimes hv=h(W_{1}\otimes v)$ for any $h\in H$.
It follows from Lemma 2.4 that $V$ is surjective over
$(W_{1}/\Sigma)\otimes_{E}W_{2}$. In particular, $V$ contains an element of
type $\sum_{i=1}^{m}a_{i}\otimes b_{i}$ for some $a_{1}\in
W_{1}\smallsetminus\Sigma$, whose projection to $W_{1}/\Sigma$ is not fixed by
$\Xi$, for some $a_{2},\dots,a_{m}\in\Sigma$ and for some
$E^{\prime}$-linearly independent $b_{1},\dots,b_{m}\in W_{2}$. Then there
exists an element $h\in\Xi$ such that $ha_{1}-a_{1}\in
W_{1}\smallsetminus\Sigma$, and therefore, $V$ contains an element of type
$\sum_{i=1}^{m}a\otimes b_{1}$ for some $a\in W_{1}\smallsetminus\Sigma$. ∎
###### Proposition 2.6.
Let $W\in\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}(E)$ be an object. For any
open subgroup $U$ of $G$, denote by $W_{(U)}$ the sum of all proper
subrepresentations of $U$ in $W$; and by $\Xi_{U}$ the pointwise stabilizer of
$W_{(U)}$ in $U$. Suppose that for any open subgroup $U$ of $G$: (i) the
$E$-representation $W/W_{(U)}$ of $U$ is absolutely irreducible and non-
trivial in restriction to $\Xi_{U}$777In particular, $W$ is absolutely
indecomposable. Any non-zero quotient of $A(F)$ for an absolutely simple
algebraic $k$-group $A$ is an example of such $W$. (Indeed, any open subgroup
$U\subset G$ contains $G_{F|L}$ for a finitely generated $L$ in $F|k$, so any
$t\in A(F)\smallsetminus A(\overline{L})$ is a cyclic vector of $A(F)$,
considered as $U$-module. Here $\overline{L}$ is the algebraic closure of $L$
in $F$. If the transcendence degree of $L|k$ is minimal then, by [10],
$\overline{L}$ is $U$-invariant, so $A(F)_{(U)}=A(\overline{L})$. ∎) and (ii)
any irreducible smooth representation of $K$ can be embedded into $W$ so that
its image does not meet $W_{(U)}$. Then ${\mathcal{I}}_{S}$ annihilates any
quotient of $W\otimes_{E}V$ for any
$V\in\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}(E)$.
Proof. It suffices to check the vanishing of
${\mathcal{I}}_{S}(W\otimes_{E}V)$. Extending the coefficients if needed, we
may assume that $E$ is big enough (i.e., algebraically closed and
$\\#E>\\#k$), so that any smooth irreducible $E$-representation of any open
subgroup of $G$ is absolutely irreducible.888Schur’s lemma=[2, Claim 2.11]:
Let $H$ be a totally disconnected group and $E$ be a field of cardinality
greater than the cardinality of $H/U$ for any open subgroup $U$ of $H$. Then
the endomorphisms of the smooth irreducible $\overline{E}$-representation of
$H$ are scalar.
The vanishing holds if the $G$-module $W\otimes_{E}V$ is spanned by the
elements $g\xi-\xi$ for all $\xi\in(W\otimes_{E}V)^{K}$ and all $g\in G$.
Equivalently, as the restriction of $V$ to $K$ is semisimple, the $G$-span of
such elements $g\xi-\xi$ contains $W\otimes_{E}\rho$ for any irreducible
$E$-subrepresentation $\rho$ of $K$ in $V$. By (ii), $W$ contains a
$E$-subrepresentation of $K$ which is (a) dual to $\rho$ and (b) outside of
$W_{(U)}$, where $U\subset G$ is the pointwise stabilizer of $\rho$. Then
there is an element $\xi\in(W\otimes_{E}\rho)^{K}$, which is not in
$W_{(U)}\otimes_{E}\rho$.
As the $\Xi_{U}$-module $W/W_{(U)}$ is non-trivial, there exists an element
$u\in\Xi_{U}$ such that $\eta:=u\xi-\xi$ is not in $W_{(U)}\otimes_{E}\rho$.
Denote by $\widetilde{U}$ the subgroup in $G$ generated by $U$ and $K$. Then
$\widetilde{U}$ contains $U$ as a normal subgroup of finite index;
$\widetilde{U}$ acts on $W_{(U)}$; $\rho$ can be viewed as a representation of
$\widetilde{U}$ via the identification $\widetilde{U}/U=K/U\cap K$. By Lemma
2.5 (with $H=\widetilde{U}$), the element $\eta$ generates the
$E[\widetilde{U}]$-module $W\otimes_{E}\rho$. ∎
###### Lemma 2.7 (A source of representations of $G$ containing all
irreducible smooth representations of $K$).
If a subrepresentation $W$ of $G$ in
$\bigotimes^{\bullet}_{F}\Omega^{1}_{F|k}$ does not contain regular
forms,999Examples of such $W$ are subrepresentations of ${\rm
Sym}^{2}_{F}\Omega^{1}_{F|k}$, of $\Omega^{\bullet}_{F|k,\text{exact}}$, or of
the image in $\Omega^{j}_{F|k}$ of $\wedge^{j}\Omega^{1}_{F|k,\log}$, where
$d\log:F^{\times}/k^{\times}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\Omega^{1}_{F|k,\log}$,
for any $j\geq 1$. It follows directly from Hilbert’s Satz 90 that the
representation $F$ (and therefore, the irreducible representation
$d:F/k\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\Omega^{1}_{F|k,\text{exact}}$)
of $G$ contains all irreducible smooth (and thus, finite-dimensional)
representations of $K$. i.e., forms from $\Gamma(X,\Omega^{\bullet}_{X|k})$
for a smooth proper $k$-variety $X$ with $k(X)\subset F$, then $W$ contains
each irreducible smooth representation $\rho$ of $K$.
As mentioned in §2.1, if no non-zero element of ${\mathbb{D}}_{E}(G)$
annihilates a smooth representation $W$ then $W$ contains all irreducible
smooth representations of $K$. The vanishing of the annihilators of $F/k$ and
$F^{\times}/k^{\times}$ is shown in [7, Prop.4.2]. Assume for simplicity that
$F^{K}|k$ is purely transcendental.
Proof. Let $p_{\rho}$ be the central projector in the group algebra of
$Q=K/\ker\rho$ onto the $\rho$-isotypical part. As explained in [8, Prop.7.6],
$W$ contains a non-zero element $\omega$ fixed by the group
$G_{F|k({\mathbb{P}}^{M})}$ for an appropriate $M\geq 1$ and an embedding
$k({\mathbb{P}}^{M})\hookrightarrow F$. The finite field extension
$F^{\ker\rho}|F^{K}$ can be considered as a purely transcendental extension of
a function field extension $k(Y)|k(Y)^{Q}$ of smooth projective $k$-varieties
of dimension $\geq M$. Consider $\omega$ as a differential form with poles on
${\mathbb{P}}^{M}_{k}$. Fix a sufficiently general finite morphism
$f:Y\longrightarrow{\mathbb{P}}^{M}_{k}$, unramified above the poles of
$\omega$, and such that the poles of $f^{\ast}\omega$ pass through a fixed
point of $Y$, but not through another point of its $Q$-orbit. Then, as $Q$
acts freely on the set of ‘sufficiently general’ divisors on $Y$, the form
$p_{\rho}f^{\ast}\omega$ is non-zero, and thus, $p_{\rho}f^{\ast}\omega$ spans
a $K$-submodule in $W$ isomorphic to $\rho$. ∎
Remark. The vanishing of ${\mathcal{I}}_{S}$ on any smooth semilinear
representation $V$ of $G$ is evident: Let $L$ be the function field of an
affine $k$-space embedded into $F$. For any $v\in V^{G_{F|L}}$ and any $x\in
F$ transcendental over $L$ the vector $xv$ belongs to $V^{G_{F|L(x)}}$, so its
image ${\mathcal{I}}_{S}V$ should be fixed by $G$. In particular, the image of
$xv$ in ${\mathcal{I}}_{S}V$ coincides with the image of $2xv$, and thus, $xv$
becomes $0$ in ${\mathcal{I}}_{S}V$. Such vectors $xv$ span $V$, so
${\mathcal{I}}_{S}V=0$.
###### Corollary 2.8.
For any $k$-variety $U$ and any rational closed form $\eta$ on
$U\times{\mathbb{A}}^{1}$ there exist an affine variety $Y$, dominant
morphisms $\pi:Y\to U\times{\mathbb{A}}^{1}$,
$\pi_{1},\dots,\pi_{m}:Y\to{\mathbb{A}}^{N}_{k}$ and rational closed forms
$\eta_{1},\dots,\eta_{m}$ on ${\mathbb{A}}^{N}_{k}$ and $\eta_{0}$ on $U$ such
that $\pi^{\ast}\eta=({\rm
pr}_{U}\circ\pi)^{\ast}\eta_{0}+\pi_{1}^{\ast}\eta_{1}+\dots+\pi_{m}^{\ast}\eta_{m}$.
Proof. We consider $\eta$ as a section of the sheaf
$\mathop{\underline{\Omega}}^{\bullet}_{|k,\text{closed}}:X\mapsto\Omega^{\bullet}_{k(X)|k,\text{closed}}$
over $U\times{\mathbb{A}}^{1}$. Proposition 3.3 describes the kernel of
$\mathop{\underline{\Omega}}^{\bullet}_{|k,\text{closed}}\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}{\mathcal{I}}\mathop{\underline{\Omega}}^{\bullet}_{|k,\text{closed}}$
as the ideal generated by the exact and the logarithmic differentials. By
Proposition 2.6, applied to $W=F^{\times}/k^{\times}$, ${\mathcal{I}}_{S}$
annihilates the kernel of $\alpha$. Thus, modulo closed forms coming from
projective spaces, $\eta$ comes from $U$. ∎
Let $X$ be a smooth proper $k$-variety and
$W:={\mathbb{Q}}[\\{k(X)\stackrel{{\scriptstyle/k}}{{\hookrightarrow}}F\\}]$
be the module of generic 0-cycles on $X$. The space $W^{K}$ is the image of
the projector defined by the Haar measure of $K$. As the generators of $W$ are
generic points of $X$, the space $W^{K}$ is spanned by the 0-cycles of type
$p_{\ast}\pi^{\ast}q$ for all diagrams of dominant $k$-morphisms
$X\stackrel{{\scriptstyle
p}}{{\leftarrow\longleftarrow}}Y\stackrel{{\scriptstyle\pi}}{{\longrightarrow\to}}{\mathbb{P}}^{N}_{k}$,
where $\pi$ is generically finite, and all generic points
$q\in{\mathbb{P}}^{N}(F^{K})$. (Indeed, for any generic $F$-point
$\sigma:k(X)\stackrel{{\scriptstyle/k}}{{\hookrightarrow}}F$ of $X$ the orbit
$K\sigma$ is finite, so the compositum $L_{1}$ of the images of the elements
of $K\sigma$ is finitely generated over $k$. Let $L_{0}\subset F^{K}$ be a
finitely generated and purely transcendental extension of $k$ containing
$L_{1}^{K}$. Let $Y$ be a $K$-equivariant smooth $k$-model of $L_{0}L_{1}$.
Then $p$ and $\pi$ are induced by the inclusions $k(X)\subset k(Y)\supset
L_{0}$.) Thus, the module ${\mathcal{I}}_{S}W$ is the quotient of $W$ by the
${\mathbb{Q}}$-span of 0-cycles of type
$p_{\ast}\pi^{\ast}q_{1}-p_{\ast}\pi^{\ast}q_{2}$ for all dominant
$k$-morphisms $p:Y\to X$, generically finite $k$-morphisms
$\pi:Y\to{\mathbb{P}}^{N}_{k}$ and all generic points
$q_{1},q_{2}\in{\mathbb{P}}^{N}(F)$.
###### Lemma 2.9.
Let $X$ be a smooth proper curve over $k$ of genus $g$. Then the $G$-module
$Z^{{\rm
rat}}_{0}(k(X)\otimes_{k}F):=\ker[Z_{0}(k(X)\otimes_{k}F)\longrightarrow
CH_{0}(X\times_{k}F)]$ is generated by
$w_{N}=\sum^{N}_{j=1}\sigma_{j}-\sum^{N}_{j=1}\tau_{j}$ for all $N>g$, where
$(\sigma_{1},\dots,\sigma_{N};\tau_{1},\dots,\tau_{N})$ is a generic $F$-point
of the fibre over $0$ of the map $X^{N}\times_{k}X^{N}\stackrel{{\scriptstyle
p_{N}}}{{\longrightarrow}}\mathop{\mathrm{Pic}}\nolimits^{0}X$ sending
$(x_{1},\dots,x_{N};y_{1},\dots,y_{N})$ to the class of
$\sum^{N}_{j=1}x_{j}-\sum^{N}_{j=1}y_{j}$.
Proof. Let
$\gamma_{1},\dots,\gamma_{s}:k(X)\stackrel{{\scriptstyle/k}}{{\hookrightarrow}}F$
and
$\delta_{1},\dots,\delta_{s}:k(X)\stackrel{{\scriptstyle/k}}{{\hookrightarrow}}F$
be generic points of $X$ such that
$\sum^{s}_{j=1}\gamma_{j}-\sum^{s}_{j=1}\delta_{j}$ is the divisor of a
rational function on $X_{F}$.
We need to show that $\sum^{s}_{j=1}\gamma_{j}-\sum^{s}_{j=1}\delta_{j}$
belongs to the $G$-submodule in $Z^{{\rm rat}}_{0}(k(X)\otimes_{k}F)$
generated by $w_{N}$’s.
There is a collection
$\alpha_{1},\dots,\alpha_{g}:k(X)\stackrel{{\scriptstyle/k}}{{\hookrightarrow}}F$
of generic points of $X$ such that the class of
$\sum^{s}_{j=1}\gamma_{j}+\sum^{g}_{j=1}\alpha_{j}$ in
$\mathop{\mathrm{Pic}}\nolimits^{s+g}X$ is a generic point. Then there is a
collection
$\xi_{1},\dots,\xi_{s+g}:k(X)\stackrel{{\scriptstyle/k}}{{\hookrightarrow}}F$
of generic points of $X$ in general position such that
$\sum^{s}_{j=1}\gamma_{j}+\sum^{g}_{j=1}\alpha_{j}-\sum^{s+g}_{j=1}\xi_{j}$ is
divisor of a rational function on $X_{F}$ (so the same holds also for
$\sum^{s}_{j=1}\delta_{j}+\sum^{g}_{j=1}\alpha_{j}-\sum^{s+g}_{j=1}\xi_{j}$).
We may, thus, assume that $\delta_{1},\dots,\delta_{s}$ are in general
position.
Fix a collection $\\{\varkappa_{ij}\\}_{1\leq i\leq g,1\leq j\leq s}$ of
generic points of $X$ in general position, also with respect to
$\gamma_{1},\dots,\gamma_{s}$ and to $\delta_{1},\dots,\delta_{s}$, such that
the classes of
$\gamma_{1}+\sum^{g}_{i=1}\varkappa_{i1},\dots,\gamma_{s}+\sum^{g}_{i=1}\varkappa_{is}$
in $\mathop{\mathrm{Pic}}\nolimits^{g+1}X$ are generic points in general
position. Then one can choose a collection $\\{\xi_{ij}\\}_{0\leq i\leq
g,1\leq j\leq s}$ of generic points of $X$ in general position such that
$\gamma_{j}+\sum^{g}_{i=1}\varkappa_{ij}-\sum^{g}_{i=0}\xi_{ij}$ is divisor of
a rational function on $X_{F}$ (so the same holds also for
$\sum^{s}_{j=1}\sum^{g}_{i=0}\xi_{ij}-\left(\sum^{s}_{j=1}\delta_{j}+\sum^{s}_{j=1}\sum^{g}_{i=1}\varkappa_{ij}\right)$).
We may, thus, assume that both $\gamma_{1},\dots,\gamma_{s}$ and
$\delta_{1},\dots,\delta_{s}$ are in general position.
Then there is a collection of generic points
$\xi_{1},\dots,\xi_{s}:k(X)\stackrel{{\scriptstyle/k}}{{\hookrightarrow}}F$
such that the points $(\gamma_{1},\dots,\gamma_{s};\xi_{1},\dots,\xi_{s})$ and
$(\delta_{1},\dots,\delta_{s};\xi_{1},\dots,\xi_{s})$ are generic on
$p^{-1}_{s}(0)$. Then $\sum^{s}_{j=1}\gamma_{j}-\sum^{s}_{j=1}\xi_{j}$ and
$\sum^{s}_{j=1}\delta_{j}-\sum^{s}_{j=1}\xi_{j}$ are divisors of rational
functions on $X_{F}$. Clearly, such elements belong to the $G$-orbit of
$w_{s}$. ∎
Remark. The $G$-module $Z^{{\rm rat}}_{0}(k(X)\otimes_{k}F)$ from Lemma 2.9 is
generated by $w_{g+1}$. Proof. There exists an effective divisor $D$ (of
degree $g$) in the linear equivalence class of
$\sum^{N}_{j=2}\sigma_{j}-\sum^{N}_{j=g+2}\tau_{j}$, so
$w_{N}=[\sum^{N}_{j=2}\sigma_{j}-D-\sum^{N}_{j=g+2}\tau_{j}]+[\sigma_{1}+D-\sum^{g+1}_{j=1}\tau_{j}]$
is a sum of an element in the $G$-orbit of $w_{N-1}$ and an element in the
$G$-orbit of $w_{g+1}$. ∎
###### Proposition 2.10.
${\mathcal{I}}_{S}{\mathbb{Q}}[\\{k(X)\stackrel{{\scriptstyle/k}}{{\hookrightarrow}}F\\}]=\mathop{\mathrm{Pic}}\nolimits(X_{F})_{{\mathbb{Q}}}$
for any smooth proper curve $X$ over $k$.
Proof. By Lemma 2.9, it suffices to show that the images of the generators
$w_{N}$ in ${\mathcal{I}}_{S}W$ are zero. Denote by $g\geq 0$ the genus of
$X$, by $\psi_{N}$ a generic effective divisor on $X$ of degree $N$ with a
special class in $\mathop{\mathrm{Pic}}\nolimits^{N}X$. Then
$w_{N}=\sigma\psi_{N+g}-\tau\psi_{N+g}$ for some $\sigma,\tau\in G$, so it
suffices to show that the images of $\psi_{N}$’s in ${\mathcal{I}}_{S}W$ are
fixed by $G$. Denote by $X^{N}\stackrel{{\scriptstyle
s}}{{\longrightarrow}}\Sigma^{N}X\stackrel{{\scriptstyle
r}}{{\longrightarrow}}\mathop{\mathrm{Pic}}\nolimits^{N}(X)$ the natural
morphisms and set $Y:=(rs)^{-1}(\ast)$. Let $p:Y\subseteq X^{N}\longrightarrow
X$ be the projection to the first multiple; set $\pi=s|_{Y}:Y\longrightarrow
r^{-1}(\ast)$. The projection to the first $N-g$ multiples $Y\longrightarrow
X^{N-g}$ is generically finite of degree $g!$. If $N\geq 2g-1$ then
$r^{-1}(\ast)\cong{\mathbb{P}}^{N-g}$. Assume also that $N\geq g+1$ (i.e.
$N\geq\max(2g-1,g+1)$). As $s$ is generically finite of degree $N!$, one has
$(N-1)!\psi_{N}=p_{\ast}\pi^{\ast}q$ for a generic point $q$ of
$r^{-1}(\ast)$. ∎
Denote by ${\mathcal{I}}_{G}$ the full subcategory in
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}$ of ‘‘homotopy invariant’’
representations: $W^{G_{F|L^{\prime}}}=W^{G_{F|L}}$ for any purely
transcendental subextension $L^{\prime}|L$ in $F|k$.
###### Theorem 2.11.
A dominant sheaf is ${\mathbb{A}}^{1}$-invariant if and only if all its simple
subquotients are.
Proof. Let $S$ be the collection of pairs of type $(G_{F|L(x)}\subset
G_{F|L})$ for all subfields $L$ in $F|k$ of finite type and elements $x\in F$
transcendental over $L$. The following conditions are equivalent:
1. (1)
a smooth representation $W$ of $G$ is ‘‘homotopy invariant’’;
2. (2)
$W^{U_{1}}=W^{U_{2}}$ for all pairs $(U_{1}\subset U_{2})\in S$;
3. (3)
$W^{G_{F|L}}=W^{G_{F|L^{\prime}}}$ for all subfields $L$ in $F|k$ of finite
type and purely transcendental extensions $L^{\prime}|L$ in $F$ such that $F$
is algebraic over $L^{\prime}$.
(1)$\Leftrightarrow$(3) and (1)$\Leftrightarrow$(2) are evident;
(2)$\Leftrightarrow$(1) is proved in [7, Corollary 6.2]. This verifies the
condition (2) of Proposition 2.2. For each pair $(G_{F|L(x)}\subset
G_{F|L})\in S$ fix some $\sigma\in G_{F|L}$ with $x$ and $\sigma x$
algebraically independent over $L$. Then the condition (1)(i) is obvious:
$G_{F|L(x)}\cap G_{F|L(\sigma x)}=G_{F|L(x,\sigma x)}$ and $(G_{F|L(x,\sigma
x)}\subset G_{F|L(x)})\in S$; the condition (1)(ii) follows from [7, Lemma
2.16]: the subgroups $G_{F|L(x)}$ and $G_{F|L(\sigma x)}$ generate $G_{F|L}$.
∎
### 2.5. Summary of equivalences
The following categories are equivalent:
1. (1)
the category of dominant ${\mathbb{A}}^{1}$-invariant sheaves of $E$-vector
spaces;
2. (2)
the category of dominant ${\mathbb{A}}^{1}$-invariant presheaves of $E$-vector
spaces with the Galois descent property;
3. (3)
the category $\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}^{S}(E)$, where $S$
consists of the pairs of type $(G_{F|L^{\prime}}\subset G_{F|L})$ with purely
transcendental $L^{\prime}|L$ in $F|k$.
These equivalences restrict to equivalences of corresponding subcategories:
(1) of sheaves of finite-dimensional spaces, (2) of presheaves of finite-
dimensional spaces, (3) of admissible representations of $G$.101010A
representation of a totally disconnected group is admissible if it is smooth
and the fixed subspaces of all open subgroups are finite-dimensional.
Consider the following properties of a smooth representation $W$ of $G$:
1. (1)
$W\in\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}^{S}(E)$, where $S$ consists
of the pairs of type $(G_{F|L^{\prime}}\subset G_{F|L})$ with purely
transcendental $L^{\prime}|L$ in $F|k$;
2. (2)
the restriction of $W$ to a compact subgroup $U$ does not contain all smooth
irreducible representations of $U$;
3. (3)
the annihilator of $W$ in the algebra ${\mathbb{D}}_{{\mathbb{Q}}}(G)$ is non-
zero.
One has (1)$\Rightarrow$(2)$\Rightarrow$(3). [(2)$\Rightarrow$(3) is explained
in §2.1. (1)$\Rightarrow$(2): If $F^{U}$ is purely transcendental over $k$,
there are many irreducible smooth representations of $U$, entering in no
object of $\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}^{S}(E)$. Any non-
trivial smooth irreducible representation $\tau$ of $U$ such that
$F^{\ker\tau}$ is unirational (e.g., purely transcendental) over $k$ is an
example of such representation. Clearly, for any such $\tau$ the natural
projector $p_{\tau}\in{\mathbb{D}}_{{\mathbb{Q}}}(G)$ onto the
$\tau$-isotypical part belongs to the common annihilator of the objects of
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}^{S}(E)$.]
Remark. For a discrete valuation $v$ of rank $1$ on $F$, trivial on
$k$,111111By definition, this means that any maximal system of elements of
$F^{\times}$ with independent images in the valuation group, should be a
transcendence base of $F$ over a lift of a subfield of the residue field. and
a smooth representation $W$ of $G$ set $W_{v}:=\sum_{L}W^{G_{F|L}}\subseteq
W$, where $L$ runs over the subfields in the valuation ring of $v$. The
intersection $\Gamma(W):=\bigcap_{v}W_{v}$ over all such $v$’s is again in
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}$. As shown in [10, Cor.4.7], the
property (1) for $W$ implies that $W=W_{v}$ (and also $W=\Gamma(W)$, since all
$v$’s as above form a $G$-orbit, cf. [10]).
## 3\. Differential forms
Let $H^{\bullet}=\bigoplus_{q\geq 0}H^{q}$ be a cohomology theory, considered
as a dominant ${\mathbb{A}}^{1}$-presheaf. Denote by
$\underline{H}_{c}^{\bullet}$ the dominant ${\mathbb{A}}^{1}$-sheaf $X\mapsto
H^{\bullet}(X)/N^{1}$ for smooth proper $k$-varieties $X$, which is a subsheaf
of $\underline{H}^{\bullet}$, e.g., $\underline{H}^{1}_{c}:X\mapsto
H^{1}(\overline{X})$. Clearly, $\underline{H}^{\bullet}_{c}$ is a sheaf of
finite $H^{\bullet}(k)$-modules. It would follow from the standard
semisimplicity conjecture that the sheaf $\underline{H}^{\bullet}_{c}$ is
semisimple if $H^{\bullet}(k)$ is a field.
We shall be interested in the case of de Rham cohomology
$H^{\bullet}=H^{\bullet}_{{\rm dR}/k}:X\mapsto H^{\bullet}_{{\rm
dR}/k}(X):={\mathbb{H}}^{\bullet}(X,\Omega^{\bullet}_{X|k})$, where
$H^{\bullet}(k)=k$, cf. [3]. Clearly, $\underline{H}^{q}_{{\rm
dR}/k}=\mathop{\underline{\Omega}}^{q}_{|k,\text{{\rm
closed}}}/\mathop{\underline{\Omega}}^{q}_{|k,\text{{\rm exact}}}$, where
$\Omega^{q}_{|k,\text{{\rm
closed}}}:Y\mapsto\ker(d|\Gamma(Y,\Omega^{q}_{Y|k}))$ and
$\Omega^{q}_{|k,\text{{\rm exact}}}:Y\mapsto d\Gamma(Y,\Omega^{q-1}_{Y|k})$,
so
$d:{\mathcal{H}}^{{\mathbb{G}}_{a}}_{1}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathop{\underline{\Omega}}^{1}_{|k,\text{{\rm
exact}}}$. The sheaf $\underline{H}^{1}_{{\rm dR}/k,c}$ is semisimple. It is
described in Lemma 1.1.
### 3.1. Maximal ${\mathbb{A}}^{1}$-subsheaf and the
${\mathbb{A}}^{1}$-quotient of (closed) forms
Recall (§1.2) that the inclusion functor
${\mathcal{I}}_{G}\to\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}$ admits a
right adjoint $W\mapsto W^{(0)}$, the maximal subobject in
${\mathcal{I}}_{G}$. The following fact points out once more the cohomological
nature of the objects of ${\mathcal{I}}_{G}$.
###### Proposition 3.1 ([8], Prop.7.6).
The maximal subobject in ${\mathcal{I}}_{G}$ of the sheafification of
$\bigotimes^{\bullet}_{{\mathcal{O}}}\Omega^{1}_{|k}$ is
$\Omega^{\bullet}_{|k,\text{{\rm reg}}}$.
For any smooth proper $k$-variety $Y$ there are the following canonical
isomorphisms
(3)
$\mathop{\mathrm{Hom}}\nolimits_{{\mathcal{I}}_{G}}(C_{k(Y)},\Omega^{q}_{|k,\text{{\rm
reg}}})=\mathop{\mathrm{Hom}}\nolimits_{\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}}(\Psi_{Y},\Omega^{q}_{|k,\text{{\rm
reg}}})\stackrel{{\scriptstyle\sim}}{{\longleftarrow}}\Gamma(Y,\Omega^{q}_{Y|k})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathop{\mathrm{Hom}}\nolimits_{\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}}(CH_{0}(Y_{F}),\Omega^{q}_{|k,\text{{\rm
reg}}}).$
The first isomorphism is functorial with respect to the dominant morphisms
$Y\longrightarrow Y^{\prime}$, the second one is functorial with respect to
arbitrary morphisms $Y\longrightarrow Y^{\prime}$.
###### Lemma 3.2.
Let $L$ be an algebraically closed extension of $k$ and $x$ be an
indeterminant. Then there are isomorphisms $id+\sum_{\alpha\in
L}\wedge\frac{d(x-\alpha)}{x-\alpha}:(L(x)\otimes_{L}\Omega^{q}_{L|k})/\Omega^{q}_{L|k,\text{{\rm
exact}}}\oplus\bigoplus_{\alpha\in
L}(\Omega^{q-1}_{L|k}/\Omega^{q-1}_{L|k,\text{{\rm
exact}}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\Omega^{q}_{L(x)|k}/\Omega^{q}_{L(x)|k,\text{{\rm
exact}}}$ and $d+\sum_{\alpha\in
L}\wedge\frac{d(x-\alpha)}{x-\alpha}:(L(x)\otimes_{L}\Omega^{q}_{L|k})/\Omega^{q}_{L|k,\text{{\rm
closed}}}\oplus\bigoplus_{\alpha\in L}\Omega^{q}_{L|k,\text{{\rm
exact}}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\Omega^{q+1}_{L(x)|k,\text{{\rm
exact}}}$ for any $q\geq 1$. The former isomorphism restricts to an
isomorphism $H^{q}_{{\rm dR}/k}(L)\oplus\bigoplus_{\alpha\in L}H^{q-1}_{{\rm
dR}/k}(L)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{q}_{{\rm
dR}/k}(L(x))$.
Proof. As $\Omega^{q}_{L(x)|k}=L(x)\otimes_{L}\Omega^{q}_{L|k}\oplus
L(x)\otimes_{L}\Omega^{q-1}_{L|k}\wedge dx$, for any
$\omega\in\Omega^{q}_{L(x)|k}$ one has $\omega\equiv\eta\wedge
dx\pmod{L(x)\otimes_{L}\Omega^{q}_{L|k}}$ for a unique $\eta\in
L(x)\otimes_{L}\Omega^{q-1}_{L|k}$. Using partial fraction decomposition of
rational functions in $L(x)$, we get a presentation $\eta=\sum_{j\geq
0}x^{j}\eta_{j}+\sum_{\alpha\in L,~{}j\geq
1}\frac{\eta_{j,\alpha}}{(x-\alpha)^{j}}$, where
$\eta_{j},\eta_{j,\alpha}\in\Omega^{q-1}_{L|k}$. Then $\eta\wedge
dx\equiv\sum_{\alpha\in
L}\eta_{1,\alpha}\wedge\frac{d(x-\alpha)}{x-\alpha}\pmod{L(x)\otimes_{L}\Omega^{q}_{L|k}+\Omega^{q}_{L(x)|k,\text{{\rm
exact}}}}$, so $\omega\equiv\sum_{i}\phi_{i}(x)\eta_{i}+\sum_{\alpha\in
L}\eta_{1,\alpha}\wedge\frac{d(x-\alpha)}{x-\alpha}\pmod{\Omega^{q}_{L(x)|k,\text{{\rm
exact}}}}$, and thus,
$d\omega=\sum_{i}d\phi_{i}(x)\wedge\eta_{i}+\sum_{i}\phi_{i}(x)d\eta_{i}+\sum_{\alpha\in
L}d\eta_{1,\alpha}\wedge\frac{d(x-\alpha)}{x-\alpha}\equiv\sum_{i}\phi^{\prime}_{i}(x)dx\wedge\eta_{i}+\sum_{\alpha\in
L}d\eta_{1,\alpha}\wedge\frac{dx}{x-\alpha}\pmod{L(x)\otimes_{L}\Omega^{q}_{L|k}}$
for some $\phi_{i}(x)\in L(x)$ and $\eta_{i}\in\Omega^{q}_{L|k}$ (and we may
assume that $\eta_{i}$ are $L$-linearly independent). Using partial fraction
decomposition of the rational functions $\phi_{i}\in L(x)$, we see that if
$\omega$ is closed then $d\eta_{1,\alpha}=0$, $\phi_{i}\in L$ and
$\sum_{i}\phi_{i}\eta_{i}\in\Omega^{q}_{L|k}$ is closed. ∎
###### Proposition 3.3.
Let $M_{q}$ be the sheaf associated with the presheaf
$\Omega^{q}_{|k,\text{{\rm exact}}}+\Omega^{q-1}_{|k,\text{{\rm
closed}}}\wedge d\log{\mathbb{G}}_{m}\subset\Omega^{q}_{|k}$ for any $q\geq
1$. Then (i) $\Omega^{q}_{k(X\times{\mathbb{A}}^{n})|k,\text{{\rm
closed}}}=\Omega^{q}_{k(X)|k,\text{{\rm
closed}}}+M_{q}(X\times{\mathbb{A}}^{n})$ for any $n\geq 1$; (ii) $M_{q}$ is
the kernel of the natural projection
$\pi_{q}:\mathop{\underline{\Omega}}^{q}_{|k,\text{{\rm closed}}}\to
V^{q}:={\mathcal{I}}(\mathop{\underline{\Omega}}^{q}_{|k,\text{{\rm
closed}}})={\mathcal{I}}(\underline{H}^{q}_{{\rm dR}|k})$; (iii) for $q\geq
2$, $M_{q}$ is the sheaf associated with the presheaf
$\Omega^{q-1}_{|k,\text{{\rm closed}}}\wedge d\log{\mathbb{G}}_{m}$ and
$d+d\log:{\mathcal{H}}^{{\mathbb{G}}_{a}}_{1}\oplus
k\otimes{\mathcal{H}}^{{\mathbb{G}}_{m}}_{1}\to M_{1}$ is an isomorphism.
In particular, the natural projections
$\mathop{\underline{\Omega}}^{\bullet}_{|k,\text{{\rm
closed}}}\stackrel{{\scriptstyle p_{1}}}{{\to}}\underline{H}^{\bullet}_{{\rm
dR}|k}\stackrel{{\scriptstyle
p_{2}}}{{\to}}V^{\bullet}:={\mathcal{I}}(\underline{H}^{\bullet}_{{\rm
dR}|k})={\mathcal{I}}(\mathop{\underline{\Omega}}^{\bullet}_{|k,\text{{\rm
closed}}})$ are morphisms of sheaves of supercommutative $k$-algebras. (The
kernel of $p_{1}$, i.e. $\mathop{\underline{\Omega}}^{\bullet}_{|k,\text{{\rm
exact}}}$, is the ideal generated by
$\mathop{\underline{\Omega}}^{1}_{|k,\text{{\rm exact}}}$,121212More
generally, let $\omega\in\Omega^{\geq i}_{F|k}\smallsetminus\Omega^{\geq
i+1}_{F|k}$ be a closed form for some $i\geq 0$. Then any ideal in
$\mathop{\underline{\Omega}}^{\bullet}_{|k,\text{{\rm closed}}}$ containing
the $G$-orbit of $\omega$ contains $\mathop{\underline{\Omega}}^{\geq
i+1}_{|k,\text{{\rm exact}}}$. Proof. By [8, Lemma 7.7], the semilinear
representation $\Omega^{j}_{F|k}$ is irreducible for any $j\geq 0$. In
particular, $F$-linear envelope of the $G$-orbit of $\omega$ is the direct sum
of $\Omega^{j}_{F|k}$ over all $j\geq i$ such that the homogeneous component
of $\omega$ of degree $j$ is non-zero. Then
$dz\wedge\sigma\omega=d(z\cdot\sigma\omega)$ for all $z\in F$ and all
$\sigma\in G$ span the direct sum of
$\mathop{\underline{\Omega}}^{j}_{|k,\text{{\rm exact}}}$ over all $j\geq i$
as above. ∎ the kernel of $p_{2}$ is the ideal generated by
$d\log{\mathbb{G}}_{m}$.) They are surjective even as morphisms of presheaves.
Proof. Let us show that $\ker\pi_{q}$ contains $M_{q}$. For any irreducible
smooth $k$-variety $X$, any $\eta\in\Omega^{q-1}_{k(X)|k,\text{{\rm closed}}}$
and a generator $t$ of the field $k(X\times{\mathbb{G}}_{m})$ over $k(X)$ the
closed $q$-forms $\omega_{m}=\eta\wedge d\log t$ and $\omega_{a}=\eta\wedge
dt$ are sections of the sheaf $\Omega^{q}_{|k,\text{{\rm closed}}}$ over
$X\times{\mathbb{G}}_{m}$, so their images in
${\mathcal{I}}(\Omega^{q}_{|k,\text{{\rm closed}}})$ should be sections over
$X$. As there are endomorphisms $g_{m},g_{a}$ of $X\times{\mathbb{G}}_{m}|X$
such that $g_{m}t=t^{2}$ and $g_{a}t=2t$ (so $g_{?}\omega_{?}=2\omega_{?}$),
the images of $\omega_{?}$ in ${\mathcal{I}}(\Omega^{q}_{|k,\text{{\rm
closed}}})$ should be zero. The elements of type $\eta\otimes d\log t$ (resp.,
$\eta\otimes dt$) span the sheaf $\Omega^{q-1}_{|k,\text{{\rm
closed}}}\otimes{\mathcal{H}}^{{\mathbb{G}}_{m}}_{1}$ (resp.,
$\Omega^{q-1}_{|k,\text{{\rm
closed}}}\otimes{\mathcal{H}}^{{\mathbb{G}}_{a}}_{1}$, which is surjective
over $\Omega^{q}_{|k,\text{{\rm exact}}}$).
By [7, Lemma 6.3, p.200], to show that $\ker\pi_{q}=M_{q}$ it suffices to
check that, for any algebraically closed extension $F^{\prime}|k$ in $F$ and
any $t\in F\smallsetminus F^{\prime}$, any
$\omega\in\Omega^{q}_{F^{\prime}(t)|k,\text{{\rm closed}}}$ belongs in fact to
$\Omega^{q}_{F^{\prime}|k,\text{{\rm closed}}}+M_{q}$.
By Lemma 3.2, $\omega\equiv\xi+\sum_{\alpha\in
F^{\prime}}\eta_{\alpha}\wedge\frac{d(t-\alpha)}{t-\alpha}\pmod{\Omega^{q}_{F^{\prime}(t)|k,\text{{\rm
exact}}}}$, where $\xi\in\Omega^{q}_{F^{\prime}|k,\text{{\rm closed}}}$ and
$\eta_{\alpha}\in\Omega^{q-1}_{F^{\prime}|k,\text{{\rm closed}}}$, which means
that $\omega\in\Omega^{q}_{F^{\prime}|k,\text{{\rm closed}}}+M_{q}$. ∎
###### Conjecture 3.4.
The sheaf $V^{\bullet}$ is semisimple.
Remarks. 1\. It follows from Proposition 3.3 that the natural morphism
$\underline{H}^{\bullet}_{{\rm dR}/k,c}\to V^{\bullet}$ is injective.
2\. As explained in Remark on p.2.7, ${\mathcal{I}}V=0$ for any semilinear
smooth representation $V$: if $v\in V^{G_{F|L}}$ and $f\in F$ is
transcendental over $L$ then $v=fv-(f-1)v$ becomes zero in any quotient of $V$
in ${\mathcal{I}}_{G}$.
3\. For an algebra $A\in\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}$ it is not
always true that the kernel $A^{\circ}$ of the projection $A\to{\mathcal{I}}A$
is an ideal. E.g., let $A=A_{\bullet}$ be the (graded) tensor, symmetric or
skew-symmetric algebra of $A_{1}={\mathbb{Q}}[F\smallsetminus k]$. Then
${\mathcal{I}}A_{1}={\mathbb{Q}}$, so $A_{1}^{\circ}\otimes A_{1}+A_{1}\otimes
A_{1}^{\circ}$ consists of all sums in ${\mathbb{Q}}[(F\smallsetminus
k)\times(F\smallsetminus k)]$ of degree 0\. On the other hand,
${\mathcal{I}}(A_{1}\otimes
A_{1})=\bigoplus_{x\in\mathop{\mathbf{Spec}}(k({\mathbb{P}}^{1})\otimes_{k}k({\mathbb{P}}^{1}))}C_{k(x)}$,
and therefore, $A_{1}^{\circ}\otimes A_{1}+A_{1}\otimes A_{1}^{\circ}$ is
strictly bigger than $A_{2}^{\circ}$.
### 3.2. The semisimplicity of the regular forms of top degree
Let $L$ be an algebraically closed extension of $k$ with $1\leq
q=\mathop{\mathrm{tr.deg}}(L|k)<\infty$. Define a representation
$\Omega^{q}_{L|k,\text{{\rm reg}}}$ as the union in $\Omega^{q}_{L|k}$ of all
spaces $\Gamma(X,\Omega^{q}_{X|k})$ over all smooth proper varieties $X$ over
$k$ with the function field embedded into $L$ over $k$.
The naïve truncation filtration on $\Omega^{\bullet}_{\overline{U}|k}$ gives
the descending Hodge filtration $F^{\bullet}$ on $H^{q}_{{\rm
dR}/k}(\overline{U})$. The Hodge filtrations on $H^{q}_{{\rm
dR}/k}(\overline{U})$ for all $U$’s induce a canonical filtration
$F^{\bullet}$ on $\underline{H}^{q}_{{\rm dR}/k,c}$ by subsheaves of
$k$-vector spaces with associated graded quotients
$H^{p,q-p}_{|k}:Y\mapsto{\rm
coker}[\bigoplus_{D}H^{p-1}(D,\Omega^{q-p-1}_{D|k})\longrightarrow
H^{p}(\overline{Y},\Omega^{q-p}_{\overline{Y}|k})]$, where $D\to\overline{Y}$
runs over all resolutions of the divisors on $\overline{Y}$. In particular,
$H^{q,0}_{|k}=F^{q}\underline{H}^{q}_{{\rm
dR}/k,c}=\Omega^{q}_{|k,\text{reg}}:Y\mapsto\Gamma(\overline{Y},\Omega^{q}_{\overline{Y}|k})$
is the dominant subsheaf of $\underline{H}^{q}_{{\rm dR}/k,c}$ consisting of
regular differential $q$-forms.
###### Proposition 3.5 ([4]).
Suppose that the cardinality of $k$ is at most continuum. The representation
$H^{q}_{{\rm dR}/k,c}(L)$ $($and therefore, $\Omega^{q}_{L|k,\text{{\rm
reg}}})$ of $G_{L|k}$ is semisimple. Any embedding
$\iota:k\hookrightarrow{\mathbb{C}}$ into the field of complex numbers
determines
* •
a ${\mathbb{C}}$-antilinear isomorphism
$H^{s,t}_{L|k}\otimes_{k,\iota}{\mathbb{C}}\cong
H^{t,s}_{L|k}\otimes_{k,\iota}{\mathbb{C}}$,
* •
a positive definite $G_{L|k}$-equivariant hermitian form
$({\mathbb{C}}\otimes_{k,\iota}H^{q}_{{\rm
dR}/k,c}(L))\otimes_{id,{\mathbb{C}},\sigma}({\mathbb{C}}\otimes_{k,\iota}H^{q}_{{\rm
dR}/k,c}(L))\longrightarrow{\mathbb{C}}(\chi)$, where $\sigma$ is the complex
conjugation and $\chi$ is the modulus of $G_{L|k}$.
There exists a non-canonical ${\mathbb{Q}}$-linear isomorphism
$H^{s,t}_{L|k}\cong H^{t,s}_{L|k}$.
Proof. For any smooth projective $k$-variety $X$ the complexified projection
$F^{p}H^{p+q}_{{\rm dR}/k}(X)\to H^{q}(X,\Omega^{p}_{X|k})$ identifies
$F^{p}H^{p+q}_{{\rm
dR}/k}(X)\otimes_{k,\iota}{\mathbb{C}}\cap\overline{F^{q}H^{p+q}_{{\rm
dR}/k}(X)\otimes_{k,\iota}{\mathbb{C}}}$ with
$H^{q}(X,\Omega^{p}_{X|k})\otimes_{k,\iota}{\mathbb{C}}$. This gives a
decomposition ${\mathbb{C}}\otimes_{k,\iota}H^{q}_{{\rm
dR}/k,c}(L)=\bigoplus_{s+t=q}{\mathbb{C}}\otimes_{k,\iota}H^{s,t}_{L|k}$. Then
the complex conjugation on
$H^{p+q}(X_{\iota}({\mathbb{C}}),{\mathbb{C}})=H^{p+q}(X_{\iota}({\mathbb{C}}),{\mathbb{R}})\otimes_{{\mathbb{R}}}{\mathbb{C}}$
identifies $H^{q}(X,\Omega^{p}_{X|k})\otimes_{k,\iota}{\mathbb{C}}$ with
$H^{p}(X_{\iota}({\mathbb{C}}),\Omega^{q}_{X_{\iota}({\mathbb{C}})})=H^{p}(X,\Omega^{q}_{X|k})\otimes_{k,\iota}{\mathbb{C}}$.
The semisimplicity of the $k$-representation $H^{q}_{{\rm dR}/k,c}(L)$ of
$G_{L|k}$ is equivalent to the semisimplicity of its complexification. For the
latter note that there is a positive definite $G_{L|k}$-equivariant hermitian
form
$({\mathbb{C}}\otimes_{k,\iota}H^{s,t}_{L|k})\otimes_{id,{\mathbb{C}},\sigma}({\mathbb{C}}\otimes_{k,\iota}H^{s,t}_{L|k})\longrightarrow{\mathbb{C}}(\chi)$,
given by
$(\omega,\eta)=\int_{X_{\iota}({\mathbb{C}})}i^{q^{2}+2t}\omega\wedge\overline{\eta}\cdot[G_{L|k(X)}]$
for any $\omega,\eta\in H^{s,t}_{{\rm
prim}}(X_{\iota}({\mathbb{C}}))=H^{t}_{{\rm
prim}}(X,\Omega^{s}_{X|k})\otimes_{k,\iota}{\mathbb{C}}\subset{\mathbb{C}}\otimes_{k,\iota}H^{s,t}_{L|k}$.
Here $H^{s,t}_{{\rm prim}}(X_{\iota}({\mathbb{C}}))$ denotes the subspace
orthogonal to the sum of all Gysin maps $H^{s-1,t-1}(D)\longrightarrow
H^{s,t}(X_{\iota}({\mathbb{C}}))$ for all desingularizations $D$ of all
divisors on $X_{\iota}({\mathbb{C}})$, as in the definition of
$\Omega^{q}_{L|k,\text{{\rm reg}}}$, $X$ runs over all smooth proper
$k$-varieties with the function field embedded into $L|k$. ∎
### 3.3. Structure of closed 1-forms
Let ${\rm Div}^{\circ}_{{\mathbb{Q}}}:Y\mapsto{\rm Div}_{{\rm
alg}}(\overline{Y})_{{\mathbb{Q}}}$ be the presheaf of algebraically trivial
divisors. It is a sheaf.
###### Lemma 3.6.
The residue homomorphism ${\rm Res}_{Y}:H^{1}_{{\rm dR}/k}(k(Y))\to
k\otimes{\rm Div}(\overline{Y})$, $\omega\mapsto({\rm
res}_{x}\omega)_{x\in\overline{Y}^{1}}$, defines a morphism of sheaves ${\rm
Res}:\underline{H}^{1}_{{\rm dR}/k}\to k\otimes{\rm
Div}^{\circ}_{{\mathbb{Q}}}$. The short sequence $0\to\underline{H}^{1}_{{\rm
dR}/k,c}\to\underline{H}^{1}_{{\rm dR}/k}\stackrel{{\scriptstyle{\rm
Res}}}{{\longrightarrow}}{\rm Div}^{\circ}_{{\mathbb{Q}}}\otimes k\to 0$ is
exact, even as a sequence of presheaves.
Proof. As ${\rm Res}$ commutes with the restriction to any sufficiently
general curve $C$, ${\rm Res}_{X}(\omega)\cdot C={\rm Res}_{C}(\omega|_{C})\in
CH_{0}(X)$, $\deg({\rm Res}_{X}(\omega)\cdot C)=0$ by Cauchy theorem, the
pairing ${\rm NS}(X)_{{\mathbb{Q}}}\otimes
CH_{1}(X)_{{\mathbb{Q}}}/hom\longrightarrow{\mathbb{Q}}$ is non-degenerate (by
Lefschetz hyperplane section theorem), the class of ${\rm Res}_{X}(\omega)$ in
${\rm NS}(X)_{{\mathbb{Q}}}$ is zero. Thus, ${\rm Res}_{X}$ factors through
the algebraically trivial divisors on $X$.
Clearly, the kernel of ${\rm Res}$ coincides with $\underline{H}^{1}_{{\rm
dR}/k,c}$, cf. [6].131313If the residues of $\omega\in H^{1}_{{\rm
dR}/k}(k(X))$ are zero then integration along a loop depends only on its
homology class in $H_{1}(X,{\mathbb{Q}})$. There is an element $\eta$ of
$H_{{\rm dR}/k}^{1}(X)$ with the same periods as $\omega$, so integration of
$\omega-\eta$ along a path joining a fixed (rational) point with the variable
one is independent of a chosen path, and defines a meromorphic (i.e. rational)
function. Then it remains to show that any algebraically trivial divisor on
$X$ is the residue of a closed 1-form. Any algebraically trivial divisor can
be written as $D_{1}-D_{2}$ for a pair $D_{1},D_{2}$ of algebraically
equivalent effective divisors on $X$. There is a smooth projective curve $C$,
and an effective divisor $D$ on $X\times C$, such that ${\rm pr}_{X}:D\to X$
is generically finite and $D_{P}-D_{Q}=D_{1}-D_{2}$ for some points $P,Q\in
C$. By Riemann–Roch theorem for curves, there exists a 1-form
$\omega_{P,Q}\in\Omega^{1}_{C}(P+Q)$ such that ${\rm
Res}_{C}(\omega_{P,Q})=P-Q$: there is a non-holomorphic 1-form with simple
poles in the set $\\{P,Q\\}$, since
$\dim_{k}\Gamma(C,\Omega^{1}_{C}(P+Q))=\dim_{k}\Gamma(C,\Omega^{1}_{C})+1$;
there are no 1-forms with precisely one simple pole, since
$\Gamma(C,\Omega^{1}_{C}(P))=\Gamma(C,\Omega^{1}_{C}(Q))=\Gamma(C,\Omega^{1}_{C})$.
Then ${\rm Res}_{X}({\rm pr}_{X\ast}(({\rm
pr}_{C}^{\ast}\omega_{P,Q})|_{D}))=D_{1}-D_{2}$. ∎
###### Proposition 3.7.
* •
The maximal semisimple subsheaf of
$\mathop{\underline{\Omega}}^{1}_{|k,\text{{\rm closed}}}$ is canonically
isomorphic to the direct sum
$\bigoplus_{A}\Gamma(A,\Omega^{1}_{A|k})^{A(k)}\otimes_{\mathop{\mathrm{End}}\nolimits(A)}{\mathcal{H}}^{A}_{1}={\mathcal{H}}^{{\mathbb{G}}_{a}}_{1}\oplus
k\otimes{\mathcal{H}}^{{\mathbb{G}}_{m}}_{1}\oplus\Omega^{1}_{|k,\text{{\rm
reg}}}$, where $A$ runs over the set of isogeny classes of simple commutative
algebraic $k$-groups;
$\Gamma(A,\Omega^{1}_{A|k})^{A(k)}=\mathop{\mathrm{Hom}}\nolimits_{k}({\rm
Lie}(A),k)$ denotes the space of translation invariant 1-forms on $A$. The
projection $\mathop{\underline{\Omega}}^{1}_{|k,\text{{\rm
closed}}}\to\mathop{\underline{\Omega}}^{1}_{|k,\text{{\rm
closed}}}/\Omega^{1}_{|k,\text{{\rm reg}}}$ is split (but not canonically).
* •
The maximal semisimple subsheaf of $\underline{H}^{1}_{{\rm dR}/k}$ is
canonically isomorphic to $\bigoplus_{A}H^{1}_{{\rm
dR}/k}(A)\otimes_{\mathop{\mathrm{End}}\nolimits(A)}{\mathcal{H}}^{A}_{1}=k\otimes{\mathcal{H}}^{{\mathbb{G}}_{m}}_{1}\oplus\underline{H}^{1}_{{\rm
dR}/k,c}$, where $A$ runs over the set of isogeny classes of simple
commutative algebraic $k$-groups (with the zero summand corresponding to
${\mathbb{G}}_{a}$). The projection $\underline{H}^{1}_{{\rm
dR}/k}\to\underline{H}^{1}_{{\rm dR}/k}/\underline{H}^{1}_{{\rm dR}/k,c}$ is
split (but not canonically).
* •
The sheaf $V^{1}:Y\mapsto H^{1}_{{\rm
dR}/k}(k(Y))/k\otimes(k(Y)^{\times}/k^{\times})$ from Proposition 3.3 is
canonically isomorphic to
$\bigoplus_{A}V^{1}(A)\otimes_{\mathop{\mathrm{End}}\nolimits(A)}{\mathcal{H}}^{A}_{1}$,
where $A$ runs over the set of isogeny classes of simple abelian
$k$-varieties.
For any integer $q\geq 1$, the representation $\Omega^{1}_{L|k,\text{{\rm
closed}}}$ of the group $G_{L|k}$ admits similar description (cf. §3.2).
Proof. In notation of Lemma 3.6, the sheaf ${\rm Div}^{\circ}_{{\mathbb{Q}}}$
admits a natural surjective morphism onto the Picard sheaf ${\rm
Pic}^{\circ}_{{\mathbb{Q}}}={\rm
coker}[{\mathcal{H}}^{{\mathbb{G}}_{m}}_{1}\stackrel{{\scriptstyle{\rm
div}}}{{\longrightarrow}}{\rm Div}^{\circ}_{{\mathbb{Q}}}]:Y\mapsto{\rm
Pic}^{0}(\overline{Y})_{{\mathbb{Q}}}$ with the irreducible kernel
${\mathcal{H}}^{{\mathbb{G}}_{m}}_{1}$. The Picard sheaf ${\rm
Pic}^{\circ}_{{\mathbb{Q}}}$ is semisimple and it is described in Lemma 1.1.
According to Lemma 1.1, for any simple abelian variety $A$ over $k$, any non-
zero element $\xi$ of ${\rm Pic}^{0}(A)(k)_{{\mathbb{Q}}}$ provides an
embedding of ${\mathcal{H}}^{A}_{1}$ into ${\rm Pic}^{\circ}_{{\mathbb{Q}}}$.
Let us show that the natural extension
$0\to{\mathcal{H}}^{{\mathbb{G}}_{m}}_{1}\to{\rm
Div}^{\circ}_{{\mathbb{Q}}}\to{\rm Pic}^{\circ}_{{\mathbb{Q}}}\to 0$ does not
split, even after restricting to ${\mathcal{H}}^{A}_{1}$ via $\xi$.
All elements of ${\rm Pic}^{\circ}_{{\mathbb{Q}}}(A):={\rm
Pic}^{0}(A)_{{\mathbb{Q}}}$ are fixed by translations of $A$ by torsion
elements in $A(k)$. However, as the torsion subgroup in $A(k)$ is Zariski
dense, it cannot fix a non-zero element of ${\rm
Div}^{\circ}_{{\mathbb{Q}}}(A):={\rm Div}_{\text{alg}}(A)_{{\mathbb{Q}}}$.
This implies that ${\mathcal{H}}^{{\mathbb{G}}_{m}}_{1}$ is the maximal
semisimple subsheaf of ${\rm Div}^{\circ}_{{\mathbb{Q}}}$, which proves, by
Lemma 3.6, the second assertion. It follows also that the simple subquotients
of $V^{1}$ are isomorphic to ${\mathcal{H}}^{A}_{1}$ for simple abelian
$k$-varieties $A$. There are no extensions between ${\mathcal{H}}^{A}_{1}$ and
${\mathcal{H}}^{B}_{1}$ for abelian $k$-varieties $A$ and $B$, since
${\mathcal{I}}_{G}$ is a Serre subcategory of
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}$ by Proposition 2.2 and $Y\mapsto
A(k(Y))_{{\mathbb{Q}}}$ is a projective object of ${\mathcal{I}}_{G}$ by
property 5 of §1.2 and Proposition 2.10. This means that $V^{1}$ is
semisimple, which proves the third assertion.
Once we know the simple subquotients of
$\mathop{\underline{\Omega}}^{1}_{|k,\text{{\rm closed}}}$, the first
assertion follows from Proposition 3.1 and Lemma 1.1. To see that the
projections $\mathop{\underline{\Omega}}^{1}_{|k,\text{{\rm
closed}}}\to\mathop{\underline{\Omega}}^{1}_{|k,\text{{\rm
closed}}}/\Omega^{1}_{|k,\text{{\rm reg}}}$ and $\underline{H}^{1}_{{\rm
dR}/k}\to\underline{H}^{1}_{{\rm dR}/k}/\underline{H}^{1}_{{\rm dR}/k,c}$ are
split, it is enough to notice that $V^{1}$ is semisimple, and therefore, the
compositions $\Omega^{1}_{|k,\text{{\rm
reg}}}\hookrightarrow\mathop{\underline{\Omega}}^{1}_{|k,\text{{\rm
closed}}}\to V^{1}$ and $\underline{H}^{1}_{{\rm
dR}/k,c}\hookrightarrow\underline{H}^{1}_{{\rm dR}/k}\to V^{1}$ admit
splittings. ∎
Given a subfield $L$ in $F$, define the filtration $N_{\bullet}^{(L)}$ on the
$G_{F|L}$-modules $W$ by $N_{j}^{(L)}W=\sum_{F^{\prime}}W^{G_{F|F^{\prime}}}$,
where $F^{\prime}$ runs over the subfields in $F|L$ of transcendence degree
$j$. (Clearly, $N_{0}^{(L)}W=W^{G_{F|\overline{L}}}$ and
$N_{\bullet}^{(L)}=N_{\bullet}^{(\overline{L})}\subseteq
N_{\bullet}^{(L^{\prime})}$ if $L\subset L^{\prime}\subset F$.) In particular,
define the level filtration on the $G$-modules by
$N_{\bullet}:=N_{\bullet}^{(k)}$.
It is conjectured in [7, Conj.6.9] that the graded pieces of $N_{\bullet}$ on
the objects of ${\mathcal{I}}_{G}$ are semisimple.
If $U$ is an open subgroup of $G$, contained between $G_{F|\overline{L}}$ and
the normalizer of $G_{F|\overline{L}}$ in $G$ then $N_{\bullet}^{(L)}$ is a
filtration by $U$-submodules. The forgetful functor
$\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}\to\mathop{\mathcal{S}\mathrm{m}}\nolimits_{U}$
does not preserve the irreducibility (or the semisimplicity). E.g., for any
commutative simple algebraic $k$-group $A$, the restriction to $U$ of the
irreducible $G$-module $A(F)/A(k)$ is a non-split extension of the irreducible
$U$-module $A(F)/A(\overline{L})$ by the $U$\- (in fact, $(U/U\cap
G_{F|\overline{L}})$\- ) module $A(\overline{L})/A(k)$.
Questions. Let $W\in\mathop{\mathcal{S}\mathrm{m}}\nolimits_{G}$ be
irreducible and $W=N_{q}W$. Is it true that the representation
$W/N_{q-1}^{(L)}W$ of $U$ is irreducible (or zero)?
Clearly, $N_{j}\Omega^{i}_{F|k}=\Omega^{i}_{F|k}$ for any $j>i$,
$N_{j}\Omega^{i}_{F|k}=0$ for any $j<i$ and
$N_{j}\Omega^{j}_{F|k}\subseteq\Omega^{j}_{F|k,\text{closed}}$.
###### Conjecture 3.8.
$N_{j}\Omega^{j}_{F|k}=N_{j}\Omega^{j}_{F|k,\text{{\rm
closed}}}=\Omega^{j}_{F|k,\text{{\rm closed}}}$.
A ‘‘weak’’ version, $N_{j}\Omega^{j}_{F|k,{\rm reg}}=\Omega^{j}_{F|k,{\rm
reg}}$, follows from Grothendieck’s diagonal decomposition conjecture.
The Conjecture obviously holds true for $j=0$. The case $j=1$ follows from (i)
Proposition 3.7, (ii) the fact ([4, Cor.3.8]) that $F/k$ and
$F^{\times}/k^{\times}$ are acyclic, so $\Omega^{1}_{L|k,\text{closed}}\to\to
H^{0}(G_{F|L},H^{1}_{\text{dR}/k}(F)/kd\log(F^{\times}/k^{\times}))$, (iii)
$N_{1}(A(F)/A(k))=A(F)/A(k)$ for any commutative $k$-group $A$. ∎
Acknowledgements. The project originates from my stay at the Max-Planck-
Institut in Bonn, it has reached its present state at the Institute for
Advanced Study in Princeton, and it has reached the final form at the I.H.E.S.
in Bures-sur-Yvette. I am grateful to these institutions for their hospitality
and exceptional working conditions.
## References
* [1] D.Abramovich, K.Karu, K.Matsuki, J.Włodarczyk, Torification and factorization of birational maps, J. A.M.S. 15 (2002), 531–572.
* [2] I.N.Bernstein, A.V.Zelevinsky, Representations of the group $GL(n,F),$ where $F$ is a local non-Archimedean field. Uspehi Mat. Nauk 31 (1976), no. 3(189), 5–70.
* [3] A.Grothendieck, On the de Rham cohomology of algebraic varieties, I.H.É.S. Publ. Math. 29 (1966), 95–103.
* [4] U.Jannsen, M.Rovinsky, Smooth representations and sheaves, Moscow Math. J., 10, no.1 (issue dedicated to Pierre Deligne), 189–214, math.AG/0707.3914.
* [5] S.MacLane, Categories for the working mathematician. 2nd Edition, Springer, 1998.
* [6] M.Rosenlicht, Simple differentials of second kind on Hodge manifolds. Amer. J. Math. 75, (1953), 621–626.
* [7] M.Rovinsky, Motives and admissible representations of automorphism groups of fields. Math. Zeit., 249 (2005), no. 1, 163–221, math.RT/0101170.
* [8] M.Rovinsky, Semilinear representations of PGL, Selecta Math., 11 (2005), 491–522, math.RT/0306333.
* [9] M.Rovinsky, Automorphism groups of fields, and their representations, Uspekhi Matem. Nauk, 62 (378) (2007), no. 6, 87–156, translated in Russian Math. Surveys, 62 : 6 (2007), 1121–1186.
* [10] M.Rovinsky, On maximal proper subgroups of field automorphism groups. Selecta Math., 15 (2009), 343–376, math.RT/0601028.
|
arxiv-papers
| 2010-06-28T13:38:58 |
2024-09-04T02:49:11.275245
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Rovinsky",
"submitter": "Marat Rovinsky",
"url": "https://arxiv.org/abs/1006.5348"
}
|
1006.5443
|
# Hamiltonian two-body system in special relativity
Philippe Droz-Vincent
LUTH
Meudon 111Observatoire de Paris, CNRS, Université Paris Diderot, 5 place Jules
Janssen, 92195 Meudon, France
###### Abstract
We consider an isolated system made of two pointlike bodies interacting at a
distance in the nonradiative approximation. Our framework is the covariant and
a priori Hamiltonian formalism of ”predictive relativistic mechanics”, founded
on the equal-time condition. The center of mass is rather a center of energy.
Individual energies are separately conserved and the meaning of their
positivity is discussed in terms of world-lines. Several results derived
decades ago under restrictive assumptions are extended to the general case.
Relative motion has a structure similar to that of a nonrelativistic one-body
motion in a stationnary external potential, but its evolution parameter is
generally not a linear function of the center-of-mass time, unless the
relative motion is circular (in this latter case the motion is periodic in the
center-of-mass time). Finally the case of an extreme mass ratio is
investigated. When this ratio tends to zero the heavy body coincides with the
center of mass provided that a certain first integral, related to the binding
energy, is not too large.
## 1 Introduction
Classical relativistic dynamics of pointlike bodies has a long story; without
claiming to be exhaustive let us mention the Wheeler-Feynman (WF)
electrodynamics [1] based upon the Fokker action [2], the three forms of
dynamics (front form, point form, instant form) advocated by Dirac [3], and
after the discovery of a famous No-Interaction theorem [4], various efforts
made in order to circumvent it; for instance Predictive Mechanics [5][6], the
Singular Lagrangian method [8] and Constraint Dynamics [9]. In the last decade
there were a few papers along the lines of WF [10] and also the work carried
out by Lusanna et al. [11] in order to give a covariant status to the instant
form.
Beside several $n$-body generalizations, most progresses have been devoted on
the two-body problem, which is our present subject.
A first point was the possibility of actually having (unlike WF
electrodynamics) second-order differential equations describing the motion of
a finite number of degrees of freedom interacting at a distance. According to
this view, the field that carries interaction is supposed to be eliminated,
the space of initial data has a finite number of dimensions and a point in
this space uniquely determines the subsequent motion [5] [6].
A second point was about a Hamiltonian formalism. We have in mind conservative
mechanics: radiative corrections are neglected and we focus on isolated
systems, characterized by a finite number of degrees of freedom and by
Poincaré invariance. Under these conditions the no-interaction theorem [4]
excludes the possibility of demanding that the physical positions be canonical
throughout phase space (as was always done in classical mechanics). Relaxing
this requirement implies an arbitrariness which can be removed by imposing a
relationship between physical positions and canonical variables on some
submanifold $\Sigma$.
In the a priori Hamiltonian approach of predictive mechanics [7], manifest
covariance is realized with help of considering degrees of freedom that are
geometrically natural but redundant (if compared to the classical situation).
Positions and momenta are four-component objects, phase space has sixteen
independent dimensions and we employ a multitime formalism (in a different
spirit Todorov proposed to focus on the physical degrees of freedom only, in
another formulation of dynamics which is covariant as well but includes first-
class constraints [9]).
After discussing other possibilities, we put forward a natural prescription:
physical and canonical positions must coincide when both physical positions
are simultaneous with respect to the rest frame of the system [7]. In this
approach a submanifold $\Sigma$ is obtained by selecting the configurations
where the physical coordinate times $x_{1}^{0},x_{2}^{0}$ are equal in the
center-of-mass frame. An advantage of this equal-time prescription is to
permit the contact with the constraint formalism, as shown in detail by L.
Lusanna [12].
The two-body problem immediately suggests these important issues: center of
mass and relative motion. In this article we intend to study their properties
in the context of relativistic dynamics, aiming at a method in order to
simplify the determination of the world lines.
A few exploratory results that we derived in the past [13] [14] have remained
fragmentary, as most of them have been obtained with help of restrictive
assumptions concerning the shape of the interaction potential. The present
work is free of such limitation, aiming at the possibility of dealing with
realistic interactions; such interactions cannot offer the simplicity of the
academic models we considered long time ago: see for instance the unavoidable
$P^{2}$\- dependence of an electromagnetic two-body potential proposed by
Jallouli and Sazdjian [20], in the framework of relativistic quantum
mechanics.
Before we focus on its equal-time version, we shall sketch the main lines of
the a priori Hamiltonian approach in general.
In the next section we present the isolated two-body systems and introduce the
center of mass.
In the rest of the paper we consider unipotential models (for these systems
the individual energies are separately conserved).
In Section 3 we study the evolution of the canonical variables, irrespective
of their relationship to physical positions. Later, in Section 4, we focus on
the equal-time prescription and discuss a strategy for the determination of
world-lines; the importance of circular orbits is emphasized. In 5 we consider
the case of an extreme mass ratio; a toy model is presented in Section 6, and
Section 7 is devoted to concluding remarks.
The velocity of light is taken as unity, except in the Appendix. When no
confusion is possible, tensor indices are omitted and the contraction dot is
employed also for tensors, for instance $\ J\cdot P\ $ stands for the vector
$J^{\alpha\mu}\ P_{\mu}$.
### 1.1 The a priori Hamiltonian formalism
The canonical coordinates are $q_{a}^{\alpha},p_{b\beta}$ in the sixteen-
dimensional phase space; $q_{1},q_{2}$ are points in Minkowski space $\ {\cal
M}\quad$ and $\ p_{1},p_{2}\ $ are four-vectors.
The symplectic form $dq_{1}^{\alpha}\wedge dp_{1\alpha}+dq_{2}^{\alpha}\wedge
dp_{2\alpha}$ corresponds to the Poisson brackets
$\\{q_{a}^{\alpha},p_{b\beta}\\}=\delta_{ab}\ \delta^{\alpha}_{\beta}$ (1)
Note that the particle labels $a,b,c$ are not summed over when repeated.
The Hamiltonian equations of motion
${\partial q_{a}\over\partial\tau_{b}}=\\{q_{a},H_{b}\\}\qquad\quad{\partial
p_{a}\over\partial\tau_{b}}=\\{p_{a},H_{b}\\}$ (2)
involve two Hamiltonian generators $H_{1},H_{2}$ submitted to the predictivity
condition
$\\{H_{1},H_{2}\\}=0$
The two evolution parameters $\tau_{1},\tau_{2}$ are suitable generalizations
of the proper times (normalized to the masses). Our notation is choosen such
that the generators can be identified as the half-squared masses; this allows
to consider the masses as first integrals, redeeming the fact that phase space
has redundant degrees of freedom.
For instance, in the trivial case of two free particles one is left with
$\displaystyle 2H_{a}=p_{a}^{2}$, the physical positions reduce to
$x_{a}=q_{a}$ and the evolution parameters are just $\tau_{a}=s_{a}/m_{a}$,
where $s_{1},s_{2}$ are the arc lenghts.
For interacting particles the Hamiltonians involve additional terms $\
V_{1},V_{2}\ $referred to as ”potentials”. In general $q_{1},q_{2}$ differ
from the physical positions, in fact $x_{1},x_{2}$ are determined by the
partial differential equations
$\\{x_{a},H_{b}\\}=0\qquad\qquad{\rm for}\qquad a\not=b$ (3)
and some reasonable initial data. Solving these equations provides a
correspondance between physical and canonical coordinates, say
$x,v\longleftrightarrow q,p$
which, inserted into the solutions of system (2), ensures that $\ x_{1}\ $ is
a function of $\tau_{1}$ only and $\ x_{2}\ $ is a function of $\tau_{2}$
only, in other words (3) ensures the existance of one-dimensional world-lines
[7]. This point is easily checked by introducing on phase space the
Hamiltonian vector fields $X_{1},X_{2}$ defined by
$X_{a}\ f={\partial f\over\partial\tau_{a}}=\\{f,H_{a}\\}$ (4)
for all phase-space function $f$. The predictivity condition corresponds to
the vanishing of the Lie bracket $[X_{1},X_{2}]$. Although the analytic shape
of the Hamiltonians yields some information about first integrals, symmetries,
etc, the physical meaning is fixed only once a solution to (3) has been
specified. At this stage phase space is identified with the bundle product
$T({\cal M})\times T({\cal M})$.
Notation. $\displaystyle\ (H_{a})_{\rm free}={1\over 2}p_{a}^{2}\ $, they
generate $(X_{a})_{\rm free}$.
When fixing numerical values ${1\over 2}m_{1}^{2}$ and ${1\over 2}m_{2}^{2}$
to the Hamiltonian generators, it is convenient to set
$\mu={1\over 2}(m_{1}^{2}+m_{2}^{2}),\qquad\quad\nu={1\over
2}(m_{1}^{2}-m_{2}^{2})$ (5)
which amounts to
$m_{1}m_{2}=\sqrt{\mu^{2}-\nu^{2}},\qquad\qquad(m_{1}+m_{2})^{2}=2\mu+2\sqrt{\mu^{2}-\nu^{2}}$
(6)
## 2 Isolated two-body systems
The Lie algebra of the Poincaré group is spanned by $P_{\alpha},J_{\mu\nu}$.
We separate external from internal variables by setting
$P=p_{1}+p_{2}\qquad\quad Q={1\over 2}(q_{1}+q_{2})$ (7) $y={1\over
2}(p_{1}-p_{2})\qquad\quad z=q_{1}-q_{2}$ (8)
Remark in general $\ Q\ $ is not the center of mass 222even for free particles
it is center of mass only when the masses are equal..
Other definitions of $Q$ and $y$ (as conjugate to P and z respectively) were
possible, but the advantage of ours is that it does not require to a priori
fixthe numerical values of the constituent masses.
The standard Poisson brackets that do not vanish can be re-arranged as follows
$\\{Q^{\alpha},P_{\beta}\\}=\\{z^{\alpha},y_{\beta}\\}=\delta^{\alpha}_{\beta}$
(9)
and we can write
$J=Q\wedge P+z\wedge y=q_{1}\wedge p_{1}+q_{2}\wedge p_{2}$
Naturally $H_{1},H_{2}$ are supposed to be Poincaré invariant (vanishing
Poisson brackets with $P$ and $J$) and the initial conditions for solving (3)
must be invariant as well.
From (9) we can derive several useful formulas; let us list some of them. Of
course $\\{Q^{\mu},P^{2}\\}=2P^{\mu}$ and
$\\{Q^{\mu},P_{\alpha}P_{\beta}\\}=\delta^{\mu}_{\alpha}P_{\beta}+P_{\alpha}\delta^{\mu}_{\beta}$
so if we define the projector orthogonal to $P$
$\Pi=\eta-{P\otimes P\over P^{2}}$ (10)
we get
$\\{Q^{\mu},\widetilde{z}^{2}\\}=-2\widetilde{z}^{\mu}{P\cdot z\over P^{2}}$
(11)
with this notation that the tilde means an application of $\Pi,\ $ say
${\widetilde{\xi}}^{\alpha}=\Pi^{\alpha\beta}\
\xi_{\beta},\qquad\quad\forall\xi$
the r.h.s. of (11) is orthogonal to $P$; it follows that
$\\{Q\cdot P,\widetilde{z}^{2}\\}=0$ (12)
On the other hand we compute
$\\{Q\cdot P,\ {P\otimes P\over P^{2}}\\}=0$ (13)
in other words $Q\cdot P$ has a vanishing Poisson bracket with the projector
$\displaystyle\Pi$. Moreover we easily check that
$\\{Q\cdot P,{(y\cdot P)^{2}\over P^{2}}\\}=0$ (14)
It is obvious that $\\{Q\cdot P,y^{2}\\}$ vanishes and we can apply the above
formula to the identity
$\widetilde{y}^{2}=y^{2}-{(y\cdot P)^{2}\over P^{2}}$
which yields
$\\{Q\cdot P,\widetilde{y}^{2}\\}=0$ (15)
For the relative variables note that the spatial piece of the one has a
vanishing bracket with the time piece of the other (its conjugate):
$\\{\widetilde{z},\ y\cdot P\\}=\\{\widetilde{y},\ z\cdot P\\}=0$ (16)
Similarly it stems from (13) that
$\\{Q\cdot P,\widetilde{z}^{\alpha}\\}=\\{Q\cdot
P,\widetilde{y}^{\alpha}\\}=0$ (17)
Of course we already know ten first integrals, namely $P_{\alpha},J_{\mu\nu}$.
We obviously have that
${\widetilde{P\cdot J}\over P^{2}}={z\cdot P\over P^{2}}\
\widetilde{y}-\widetilde{Q}-{y\cdot P\over P^{2}}\ \widetilde{z}$ (18)
### 2.1 Center of Mass
The possibility to define a center of mass, using linear and angular momenta
as basic ingredients, has been known long time ago [16].
In a previous work [15] we proposed this canonical definition for the
components of the center of mass
$\Xi={J\cdot P\over P^{2}}+({P\cdot Q\over P^{2}})\ P$ (19)
or equivalently
$\Xi=Q+({y\cdot P\over P^{2}})z-({z\cdot P\over P^{2}})y$ (20)
This can be transformed again, if we notice that
$\displaystyle{P^{2}/2}\pm y\cdot P=P\cdot p_{1},\ ({\rm resp.\ }P\cdot
p_{2})$, we get
$\Xi={P\cdot p_{1}\over P^{2}}q_{1}+{P\cdot p_{2}\over P^{2}}q_{2}-{P\cdot
z\over P^{2}}y$ (21)
Formula (20) entails
$\Xi\cdot P=Q\cdot P$ (22)
In (19) the only quantity which is not a constant of the motion is $\ ({P\cdot
Q/P^{2}})\ $, it has the same dimension as $\tau_{1},\tau_{2}$. In contrast
setting 333 definition of $T$ has a different dimension in [15] , but the same
as here in [14] ) .
$T={P\cdot Q\over|P|}$
we give to $T$ the dimension of lenght (and time since $c=1$). One easily
computes
$\\{{(J\cdot P)^{\alpha}\over P^{2}},\ P_{\beta}\\}=\Pi^{\alpha}_{\beta}$
whence we derive the relations
$\\{\Xi^{\alpha},\
P_{\beta}\\}=\delta^{\alpha}_{\beta}\qquad\quad\\{\Xi^{\alpha},\ {P^{2}\over
2}\\}=P^{\alpha}$ (23)
Owing to the constancy of $J_{{\mu\nu}}$ and $P_{\alpha}$, we can fix these
quantities, say in particular
$P_{\alpha}=k_{\alpha},\qquad\quad{\rm timelike\ vector}$ ${\rm
define}\qquad\qquad\ k^{\alpha}k_{\alpha}=M^{2}$
Then we see that formula (19) defines the coordinates of a point which moves
on a straight line when both $\tau_{1},\tau_{2}$ run freely and independently
from $-\infty$ to $+\infty$, and the direction of this line is given by
$k^{\alpha}$.
Equation (19) becomes $\displaystyle\ \Xi^{\alpha}={\rm Const.}+({P\cdot
Q\over M})\ {k^{\alpha}\over M}$.
We can write $\displaystyle{d\Xi^{\alpha}\over dT}={k^{\alpha}\over M}$ and
consider respectively $T$ and $T/M$ as the proper time and the evolution
parameter of the center of mass.
Similarly equations (23 ) supplemented by the trivial observation that
$\displaystyle\\{P,{1\over 2}P^{2}\\}$ vanishes, can be viewed as canonical
equations of motion for $\Xi$, generated by the one-body Hamiltonian $\
{1\over 2}P^{2}\ $.
Note however that in (19) the components of $\Xi$ do not commute among
themselves; a similar situation was already encontered by Pryce [16].
From now on we shall focus on Unipotential Models characterized by the same
interaction term for both particles, say $V_{1}=V_{2}=V$. Such models permit
to write down explicit forms of the Hamiltonians. Moreover it seems that they
are still general enough for dealing with most realistic interactions.
## 3 Evolution of the canonical variables
For the moment we postpone the physical interpretation; we are interested in
the evolution of the canonical variables in terms of the parameters
$\tau_{1},\tau_{2}$, more generally we are concerned with statements that hold
true regardless to the prescription used for solving the position equations
(3). In this section, the only thing we assume about this prescription is
Poincaré invariance.
At this stage the separation of external/internal variables is just formal and
convenient for easy calculations. Because of Poincaré invariance and
predictivity, $V$ can be only a function of the five independent scalars [13]
$P^{2},{\widetilde{z}}^{2},{\widetilde{y}}^{2},\widetilde{z}\cdot\widetilde{y},y\cdot
P$ (24)
But in view of (14) it is more convenient for calculations to re-arrange them
as
$P^{2},{\widetilde{z}}^{2},{\widetilde{y}}^{2},\widetilde{z}\cdot\widetilde{y},{(y\cdot
P)^{2}\over P^{2}}$ (25)
Unless otherwise specified we consider $V$ as a function of these five
arguments. Soon it was observed [13] that
$(X_{1}-X_{2})\widetilde{z}=(X_{1}-X_{2})\widetilde{y}=0$
implying this
###### Proposition 1
In the motion, $\widetilde{z}$ and $\widetilde{y}$ will depend only on
$\tau_{1}+\tau_{2}$.
In contrast, $z\cdot P$ and $Q\cdot P$ will depend on both evolution
parameters.
For any unipotential model, the individual energies $\displaystyle\ {P\cdot
p_{1}\over|P|},\qquad{P\cdot p_{2}\over|P|}\ $ are separately conserved.
Moreover the translation invariance of $V$ implies conservation of $P^{2}$,
hence this obvious first integral
$N=H_{1}+H_{2}-{(H_{1}-H_{2})^{2}\over P^{2}}-{P^{2}\over 4}$ (26)
By elementary manipulations we find
$N=\widetilde{y}^{2}+2V$ (27)
This important function defined on phase space is intimately related with the
properties of relative motion; this can be intuitively seen as follows:
fixing numerical values to $H_{1},H_{2},P^{\alpha}$ (in particular $P\cdot
P=k\cdot k=M^{2}$) results in a numerical value for $N$, let it be
$<N>=-\Lambda$
The quantum mechanical analog of this quantity appeared, denoted as $b^{2}$,
in the work of Todorov [19].
Employing (for instance) the reduced mass of Galilean mechanics, say
$\ \displaystyle m_{0}={m_{1}m_{2}\over m_{1}+m_{2}}$
we can start from (27) and check that $\ \Lambda/2m_{0}\ $ is the leading term
in the post-Galilean development of the quantity $M-(m_{1}+m_{2})$ which is
usually considered as binding energy for the bound states of relativistic
quantum mechanic (see Appendix I).
In addition we shall see later on that ${1\over 2}N$ generates the evolution
of the spatial relative canonical variables according to a one-parameter
Hamiltonian scheme reminiscent of the nonrelativistic one-body mechanics, see
equations (44) (45) below.
From (26) and with the notation (5) we can write
$\Lambda={M^{2}\over 4}+{\nu^{2}\over M^{2}}-\mu$ (28)
This important relation between $\Lambda$ and $M$ can be solved for $M^{2}$.
We first write
$M^{4}-4(\mu+\Lambda)M^{2}+4\nu^{2}=0$ (29)
From (28) it is already clear that $\Lambda+\mu>0$. But $M^{2}$ must be real
and positive, so the possible values of $\Lambda$ are further restricted by
the condition
$|\mu+\Lambda|>|\nu|$ (30)
which ensures that $M^{2}$ is real; under this condition we can write
$M^{2}=2(\mu+\Lambda)\pm 2\ \sqrt{(\mu+\Lambda)^{2}-\nu^{2}}$ (31)
Moreover we must have at least one root of (29) positive. Since their product
is non-negative these roots cannot have opposite signs, thus it will be
sufficient to ensure that their sum is positive. Hence the condition
$\mu+\Lambda>0$ (32)
which ensures $M^{2}>0$. We can encompass both (30) and (32) by writting
$\mu+\Lambda>|\nu|$ (33)
Fortunately, the sign ambiguity in (31) can be removed, with help of the
individual positive-energy condition (remind that $P\cdot p_{a}/M$ are the
individual energies) that we assume henceforth
$P\cdot p_{1}>0,\qquad P\cdot p_{2}>0$
Indeed, the numerical values of $P\cdot p_{1}$ and $P\cdot p_{2}$ are given by
$\displaystyle{M^{2}\over 2}\pm\nu$. Requiring that both are strictly positive
amounts to the condition
$M^{2}>2|\nu|$ (34)
Let ${M^{\prime}}^{2},{M^{\prime\prime}}^{2}$ be the roots of equation (29).
If both roots were to satisfy this inequality, it would contradict the
equality ${M^{\prime}}^{2}{M^{\prime\prime}}^{2}=4\nu^{2}$ implied by the last
term in (29). It follows that
###### Proposition 2
Under the condition of positive individual energies, only one root of (29) is
admissible
( Remark For strictly equal masses, only the plus sign may be taken in (31).
Indeed this case means $\nu=0$, one root of equation (29) is obviously zero
which must be rejected, and the other root is $M^{2}=4(\mu+\Lambda)$ obtained
from (31) by choosing the plus sign).
We can assume that $m_{1}\leq m_{2}$ without loss of generality. Thus $\nu\leq
0$, and (33) becomes
$\mu+\nu+{\Lambda}>0$ (35)
or equivalently
$m_{1}^{2}+{\Lambda}>0$ (36)
We see that either $\Lambda>0$ or it satisfies $\displaystyle-m_{1}^{2}\
<\Lambda\ <0\ $. In other words,
###### Proposition 3
Under the assumption of positive individual energies, either $\Lambda>0$ or
$|\Lambda|<m_{1}^{2}$.
If we impose the individual energy conditions we additionally get
$\displaystyle{1\over 2}M^{2}>\nu$ (trivial) and also $\displaystyle{1\over
2}M^{2}>-\nu$, in other words
$M^{2}>m_{2}^{2}-m_{1}^{2}$ (37)
Thus, as soon as $m_{1}\not=m_{2}$, the collective mass of the whole system
(lenght of linear momentum) cannot be arbitrarily small.
Let us check that taking the plus sign in (31) always yields an admissible
root; eqn (35) implies that $\mu+{\Lambda}>-\nu$, thus looking at (31) we can
write
${M^{2}\over 2}>-\nu+\sqrt{(\mu+\Lambda)^{2}-\nu^{2}}>-\nu$
implying the individual energy condition (34). []
In view of the Proposition 2 above, taking the minus sign in (31)is excluded
and we can write
$M^{2}=2(\mu+\Lambda)+2\ \sqrt{(\mu+\Lambda)^{2}-\nu^{2}}$ (38)
Note that $M$ reduces to $m_{1}+m_{2}$ in the nonrelativistic limit, see
Appendix I.
We now investigate the evolution of the dynamical variables. Let us first
analyze the evolution of $\widetilde{z}$ and $\widetilde{y}$.
$(X_{1}+X_{2})\ \widetilde{z}^{\alpha}=(X_{1}+X_{2})_{\rm free}\
\widetilde{z}^{\alpha}+2\\{\widetilde{z}^{\alpha},V\\}$ (39)
$(X_{1}+X_{2})\ \widetilde{y}^{\alpha}=(X_{1}+X_{2})_{\rm free}\
\widetilde{y}^{\alpha}+2\\{\widetilde{y}^{\alpha},V\\}$ (40)
where $V$ is a function of the quantities listed in (25).
###### Proposition 4
The Poisson brackets $\\{\widetilde{z}^{\alpha},V\\}$ and
$\\{\widetilde{y}^{\alpha},V\\}$ are combinations of $\widetilde{z}^{\alpha},\
\widetilde{y}^{\alpha}$ with coefficients that are functions of the five
scalars listed in (25).
Proof.
Consider first $\widetilde{z}^{\alpha}$. Obviously
$\displaystyle\\{\widetilde{z}^{\alpha},P^{2}\\}=\\{\widetilde{z}^{\alpha},\widetilde{z}^{2}\\}=0$,
and we also have that $\displaystyle\\{\widetilde{z}^{\alpha},y\cdot P\\}=0$.
Then we compute
$\\{\widetilde{z}^{\alpha},\widetilde{y}^{2}\\}=2\widetilde{y}^{\alpha}$
$\\{\widetilde{z}^{\alpha},\widetilde{z}\cdot\widetilde{y}\\}=\widetilde{z}^{\alpha}$
hence
$\\{\widetilde{z}^{\alpha},V\\}=2{\partial V\over\partial\widetilde{y}^{2}}\
\widetilde{y}^{\alpha}+{\partial
V\over\partial(\widetilde{z}\cdot\widetilde{y})}\ \widetilde{z}^{\alpha}$ (41)
Since $V$ is a function of the scalars (25) only, its partial derivatives
involved in (41) obviously share this property.
Then consider $\widetilde{y}^{\alpha}$. Obviously
$\\{\widetilde{y}^{\alpha},P^{2}\\}=\\{\widetilde{y}^{\alpha},\widetilde{y}^{2}\\}=\\{\widetilde{y}^{\alpha},\
{(y\cdot P)^{2}\over P^{2}}\\}=0$
Then we compute
$\\{\widetilde{y}^{\alpha},\widetilde{z}^{2}\\}=-2\widetilde{z}^{\alpha}$
$\\{\widetilde{y}^{\alpha},\widetilde{z}\cdot\widetilde{y}\\}=-\widetilde{y}^{\alpha}$
hence
$\\{\widetilde{y}^{\alpha},V\\}=-2{\partial V\over\partial\widetilde{z}^{2}}\
\widetilde{z}^{\alpha}-{\partial
V\over\partial(\widetilde{z}\cdot\widetilde{y})}\ \widetilde{y}^{\alpha}$ (42)
The partial derivatives involved in this formula are functions of the scalars
(25). []
In view of Prop. 1 it is convenient to set
$\lambda=\tau_{1}+\tau_{2}$ (43)
the equations of motion for $\widetilde{z},\widetilde{y}$ are
${d\widetilde{z}\over d\lambda}=\\{\widetilde{z},{1\over
2}\widetilde{y}^{2}+V\\}$ (44) ${d\widetilde{y}\over
d\lambda}=\\{\widetilde{y},{1\over 2}\widetilde{y}^{2}+V\\}$ (45)
where the brackets can be computed as functions of
$\widetilde{z}^{\mu},\widetilde{y}^{\nu},$ and of the first integrals $P^{2},\
y\cdot P$. Once $P^{2}$ and $y\cdot P$ have been fixed, the evolution of the
spatial internal variables is given by a system of six first-order
differential equations, to solve for six unknown functions; this problem has
the structure of a nonrelativistic problem for one body in three dimensions.
The four-vectors $\widetilde{z}$ and $\widetilde{y}$ remain within the 2-plane
orthogonal to $k$ and to the (conserved) Pauli-Lubanski vector [13]. Some
solution
$\ \widetilde{z}=\zeta(\lambda,P^{2},y\cdot
P),\qquad\widetilde{y}=\eta(\lambda,P^{2},y\cdot P)\ $
of the system (44) (45) defines the evolution of the spatial relative
canonical variables $\widetilde{z},\widetilde{y}$. Interpretation in terms of
world lines, relative positions and relative orbit will be given in the next
section.
The collective evolution parameter $\lambda$ plays the role of the Newtonian
time in the analogous one-body system. But in general $\lambda$ is not the
time of any inertial observer. Therefore, in order to evaluate the schedule
of the relative motion, we should express $\lambda$ in function of $T$ and
insert the outcome into $\zeta$.
A complete knowledge of the motion also requires that we determine the
evolution of $z\cdot P$ and $Q\cdot P$ in terms of $\tau_{1},\tau_{2}$.
By sum and difference
$(X_{1}+X_{2})\ z\cdot P=2y\cdot P+2\\{z\cdot P,V\\}$ (46) $(X_{1}-X_{2})\
z\cdot P=P^{2}$ (47)
We know that $V$ depends only on the five scalars (25) and we observe that
$\\{z\cdot P,\widetilde{y}_{\alpha}\\}=0$, implying that $z\cdot P$ has a
vanishing Poisson bracket with all scalars (25) except $(y\cdot
P)^{2}/P^{2})$. We find $\\{z\cdot P,\ y\cdot P\\}=P^{2}$ hence finally
$\\{z\cdot P,V\\}$ only depends on the five scalars (25).
After integrating the system (44)(45) let us set
$\\{z\cdot P,V\\}=G(\lambda,M^{2},\nu)$ (48)
We obtain
$(X_{1}+X_{2})z\cdot P=2\nu+2G(\lambda,M^{2},\nu)$ $(X_{1}-X_{2})z\cdot
P=M^{2}$
Since
$(X_{1}+X_{2})={\partial\over\partial\tau_{1}}+{\partial\over\partial\tau_{2}}=2{\partial\over\partial\lambda}$
$(X_{1}-X_{2})={\partial\over\partial\tau_{1}}-{\partial\over\partial\tau_{2}}=2{\partial\over\partial(\tau_{1}-\tau_{2})}$
we finally have
$z\cdot P=\nu\lambda+\int Gd\lambda+{M^{2}\over 2}\ (\tau_{1}-\tau_{2})+{\rm
const.}$ (49)
Observing that
${M^{2}\over 2}\pm\nu=P\cdot p_{1}\ ({\rm resp.}P\cdot p_{2})$
we may write equivalently
$z\cdot P=(P\cdot p_{1})\tau_{1}-(P\cdot p_{2})\tau_{2}+\int Gd\lambda+{\rm
const.}$ (50)
which reduces to eq. (3.6) of [13] when $G$ vanishes.
Similarly in view of (14) and (17) we can simply write (with $\partial$
according to (25) )
$\\{Q\cdot P,V\\}={\partial V\over\partial P^{2}}\\{Q\cdot P,P^{2}\\}$
where $\displaystyle\\{Q\cdot P,P^{2}\\}=2P^{2}$ therefore
$\\{Q\cdot P,V\\}=2P^{2}\ {\partial V\over\partial P^{2}}$ (51)
which only depends on the five scalars (25).
After integration of the system (44)(45) let us set
$\\{Q\cdot P,V\\}=F(\lambda,M^{2},\nu)$ (52)
in other words we perform the substitution
$F={\rm subs.}\ (\widetilde{z}=\zeta,\ \widetilde{y}=\eta,\ P^{2}=M^{2},\
y\cdot P=\nu|\quad\\{Q\cdot P,V\\}\ )$ (53)
Now we can write
$(X_{1}-X_{2})Q\cdot P=\\{Q\cdot P,y\cdot P\\}=y\cdot P$ (54)
$(X_{1}+X_{2})Q\cdot P=(X_{1}+X_{2})_{\rm free}+2\\{Q\cdot P,V\\}$ (55)
$(X_{1}+X_{2})Q\cdot P={1\over 2}P^{2}+2\\{Q\cdot P,V\\}$ (56)
Straightforward integration yields
$Q\cdot P={\nu\over 2}\ (\tau_{1}-\tau_{2})+{M^{2}\over 4}\lambda+\int
Fd\lambda+{\rm const.}$ (57)
We see that modulo the solving of (44) (45) the evolution of $Q\cdot P$ in
terms of $\tau_{1},\tau_{2}$ will be given by a quadrature. In the very
special case where $F\equiv 0$ (with our present notation) the above formula
reduces to eq (3.8) of [13]. But most realistic potentials actually depend on
$P^{2}$ which implies that $F$ differs from zero.
In contrast $G$ vanishes in several cases of interest, for instance
$G\equiv 0$ provided $V$ depends only on the dynamical variables
$\widetilde{z}^{2},P^{2},L^{2}$ .
This statement stems from (16).
To summarize: After integrating (44)(45) we got $z\cdot P$ and $Q\cdot P$ as
functions of $\tau_{1},\tau_{2}$. Since we have the first integrals
$P^{\alpha}=k^{\alpha}$ and $y\cdot P=\nu$ the only remaining dynamical
variables to be determined are ${\widetilde{Q}}^{\beta}$. According to (18)
${\widetilde{Q}}={z\cdot P\over P^{2}}\ \widetilde{y}-{y\cdot P\over P^{2}}\
\widetilde{z}-{\widetilde{P\cdot M}\over P^{2}}$
where the last term is also a first integral, thus everything in the right-
hand side is already detrmined. This observation achieves to determine the
evolution of $q_{1},q_{2},p_{1},p_{2}$, say
$q_{a}=\phi_{a}(\tau_{1},\tau_{2}),\qquad
p_{b}=\psi_{b}(\tau_{1},\tau_{2}),\qquad$ (58)
These functions define the two-dimensional ”orbits” 444To avoid confusion with
trajectories in space, we put the word orbit between quotation marks when it
is meant in the group-theoretical sense. of the evolution group in phase
space.
## 4 World lines in Unipotential Models
The solutions of system (2) can be interpreted in terms of world lines
provided we ultimately introduce the physical coordonates
$x_{1}^{\alpha},x_{2}^{\beta}$ as functions of the canonical coordonates. Our
Cauchy surface for solving the position equations (3) is $(\Sigma)$ defined by
$\ P\cdot z=0\ $, and our initial condition
$x_{a}^{\alpha}-q_{a}^{\alpha}=0\qquad\quad{\rm on\ the\
surface}\qquad(\Sigma)$ (59)
can be formulated also as
$x_{a}^{\alpha}=q_{a}^{\alpha}+O(P\cdot z)$ (60)
where $O(P\cdot z)$ symbolically represents any expression which vanishes with
$P\cdot z$. In principle formula (58) must be inserted into the solutions of
(3) and yields the worldlines in terms of the individual evolution parameters,
say $\ x_{1}(\tau_{1}),x_{2}(\tau_{2})\ $.
The above prescription offers several advantages. First of all, setting
$r^{\alpha}=x_{1}^{\alpha}-x_{2}^{\alpha}$
condition (59) implies that also $P\cdot r$ vanishes on $(\Sigma)$. In the
rest frame we can write $\ x_{1}^{0}=x_{2}^{0}=T,\ $ thus finally the manifold
$(\Sigma)$ can be called the Equal-Time Surface .
At equal times the radius-vector ${\widetilde{r}}^{\alpha}$ moves on a curve
that we may call the relative orbit. As observed long time ago [13], this
curve lies on the 2-plane mentioned in the previous section (orbital plane).
This version of the Hamiltonian formalism clarifies the formal definition (20)
written for the center of mass.
Indeed (20) is equivalent to (21), say
$\Xi={{(P\cdot p_{1})\ q_{1}+(P\cdot p_{2})\ q_{2}}\over P^{2}}+O(P\cdot z)$
In terms of the individual energies $M_{a}=(P\cdot p_{a})/|P|$, we have
$\Xi={M_{1}q_{1}+M_{2}q_{2}\over M_{1}+M_{2}}+O(P\cdot z)$ (61)
Now at equal times $P\cdot z$ vanishes; fixing $P^{\alpha}=k^{\alpha}$ we can
replace $q_{a}$ by $x_{a}$ and $P^{2}$ by $M^{2}$, so we are left with
$\Xi|_{\Sigma}={M_{1}x_{1}+M_{2}x_{2}\over M_{1}+M_{2}}$ (62)
Notice that $M_{1}+M_{2}=M$ and remember that the individual energies reduce
to the masses in the nonrelativistic limit (see Appendix II). In view of these
remarks, formula (62) is more intuitive and significant when $P\cdot p_{1}$
and $P\cdot p_{2}$ are both positive: the analogy with the Newtonian
definition of center of mass becomes obvious, which legitimates the positive-
energy condition. At this stage it is clear that definition (20) agrees with
the one proposed by Pryce [16]; similarly, formula (62) agrees with the notion
of center of energy according to Fischbach et al [17].
### 4.1 Rest-Frame description
In practice, instead of trying to describe the motion in terms of the
independent parameters $\tau_{1},\tau_{1}$ we have better to fix the linear
momentum $k^{\alpha}$, so defining a slicing of spacetime by the three-planes
orthogonal to $k^{\alpha}$. These three-planes intersect both world-lines,
which provides the rest-frame description of dynamics, as follows:
among all possible couples $\ x_{1},x_{1}\ $ the slicing selects the equal-
time configurations, characterized by $k\cdot r=0$. We just have to determine
the sequence of these configurations, that is a one-parameter set. Picking up
the equal-time configurations obviously induces a relation between $\tau_{1}$
and $\tau_{2}$, by cancellation of $\ P\cdot z\ $ in formula (49). One is left
with a (possibly nonlinear) expression of $\tau_{1}-\tau_{2}$ as a function of
$\lambda$.
After imposing (59), formula (49) permits us to express everything in terms of
$\lambda$ only. Formula (20) permits to write
$q_{1}=\Xi-({\nu\over M^{2}}-{1\over 2})\ z+O(P\cdot z)$ (63)
$q_{2}=\Xi-({\nu\over M^{2}}+{1\over 2})\ z+O(P\cdot z)$ (64)
But $z=\widetilde{z}+O(P\cdot z)$. Taking (59) into account (which implies
$\widetilde{z}=\widetilde{r}+O(P\cdot z)$), we put $P\cdot z$ equal to zero in
(63) (64) and obtain the equal-time description of the motion
$x_{1}=\Xi-({\nu\over M^{2}}-{1\over 2})\zeta(\lambda,M^{2},\nu)$ (65)
$x_{2}=\Xi-({\nu\over M^{2}}+{1\over 2})\zeta(\lambda,M^{2},\nu)$ (66)
These formulas yield a representation of both world lines in terms of the same
parameter $\lambda$. But for the sake of a better understanding of the motion
it is interesting to express $\lambda$ as a function of the center-of-mass
proper time $T$.
Cancelling $P\cdot z$ in (49), our equal-time prescription implies
${1\over 2}M^{2}(\tau_{1}-\tau_{2})=-\nu\lambda-\int Gd\lambda+{\rm const.}$
Inserting into (57) yields the common value of $\Xi^{0}$ and $Q^{0}$ in the
center-of-mass frame, say
$T=\lambda({M\over 4}-{\nu^{2}\over M^{3}})-{\nu\over M^{3}}\ \int
Gd\lambda+{1\over M}\int Fd\lambda+{\rm const.}$ (67)
In principle this equation must be solved for $\lambda$. Let us stress that
only in the very special case where $G$ and $F$ are constant, the time of the
center of mass is for all orbits a linear function of the parameter $\
\lambda\ $.
For instance, this situation is realized when both $\partial V/\partial P^{2}$
and $\partial V/\partial(y\cdot P)$ identically vanish, implying $G=F=0$. This
situation will be referred to as the Academic Case.
Otherwise, in the most general case $\lambda$ and $T$ are related in a
nonlinear way, but for exceptional orbits.
Note that for a physically admissible solution to (44) (45), $T$ should
monotonously increase as a function of $\lambda$. In the academic case this is
automatically ensured by the condition (34) of positive individual energies.
When the interaction potential $V$ is more complicated we must demand
$dT/d\lambda>0$, whereas we can write
${dT\over d\lambda}={M\over 4}-{\nu^{2}\over M^{3}}-{\nu G\over M^{3}}+{F\over
M}$ (68)
For example assume for a moment that $G\equiv 0$, we are left with a simpler
condition. If $F$ is positive no problem; otherwise the positive-energy
condition must be replaced by a more restrictive and model-dependent condition
(e.g. see the toy model of Section 6 ).
More generally, the discussion remains easy when $F$ and $G$ are bounded; as
we shall see in the following section, this circumstance arises in case of
circular motion.
### 4.2 Circular Motion
Circular motion is characterized by the constancy of $\widetilde{z}^{2}$,
which implies that at equal times ${\widetilde{r}}^{2}$ also is constant. We
can check that
###### Proposition 5
On any circular orbit the functions $G,F$ and the five dynamical variables
(25) are constant.
Proof.
The interaction potential is a function of the five dynamical variables (24)
We have four remarkable constants of the motion $P^{2},N,y\cdot P$ and the
square of angular momentum
$L^{2}=\widetilde{z}^{2}\widetilde{y}^{2}-(\widetilde{z}\cdot\widetilde{y})^{2}$
(69)
when fixed, they respectively take on the following numerical values
$M^{2},-\Lambda,\nu,l^{2}$
In the set (24) we can replace
$\widetilde{z}^{2},\widetilde{y}^{2},\widetilde{z}\cdot\widetilde{y}$ by the
equivalent set of scalars $\widetilde{z}^{2},\widetilde{y}^{2},L^{2}$. So let
$V=f(P^{2},\widetilde{z}^{2},\widetilde{y}^{2},L^{2},y\cdot P)$
Since $N=\widetilde{y}^{2}+2V$ we can write
$N-\widetilde{y}^{2}=2f(P^{2},\widetilde{z}^{2},\widetilde{y}^{2},L^{2},y\cdot
P)$
This equation implicitly defines $\widetilde{y}^{2}$ as a function of
$P^{2},\widetilde{z}^{2},N,L^{2},y\cdot P$, if we leave apart a very
exceptional case where $V$ linearly depends on $\widetilde{y}^{2}$ in a
special manner (such case is not realistic anyway).
Therefore it is sufficient that $\widetilde{z}^{2}=\rm const.\ $ for having
also $\ \widetilde{y}^{2}=\rm const.\ $, which in turn, according to (69),
implies $\widetilde{z}\cdot\widetilde{y}=\rm const.$.
Finally the five dynamical variables (24), or equivalently (25), remain
constant on the circular orbit. []
It follows that $\lambda$ is a linear function of $T$ on circular orbits.
###### Theorem 1
If the interaction is such that $\\{\widetilde{z},V\\}=0$ and
$\displaystyle{\partial V\over\partial(\widetilde{z}\cdot\widetilde{y})}=0$
there exist circular orbits; on these orbits the relative motion is periodic
in terms of the center-of-mass time.
Proof. According to (44) we have that $\displaystyle{d\widetilde{z}\over
d\lambda}=\widetilde{y}$ and acording to (45) we get
$\displaystyle{d\widetilde{y}\over d\lambda}=-2{\partial
V\over\partial\widetilde{z}^{2}}\ \widetilde{z}$. Equations (44)(45) are
similar to that of a three-dimensional two-body problem. The case when
$\\{\widetilde{z}^{\alpha},\ V\\}$ is zero corresponds to the classical
problem of motion under a central force, where circular orbits are known to
exist.
As seen above, $\widetilde{y}^{2}$ is constant. In the analogy between our
system and that of Galilean mechanics, $\lambda$ plays the role of time and
$-\widetilde{y}^{2}$ represents the squared velocity. As well as a circular
motion with a velocity of constant lenght is necessarily periodic in time,
here we have that $\widetilde{z}$ and $\widetilde{y}$ are periodic functions
of $\lambda$. Since we consider circular motion, $T$ is a linear function of
$\lambda$ and vice versa, so periodicity in $\lambda$ implies periodicity in
$T$. []
Example: any $V(P^{2},\widetilde{z}^{2})$ admits circular orbits.
## 5 Extreme mass ratio, one-body limit
We keep considering unipotential models. The case where one mass can be
neglected in front of the other one is of practical interest when one tries to
justify a resonable expression of $V$. Indeed it is naturally expected that in
the limit of an extreme mass ratio we recover a system made of particle $1$
moving in the external field created by particle $2$, the latter undergoing
rectilinear uniform motion.
Without loss of generality we assume $m_{1}\ll m_{2}$. Note that we cannot
just put $m_{2}$ to infinity. This can be seen already in the framework of
Newtonian mechanics, because the gravitational potential created around
particle $2$ could not remain finite when $m_{2}$ tends to infinity. Therefore
we shall rather put
$\ m_{1}=\gamma\ m_{2}\ $
and study the limit for $\gamma\rightarrow 0$.
In order to alleviate calculations, let us set $\ \varepsilon=\gamma^{2}\ $.
We have
$\mu={1\over 2}m_{2}^{2}(\varepsilon+1)$ (70) $\nu={1\over
2}m_{2}^{2}(\varepsilon-1)$ (71)
whence we derive
$\mu^{2}-\nu^{2}={1\over 4}\ m_{2}^{4}\
[(\varepsilon+1)^{2}-(\varepsilon-1)^{2}]$ $\mu^{2}-\nu^{2}=\varepsilon\
m_{2}^{4}$ (72)
We whish to investigate whether, looking at things from the rest frame, the
center of mass $\Xi$ and the spacetime position $x_{2}$ of the most heavy body
actually coincide in the limit $\gamma\rightarrow 0$. By formula (20) we may
write
$\Xi=Q+({y\cdot P\over P^{2}})\ z-({P\cdot z\over P^{2}})\ y$ (73)
where $Q=q_{2}+{1\over 2}z=q_{1}-{1\over 2}z$. On the mass shell we have that
$P^{2}=M^{2}$ so
$\Xi=q_{2}+({1\over 2}+{\nu\over M^{2}})z-{P\cdot z\over M^{2}}y$ (74)
We can write $\ m_{1}^{2}=\varepsilon\ m_{2}^{2}\ $, where, of course
$\quad\varepsilon=\gamma^{2}$.
We are interested in what happens when $\gamma\rightarrow 0$.
In order to consider the most general case, let us set
$\Lambda=\alpha\ m_{2}^{2}$ (75)
without assuming for the moment any restriction about the magnitude 555Here we
change the notation of [14] by suppression of a factor ${1\over 2}$. of
$\alpha$.
In view of (70) and (72) we get
$(\mu+\Lambda)^{2}-\nu^{2}=m_{2}^{2}\alpha\ (\Lambda+2\mu)+\varepsilon
m_{2}^{4}$ $(\mu+\Lambda)^{2}-\nu^{2}=m_{2}^{2}\ \alpha\
[m_{2}^{2}\alpha+m_{2}^{2}(\varepsilon+1)]+\varepsilon m_{2}^{4}$
$(\mu+\Lambda)^{2}-\nu^{2}=m_{2}^{4}\ ({\alpha}+1)({\alpha}+\varepsilon)$ (76)
Notice that $2(\mu+\Lambda)=m_{2}^{2}(1+\varepsilon+2\alpha)$. Inserting into
(38) yields the rigorous formula
$M^{2}=m_{2}^{2}[1+2\alpha+\varepsilon+2\sqrt{(1+\alpha)(\alpha+\varepsilon)}]$
(77)
valid irrespective of the order of magnitude of $\alpha$ and $\varepsilon$.
Now we are in a position to make the following statement
###### Theorem 2
Provided we can neglect $\ \sqrt{|\Lambda|}\ $ in front of $\ m_{2}\ $, we
have that $M^{2}\rightarrow-2\nu$, which entails that, at equal times, $\Xi$
and $x_{2}$ coincide in the limit $\gamma\rightarrow 0$.
Indeed neglecting $\displaystyle{\sqrt{|\Lambda|}\over m_{2}}\ $ amounts to
cancel $\alpha$ in (77), that yields $-{1\over 2}$ as the limit of the ratio
$\displaystyle\nu/M^{2}$, making the second term in the right-hand side of
(74) to vanish; remember that the third term vanishes at equal times. []
Owing to Propo. 3 the condition for this result is always satisfied for
negative $\Lambda$.
In contradistinction large positive values of $\Lambda$ forbid $\Xi$ to
coincide with the heavy body in the limit of an extreme mass ratio. In this
case $\alpha>0$ and formula (77) can be written
$M^{2}=m_{2}^{2}\
[1+2\alpha+\varepsilon+2\sqrt{(1+\alpha)\alpha}\quad\sqrt{1+\varepsilon/\alpha}]$
Define
$\beta=2\alpha+2\sqrt{\alpha^{2}+\alpha}\ $ (78)
Since $\alpha>0$ it is clear that $\beta>2\alpha$. We get
$M^{2}=m_{2}^{2}(1+\beta)+O(\varepsilon)$ (79)
now using (71) yields
${1\over 2}+{\nu\over M^{2}}={\beta\over 2(1+\beta)}+O(\varepsilon)$ (80)
which in general cannot vanish when $\varepsilon\rightarrow 0$. []
This situation can be physically interpreted as follows: Considered at equal
times, our covariant definition of the center of mass $\Xi$ reduces to that of
Fokker and Pryce [16]. See also Moeller [18]. Accordingly we notice that $\
\Xi\ $ is in fact a center of energy; therefore not only the masses but also
the energies must be taken into account. Even if $m_{1}$ is very small, it
must be understood that we cannot consider particle $1$ as a test particle
when its motion involves a too large amount of energy.
## 6 Toy Model
. Consider a harmonic potential
$V=\chi\ \sqrt{P^{2}}\widetilde{z}^{2}$ (81)
with $\chi$ a positive string constant (this potential differs from the one
considered in [13] by because it is $P^{2}$-dependent, which alows for the
correct dimension of the coupling constant).
The structure of the calculations derived from (81) is that of a
nonrelativistic problem. Our notation is such that the relativistic potential
and its nonrelativistic conterpart have opposite signs, so we have a positive
$\Lambda$.
We must compute $F$, as defined in (52), according to (51) and after solving
the reduced equations of motion (44)(45). The solution to this system is
$\widetilde{z}=A\ \sin(\Omega\lambda+C)+B\ \cos(\Omega\lambda+C)$ (82)
$\widetilde{y}=A\Omega\ \cos(\Omega\lambda+C)-B\Omega\ \sin(\Omega\lambda+C)$
(83)
where $A,B$ are mutually orthogonal spacelike constant vectors (they span the
orbital plane, their lenghts are the half-axes of an ellipse) and $C$ is a
scalar constant; moreover we have
$\Omega=\sqrt{2\chi|P|}$
Note that
$\\{Q\cdot P,V\\}=2{\partial V\over\partial P^{2}}P^{2}=\chi\
\sqrt{P^{2}}\widetilde{z}^{2}=V$
which is always negative. It is clear that $F$ will depend on $\lambda$ only
through $\widetilde{z}^{2}$. Taking (82) into account and fixing
$P^{\alpha}=k^{\alpha}$ we are left with
$\Omega=\sqrt{2\chi M}$
whence we derive
$<N>=2\chi M(A^{2}+B^{2})=-2\chi M(a^{2}+b^{2})$ $a^{2}+b^{2}={\Lambda\over
2\chi M}$
setting $\ A^{2}=-a^{2},\ B^{2}=-b^{2}\ $. We obtain
$F=-\chi M[a^{2}\sin^{2}(\Omega\lambda+C)+b^{2}\cos^{2}(\Omega\lambda+C)]$
(84)
In order to calculate $\int Fd\lambda$ we notice the primitive
$\int^{\lambda}(a^{2}\sin^{2}(\Omega\lambda+C)+(b^{2}\cos^{2}(\Omega\lambda+C))\
d\lambda=$ ${a^{2}+b^{2}\over 2}\ \lambda+{b^{2}-a^{2}\over
2\Omega}\sin(\Omega\lambda+C)\cos(\Omega\lambda+C)+{\rm const.}$
Finally we find
$\int^{\lambda}Fd\lambda=-\chi M\ [{a^{2}+b^{2}\over 2}\
\lambda+{b^{2}-a^{2}\over 4\Omega}\sin(2\Omega\lambda+2C)]+{\rm const.}$ (85)
so there is a secular (linear) term plus a periodic correction; in this
particular example $\widetilde{r}$ is a periodic function of $\lambda$.
Since $z\cdot P$ has a vanishing bracket with $P^{2}$ and $\widetilde{z}^{2}$
it is obvious that $\\{z\cdot P,V\\}$ hence $G$, vanishes.
Owing to (34) we are sure that $\displaystyle\ {M\over 4}-{\nu^{2}\over
M^{3}}>0$. But the question is about $\displaystyle\ {dT\over d\lambda}\ $. It
can be directly read off (84) that
$|F|\leq\chi M(a^{2}+b^{2})$
so the condition for $\displaystyle{dT\over d\lambda}>0$ is
${M^{2}\over 4}-{\nu^{2}\over M^{2}}>|F|$
But
$|F|\leq\chi M(a^{2}+b^{2})$
hence a sufficient condition
${M^{2}\over 4}-{\nu^{2}\over M^{2}}>{\Lambda\over 2}$
## 7 Conclusion
Center of mass and relative motion are well understood concepts for isolated
two-body systems, provided interaction is of the unipotential type and
physical positions are fixed by the equal-time prescription. A large part of
our picture can be made abstractly but most physical features show up in the
light of the equal-time description. The main scheme was put forward many
years ago [7], but several important consequences are considered here for the
first time.
In the present work we regard seriously the fact that the collective evolution
parameter which arises in the reduced equations of motion (44) (45) is
generally not a linear function of the center-of-mass time. This peculiarity
(which doesnot concern the geometry of the orbit) automatically affects the
schedule of relative motion.
This point led us to consider an important exception, the case of circular
orbits, where the relative motion is periodic in $T$.
Although the individual evolution parameters $\tau_{1},\tau_{2}$ are generally
not affine on the world lines, our equal-time treatment offers the possibility
to end up with a description in terms of the center-of-mass time.
On the other hand, examinating the case of an extreme mass ratio provides an
illustration of the nature of center of mass in relativity. Indeed the result
expressed in Theorem 2 forces one to interprete $\Xi$ as a center of energy
rather than of mass; this is in agreement with an ancient literature and with
the spirit of relativity.
In this paper we considered relative motion essentially by analogy with a
nonrelativistic one-body problem, naturally suggested by (44) (45). But of
course the question as to know whether (and how) a ficticious relativistic
one-body system can be invoked, is relevant and deserves a separate
publication.
APPENDIX
I. Reverting from $\mu,\nu$ to $m_{1},m_{2}$, formula (38) implies this useful
approximation
$M=m_{1}+m_{2}+{\Lambda\over 2m_{0}c^{2}}+O(1/c^{4})$ (86)
At first sight the appearence of $m_{0}$ in this formula seems to indicate
that the non-relativistic expression for reduced mas goes over to the
relativistic realm without modification; notice however that we could replace
$m_{0}$ by any positive $m$ in this formula, provided that
$m=m_{0}+O(1/c^{2})\ $. This remarks leaves open the possibility that the
”good” relativistic generalization of the reduced mass may coincide with
$m_{0}$ only in the nonrelativistic limit, as happens for instance with
Todorov’s [19] reduced mass $\displaystyle m_{T}={m_{1}m_{2}\over M}$.
II. Defining $\ M_{a}=P\cdot p_{a}/Mc^{2}\ $ we have
$M_{1}+M_{2}=M,\qquad\quad M_{1}-M_{2}={2y\cdot P\over Mc^{2}}={2\nu\over M}$
But $2\nu=(m_{1}^{2}-m_{2}^{2})c^{2}$ thus $\displaystyle
M_{1}-M_{2}={m_{1}^{2}-m_{2}^{2}\over M}$.
According to (86) we have $\displaystyle{1\over M}={1\over
m_{1}+m_{2}}+O(1/c^{2})$, so finally
$M_{1}-M_{2}={m_{1}^{2}-m_{2}^{2}\over
m_{1}+m_{2}}+O(1/c^{2})=m_{1}-m_{2}+O(1/c^{2})$
Finally $M_{a}=m_{a}+O(1/c^{2})$.
## References
* [1] J.A.Wheeler, R.P.Feynman, Rev. Mod. Phys. 17, 157 (1945).
* [2] A.D. Fokker Zeitsch. f. Physik, 58, 386-393 (1929).
* [3] P.A.M. Dirac, Forms of relativistic dynamics, Rev. Mod. Phys. 21, 392-9 (1949)
* [4] D.G. Currie, Journ. Math. Phys. 4, 1470 (1963), Phys.Rev. 142, 817 (1966). D.G. Currie, T.F. Jordan, E.C.G. Sudarshan, Rev.Mod.Phys. 35, 350 (1963).
* [5] Ph. Droz-Vincent, Lett. Nuov. Cim. 1 839 (1969); Physica Scripta 2, 120 (1970)
* [6] L. Bel, Ann. Inst. Henri Poincaré, 12 , 307 (1970). R. Arens, Arch. for Rat. Mech. and Analysis, 47 , 255 (1972).
* [7] Ph. Droz-Vincent, Reports in Math. Phys. 8, (1975) 79
* [8] D. Dominici, J. Gomis, G. Longhi, Nuov. Cim. 48 A , 257 (1978); Ibid. 48 B , 152 (1978); Ibid. 56 A , 263 (1980).
* [9] I.T. Todorov, JINR Report E2-10125, unpublished (1976).
* [10] D. J. Louis-Martinez, Phys. Letters B, 632, 733-739 (2006). J.L. Friedman and Koji Uryu, Phys. Rev. D 73, 104039 (2006),
* [11] D. Alba, L. Lusanna, M. Pauri, Jour. Math. Phys. 43 , 1677 (2002); D. Alba, H. Crater, L. Lusanna, Jour. of Phys. A 40, 9585 (2007)
* [12] L. Lusanna, Il Nuov. Cim. 65 B , 135 (1981).
* [13] Ph. Droz-Vincent, Ann. Inst. H. Poincaré, 27, 407 (1977)
* [14] Ph. Droz-Vincent, C. R. Acad. Sciences, Paris 290, 115 (1980)
* [15] Ph. Droz-Vincent, J.M.Ph. (1996)
* [16] Pryce, Proc. Roy. Soc. 195 A, 62 (1948)
* [17] E. Fischbach, B.S. Freeman, W-K. Cheng, Phys. Rev.D, 23, 2157-2180, (1981); see Section 3, especially eq. (3.31 a ).
* [18] C. Moeller Ann. Inst. Henri Poincaré, 11, 251 (1949).
* [19] I.T. Todorov, in ”Properties of fundamental interactions”, 9 part C, A. Zichichi Ed. Editrice Compositori, Bologna (1973). V.A. Rizov, I.T. Todorov, B. L. Aneva, Nucl. Phys. B 98, 447-471 (1975).
* [20] H. Jallouli and H. Sazdjian, Ann. of Phys. 253, 376-426, (1997).
|
arxiv-papers
| 2010-06-28T19:32:39 |
2024-09-04T02:49:11.288995
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Philippe Droz-Vincent (LUTH Observatoire de Paris-Meudon)",
"submitter": "Philippe Droz-Vincent",
"url": "https://arxiv.org/abs/1006.5443"
}
|
1006.5683
|
# Estimates for constant mean curvature graphs in $M\times\mathbb{R}$
José M. Manzano José Miguel Manzano, Dpto. Geometría y Topología, Universidad
de Granada. Email address: jmmanzano@ugr.es
###### Abstract.
We will discuss some sharp estimates for CMC graphs $\Sigma$ in a Riemannian
3-manifold $M\times\mathbb{R}$ whose boundary $\partial\Sigma$ is contained in
a slice $M\times\\{t_{0}\\}$. We will start by giving sharp lower bounds for
the geodesic curvature of the boundary and improve these bounds when assuming
additional restrictions on the maximum height that such a surface reaches in
$M\times\mathbb{R}$. We will also give a bound for the distance from an
interior point to the boundary in terms of the height at that point, and
characterize when these bounds are attained.
###### Key words and phrases:
product manifolds, constant mean curvature, invariant surfaces, boundary
curvature estimates, height estimates
###### 2000 Mathematics Subject Classification:
Primary 53A10, Secondary 49Q05, 53C42
Research partially supported by a Spanish MEC-FEDER Grant no. MTM2007-61775
and a Regional J. Andalucía Grant no. P06-FQM-01642.
## 1\. Introduction
Constant mean curvature surfaces in several $3$-manifolds have been
extensively studied in recent times. One of the most important families of
such $3$-manifolds are product spaces $M\times\mathbb{R}$, $M$ being a
Riemannian surface, which includes some homogeneous spaces as
$\mathbb{R}^{3}$, $\mathbb{H}^{2}\times\mathbb{R}$ and
$\mathbb{S}^{2}\times\mathbb{R}$. It was Rosenberg in [12] who started the
study of minimal surfaces in $M\times\mathbb{R}$ and, since then, many papers
in this setting have appeared.
We will focus on constant mean curvature $H>0$ graphs $\Sigma$ in
$M\times\mathbb{R}$ whose boundary $\partial M$ lies in some slice
$M\times\\{t_{0}\\}$ and, if we denote by $c$ the infimum of the Gaussian
curvature of the domain of $M$ over which $\Sigma$ is a graph, we will assume
the hypothesis $4H^{2}+c>0$. As CMC graphs are stable, it is possible to apply
Theorem 2.8 in [6] to conclude that the distance function
$d(p,\partial\Sigma)$, $p\in\Sigma$, is bounded, so the height function is
also bounded. In the case $4H^{2}+c\leq 0$, this property fails to be true as
invariant examples in $\mathbb{H}^{2}\times\mathbb{R}$ given in [7] and [13]
show.
To fix notation, let $M$ be a Riemannian surface without boundary and consider
$\Sigma\subseteq M\times\mathbb{R}$ an embedded constant mean curvature $H>0$
surface (an $H$-surface in the sequel). Let us also consider the height
function $h\in C^{\infty}(\Sigma)$ given by $h(p,t)=t$ and the angle function
$\nu\in C^{\infty}(\Sigma)$ given by $\nu=\langle N,E_{3}\rangle$, where $N$
is the unit normal vector field to $\Sigma$ for which the mean curvature of
$\Sigma$ is $H$, and $E_{3}=\partial_{t}$ is the vertical Killing vector
field. Throughout this paper, we will denote by $K$ the intrinsic curvature of
$\Sigma$ and $K_{M}$ will stand for the intrinsic curvature of $M$ extended to
$M\times\mathbb{R}$ by making it constant along the vertical geodesics.
Besides, $\sigma$ and $A$ will be the second fundamental form and the shape
operator of $\Sigma$, respectively. In this situation, the Gauss equation
reads $\det(A)=K-K_{M}\nu^{2}$.
Aledo, Espinar and Gálvez proved in [1] that if $\Sigma\subseteq
M\times\mathbb{R}$ is a constant mean curvature $H>0$ graph (or $H$-graph for
short) over a compact open domain that extends to its boundary with $h=0$ and
$\nu=\nu_{0}$ in $\partial\Omega$, and we denote by
$c=\inf\\{K_{M}(p):p\in\Omega\\}>-4H^{2}$, then $\Sigma$ can reach at most
height $\alpha(c,H,\nu_{0})$, where
(1)
$\alpha(c,H,\nu_{0})=\begin{cases}\frac{4H}{\sqrt{-4cH^{2}-c^{2}}}\left(\arctan\left(\frac{\sqrt{-c}}{\sqrt{c+4H^{2}}}\right)+\arctan\left(\frac{\nu_{0}\sqrt{-c}}{\sqrt{c+4H^{2}}}\right)\right)&\text{if
}c<0,\\\ \frac{1+\nu_{0}}{H}&\text{if }c=0,\\\
\frac{4H}{\sqrt{4cH^{2}+c^{2}}}\left(\operatorname{arctanh}\left(\frac{\sqrt{c}}{\sqrt{c+4H^{2}}}\right)+\operatorname{arctanh}\left(\frac{\nu_{0}\sqrt{c}}{\sqrt{c+4H^{2}}}\right)\right)&\text{if
}c>0.\end{cases}$
Indeed, they gave the estimate for the case $\nu_{0}=0$ but their argument can
be directly generalized to this more general case. They also proved that this
bound is the best one in terms of $c$ and $H$ in the sense that in the
homogeneous space $\mathbb{M}^{2}(c)\times\mathbb{R}$ the only such $H$-graphs
for which equality holds are rotationally invariant spheres for $\nu_{0}=0$,
and in the general case they are spherical caps of rotationally invariant
spheres which meet the boundary with constant angle function $\nu_{0}$.
We will restrict ourselves to the capillarity problem, i.e. when the surface
has constant angle function $\nu=\nu_{0}$ for some $-1<\nu_{0}\leq 0$ along
its boundary. This situation includes compact embedded CMC surfaces for
$\nu_{0}=0$ because of the Alexandroff reflection principle, and the more
general case of embedded $H$-bigraphs, that is to say, (not necessarily
compact) connected embedded $H$-surfaces which are made up of two graphs,
symmetric with respect to some slice $M\times\\{t_{0}\\}$. Ritoré [10] and
Große-Brauckmann [2] constructions are examples of this kind of surfaces in
$\mathbb{R}^{3}$. We will prove the following results, where we denote by
$\mathbb{M}^{2}(c)$ the simply-connected constant curvature $c$ surface.
* •
The geodesic curvature $\kappa_{g}$ of $\partial\Omega$ in $M$, with respect
to the outer conormal vector field, satisfies the lower bound
$\kappa_{g}\geq\frac{-4H^{2}+c(1-\nu_{0})^{2}}{4H\sqrt{1-\nu_{0}^{2}}},$
and, when $M=\mathbb{M}^{2}(c)$, equality holds only for rotationally
invariant spheres.
* •
If we additionally suppose that $|h|\leq m\cdot\alpha(c.H,\nu_{0})$ for some
constant $0<m\leq\frac{1}{2}$, then the previous bound is improved to the
following one:
$\kappa_{g}\geq\frac{(4-8m)H^{2}+c(1-\nu_{0})^{2}}{4mH\sqrt{1-\nu_{0}^{2}}}.$
In this case, when $M=\mathbb{M}^{2}(c)$, equality holds if, and only if,
$m=\frac{1}{2}$ and $\Sigma$ is the boundary of some neighborhood of a
geodesic of $\mathbb{M}^{2}(c)\times\\{0\\}$, examples which will be fully
described in section 2.
* •
In the last section, we will give an application of the techniques used in the
above two items to obtain a sharp lower bound for the distance from a point in
$\Sigma$ to $\partial\Sigma$. As in the results above, equality holds when the
surface is rotationally invariant.
The author would like to thank Joaquín Pérez, Magdalena Rodríguez and
Francisco Torralbo for some helpful conversations.
## 2\. Invariant surfaces in $\mathbb{H}^{2}\times\mathbb{R}$ and
$\mathbb{S}^{2}\times\mathbb{R}$
In this section, we will study surfaces that are invariant by $1$-parameter
groups of isometries in $\mathbb{M}^{2}(c)\times\mathbb{R}$ which act
trivially on the vertical lines. In fact, among these, we are interested in
surfaces which are $H$-bigraphs (i.e. embedded $H$-surfaces symmetric with
respect to a horizontal slice), for $H>0$ and $4H^{2}+c>0$. Thus, these groups
of isometries can be identified with $1$-parameter groups of isometries of the
base $\mathbb{M}^{2}(c)$.
In $\mathbb{H}^{2}$, there exist three different types of $1$-parameter groups
of isometries, namely, rotations around a point, parabolic translations (i.e.
rotations about a point at infinity) and hyperbolic translations. The family
of rotationally invariant CMC surfaces in $\mathbb{H}^{2}\times\mathbb{R}$ was
studied by Hsiang and Hsiang [5] and those invariant by the other two families
(including screw motion) were also studied by Sa Earp [13] but it was Onnis
[7] who gave a full classification of all invariant CMC surfaces in
$\mathbb{H}^{2}\times\mathbb{R}$. The case of $\mathbb{S}^{2}$ is quite
different, because the only $1$-parameter groups of isometries of
$\mathbb{S}^{2}$ are the rotations around a certain point and, up to
conjugation, this point can be supposed to be the north pole. Such
rotationally invariant $H$-surfaces were classified by Pedrosa [9].
Finally, the only $1$-parameter groups of isometries of $\mathbb{R}^{2}$ are
rotations around a point and translations; the former give rise in
$\mathbb{R}^{3}=\mathbb{R}^{2}\times\mathbb{R}$ to Euclidean spheres of radius
$\frac{1}{H}$, the latter to horizontal cylinders of radius $\frac{1}{2H}$.
For the sake of completeness, we will now derive the parametrizations and
formulas that we will need in each of these situations. We will begin with
rotations in both $\mathbb{H}^{2}\times\mathbb{R}$ and
$\mathbb{S}^{2}\times\mathbb{R}$ and then proceed to parabolic and hyperbolic
translations in $\mathbb{H}^{2}\times\mathbb{R}$. Let us recall that, up to a
homothety, we can suppose $c\in\\{-1,0,1\\}$ and, in the cases $c=1$ and
$c=0$, the condition $4H^{2}+c>0$ is meaningless (as $H>0$) but, for $c=-1$,
it implies that $H>\frac{1}{2}$.
### 2.1. Rotationally invariant surfaces in $\mathbb{H}^{2}\times\mathbb{R}$
and $\mathbb{S}^{2}\times\mathbb{R}$
To start with, let us consider the model
$\mathbb{H}^{2}\times\mathbb{R}=\\{(x,y,z,t)\in\mathbb{R}^{4}:x^{2}+y^{2}-z^{2}=-1,z>0\\}$
endowed with the metric $dx^{2}+dy^{2}-dz^{2}+dt^{2}$. It was shown by Hsiang
and Hsiang that, for any $H>\frac{1}{2}$, the only rotationally invariant
$H$-bigraphs are the rotationally invariant CMC spheres. If we suppose the
axis of rotation to be $\\{(0,0,1)\\}\times\mathbb{R}$, the upper half of such
a sphere is parametrized by $X(r,u)=\left(\sinh r\cos u,\sinh r\sin u,\cosh
r,h(r)\right)$, where $u\in\mathbb{R}$ and
$h(r)=\frac{4H}{\sqrt{4H^{2}-1}}\arcsin\sqrt{\frac{1-(4H^{2}-1)\sinh^{2}\frac{r}{2}}{4H^{2}}},\quad
r\in\left[0,2\mathop{\rm arcsinh}\nolimits\frac{1}{\sqrt{4H^{2}-1}}\right]$
(see figure 1 where some examples have been depicted).
Figure 1. On the left, rotationally invariant CMC spheres (the horizontal axis
represents the intrinsic length in $\mathbb{H}^{2}$ and the vertical one is
the real line) and, on the right, CMC cylinders invariant under hyperbolic
translations, where we see their intersection with the plane $y=1$ in the
halfspace model. In both cases, the represented values of $H$ are 0.54, 0.6,
0.7, 0.8, 0.9 and 1.
On the other hand, we will consider the standard model of
$\mathbb{S}^{2}\times\mathbb{R}$ as a submanifold of $\mathbb{R}^{4}$, given
by
$\mathbb{S}^{2}\times\mathbb{R}=\\{(x,y,z,t)\in\mathbb{R}^{4}:x^{2}+y^{2}+z^{2}=1\\}$
with the induced Riemannian metric. It is well-known that every $1$-parameter
group of ambient isometries consists only of rotations so, up to an isometry,
they may be supposed to be rotations around the axis
$\\{(0,0,1,t):t\in\mathbb{R}\\}$. Hence, the orbit space can be identified
with the totally geodesic surface
$\\{(x,y,z,t)\in\mathbb{S}^{2}\times\mathbb{R}:x=0\\}\cong\mathbb{S}^{1}\times\mathbb{R}$,
and we will take the generating curve as
$\gamma(t)=\left(0,\sin r(t),\cos r(t),h(t)\right)$
for some functions $r,h$ defined on some interval of the real line. Pedrosa
[9] showed that the generated surface has constant mean curvature
$H\in\mathbb{R}$ if, and only if, certain ODE system is satisfied. In fact, he
proved that in the intervals where $r$ is invertible, we can take it as the
parameter and the corresponding ODE system becomes
(2) $\left\\{\begin{array}[]{l}h^{\prime}(r)=\cot(\sigma(r)),\\\
\sigma^{\prime}(r)=\frac{2H+\cot(r)\cos(\sigma(r))}{\operatorname{sen}(\sigma(r))},\end{array}\right.$
for an auxiliary function $\sigma$. The second equation can be easily solved
as it only depends on $r$ and $\sigma$ and we obtain
$\sigma(r)=\arccos(2H(c_{0}+\cos r)\csc r)$
for some $c_{0}\in\mathbb{R}$, where
$r\in[a(c_{0}),b(c_{0})]\subseteq[-\pi,\pi]$ is the maximal interval in which
$\sigma$ is defined. By plugging this expression into the first equation in
(2), we arrive to
(3) $h(r)=\int_{a(c_{0})}^{r}\frac{2H(c_{0}+\cos s)\csc
s}{\sqrt{1-4H^{2}(c_{0}+\cos s)^{2}\csc^{2}s}}\,\mathrm{d}s.$
The only two cases which lead to $H$-bigraphs are the following:
* •
For $c_{0}=-1$, rotationally invariant spheres are obtained. More explicitly,
$h(r)=\frac{4H}{\sqrt{1+4H^{2}}}\operatorname{arccosh}\left(\frac{\sqrt{1+4H^{2}}}{4H}\cos\frac{r}{2}\right)$
where $r$ lies in the interval $[-2\arctan\frac{1}{H},2\arctan\frac{1}{H}]$.
Thus, the maximum height is attained for $r=0$ and the sphere is a bigraph
over a domain whose boundary has constant geodesic curvature in
$\mathbb{S}^{2}$ with respect to the outer conormal vector field, equal to
$-H+\frac{1}{4H}$.
* •
For $c_{0}=0$, we obtain rotationally invariant tori instead. In this case,
$h(r)=\frac{2H}{\sqrt{1+4H^{2}}}\operatorname{arccosh}\left(\frac{\sqrt{1+4H^{2}}}{2H}\sin
r\right)$
where
$r\in[\frac{\pi}{2}-\arctan\frac{1}{2H},\frac{\pi}{2}+\arctan\frac{1}{2H}]$.
The maximum height is attained when $r=\frac{\pi}{2}$ and the boundary of the
domain over which the torus is a bigraph has two connected components which
have constant geodesic curvature $\frac{1}{2H}$ in
$\mathbb{S}^{2}\times\mathbb{R}$ (with respect to the outer conormal vector
field).
These two families are represented in figure 2. We remark that the maximum
height of a CMC torus is exactly a half of that of the corresponding sphere
for the same mean curvature.
Figure 2. On the left, rotationally invariant CMC spheres and, on the right,
rotationally invariant CMC tori, in $\mathbb{S}^{2}\times\mathbb{R}$. In both
cases, the represented values of $H$ are 0.05, 0.12, 0.331372, 0.6, 1 and 2.
The horizontal axis measures the intrinsic distance in $\mathbb{S}^{2}$ while
the vertical one is the real line. The maximum height is attained for
$H\approx 0.331372$, which is drawn as a dashed line.
### 2.2. Invariant surfaces under hyperbolic translations in
$\mathbb{H}^{2}\times\mathbb{R}$
In this section, we will work in the upper halfplane model
$\mathbb{H}^{2}\times\mathbb{R}=\\{(x,y,t)\in\mathbb{R}^{3}:y>0\\}$ endowed
with the metric $(dx^{2}+dy^{2})/y^{2}+dt^{2}$. Up to conjugation by an
ambient isometry, the $1$-parameter group of hyperbolic translations may be
considered to be $\\{\Phi^{h}_{s}\\}_{s\in\mathbb{R}}$, where
$\Phi_{s}^{h}:\mathbb{H}^{2}\times\mathbb{R}\rightarrow\mathbb{H}^{2}\times\mathbb{R},\quad\quad\Phi_{s}^{h}(x,y,t)=(xe^{s},ye^{s},t).$
First of all, we observe that, as the orbit of any point is the horizontal
Euclidean straight line which joins the point to a point in the axis $x=y=0$,
we can consider the plane $y=1$ as the orbit space of this group of
transformations.
Let us take a curve $\gamma(t)=(x(t),1,h(t))$ for some $C^{2}$ functions $x,h$
defined in some interval of the real line. Thus, a surface invariant by
$\\{\Phi^{h}_{s}\\}_{s\in\mathbb{R}}$ can be parametrized as
(4) $X(u,t)=\left(x(t)e^{u},e^{u},h(t)\right).$
It is a straightforward computation to check that the mean curvature of this
parametrization is given by
(5)
$H=\frac{-x^{3}(h^{\prime})^{3}-xh^{\prime}((h^{\prime})^{2}+2(x^{\prime})^{2})+x^{2}(h^{\prime}x^{\prime\prime}-x^{\prime}h^{\prime\prime})-x^{\prime}h^{\prime\prime}+h^{\prime}x^{\prime\prime}}{2((1+x^{2})(h^{\prime})^{2}+(x^{\prime})^{2})^{3/2}}.$
In order to simplify this equation, we will reparametrize the curve $\gamma$
in such a way the denominator simplifies. Observe that we can suppose that
$(h^{\prime})^{2}+(x^{\prime})^{2}/(1+x^{2})=1$ so there exists a $C^{1}$
function $\alpha$ such that $h^{\prime}=\cos\alpha$ and
$x^{\prime}=\sqrt{1+x^{2}}\sin\alpha$. Now, we can obtain expressions for
$x^{\prime\prime}$ and $h^{\prime\prime}$ just by taking derivatives in these
identities. If we substitute the results in equation (5), we get
$H=\frac{\sqrt{1+x^{2}}\alpha^{\prime}-x\cos\alpha}{2\sqrt{1+x^{2}}}.$
The proof of the following lemma is now trivial.
###### Lemma 2.1.
The parametrized surface defined in (4) has constant mean curvature
$H\in\mathbb{R}$ if, and only if, the functions $(x,h,\alpha)$ satisfy the
following ODE system
(6) $\left\\{\begin{array}[]{l}h^{\prime}=\cos\alpha\\\
x^{\prime}=\sqrt{1+x^{2}}\sin\alpha\\\
\alpha^{\prime}=2H+\frac{x\cos\alpha}{\sqrt{1+x^{2}}}\end{array}\right.$
Furthermore, the energy function $E=-2Hx-\sqrt{1+x^{2}}\cos\alpha$ is constant
along any solution.
We will restrict ourselves to the case $H>1/2$. Plugging the expression of the
energy into the second equation in (6), it is not difficult to conclude that
$x$ verifies the equation
(7) $(x^{\prime})^{2}=(1-E^{2})+4HEx+(1-4H^{2})x^{2}.$
As $H>\frac{1}{2}$, the RHS has two different real roots as a polynomial in
$x$ and, if we factor it, the equation can be expressed, up to a sign, as
$\frac{x^{\prime}}{\sqrt{(4H^{2}+E^{2}-1)-\left((4H^{2}-1)x-2HE\right)^{2}}}=\frac{\pm
1}{\sqrt{4H^{2}-1}},$
from where it is easy to deduce that there exists $c_{0}\in\mathbb{R}$ such
that
(8)
$x(t)=\frac{2HE}{4H^{2}-1}+\frac{\sqrt{4H^{2}+E^{2}-1}}{4H^{2}-1}\sin\left(\pm
t\sqrt{4H^{2}-1}+c_{0}\right).$
After a translation and a reflection in the parameter $t$, we will suppose
without loss of generality that $c_{0}=0$ and the $\pm$ sign is positive. Now,
we can integrate $h$ by taking into account the identity
$h^{\prime}=\cos\alpha=(E+2H)/\sqrt{1+x^{2}}$, and we get
(9)
$h(t)=h(0)+\int_{0}^{t}\frac{(8H^{2}-1)E+2H\sqrt{4H^{2}+E^{2}-1}\sin\left(s\sqrt{4H^{2}-1}\right)}{\sqrt{\left(1-4H^{2}\right)^{2}+\left(\sqrt{4H^{2}+E^{2}-1}\sin(s\sqrt{4H^{2}-1})+2HE\right)^{2}}}\,\mathrm{d}s.$
Finally, we are able to characterize the surfaces we were looking for. Some
pictures of them are drawn in Figure 1.
###### Proposition 2.2.
Let $(x,h,\alpha)$ be a solution of (6) with energy $E\in\mathbb{R}$ for some
$H>\frac{1}{2}$. Then, the generated invariant surface can be extended to an
$H$-bigraph if, and only if, $E=0$. In this case, the generating curve can be
reparametrized, up to an ambient isometry, as
$\left.\begin{array}[]{l}x(r)=\displaystyle\frac{1}{\sqrt{4H^{2}-1}}\sin r\\\
h(r)=\displaystyle\frac{2H}{\sqrt{4H^{2}-1}}\arctan\frac{\cos
r}{\sqrt{4H^{2}-1+\sin^{2}r}}\end{array}\right\\},\quad r\in\mathbb{R}.$
###### Proof.
Let us split $h(t)=h_{1}(t)+h_{2}(t)$ in (9) by splitting the integrand in two
additive terms which correspond to the two terms in its numerator. The first
term does not vanish unless $E=0$ so $h_{1}$ is monotonic and the second one
is an odd periodic function in $s$ which vanishes at $s=k\pi/\sqrt{4H^{2}-1}$
for any $k\in\mathbb{Z}$. On the other hand, if the parametrization interval
contains $t=0$, from (8) we deduce that $|t|\leq\pi/(2\sqrt{4H^{2}-1})$ so the
surface is a graph and, furthermore, the points at which the normal vector
field is horizontal must satisfy $x^{\prime}=0$, so the parametrization
interval must be $|t|\leq\pi/(2\sqrt{4H^{2}-1})$. Now, as the integral of
$h_{2}^{\prime}$ over $[-\pi/(2\sqrt{4H^{2}-1}),\pi/(2\sqrt{4H^{2}-1})]$
vanishes, we have
$h\left(\frac{\pi}{2\sqrt{4H^{2}-1}}\right)-h\left(\frac{-\pi}{2\sqrt{4H^{2}-1}}\right)=h_{1}\left(\frac{\pi}{2\sqrt{4H^{2}-1}}\right)-h_{1}\left(\frac{-\pi}{2\sqrt{4H^{2}-1}}\right).$
The RHS term vanishes if, and only if, $h_{1}$ identically vanishes as it is
monotonic and $h_{1}$ vanishes if, and only if, $E=0$. The expressions given
in the statement follow from a direct computation in (8) and (9) for $E=0$ and
from the substitution $r=t\sqrt{4H^{2}-1}$. Observe that there is no
restriction in taking $r\in\mathbb{R}$ because this parametrization generates
the whole bigraph. ∎
In the parametrization given in the statement of Proposition 2.2, observe that
the maximum height is attained for $r=0$ and the surface is a bigraph over a
domain whose boundary consists in two hypercycles which have constant geodesic
curvature in $\mathbb{H}^{2}$ with respect to the outer conormal vector field,
equal to $\frac{-1}{2H}$. Furthermore, the maximum height is exactly a half of
that of the corresponding CMC sphere.
### 2.3. Invariant surfaces under parabolic translations
In this case, we will also consider the upper halfplane model for
$\mathbb{H}^{2}$ so, up to conjugation by an ambient isometry, the
$1$-parameter group of parabolic translations is
$\\{\Phi_{s}^{p}\\}_{s\in\mathbb{R}}$, where
$\Phi_{s}^{p}:\mathbb{H}^{2}\times\mathbb{R}\rightarrow\mathbb{H}^{2}\times\mathbb{R},\quad\quad\Phi_{s}^{p}(x,y,t)=(x+s,y,t).$
Hence, the orbit of any point in $\mathbb{H}^{2}$ is a horizontal Euclidean
line parallel to the plane $y=0$. Thus, the orbit space may be considered to
be the Euclidean plane $x=0$ so the generating curve can be thought as
$\gamma(t)=(0,y(t),h(t))$ and a surface invariant by
$\\{\Phi^{p}_{s}\\}_{s\in\mathbb{R}}$ can be parametrized as
$X(u,t)=\left(s,y(t),h(t)\right).$
It is straightforward to check that the mean curvature of this parametrization
is
(10)
$H=-\frac{y^{2}\left(-h^{\prime\prime}y^{\prime}+h^{\prime}y^{\prime\prime}+y(h^{\prime})^{3}\right)}{2\left(y^{2}(h^{\prime})^{2}+(y^{\prime})^{2}\right)^{3/2}}.$
Furthermore, there is no loss of generality in supposing that the curve
$\gamma$ is parametrized by its arc-length, i.e.
$1=\|\alpha^{\prime}\|^{2}=(y^{\prime})^{2}/y^{2}+(h^{\prime})^{2}$. Hence, we
can take an auxiliary function $\alpha$, determined by
$y^{\prime}=y\sin\alpha$, $h^{\prime}=\cos\alpha$. Substituting these
equalities in (10), it simplifies to the following ODE system
(11) $\left\\{\begin{array}[]{l}y^{\prime}=y\sin\alpha\\\
h^{\prime}=\cos\alpha\\\ \alpha^{\prime}=-2H-\cos\alpha.\end{array}\right.$
Observe that, if we assume an initial condition
$\alpha(0)=\alpha_{0}\in[0,2\pi]$, the third equation has a unique solution.
Let us focus in the case $H>\frac{1}{2}$ which is the most interesting for our
purposes and allows us to integrate the function $\alpha$ as
(12)
$\alpha(t)=2\arctan\left(\frac{(2H+1)}{\sqrt{4H^{2}-1}}\tan\left(\frac{1}{2}\sqrt{4H^{2}-1}(t-c_{0})\right)\right)$
for some $c_{0}\in\mathbb{R}$ depending on $\alpha_{0}$. We emphasize that
this formula defines $\alpha:\mathbb{R}\rightarrow\mathbb{R}$ as a strictly
increasing diffeomorphism by considering all the branches of the function
$\arctan$ and extending it by continuity, so the uniqueness of solution
guarantees that every solution is considered in (12). We will suppose, after a
translation in the parameter $t$, that $c_{0}=0$. By plugging expression (12)
into the first two equations in (11), we can integrate $h$ and $y$ to obtain
(13)
$\begin{array}[]{l}y(t)=c_{1}\left(\cos\left(t\sqrt{4H^{2}-1}\right)+2H\right)\\\
h(t)=\alpha(t)+2Ht+c_{2},\end{array}$
for some constants $c_{1}>0$ and $c_{2}\in\mathbb{R}$ which can be supposed to
be $c_{1}=1$ (after a hyperbolic translation) and $c_{2}=0$ (after a vertical
translation).
###### Proposition 2.3.
There are no invariant embedded bigraphs under parabolic translations with
constant mean curvature $H>\frac{1}{2}$.
###### Proof.
Observe that such a graph must be given by a triple $(y,h,\alpha)$ satisfying
(11) so (12) and (13) are also satisfied. The values of $t\in\mathbb{R}$ for
which $y^{\prime}=0$ are $t_{k}=k\pi/\sqrt{4H^{2}+1}$ for any $k\in\mathbb{Z}$
(these ones correspond to the points in the surface whose tangent plane is
vertical). Now from (12) and (13) it is easy to check that $h(t_{k})\neq
h(t_{k+1})$ for every $k\in\mathbb{Z}$, which makes impossible the surface to
be a bigraph. ∎
## 3\. Boundary curvature estimates
Let us suppose along this section that $\Sigma\subseteq M\times\mathbb{R}$ is
a graph over a domain $\Omega\subseteq M$ with constant mean curvature $H>0$.
Following the ideas given in [1], for any given $c\in\mathbb{R}$ with
$c+4H^{2}>0$, we will consider the function $g:[-1,1]\rightarrow\mathbb{R}$
determined by
(14) $g^{\prime}(t)=\frac{4H}{4H^{2}+c(1-t^{2})},\quad\quad g(0)=0,$
which is strictly increasing and allows us to define the smooth function
$\psi=h+g(\nu)\in C^{\infty}(\Sigma)$, where $h$ and $\nu$ are the height and
angle functions, respectively. We are interested in applying the boundary
maximum principle for the laplacian to $\psi$ so we will need to work out
$\Delta\psi$ (where the laplacian is computed in the surface $\Sigma$) and
$\frac{\partial\psi}{\partial\eta}$, where $\eta$ is some outer conormal
vector field to $\partial\Sigma$. The following formulas will be useful.
###### Lemma 3.1.
In the previous situation, the following equalities hold.
* i)
$\nabla h=E_{3}^{\top}$,
* ii)
$\Delta h=2H\nu$,
* iii)
$\nabla\nu=-AE_{3}^{\top}$,
* iv)
$\Delta\nu=\left(2K-4H^{2}-K_{M}(1+\nu^{2})\right)\nu$.
###### Proof.
The identities for the gradient and the laplacian of $h$ are easy to check as
$h$ is the restriction to $\Sigma$ of the height function in
$M\times\mathbb{R}$ (see also [12, Lemma 3.1]). On the other hand, the
gradient of $\nu=\langle N,E_{3}\rangle$ satisfies
$\langle\nabla\nu,X\rangle=X(\langle
N,E_{3}\rangle)=\langle\nabla_{X}N,E_{3}\rangle=\langle-
AX,E_{3}\rangle=\langle X,-AE_{3}^{\top}\rangle$
for any vector field $X$ on $\Sigma$, so $\nabla\nu=-AE_{3}^{\top}$. Finally,
since the vertical translations are isometries of $M\times\mathbb{R}$, $\nu$
is a Jacobi function, i.e. $\nu$ lies in the kernel of the linearized mean
curvature operator $L=\Delta-[|\sigma|^{2}+\mathop{\rm Ric}\nolimits(N)]$ on
$\Sigma$ so we can compute its laplacian from $L\nu=0$ and obtain
$\Delta\nu=\left(|\sigma|^{2}+\mathop{\rm
Ric}\nolimits(N)\right)\nu=\left(2K-4H^{2}-K_{M}(1+\nu^{2})\right)\nu,$
where we have used the Gauss equation and the well-known identities
$|\sigma|^{2}=4H^{2}-2\det(A)$ and $\mathop{\rm
Ric}\nolimits(N)=K_{M}(1-\nu^{2})$. ∎
On the other hand, we need to obtain some suitable expression for the modulus
of the Abresch-Rosenberg differential, in the case $M=\mathbb{M}^{2}(c)$. If
we take a conformal parametrization $(U,z)$ in $\Sigma$, this quadratic
differential can be written as
$Q=(2Hp-ch_{z}^{2})\,\mathrm{d}z^{2}$
(see [4]), where
$p\,\mathrm{d}z^{2}=\langle-\nabla_{\partial_{z}}N,\partial_{z}\rangle\,\mathrm{d}z^{2}$
is the Hopf differential and $h_{z}=\frac{\partial h}{\partial z}$. Although
this expression depends on the parametrization, we may consider the function
$\displaystyle q=\frac{4}{\lambda^{2}}|Q|^{2}$
$\displaystyle=\tfrac{4}{\lambda^{2}}\left(4H^{2}|p|^{2}+c^{2}|h_{z}|^{4}-2cH(ph_{\bar{z}}^{2}-\bar{p}h_{z}^{2})\right)$
(15)
$\displaystyle=4H^{2}(H^{2}-\det(A))+\tfrac{c^{2}}{4}(1-\nu^{2})^{2}-c(\|\nabla\nu\|^{2}-(2H^{2}-\det(A))(1-\nu^{2})),$
which is well-defined and smooth on the whole $\Sigma$.
Back to the computation of $\Delta\psi$ and taking into account the formulas
in Lemma 3.1 and identity (15), we get
(16) $\Delta\psi=\Delta
h+g^{\prime}(\nu)\Delta\nu+g^{\prime\prime}(\nu)\|\nabla\nu\|^{2}=\frac{-8q\nu}{\left(4H^{2}+c(1-\nu^{2})\right)^{2}}-\frac{4H\nu\left(1-\nu^{2}\right)(K_{M}-c)}{4H^{2}+c(1-\nu^{2})}.$
Finally, we are interested in working out $\frac{\partial\psi}{\partial\eta}$
along $\partial\Sigma$, where we have considered the outer conormal vector
field to $\partial\Sigma$ in $\Sigma$ given by $\eta=-E_{3}^{\top}$ (it does
not matter which outer conormal vector field is chosen as the only needed
information is the sign of $\frac{\partial\psi}{\partial\eta}$). Hence,
$\displaystyle\frac{\partial h}{\partial\eta}$ $\displaystyle=\langle\nabla
h,\eta\rangle=\langle
E_{3}^{\top},-E_{3}^{\top}\rangle=-\|E_{3}^{\top}\|^{2},$
$\displaystyle\frac{\partial\nu}{\partial\eta}$
$\displaystyle=\langle\nabla\nu,\eta\rangle=\langle-
AE_{3}^{\top},-E_{3}^{\top}\rangle=\langle\overline{\nabla}_{E_{3}^{\top}}E_{3}^{\top},N\rangle.$
However, if we parametrize $\partial\Sigma$ by $\gamma$ with
$\|\gamma^{\prime}\|=1$, then
$\\{E_{3}^{\top}/\|E_{3}^{\top}\|,\gamma^{\prime}\\}$ is an orthonormal basis
of $T\Sigma$, and it is clear that
$2H=\left\langle\frac{1}{\|E_{3}^{\top}\|^{2}}\overline{\nabla}_{E_{3}^{\top}}E_{3}^{\top}+\overline{\nabla}_{\gamma^{\prime}}\gamma^{\prime},N\right\rangle$
so
$\frac{\partial\nu}{\partial\eta}=2H\|E_{3}^{\top}\|^{2}+\|E_{3}^{\top}\|^{3}\kappa_{g}$.
Here, $\kappa_{g}$ denotes the geodesic curvature of
$\partial\Omega=\partial\Sigma$ in the base $M$ with respect to
$\|E_{3}^{\top}\|^{-1}(N-\nu E_{3})$, the outer conormal vector field to
$\partial\Omega$ in $M$. Finally, we obtain
(17) $\frac{\partial\psi}{\partial\eta}=\frac{\partial
h}{\partial\eta}+g^{\prime}(\nu)\frac{\partial\nu}{\partial\eta}=\|E_{3}^{\top}\|^{2}\left(-1+g^{\prime}(\nu)(2H+\|E_{3}^{\top}\|\kappa_{g})\right).$
Note that any outer conormal vector field to $\partial\Omega$ in $M$ is a
linear combination of $N$ and $E_{3}$, which is the key property to relate the
geometries of $\Sigma$ and $M$. Moreover, these computations allow us to give
an optimal bound for the geodesic curvature of the boundary of the domain of a
compact $H$-graph with a capillarity boundary condition.
###### Theorem 3.2.
Let $\Sigma\subseteq M\times\mathbb{R}$ be a constant mean curvature $H>0$
graph over a compact regular domain $\Omega\subseteq M$ with zero values in
$\partial\Omega$. Let us consider $c=\inf\\{K_{M}(p):p\in\Omega\\}$ and
suppose that $c+4H^{2}>0$ and $\nu=\nu_{0}$ in $\partial\Omega$ for some
$-1<\nu_{0}\leq 0$. Then
(18) $\kappa_{g}\geq\frac{-4H^{2}+c(1-\nu_{0}^{2})}{4H\sqrt{1-\nu_{0}^{2}}},$
where $\kappa_{g}$ is the geodesic curvature of $\partial\Omega$ in $M$ with
respect to the outer conormal vector field.
Furthermore, if there exists $p\in\partial\Omega$ such that equality holds in
(18), then $\Omega$ has constant curvature and $\Sigma$ is invariant by a
$1$-parameter group of isometries.
###### Proof.
Let us consider the function $\psi=h+g(\nu)$, defined in terms of (14). Since
$\nu\leq 0$ and $K_{M}\geq c$ in $\Sigma$, equation (16) insures that
$\Delta\psi\geq 0$ in $\Sigma$. Let $p_{0}\in\Sigma$ a point where $\psi$
attains its maximum. If $p_{0}$ is interior to $\Sigma$, then the maximum
principle for the laplacian guarantees that $\psi$ is constant, so from (16),
we get that $q=0$ and $K_{M}=c$ in $\Omega$. These conditions imply that
$\Omega$ has constant curvature $c$ and that $\Sigma$ is invariant by a
1-parameter subgroup of isometries which acts trivially on the vertical lines
(see [3, Lemma 6.1]).
If, on the contrary, $p_{0}\in\partial\Omega$ and $\psi(p_{0})>\psi(p)$ for
every interior point $p\in\Omega$, then such maximum is attained in the whole
boundary as $\psi$ is constant along it, so the maximum principle in the
boundary insures that $\frac{\partial\psi}{\partial\eta}>0$ in
$\partial\Omega$. Then the desired strict inequality for $\kappa_{g}$ can be
deduced from (17). ∎
###### Remark 3.3.
In the situation of the statement of Theorem 3.2, if we suppose that
$-1<\nu\leq\nu_{0}$ in $\partial\Omega$ for some $\nu_{0}\leq 0$ (instead of
$\nu=\nu_{0}$ in $\partial\Omega$) and there exists a point
$p\in\partial\Omega$ such that $\nu(p)=\nu_{0}$ and at which the inequality
for $\kappa_{g}$ becomes and equality, then $\Omega$ has constant curvature
and $\Sigma$ is a spherical cap of a standard rotational sphere. This can be
easily seen as a consequence of the maximum principle in the boundary.
Observe that, in case $\Sigma\subseteq M\times\mathbb{R}$ is a compact
embedded constant mean curvature $H>0$ surface, it is possible to apply the
Alexandrov reflection principle to vertical reflections and conclude that
$\Sigma$ is a symmetric bigraph with respect to some slice
$M\times\\{t_{0}\\}$ which, after a vertical translation, may be supposed to
be $M\times\\{0\\}$. In this setting, it is obvious that $\Sigma$ intersects
orthogonally such a slice.
###### Corollary 3.4.
Let $H>0$ and $\Sigma\subseteq M\times\mathbb{R}$ be a compact embedded
$H$-bigraph over a domain $\Omega\subseteq M$, symmetric with respect to
$M\times\\{0\\}$. If we suppose that
$c=\inf\\{K_{M}(p):p\in\Omega\\}>-4H^{2}$, then $\partial\Omega$ is a curve
whose geodesic curvature in $M$ with respect to the outer conormal vector
field satisfies
$\kappa_{g}\geq-H+\frac{c}{4H}.$
Furthermore, if equality holds at some point in $\partial\Omega$, then
$\Omega$ has constant curvature and $\Sigma$ is invariant under a
$1$-parameter isometry group.
We now adjust the value of $H$ for which the lower bound is exactly zero.
###### Corollary 3.5.
Let $M$ be a orientable complete Riemannian surface with $K_{M}\geq c>0$ in
$M$. Then, each compact embedded $H$-surface in $M\times\mathbb{R}$ with
$0<H<\frac{1}{2}\sqrt{c}$ is an $H$-bigraph over a connected domain $\Omega$
and $M\smallsetminus\Omega$ is a finite union of disjoint convex disks.
Furthermore, either
* •
these disks are strictly convex or
* •
$M=\mathbb{S}^{2}(c)$, $\Omega$ is a closed hemisphere and $\Sigma$ is a
rotationally invariant $H$-sphere for $H=\frac{1}{2}\sqrt{c}$ (in this case
$\kappa_{g}$ identically vanishes).
We recall that, in the case $M=\mathbb{S}^{2}(c)$, each of these disks must
lie in an open hemisphere because they are convex.
## 4\. Further boundary curvature estimates
In this section, we will obtain better estimates for the geodesic curvature of
the boundary by assuming restrictions on the maximum height that the surface
can reach. In order to achieve this, we will use a technique which has its
origins in a paper by Payne and Philippin [8] and which has also been used by
Ros and Rosenberg in [11].
Let $\Sigma\subseteq M\times\mathbb{R}$ be a constant mean curvature $H>0$
surface which is a graph over a domain $\Omega\subseteq M$ and extends
continuously to the boundary of $\Omega$ with zero values. For any given
$m>0$, let consider the function $g_{m}:[-1,1]\rightarrow\mathbb{R}$
determined by
(19) $g_{m}^{\prime}(t)=\frac{4mH}{4H^{2}+c(1-t^{2})},\quad\quad g(0)=0.$
This function allows us to take
$X=\frac{2H\nu(2m-1)}{m(1-\nu)^{2}}E_{3}^{\top}-\frac{2H\nu
g_{m}^{\prime}(\nu)}{m(1-\nu^{2})}AE_{3}^{\top},$
which is a smooth vector field defined on $\Sigma\smallsetminus V$, where
$V=\\{p\in\Sigma:\nu(p)=-1\\}$ is the subset of $\Sigma$ with vertical Gauss
map. We will consider the second order elliptic operator $L$ on
$C^{\infty}(\Sigma\smallsetminus V)$ given by $Lf=\Delta f+X(f)$, and the
function $\psi_{m}=h+g_{m}(\nu)\in C^{\infty}(\Sigma)$. We are now interested
in working out $L\psi_{m}$. By using Lemma 3.1, we obtain
(20)
$L\psi_{m}=-\frac{(m-1)(m-\frac{1}{2})H\nu}{m}-\frac{4Hm\nu(1-\nu^{2})(K_{M}-c)}{4H^{2}+c(1-\nu^{2})}.$
We observe that the second term in the RHS is positive because $K_{M}\geq c$
is satisfied. Moreover, for $m\geq 1$ or $m\leq\frac{1}{2}$, the first term is
also positive so the function $\psi_{m}$ verifies $L\psi_{m}\geq 0$ in
$\Sigma\smallsetminus V$. Thus, it is possible to apply the maximum principle
for the operator $L$ in $\Sigma\smallsetminus V$, which insures that
$\psi_{m}$ cannot achieve an interior maximum in $\Sigma\smallsetminus V$
unless it is constant.
###### Lemma 4.1.
Let $\Sigma\subseteq M\times\mathbb{R}$ be a constant mean curvature $H>0$
graph over a (not necessarilly compact) domain $\Omega\subseteq M$ which
extends continuously to $\partial\Omega$ with zero values and suppose that
$c=\inf\\{K_{M}(p):p\in\Omega\\}>-4H^{2}$. Aditionally, if $\psi_{m}$ is
constant in $\Sigma$ for some $m\leq\frac{1}{2}$, then
* a)
$m=\frac{1}{2}$ and $K_{M}$ is constant in $\Omega$,
* b)
$\Sigma$ is invariant by a $1$-parameter group of isometries.
In particular, if $c>0$ and $M=\mathbb{S}^{2}(c)$, then $\Sigma$ is a compact
rotationally invariant torus and, if $c\leq 0$ and $M=\mathbb{H}^{2}(c)$, then
$\Sigma$ is an invariant horizontal cylinder, both described in Section 2.
###### Proof.
If $\psi_{m}$ is constant, then $L\psi_{m}=0$ so from (20) we get that
$(m-1)(m-\frac{1}{2})\leq 0$ which is only possible if $m=\frac{1}{2}$. Then,
as equality in (20) holds, $K_{M}$ must be constant in $\Sigma\smallsetminus
V$ so it is constant in $\Sigma$ as $K_{M}$ is continuous and $V$ has empty
interior.
Now, suppose that $m=\frac{1}{2}$ and $\psi_{m}$ is constant. On one hand,
from $\nabla\psi_{m}=0$ we obtain
$AE_{3}^{\top}=\frac{1}{g^{\prime}(\nu)}E_{3}^{\top}$ so $E_{3}^{\top}$ must
be a principal direction and $\frac{1}{g^{\prime}(\nu)}$ its corresponding
principal curvature. We also deduce the following expressions:
(21) $\displaystyle\det(A)$
$\displaystyle=\frac{1}{g^{\prime}(\nu)}\left(2H-\frac{1}{g^{\prime}(\nu)}\right),$
$\displaystyle\|\nabla\nu\|^{2}$ $\displaystyle=\langle
AE_{3}^{\top},AE_{3}^{\top}\rangle=\frac{1-\nu^{2}}{g^{\prime}(\nu)^{2}}.$
If we consider the differentiable function $f:\ ]-1,1[\ \rightarrow\mathbb{R}$
determined by
$f^{\prime}(t)=\frac{1}{\sqrt{(1-t^{2})(4H^{2}+c(1-t^{2}))}},\quad\quad
f(0)=0,$
and take into account (21) and Lemma 3.1, it is easy to check that
$\displaystyle\Delta(f(\nu))$
$\displaystyle=f^{\prime\prime}(\nu)\|\nabla\nu\|^{2}+f^{\prime}(\nu)\Delta\nu$
$\displaystyle=\frac{f^{\prime\prime}(\nu)}{g^{\prime}(\nu)^{2}}(1-\nu^{2})+f^{\prime}(\nu)(2\det(A)-4H^{2}-c(1-\nu^{2}))\nu=0$
where we have also used that $K_{M}$ is constant by item (a). As $f(\nu)$ is a
non-constant harmonic function on $\Sigma\smallsetminus V$, we can (at least
locally) take a conformal parameter $z$ on $\Sigma\smallsetminus V$ such that
$\mathop{\rm Re}\nolimits(z)=f(\nu)$. Now, we can repeat the arguments given
in [3, Lemma 6.1] to conclude that $\Sigma$ is invariant by a $1$-parameter
group of isometries of $\mathbb{M}^{2}(c)\times\mathbb{R}$. As $\Sigma$ is an
embedded $H$-graph over a domain $\Omega\subseteq M$ which continuously
extends to $\partial\Omega$ with zero values, the isometries in this group act
trivially on the vertical lines so, if $c\neq 0$, we are in the situation
studied in Section 2 and the only posibilities for $\Sigma$ are those
mentioned in the statement. If $c=0$, it is well-known that $\Sigma$ must be a
horizontal cylinder. ∎
###### Theorem 4.2.
Let $\Sigma\subseteq M\times\mathbb{R}$ be a constant mean curvature $H>0$
graph over a compact regular domain $\Omega$ with zero values in
$\partial\Omega$ and suppose that $c=\inf\\{K_{M}(p):p\in\Omega\\}$ satisfies
that $4H^{2}+c>0$ and $\nu=\nu_{0}$ in $\partial\Omega$ for some
$-1<\nu_{0}\leq 0$.
If there exists $0<m\leq\frac{1}{2}$ such that $|h|\leq
m\cdot\alpha(c,H,\nu_{0})$, then the following lower bound for the geodesic
curvature of $\partial\Omega$ in $M$ (with respect to the outer conormal
vector field) holds:
$\kappa_{g}\geq\frac{(4-8m)H^{2}+c(1-\nu_{0}^{2})}{4mH\sqrt{1-\nu_{0}^{2}}}.$
###### Proof.
Let us consider the function $\psi_{m}=h+g_{m}(\nu)\in C^{\infty}(\Sigma)$,
which verifies that $L\psi_{m}\geq 0$ in view of (20). As $\Sigma$ is compact,
there exists a point $p_{0}\in\Sigma$ where $\psi_{m}$ attains its maximum. We
distinguish three possibilities:
* •
If $p_{0}$ is an interior point of $\Sigma\smallsetminus V$, then $\psi_{m}$
is constant in $\Sigma$, which implies that $m=\frac{1}{2}$, $K_{M}$ is
constant in $\Omega$ and $\Sigma$ is invariant by a $1$-parameter isometry
group because of Lemma 4.1.
* •
If $p_{0}\in\partial\Omega$, then such a maximum is attained in the whole
boundary $\partial\Omega$ since $(\psi_{m})_{|\partial\Omega}$ is constant.
Then the boundary maximum principle for the operator $L$ guarantees that
$\frac{\partial\psi_{m}}{\partial\eta}\geq 0$ along $\partial\Omega$. It is
straightforward to check from (17) that this is equivalent to the inequality
in the statement above.
* •
If $p_{0}\in V$, then $\nu(p_{0})=-1$. The inequality we are looking for would
be proved if we discarded this case, but it turns out that $h\leq
m\cdot\alpha(c,H,\nu_{0})=-g_{m}(-1)$ so
$\psi_{m}\leq\psi_{m}(p_{0})=h(p_{0})+g_{m}(-1)=h(p_{0})-m\cdot\alpha(c,H,\nu_{0})+g_{m}(\nu_{0})\leq
g_{m}(\nu_{0})$ and, since $\psi_{m}$ is equal to $g_{m}(\nu_{0})$ in
$\partial\Omega$, the maximum is also attained in the boundary, which reduces
this case to the previous one.
∎
We now apply the theorem to the compact embedded case, where we lay in the
same situation of Corollary 3.4.
###### Corollary 4.3.
Let $\Sigma\subseteq M\times\mathbb{R}$ be a compact constant mean curvature
$H>0$ surface, symmetric with respect to $M\times\\{0\\}$, and suppose that
$c=\inf\\{K_{M}(p):p\in\Omega\\}$ satisfies $4H^{2}+c>0$. If there exists
$0<m\leq\frac{1}{2}$ such that $h\leq m\cdot\alpha(c,H,0)$, then the following
lower bound for the geodesic curvature of $\partial\Omega$ in $M$ (with
respect to the outer conormal vector field) holds:
$\kappa_{g}\geq\frac{(4-8m)H^{2}+c}{4mH}.$
We now adjust the constant $0<m<\frac{1}{2}$ to guarantee the convexity of the
boundary, as we did in Corollary 3.5.
###### Corollary 4.4.
Let $\Sigma\subseteq M\times\mathbb{R}$ be an $H$-graph over a compact regular
domain $\Omega$ with zero boundary values. Suppose that
$c=\inf\\{K_{M}(p):p\in\Omega\\}$ satisfies that $4H^{2}+c>0$ and
$\nu=\nu_{0}$ in $\partial\Omega$ for some $-1<\nu_{0}\leq 0$. In any of the
following two situations,
* i)
$c\geq 0$ and $h\leq\frac{1}{2}\alpha(c,H,\nu_{0})$ in $\Sigma$ or
* ii)
$c<0$ and $h\leq\frac{4H^{2}+c(1-\nu_{0})^{2}}{8H^{2}}\alpha(c,H,\nu_{0})$ in
$\Sigma$,
the boundary $\partial\Omega$ is convex in $M$ with respect to the outer
conormal vector field.
We finally wonder if the compactness hypothesis for the domain of the graph
can be removed. In order to achieve this, we will restrict ourselves to the
ambient space $\mathbb{M}^{2}(c)\times\mathbb{R}$ and $\nu_{0}=0$ (that is,
$\Sigma$ is an $H$-bigraph), where the technique developed by Ros and
Rosenberg in [11] can be easily adapted.
###### Theorem 4.5.
Let $\Sigma\subseteq M(c)\times\mathbb{R}^{+}$ be a properly embedded
$H$-bigraph over a domain $\Omega\subseteq M$ with $4H^{2}+c>0$, symmetric
with respect to $M\times\\{0\\}$, and suppose that there exists
$0<m\leq\frac{1}{2}$ such that $|h|\leq m\cdot\alpha(c,H,\nu_{0})$ in
$\Sigma$. Then, the following lower bound for the geodesic curvature of
$\partial\Omega$ in $M$ (with respect to the outer conormal vector field)
holds:
$\kappa_{g}\geq\frac{(4-8m)H^{2}+c}{4mH}.$
Furthermore, if there exists $p\in\partial\Omega$ for which equality holds,
then
* i)
$\Sigma$ is a rotationally invariant torus (described in Section 2.1) if
$c>0$,
* ii)
$\Sigma$ is a invariant cylinder under horizontal translations if $c=0$, and
* iii)
$\Sigma$ is a invariant cylinder under hyperbolic translations (described in
Section 2.2) if $c<0$.
###### Proof.
Let us consider the same function $\psi_{m}\in C^{\infty}(\Sigma)$ as before.
If $\psi_{m}$ attained its maximum or $\sup_{\Sigma}\psi_{m}\leq 0$, we could
reason in the same way we did for the compact case and the proof would be
finished. Otherwise, let us take a sequence $\\{p_{n}\\}\subseteq\Sigma$ such
that $\\{\psi_{m}(p_{n})\\}$ converges to $\sup\psi_{m}$, and distinguish two
cases.
* •
If $\lim\\{h(p_{n})\\}=0$, then
$\psi_{m}(p_{n})=h(p_{n})+g_{m}(\nu(p_{n}))\leq h(p_{n})\rightarrow 0$ from
where $\sup_{\Sigma}\psi_{m}\leq 0$ and we are done.
* •
If $\\{h(p_{n})\\}$ does not converge to zero, we can suppose that
$\\{h(p_{n})\\}\rightarrow a>0$ without loss of generality. Now, ambient
isometries allow us to translate $\Sigma$ horizontally so that $p_{n}$ is over
some fixed point $q_{0}\in M$ and standard convergence arguments make possible
to consider $\Sigma_{\infty}$, the limit $H$-graph of these translated
surfaces (note that $\Sigma$ has uniform curvature estimates around $p_{n}$ by
stability, since the distance from $p_{n}$ to $\partial\Sigma$ is bounded away
from zero). The corresponding function in $\Sigma_{\infty}$, given by
$\psi_{m,\infty}=h_{\infty}+g_{m}(\nu_{\infty})\in
C^{\infty}(\Sigma_{\infty})$, attains its maximum at the interior point
$p_{\infty}=\lim\\{p_{n}\\}$ (which is not in the boundary of
$\Sigma_{\infty}$ because $h_{\infty}(p_{\infty})=a>0$) , so it identically
vanishes since $\psi_{\infty}$ vanishes at $\partial\Sigma_{\infty}$. This
implies that $\sup_{\Sigma}\psi_{m}=\psi_{m,\infty}(p_{\infty})=0$ and we are
also done.
In case that equality holds, the description in the statement follows from
Lemma 4.1. ∎
Observe that, if the maximum heights of a sequence $\\{\Sigma_{n}\\}$ of such
$H$-bigraphs tend to zero, then Theorem 4.5 insures that the geodesic
curvatures of the boundaries diverge uniformly, in the sense that the bound
only depends on that maximum height. Thus, the sequence of domains
$\Omega_{n}\subseteq M$ over which $\Sigma_{n}$ is a graph cannot eventually
omit any set in $M$ with nonempty interior.
## 5\. Intrinsic length estimates
Let $\Sigma\subseteq M\times\mathbb{R}$ be an $H$-graph over a compact domain
$\Omega\subseteq M$ which extends continuously to the boundary with zero
values. Suppose that $K_{M}\geq c>-4H^{2}$ in $\Sigma$ for some $c>0$. In
Section 3 we proved that $\psi=h+g(\nu)$ is subharmonic in $\Sigma$, where $g$
is defined in (14) so, if we suppose that $\nu\leq\nu_{0}$ along
$\partial\Sigma$, then $h+g(\nu)\leq g(\nu_{0})$, as a consequence of that $g$
is strictly increasing and $h$ vanishes on $\partial\Sigma$. Therefore, as $g$
is also an odd function, we derive that $g(-\nu)\geq h-g(\nu_{0})$. Now we can
invert the function $g$ and square both sides to obtain
(22)
$\nu^{2}\geq\zeta(h,\nu_{0}):=\begin{cases}\frac{c+4H^{2}}{c}\tanh^{2}\left(\frac{\sqrt{c^{2}+4H^{2}c}}{4H}(h-g(\nu_{0}))\right)&\text{if
}c<0,\\\ H^{2}(h-g(\nu_{0}))^{2}&\text{if }c=0,\\\
\frac{c+4H^{2}}{-c}\tan^{2}\left(\frac{\sqrt{-c^{2}-4H^{2}c}}{4H}(h-g(\nu_{0}))\right)&\text{if
}c>0.\\\ \end{cases}$
Let $\gamma:[a,b]\rightarrow\Sigma$ be a smooth curve which is parametrized by
arc-length and let $\eta$ be a smooth unit vector field along $\gamma$,
orthogonal to $\gamma^{\prime}$. Then, as $\\{N,\alpha^{\prime},\eta\\}$ is an
orthonormal frame, we have
$E_{3}=\langle N,E_{3}\rangle
E_{3}+\langle\gamma^{\prime},E_{3}\rangle\gamma^{\prime}+\langle\eta,E_{3}\rangle\eta,$
and, since $\langle N,E_{3}\rangle=\nu$ and
$\langle\gamma^{\prime},E_{3}\rangle=h^{\prime}(\gamma)$, we deduce that
$1=\nu^{2}+h^{\prime}(\gamma)^{2}+\langle\eta,E_{3}^{\top}\rangle^{2}$. Taking
into account that $\langle\eta,E_{3}^{\top}\rangle^{2}\geq 0$, we finally get
$|h^{\prime}|\leq 1-\nu^{2}$. Thus, plugging (22) into this inequality, we
have
(23) $\mathop{\rm
Long}\nolimits(\gamma)\geq\int_{0}^{a}\frac{|h^{\prime}|}{\sqrt{1-\nu^{2}}}\,\mathrm{d}t\geq\int_{0}^{a}\frac{-h^{\prime}}{\sqrt{1-\zeta(h,\nu_{0})}}\,\mathrm{d}t=\int_{h(a)}^{h(0)}\frac{ds}{\sqrt{1-\zeta(s,\nu_{0})}}.$
Considering all the curves that join a point $p$ with the boundary (along
which the height vanishes), we obtain the following result:
###### Theorem 5.1.
Let $\Sigma\subseteq M\times\mathbb{R}$ be a constant mean curvature $H>0$
graph over a compact domain $\Omega\subseteq M$ which extends continuously to
the boundary with zero values and suppose that
$c=\inf\\{K_{M}(p):p\in\Sigma\\}>-4H^{2}$. If $\nu\leq\nu_{0}$ in
$\partial\Omega$ for some $-1<\nu_{0}\leq 0$, then
$\mathop{\rm
dist}\nolimits(p,\partial\Sigma)\geq\int_{0}^{h(p)}\frac{ds}{\sqrt{1-\zeta(s,\nu_{0})}}.$
Furthermore, if there exists $p\in\Sigma$ such that equality holds, then
$\Omega$ has constant curvature and $\Sigma$ is a spherical cap of a
rotationally invariant sphere.
In other words, Theorem 5.1 is a comparison result which claims that
rotationally invariant spheres in the corresponding homogeneous space
$\mathbb{M}^{2}(c)\times\mathbb{R}$ minimize the distance from a point to the
boundary in terms of the height of that point.
We remark that the bound given in the statement can be worked out explicitly
in terms of elementary functions, but the result of that computation is a
large formula which does not contribute to a better understanding so we have
preferred to leave it in this way.
## References
* [1] J. Aledo, J. Espinar, and J. Gálvez. Height estimates for surfaces with positive constant mean curvature in $\mathbb{M}^{2}\times\mathbb{R}$. Illinois J. Math., 52(1):203–211, 2008.
* [2] K. Große Brauckmann. New surfaces of constant mean curvature. Math. Z., 214:527–565, 1993. MR1248112, Zbl 0806.53005.
* [3] J. M. Espinar and H. Rosenberg. Complete constant mean curvature surfaces in homogeneous spaces. To appear in Comment. Math. Helv., 2009.
* [4] I. Fernandez and P. Mira. A characterization of constant mean curvature surfaces in homogeneous 3-manifolds. Diff. Geom. Appl., 25:281–289, 2007.
* [5] W. T. Hsiang and W. Y. Hsiang. On the uniqueness of isoperimetric solutions and imbedded soap bubbles in noncompact symmetric spaces. I. Invent. Math., 98:39–58, 1989. MR1010154 (90h:53078).
* [6] W. H. Meeks III, J. Pérez, and A. Ros. Stable constant mean curvature surfaces. In Handbook of Geometrical Analysis, volume 1, pages 301–380. International Press, edited by Lizhen Ji, Peter Li, Richard Schoen and Leon Simon, ISBN: 978-1-57146-130-8, 2008. MR2483369, Zbl 1154.53009.
* [7] Irene I. Onnis. Invariant surfaces with constant mean curvature in $\mathbb{H}^{2}\times\mathbb{R}$. Ann. Mat. Pura Appl., 187(4):667–682, 2008.
* [8] L. E. Payne and G. A. Philippin. Some maximum principles for nonlinear elliptic equations in divergence form with applications to capillary surfaces and to surfaces of constant mean curvature. Nonlinear Analysis: Theory, Methods & Applications, 3(2):193–211, 1979.
* [9] R. Pedrosa. The isoperimetric problem in spherical cylinders. Annals of Global Analysis and Geometry, 26(4):333–354, 2004.
* [10] M. Ritoré. Examples of constant mean curvature surfaces obtained from harmonic maps to the two sphere. Math. Z., 226:127–146, 1997.
* [11] A. Ros and H. Rosenberg. Properly embedded surfaces with constant mean curvature. Preprint.
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|
arxiv-papers
| 2010-06-29T17:20:52 |
2024-09-04T02:49:11.305849
|
{
"license": "Public Domain",
"authors": "Jos\\'e M. Manzano",
"submitter": "Jos\\'e Miguel Manzano",
"url": "https://arxiv.org/abs/1006.5683"
}
|
1006.5692
|
# Corrugated single layer templates for molecules: From $h$-BN Nanomesh to
Graphene based Quantum dot arrays
Haifeng Ma Physik-Institut, Universität Zürich, Winterthurerstrasse 190,
CH-8057 Zürich, Switzerland Mario Thomann Physik-Institut, Universität
Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland Jeanette
Schmidlin Physik-Institut, Universität Zürich, Winterthurerstrasse 190,
CH-8057 Zürich, Switzerland Silvan Roth Physik-Institut, Universität Zürich,
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland Martin Morscher Physik-
Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich,
Switzerland Thomas Greber greber@physik.uzh.ch Physik-Institut, Universität
Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
###### Abstract
Functional nano-templates enable self-assembly of otherwise impossible
arrangements of molecules. A particular class of such templates is that of
$sp^{2}$ hybridized single layers of hexagonal boron nitride or carbon
(graphene) on metal supports. If the substrate and the single layer have a
lattice mismatch, superstructures are formed. On substrates like rhodium or
ruthenium these superstructures have unit cells with $\sim$3 nm lattice
constant. They are corrugated and contain sub-units, which behave like traps
for molecules or quantum dots, which are small enough to become operational at
room temperature. For graphene on Rh(111) we emphasize a new structural
element of small extra hills within the corrugation landscape. For the case of
molecules like water it is shown that new phases assemble on such templates,
and that they can be used as ”nano-laboratories” where many individual
processes are studied in parallel. Furthermore, it is shown that the
$h$-BN/Rh(111) nanomesh displays a strong scanning tunneling microscopy
induced luminescence contrast within the 3 nm unit cell which is a way to
address trapped molecules and/or quantum dots.
hexagonal boron nitride, graphene, nano-template, quantum dot, nano-ice,
nanomesh, electroluminescence
## I Introduction
Graphene resounds throughout the land Geim and Novoselov (2007). It is a
single layer of $sp^{2}$ hybridized carbon, has remarkable chemical stability
and physical properties like that of a conductor with high charge carrier
mobility. Her polar sister, hexagonal boron nitride, has similar chemical
stability, though is an insulator.
Single layer hexagonal boron nitride ($h$-BN) and graphene ($g$) also have
great potential as templates for molecular self-assembly. The layers are grown
and supported on transition metal surfaces Oshima and Nagashima (1997); Greber
(2010a). Here we focus on _corrugated_ single layers. The corrugation is a
vertical deformation of the surface that can be described as a static
distortion wave. The physical origin of these distortions are the mismatch and
the concomitant epitaxial stress between the overlayer and the substrate,
where the anisotropic bonding or lock in energy imposes this kind of
dislocations. The much softer out of plane modulus of the $sp^{2}$ layers
causes a large vertical distortion compared to the in-plane straining. The
wavelengths, or superlattice constants, of these static distortion waves can
be calculated from the lattice mismatch between the overlayer and the
substrate. For rhodium and ruthenium it is in the order of 3 nm and the
corrugation, or peak to peak amplitude (between 0.05 and 0.15 nm) are the
essential features and determine the template function. It has been shown that
the corrugation imposes _lateral_ electric fields, which can guide charged or
polarizable media Dil et al. (2008); Brugger et al. (2009). This property
leads to 3 bond hierarchy levels. The $\sigma$-bonds, in the order of 10 eV
provide chemical stability and robustness, the $\pi$-bonds, in the order of 1
eV, the adsorption energies, and the $\alpha$-bonds(named after the label
$\alpha$ for the polarizability $\alpha$), in the order of 100 meV, are
responsible for the lateral trapping of molecules Greber (2010b).
This article covers the basic ingredients of the geometric structure of
lattice mismatched $sp^{2}$ layers on transition metals, their potential as
”nano-laboratories”, where an example of the behavior of nano-ice clusters as
a function of temperature is given. Finally, the potential of such
superstructures as quantum dot arrays is outlined. It is shown that such
quantum dots can be addressed by electroluminescence, where the yield varies
one order of magnitude within the 3 nm unit cell of $h$-BN/Rh(111).
## II Geometric structure
When the lattice mismatch $M$ of an overlayer with the substrate exceeds a
critical value, superstructures with large lattice constants are formed. For
parallel epitaxy we write:
$M=\frac{a_{ovl}-a_{sub}}{a_{sub}}$ (1)
where $a_{ovl}$ and $a_{sub}$ are the overlayer and the substrate lattice
constants, respectively. In this notation positive (negative) $M$ indicate
compressive (tensile) stress in the overlayer, and vice versa. For most
transition metal substrates with $h$-BN or $g$ overlayers the mismatch is
negative, i.e. tensile stress in the overlayer prevails.
If the lattice of the overlayer and the substrate are rigid and parallel, the
superstructure lattice constant gets $a_{ovl}/|M|$, where $a_{ovl}$ is the
1$\times$1 lattice constant of $h$-BN or graphene ($\sim$ 0.25 nm). Besides
the mismatch, the balance between the lock in energy and the strain energy is
decisive for the resulting morphology of the systems. Lock in energy has to be
paid when the over-layer atoms are moved parallel to the surface, away from
the lowest energy sites. For systems with small lock in energy compared to the
strain energy, we expect flat floating layers, reminiscent to incommensurate
moiré patterns without a lock in to a high symmetry direction of the
substrate. Such examples of moiré’s are e.g. $h$-BN/Pd(111)Morscher et al.
(2006); Greber et al. (2009) or $g$/Ir(111) N’Diaye et al. (2006). However, if
the unit cells of the superstructure contain regions with distinct electronic
structure, as it is e.g. the case for $h$-BN/Rh(111) Corso et al. (2004);
Berner et al. (2007) or $g$/Ru(0001) Brugger et al. (2009); Zhang et al.
(2009), it is appropriate to use a term distinct from moiré, like it is
’nanomesh’. Preobrajenski et al. were the first who also used the term
nanomesh for $g$-systems where two distinct carbon core levels have been found
Preobrajenski et al. (2008). The energy difference in the core level binding
energies was assigned to the corrugation, i.e. different ’elevations’ or
distances of the $sp^{2}$ layers from the substrate. For a superstructure of a
honeycomb lattice of $sp^{2}$ hybridized layers on a hexagonal closed packed
substrate as Rh(111) or Ru(0001), it is convenient to describe different
locations along the notation used by Auwärter and Grad et al. Auwärter et al.
(1999); Grad et al. (2003). The honeycomb lattice is made of a base with two
atoms (B,N) or (CA,CB). These atoms may sit on $top$, on $fcc$ or on $hcp$
sites within the substrate unit cell (see Figure 1). The $top$ site is
occupied if an atom of the honeycomb sits on top of a substrate atom, below
the $fcc$ hollow site no atom is found in the second substrate layer, while
there is one for the $hcp$ hollow site.
Figure 1: Scheme for the classification of the adsorption sites on a hexagonal
closed packed surface: $top$, $fcc$, and $hcp$. The dashed line shows the
(1$\times$1) unit cell. After Ref. Auwärter et al. (1999)
We note that for single domain $h$-BN structures only three of the six
combinations of the honeycomb base like ($top$,$fcc$), ($hcp$,$top$),
($fcc$,$hcp$) or ($fcc$,$top$), ($top$,$hcp$), ($hcp$,$fcc$) occur. For
graphene, where the two carbon atoms are distinct by their coordination to the
substrate only, no such domains are expected. When the honeycomb is
mismatched, i.e. does not have the lattice constant of the substrate, the
assignments (B,N)=($fcc$,$top$),($hcp$,$fcc$) etc. are only approximately
valid and we write e.g. (CA,CB)$\sim$($top$, $hcp$). With this scheme in mind
we may understand the complementarity of mismatched $h$-BN and $g$
’nanomeshes’, where it is found that about one third of the super cell
(B,N)$\sim$($hcp$,$fcc$) and (CA,CB)$\sim$($hcp$,$fcc$) do not bind to the
substrate and consequently belong to the elevated regions. For $h$-BN there is
also no bonding for boron on $top$ sites i.e. (B,N)$\sim$($top$,$hcp$), and
consequently two thirds of the mismatched $h$-BN layers are elevated and form
the connected ’wire’ network. Graphene is complementary i.e. the
(CA,CB)$\sim$($top$,$hcp$) and the (CA,CB)$\sim$($fcc$,$top$) sites bind to
the substrate and form a connected hexagonal ’valley’ network, with graphene
in close contact to the substrate Brugger et al. (2009). It has to be
emphasized that the substrate breaks the symmetry between the sublattice made
of CA and CB atoms, respectively, and disables the formation of Dirac cones,
which are the attribute of freestanding graphene. For $h$-BN this symmetry
breaking is intrinsic, since B and N are different and induce polarity with
electron transfer to nitrogen.
Figure 2 shows room temperature scanning tunneling microscopy (STM) pictures
of $g$/Rh(111) and $h$-BN/Rh(111). The relief views in a) and c) are extracted
by the WSxM Scanning Probe Microscopy software Horcas et al. (2007) from the
scanning tunneling microscopy data in b) $h$-BN/Rh(111) and d) $g$/Rh(111),
respectively. Clearly, the inverted topographies of the two layer systems can
be seen. The $h$-BN/Rh(111) nanomesh has a 12$\times$12 periodicity where 13
BN units sit on 12 Rh substrate units, which corresponds to a 3.2 nm
superlattice constant Corso et al. (2004). The labels for the two topographic
elements are ’holes’, ’pores’, ’cavities’ or ’cells’ for the (B,
N)=($fcc$,$top$) regions with close binding and ’wires’ for the (B,
N)=($hcp$,$fcc$) & ($top$,$hcp$) regions, which are elevated by about 0.1 nm.
The $g$/Rh(111) ’nanomesh’ has a periodicity of about 11$\times$11, where 12
$g$ units sit on 11 Rh substrate units, which corresponds to a 2.9 nm
superlattice constant Müller et al. (2009). The slightly smaller unit cell is
due to the smaller lattice constant of graphene compared to that of hexagonal
boron nitride. The labels for the two topographic elements are ’mounds’,
’hills’ or ’ripples’ for the ($C_{A}$, $C_{B}$)=($hcp$,$fcc$) protrusions with
loose binding and ’valleys’ for the ($C_{A}$, $C_{B}$)=($fcc$,$top$) &
($top$,$hcp$) regions, which are about 0.1 nm closer to the substrate. For the
case of $g$/Rh(111) we would like to mention a difference, compared to the
related $g$/Ru(0001) system. It can be seen that the strongest bonding does
not coincide with the ($fcc$,$top$) or ($top$,$hcp$) sites but 3 small extra
dips in the valleys where carbon atoms are closer to bridge sites are binding
strongest to the substrate Iannuzzi . This, compared to $g$/Ru(0001), new
structural element might impose extra effects in the template function and
should be further explored.
Figure 2: Scanning tunneling microscopy data of $sp^{2}$ single layers on
Rh(111). a) Relief view of $g$/Rh(111). Note the hills and the valleys, and
the small extra hills at the ($hcp$,$fcc$) sites. b) corresponding STM picture
($I_{t}=0.8$ nA, $U_{t}=-0.8$ V , 11$\times$14 nm2). c) Relief view of
$h$-BN/Rh(111) nanomesh. The elevated regions form the so called wires of the
nanomesh. d) corresponding STM picture ($I_{t}=1$ nA, $U_{t}=1$ V,
11$\times$14 nm2).
As it was inferred by photoemission from adsorbed xenon for $h$-BN/Rh(111) Dil
et al. (2008) and $g$/Ru(0001) Brugger et al. (2009), in both systems ’high’
or ’elevated’ regions have a high local work function while low regions have a
lower local work function Brugger et al. (2009). These physically and
electronically corrugated landscapes form templates for the self-assembly of
molecular arrays as it was shown for $h$-BN nanomesh Corso et al. (2004);
Berner et al. (2007); Ma et al. (2010), $g$/Rh(111) Pollard et al. (2010) or
$g$/Ru(0001) Mao et al. (2009). Also it was demonstrated that these substrates
may be used for the growth of metal nano-clusters Zhang et al. (2008); Pan et
al. (2009).
## III Nano-laboratories: Water on the $h$-BN nanomesh
Here we want to highlight the opportunity to use a template like the $h$-BN
nanomesh as a nano-laboratory, where processes may be studied in a parallel
fashion, i.e. in an ensemble, at the same time, under the same temperature and
pressure conditions. For scanning probes this also comprises the added value
that the data are recorded with the same tip. These features increase the data
flux from the experiment by orders of magnitude. In particular we expect that
e.g. the diversification in growth processes may be studied. This nano-
laboratory assay allows, to study equilibrium as well as non equilibrium
processes. If we want to lend a picture from biology the $h$-BN nanomesh can
be considered as a ”cell culture”, though at a 3 orders of magnitude smaller
length scale. In order to illustrate the ”nano-laboratory” we show data on the
temperature evolution of water clusters in the $h$-BN nanomesh. Recently it
was shown that water self-assembles in the ’holes’ of $h$-BN nanomesh in
forming ice nanoclusters containing about 40 water molecules. The clusters
consist in a bilayer of ice, reminiscent to the basal plane of hexagonal ice,
and display proton disorder that was accessed with tunneling barrier height
measurements Ma et al. (2010). In Figure 3 the development of the ice clusters
as a function of temperature is shown, for 5 different temperatures, About 7
nanomesh ”cells” with a diameter of 2 nm contain each one ice cluster. It has
to be mentioned that the results for different temperatures do not show the
same cells, because the thermal drift in the present set up during the warm up
from 34 to 151 K does not yet allow to track individual cells. The
superstructure periodicity does, however, allow an almost perfect drift
correction at a given temperature. It can be clearly seen that the rims of the
ice clusters have a distinct behavior with respect to the bulk. Figure 3 a)
shows the ordered ice clusters made by about 40 water molecules at 34 K. Every
second water molecule shows up as a protrusion inside the $h$-BN nanomesh. At
101 K the edge of the clusters start to display disorder (Figure 3 b). Further
increase of the temperature induces an increase of the cluster height (Figure
3 c) and d)) and at 151 K the clusters sublimated (Figure 3 e) and bare
nanomesh is observed.
Figure 3: Behavior of nano-ice clusters in the temperature range between 34 K
and 151 K as recorded by variable temperature scanning tunneling microscopy
(VT-STM). The 5 hexagonal frames with a side length of 4.2 nm show the
constant current feed-back signal, after drift correction. (a) Nano-ice
clusters at 34 K. $V_{t}=-0.05$ V, $I_{t}=100$ pA. (b) At 101 K water
molecules at the rims of the nano-ice cluster become mobile, the core remains
frozen. $V_{t}=-0.05$ V, and $I_{t}=100$ pA. (c) Low coordinated water induced
protrusions at the cluster boundary are observed at 139 K, the core features
weaken. $V_{t}=-0.01$ V, and $I_{t}=50$ pA. (d) as (c), the rims show higher
contrast than the core molecules which still show crystallinity at 143 K.
$V_{t}=-0.01$ V, and $I_{t}=50$ pA. (e) Empty nanomesh after water desorbed
from the surface. $V_{t}=-0.05$ V, and $I_{t}=40$ pA.
## IV Addressing Room temperature Quantum dots
A quantum dot is a small physical object that is confined in 3 dimensions,
where nuclei and atoms are the most prominent examples. The localization to a
’point’ implies the absence of dispersion of the electronic states. The size
of a quantum dot determines the energy level spacing. If this spacing $\Delta
E$ is compared to $k_{B}T$ we get a measure for the temperature below which we
expect the ’dots’ to behave like quantum objects, or above which the occupancy
of different levels fluctuates. The Rydberg energy (13.6 eV), which is the
scale for the electronic level spacing in a Coulomb potential of a proton is
proportional to ${a_{o}}^{-1}$, where $a_{0}$=0.05 nm is the Bohr radius. If
we are interested in quantum dots that are operational at room temperature
($k_{B}T$=25 meV), this limits the size of quantum dots to below 500 $a_{o}$.
Though, for practical purposes the level spacing should be at least one order
of magnitude larger than $k_{B}T$ and thus room temperature quantum dots
should be smaller than 5 nm.
The $sp^{2}$ templates discussed in this paper do match this condition. Indeed
quantum dot behavior was found for graphene on ruthenium Zhang et al. (2009).
For $g$/Ru(0001) photoemission showed one set of of $sp^{2}$ valence bands,
while $h$-BN/Ru(0001) and $h$-BN/Rh(111) do show two $sp^{2}$ valence band
structures split by about 1 eV Goriachko et al. (2007). The two band structure
systems were assigned to the two regions within the super cells, where the
corrugation of the $h$-BN imposed, a mainly electrostatically driven splitting
Laskowski et al. (2007); Berner et al. (2007). As expected this splitting is
also observed with high resolution B 1s, C 1s and N 1s core level
spectroscopies Preobrajenski et al. (2008), and photoemissions from adsorbed
Xe Dil et al. (2008); Brugger et al. (2009). Interestingly the valence band
splitting was not observed for $g$/Ru(0001) Brugger et al. (2009). The obvious
difference between graphene and $h$-BN is that graphene on ruthenium has a
Fermi surface, while $h$-BN on Ru(0001) Brugger et al. (2009) or on Rh(111)
Greber et al. (2009) has not, may not explain this with a screening argument,
since the C 1s core level is still split on $g$/Ru(0001) Preobrajenski et al.
(2008). The seeming paradox can be resolved, when we assign to the hills in
the $g$/Ru(0001) superstructure an isolated, molecule like behavior, without
dispersion, which qualifies them as quantum dots arranged on a hexagonal array
with 3 nm spacing. For the case of the $h$-BN, the holes might also be
identified as quantum dots, however, angular resolved photoemission shows
dispersion of the $h$-BN valence bands, also for the bands assigned to the
holes that are separated by the superlattice constant of 3 nm Brugger et al.
(2009). A difference between the hills of $g$/Ru(0001) and the holes of
$h$-BN/Ru(0001) is the fact that $h$-BN holes are in close contact to the
substrate, while graphene hills are decoupled. This imposes a lateral vacuum
tunneling barrier for electrons on the hills, while this barrier is much lower
for the case of the $h$-BN holes that are in close proximity to the metal
substrate.
If the $sp^{2}$ templates are decorated with molecules (or clusters), the
quantum dots change and the coupling between them will be affected. It will be
interesting to further explore this coupling and to try to control it.
Electroluminescence could serve as a tool to access such information. With
scanning tunneling microscopy induced electroluminescence, light emission can
be probed as a function of the tunneling site with sub-wavelength resolution.
As we show here it is one possible road to access single unit cells and is
considered to be a realization of a nano-device, where it comes to the
transport of information localized at the nanometer scale to the macroscopic
millimeter scale.
Electroluminescence in scanning tunneling microscopy was pioneered by
Gimzewski et al. Gimzewski et al. (1988), and is a way to record inelastic
scattering in tunneling junctions, where e.g. molecular vibrations may be
resolved, if the wavelength of the emitted photons is measured Wu et al.
(2006). Figure 4 shows the correlation between topography and light emission
from $h$-BN nanomesh. Light was collected by a lens system connecting the
tunneling junction with a cooled red sensitive Hamamatsu R5929 photomultiplier
tube operating in the wavelength window between 300 and 850 nm. The tunneling
voltage was set to -2.5 V (tunneling electrons from the substrate to the tip)
and tungsten tips with a 80 nm gold coating were used. For isotropic emission
into the $2\pi$ half space above a surface, the detection probability of a 2
eV photon is 0.4 %, and the dark count rate was about 25 counts/s. The photon
map in Figure 4 b) and the corresponding cut in Figure 4 c) show strong
electroluminescence from the nanomesh ’holes’, which is more than one order of
magnitude larger than that from the wires. Though, the average quantum
efficiency (detected photons per scan line) varies in the shown data by one
order of magnitude. This variation must be ascribed to changes in the gold tip
where plasmon excitations/deexcitations cause photon emission. The arrows on
the right of Figure 4a) and b) indicate three distinct tip changes $A,B,C$. It
can be seen that the topography image before change $A$ is inverted after
change $C$. The enhancement of the quantum efficiency at change $B$ does
neither coincide with $A$ nor $C$. However, the results suggest that the
tunneling junction with the tip on top of a hole of the $h$-BN nanomesh
imposes more inelastic scattering events on the tip. The high electron
affinity and the concave form of the holes lets them act like a resonator
cavity, where the electrons are focused on the tip, and where the probability
for a plasmon excitation increases. It has to be mentioned that the inverse
situation is observed for graphene on Ru, where the poor electron affinity and
the convexity of the hills defocus electrons in a tunneling junction with the
tip on top of the hill Zhang et al. (2009).
The data shown in Figure 4 indicate that scanning tunneling microscopy induced
luminescence can be used for the identification of sites within the
12$\times$12 super cell of $h$-BN/Rh(111) with sub-nanometer resolution. Also
the experiments indicate that the control of the tip parameters is crucial for
a successful application of this effect.
Figure 4: Room temperature scanning tunneling microscopy and photon emission
scanning tunneling microscopy from $h$-BN nanomesh with a gold coated tungsten
tip. (30$\times$15 nm, $I_{t}=2.6$ nA, $V_{t}=-2.5$ V, scan time 110 s with
128 horizontal scanning lines from bottom to top). The labels $A,B,C$ indicate
tip changes (for details see text). a) Topography, note the 3 nm periodicity
of $h$-BN nanomesh, and the tip changes $A$, $B$. b) Light map, i.e.
simultaneously to a) recorded photoncurrent. For a certain line series the
luminescence is particularly high, and the periodicity of the $h$-BN nanomesh
lights up. c) Cut across the light map, along the red line in b). The
polychromatic light current is given in photons/s.
## V Summary
In summary it is recalled that lattice mismatched $sp^{2}$ hybridized single
layers of $h$-BN and graphene may be used as templates for the self-assembly
of molecular structures. For $g$/Rh(111) a new structural element, extra
”hills” in the valleys, are emphasized. In a second section it is shown that
$h$-BN/Rh(111) can be used as a ”nano-laboratory”, where molecular processes
in individual nanomesh cells may be studied. Finally it is outlined that these
superstructures have the features of quantum dots, which are small enough to
become operational at room temperature. As an example on how such objects can
be addressed the luminescence as induced by a scanning tunneling microscope is
demonstrated to have a resolution better than one nanometer.
## VI Acknowledgements
Financial support by the Swiss National Science Foundation is gratefully
acknowledged.
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|
arxiv-papers
| 2010-06-29T17:57:19 |
2024-09-04T02:49:11.314223
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Haifeng Ma, Mario Thomann, Jeanette Schmidlin, Silvan Roth, Martin\n Morscher, and Thomas Greber",
"submitter": "Haifeng Ma",
"url": "https://arxiv.org/abs/1006.5692"
}
|
1006.5725
|
# Nonstable $K$–theory for extension algebras of the simple purely infinite
$C^{*}$–algebra by certain $C^{*}$–algebras
Zhihua Li ∗ and Yifeng Xue ∗∗
###### Abstract.
Let $0\longrightarrow\mathcal{B}\stackrel{{\scriptstyle
j}}{{\longrightarrow}}E\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}\mathcal{A}\longrightarrow
0$ be an extension of $\mathcal{A}$ by $\mathcal{B}$, where $\mathcal{A}$ is a
unital simple purely infinite $C^{*}$–algebra. When $\mathcal{B}$ is a simple
separable essential ideal of the unital $C^{*}$–algebra $E$ with
$\mathrm{RR}(\mathcal{B})=0$ and (PC), $K_{0}(E)=\\{[p]\mid p$ is a projection
in $E\setminus B\\}$; When $B$ is a stable $C^{*}$–algebra,
$\mathfrak{U}(C(X,E))/\mathfrak{U}_{0}(C(X,E))\cong K_{1}(C(X,E))$ for any
compact Hausdorff space $X$.
Keywords $K$-groups; simple purely infinite $C^{*}$–algebra; real rank zero.
2000 MR Subject Classification 46L05.
Department of Mathematics and Computer Science, Yichun University, Yichun,
Jiangxi, 336000
Department of Mathematics, East China Normal University, Shanghai 200241.
email: yfxue@math.ecnu.edu.cn
Project supported by Natural Science Foundation of China (no.10771069) and
Shanghai Leading Academic Discipline Project(no.B407)
## 1\. Introduction
Let $\mathcal{E}$ be a $C^{*}$–algebra. Denote by
$\mathrm{M}_{n}(\mathcal{E})$ the $C^{*}$–algebra of all $n\times n$ matrices
over $\mathcal{E}$. If $\mathcal{E}$ is unital, write
$\mathfrak{U}(\mathcal{E})$ to denote the unitary group of $\mathcal{E}$ and
$\mathfrak{U}_{0}(\mathcal{E})$ to denote the connected component of the unit
in $\mathfrak{U}(\mathcal{E})$. Put
$U(\mathcal{E})=\mathfrak{U}(\mathcal{E})/\mathfrak{U}_{0}(\mathcal{E})$. If
$\mathcal{E}$ has no unit, we set
$U(\mathcal{E})=\mathfrak{U}(\mathcal{E}^{+})/\mathfrak{U}_{0}(\mathcal{E}^{+})$,
where $\mathcal{E}^{+}$ is the $C^{*}$–algebra obtained by adding a unit to
$\mathcal{E}$. Two projections $p,\ q$ in $\mathcal{E}$ are equivalent,
denoted $p\sim q$, if $p=v^{*}v,q=vv^{*}$ for some $v\in\mathcal{E}$. Let
$[p]$ denote the equivalence of $p$ with respect to “$\sim$”. Let $p,\ r$ be
projections in $\mathcal{E}$. $[p]\leq[r]$ (resp. $[p]<[r]$) means that there
is projection $q\leq r$ (resp. $q<r$) such that $p\sim q$. A projection $p$ in
$\mathcal{E}$ is called to be infinite, if $[p]<[p]$. The simple
$C^{*}$–algebra $\mathcal{E}$ is called to be purely infinite if every nonzero
hereditary subalgebra of $\mathcal{E}$ contains an infinite projection.
Let $K_{0}(\mathcal{E})$ and $K_{1}(\mathcal{E})$ be the $K$–groups of the
$C^{*}$–algebra $\mathcal{E}$ and let $i_{\mathcal{E}}\colon
U(\mathcal{E})\rightarrow K_{1}(\mathcal{E})$ be the canonical homomorphism
(cf. [1]).
The main tasks in non–stable $K$–theory are how to use the projection in
$\mathcal{E}$ to represent $K_{0}(\mathcal{E})$ and how to show
$i_{\mathcal{E}}$ is isomorphic. Cuntz showed in [2] that
$K_{0}(\mathcal{E})\cong\\{[p]|\,p\in\mathcal{E}\ \text{nonzero
projection}\\}$ and $i_{\mathcal{E}}$ is isomorphic, when $\mathcal{E}$ is a
simple unital purely infinite $C^{*}$–algebra. Rieffel and Xue proved that
under some restrictions of stable rank on the $C^{*}$–algebra $\mathcal{E}$,
$i_{\mathcal{E}}$ may be injective, surjective or isomorphic (cf. [6, 7],
[12]).
Let $\mathcal{B}$ be a closed ideal of a unital $C^{*}$–algebra $E$. Let
$\pi\colon E\rightarrow E/\mathcal{B}=\mathcal{A}$ be the quotient map. We
will use these symbols $E$, $\mathcal{B}$, $\mathcal{A}$ and $\pi$ throughout
the paper. Liu and Fang proved in [5] that
1. (1)
$K_{0}(E)=\\{[p]|\,p\ \text{is a projection in}\ E\backslash\mathcal{B}\\}$
and
2. (2)
$i_{E}\colon U(E)\rightarrow K_{1}(E)$ is isomorphic.
when $\mathcal{B}=\mathcal{K}$ (the algebra of compact operators on some
separable Hilbert space) and $\mathcal{A}$ is a unital simple purely infinite
$C^{*}$–algebra. Visinescu showed in [10] that the above results are also true
when $\mathcal{B}$ is purely infinite.
In this short note, we show that (1) is true when $\mathcal{B}$ is a separable
simple $C^{*}$–algebra with $\mathrm{RR}(\mathcal{B})=0$ and (PC) (see §2
below) and $\mathcal{A}$ is unital simple purely infinite; We also prove that
$i_{C(X,E)}$ is isomorphic for any compact Hausdorff space $X$ when
$\mathcal{B}$ is stable and $\mathcal{A}$ is unital simple purely infinite.
## 2\. $K_{0}$–group of the extension algebra
Let $\mathcal{E}$ be a $C^{*}$–algebra. $\mathcal{E}$ is of real rank zero,
denoted by $\mathrm{RR}(\mathcal{E})=0$, if every self–adjoint element in
$\mathcal{E}$ can be approximated by an self–adjoint element in $\mathcal{E}$
with finite spectra (cf. [3]). A non–unital, $\sigma$–unital $C^{*}$–algebra
$\mathcal{E}$ with $\mathrm{RR}(\mathcal{E})=0$ is said to have property (PC)
if it $\mathcal{E}$ has finitely many (densely defined) traces, say
$\\{\tau_{1},\cdots,\tau_{k}\\}$ such that following conditions are satisfied:
1. (1)
there is an approximate unit $\\{e_{n}\\}$ of $\mathcal{E}$ consisting of
projections such that $\lim\limits_{n\to\infty}\tau_{i}(e_{n})=\infty$,
$i=1,\cdots,k;$
2. (2)
for two projections $p,\,q\in\mathcal{E}$, if $\tau_{i}(p)<\tau_{i}(q)$,
$i=1,\cdots,k$, then $[p]\leq[q]$.
Obviously, stable simple AF–algebras with only finitely many extremal traces
have (PC) and $\mathcal{A}_{\theta}\otimes\mathcal{K}$ also has (PC), where
$\mathcal{A}_{\theta}$ is the irrational rotation algebra and $\mathcal{K}$ is
the algebra of compact operators on some complex separable Hilbert space.
###### Remark 2.1.
Let $\mathcal{E}$ be a non–unital, $\sigma$–unital $C^{*}$–algebra with
$\mathrm{RR}(\mathcal{E})=0$ and (PC). Let $\\{f_{n}\\}$ be an approximate
unit of $\mathcal{E}$ consisting of increased projections. Suppose
$\lim\limits_{n\to\infty}\tau_{i}(e_{n})=\infty$, $i=1,\cdots,k$, for some
approximate unit $\\{e_{n}\\}$ of $\mathcal{E}$ consisting of projections.
Then there $\\{e_{n_{j}}\\}\subset\\{e_{n}\\}$ such that
$\tau_{i}(e_{n_{j}})>j$, $j\geq 1$, $i=1,\cdots,k$. Since
$\lim\limits_{s\to\infty}\|f_{s}e_{n_{j}}f_{s}-e_{n_{j}}\|=0$, $j\geq 1$, we
can find projections $f_{s_{j}}\leq f_{s}$ for $s$ large enough such that
$f_{s_{j}}\sim e_{n_{j}}$, $j\geq 1$. Then
$\tau_{i}(f_{s})\geq\tau_{i}(f_{s_{j}})=\tau_{i}(e_{n_{j}})>j,\quad
i=1,\cdots,k,$
so that $\lim\limits_{n\to\infty}\tau_{i}(f_{n})=\infty$, $i=1,\cdots,k$.
With symbols as above, we can extend $\tau_{i}$ to $M(\mathcal{E})$ by
$\tau_{i}(x)=\sup\limits_{n\geq 1}\tau_{i}(f_{n}xf_{n})$ for positive element
$x\in M(\mathcal{E})$ (cf. [4, P324]), $i=1,\cdots,k$, where $M(\mathcal{E})$
is the multiplier algebra of $\mathcal{E}$.
###### Lemma 2.2.
Suppose that $\mathcal{B}$ is an essential ideal of $E$ and
$\mathcal{A},\,\mathcal{B}$ are simple. Then every positive element in
$E\backslash\mathcal{B}$ is full.
###### Proof.
Let $a\in E\backslash\mathcal{B}$ with $a\geq 0$ and let $I(a)$ be closed
ideal generated by $a$ in $E$. Since $\pi(I(a))$ is a nonzero closed ideal in
$\mathcal{A}$ and $\mathcal{A}$ is simple, we get that
$1_{\mathcal{A}}\in\pi(I(a))$ and hence there is $x\in\mathcal{B}$ such that
$1_{E}+x\in I(a)$. Since $\mathcal{B}$ is an essential ideal, it follows that
$a\mathcal{B}a\not=\\{0\\}$. Choose a nonzero element
$b\in\overline{a\mathcal{B}a}\subset I(a)$. Since $\mathcal{B}$ is simple, $x$
is in the closed ideal of $\mathcal{B}$ generated by $b$. Thus, $x\in I(a)$
and consequently, $1_{E}\in I(a)$. ∎
The following lemma slightly improves Lemma 2.1 of [10], whose proof is
essentially same as it in [11, Lemma 3.2] and [10, Lemma 2.1].
###### Lemma 2.3.
Suppose that $\mathrm{RR}(\mathcal{B})=0$. Let $p,\,q$ be projections in $E$
and assume that there is $v\in\mathcal{A}$ such that $\pi(p)=v^{*}v$ and
$vv^{*}\leq\pi(q)$ in $\mathcal{A}$. Then there is a projection $e\in
p\mathcal{B}p$ and a partial isometry $u\in E$ such that $p-e=u^{*}u$,
$uu^{*}\leq q$ and $\pi(u)=v$.
###### Proof.
Let $v\in\mathcal{A}$ such that $\pi(p)=v^{*}v,\ vv^{*}\leq\pi(q)$. Choose
$u_{0}\in E$ such that $\pi(u_{0})=v$ and set $w=qu_{0}p$. Then
$\pi(w^{*}w)=\pi(p),\ \pi(w)=v$. Thus, $p-w^{*}w\in p\mathcal{B}\,p$. Since
$\mathrm{RR}(\mathcal{B})=0$, $p\mathcal{B}p$ has an approximate unit
consisting of projections. So there is a projection $e\in p\mathcal{B}p$ such
that
$\|(p-e)(p-w^{*}w)(p-e)\|=\|(p-e)-(p-e)w^{*}w(p-e)\|<1.$
Then $z=(p-e)w^{*}w(p-e)$ is invertible in $(p-e)E(p-e)$ and $\pi(z)=\pi(p)$.
Let $s=\big{(}(p-e)w^{*}w(p-e)\big{)}^{-1}$, i.e., $zs=sz=p-e$. Then
$\pi(s)=\pi(p)$. Put $u=ws^{\frac{1}{2}}$. Then $uu^{*}=wsw^{*}\leq q$,
$\pi(u)=v$ and
$\displaystyle u^{*}u=$ $\displaystyle
s^{\frac{1}{2}}w^{*}ws^{\frac{1}{2}}=s^{\frac{1}{2}}(p-e)w^{*}w(p-e)s^{\frac{1}{2}}$
$\displaystyle=$ $\displaystyle(p-e)w^{*}w(p-e)s=p-e.$
∎
###### Lemma 2.4.
Suppose that $\mathcal{A}$ is unital simple purely infinite and $\mathcal{B}$
is an essential ideal of a unital $C^{*}$–algebra $E$, moreover $\mathcal{B}$
is separable simple with $\mathrm{RR}(\mathcal{B})=0$ and (PC). Let $p,\,q$ be
projections in $E\backslash\mathcal{B}$ and let $r$ be a nonzero projection in
$p\mathcal{B}p$. Then there is a projection $r^{\prime}$ in $q\mathcal{B}q$
such that $[r]\leq[r^{\prime}]$.
###### Proof.
Since $\mathcal{B}$ has (PC), there are densely defined traces
$\tau_{1},\cdots,\tau_{k}$ on $\mathcal{B}$ and an approximate unit
$\\{f_{n}\\}$ of $\mathcal{B}$ consisting of increased projections such that
$\lim\limits_{n\to\infty}\tau_{i}(f_{n})=\infty$, $i=1,\cdots,k$ and
$\tau_{i}(e)<\tau_{i}(f)$, $i=1,\cdots,k$ implies that $[e]\leq[f]$ for any
two projections $e,\,f$ in $\mathcal{B}$.
By Lemma 2.2, there are $x_{1},\cdots,x_{m}\in\mathcal{B}$ such that
$\sum\limits^{m}_{i=1}x_{i}^{*}qx_{i}=1_{E}$. We regard $E$ as a
$C^{*}$–subalgebra of $M(\mathcal{B})$ for $\mathcal{B}$ is essential. Thus,
$\infty=\tau_{i}(1_{E})=\sum\limits^{m}_{j=1}\tau_{i}(x^{*}_{j}qx_{j})\leq\sum\limits^{m}_{j=1}\tau_{i}(\|x_{j}\|^{2}q),$
i.e., $\tau_{i}(q)=\infty$, $i=1,\cdots,k$. Let $r$ be a nonzero projection in
$p\mathcal{B}p$. Let $\\{g_{n}\\}$ be an approximate unit for $q\mathcal{B}q$
consisting of increased projections. Since $\sup\limits_{n\geq
1}\tau_{i}(g_{n})=\tau_{i}(q)=\infty$, $i=1,\cdots,k$, it follows that there
is $n_{0}$ such that $\tau_{i}(g_{n_{0}})>\tau_{i}(r)$, $i=1,\cdots,k$. Put
$r^{\prime}=g_{n_{0}}$. Then we get $[r]\leq[r^{\prime}]$. ∎
Now we can prove the main result of the section as follows:
###### Theorem 2.5.
Suppose that $\mathcal{A}$ is unital simple purely infinite and $\mathcal{B}$
is an essential ideal of $E$, moreover $\mathcal{B}$ is separable simple with
$\mathrm{RR}(\mathcal{B})=0$ and (PC). Then
$K_{0}(E)=\\{[p]|\,p\ \text{is a projection in}\ E\backslash\mathcal{B}\\}.$
###### Proof.
Set $\mathcal{P}(E)=\\{p\ \text{is a projection in}\
E\backslash\mathcal{B}\\}$. By [2, Theroem 1.4], when $\mathcal{P}(E)$
satisfies following conditions:
1. $(\Pi_{1})$
If $p,\ q\in\mathcal{P}(E)$ and $pq=0$, then $p+q\in\mathcal{P}(E);$
2. $(\Pi_{2})$
If $p\in\mathcal{P}(E)$ and $p^{\prime}$ is a projection in $E$ such that
$p\sim p^{\prime}$, then $p^{\prime}\in\mathcal{P}(E);$
3. $(\Pi_{3})$
For any $p,q\in\mathcal{P}(E)$, there is $p^{\prime}$ such that
$p^{\prime}\sim p,\ p^{\prime}<q$ and $q-p^{\prime}\in\mathcal{P}(E);$
4. $(\Pi_{4})$
If $q$ is a projection in $E$ and there is $p\in\mathcal{P}(E)$ such that
$p\leq q$, then $p\in\mathcal{P}(E)$,
then $K_{0}(E)=\\{[p]|\,p\in\mathcal{P}(E)\\}$. Therefore, we need only check
that $\mathcal{P}(E)$ satisfies above conditions.
Let $\mathcal{P}(\mathcal{A})$ be the set of all nonzero projections in
$\mathcal{A}$. By [2, Proposition 1.5], $\mathcal{P}(\mathcal{A})$ satisfies
$(\Pi_{1})\sim(\Pi_{4})$. Clearly, $\mathcal{P}(E)$ satisfies $(\Pi_{1})$,
$(\Pi_{2})$ and $(\Pi_{4})$. We now show that $\mathcal{P}(E)$ satisfies
$(\Pi_{3})$.
Let $p,\,q\in\mathcal{P}(E)$. Then there exists a projection
$f\in\mathcal{P}(\mathcal{A})$, such that $f\sim\pi(p)$, $f<\pi(q)$ and
$\pi(q)-f\in\mathcal{P}(\mathcal{A})$, that is, there is a partial isometry
$v\in\mathcal{A}$ such that $f=v{v}^{*}<\pi(q)$ and $\pi(p)={v}^{*}v$. Thus,
there are $u\in E$ and a projection $r\in p\mathcal{B}p$ such that
$p-r=u^{*}u$, $uu^{*}\leq q$ and $\pi(u)=v$ by Lemma 2.3. Note that
$q-uu^{*}\not\in\mathcal{B}$ and $(q-uu^{*})\mathcal{B}(q-uu^{*})\not=\\{0\\}$
($\mathcal{B}$ is an essential ideal). Then by Lemma 2.4, there is
$w_{0}\in\mathcal{B}$ such that $r=w_{0}^{*}w_{0}$,
$w_{0}w_{0}^{*}\in(q-uu^{*})\mathcal{B}(q-uu^{*})$. Put $\hat{u}=u+w_{0}$.
Then $p=\hat{u}^{*}\hat{u}$, $\hat{u}\hat{u}^{*}\leq q$ and
$\pi(q-\hat{u}\hat{u}^{*})=\pi(q)-f\not=0$, i.e.,
$q-\hat{u}\hat{u}^{*}\in\mathcal{P}(E)$. ∎
## 3\. $K_{1}$-group of the extension algebra
Recall from [12] that a unital $C^{*}$–algebra $\mathcal{E}$ has
$1$–cancellation, if a projection $p\in\mathrm{M}_{2}(\mathcal{E})$ satisfies
$\mathrm{diag}(p,1_{k})\sim\mathrm{diag}(p_{1},1_{k})$ for some $k$, then
$p\sim p_{1}$ in $\mathrm{M}_{2}(\mathcal{E})$, where
$p_{1}=\mathrm{diag}(1,0)$. If $\mathcal{E}$ has no unit and $\mathcal{E}^{+}$
has $1$–cancellation, we say $\mathcal{E}$ has $1$–cancellation. It is known
that when $\mathcal{B}$ has $1$–cancellation, we have following exact sequence
of groups:
$U(\mathcal{B})\stackrel{{\scriptstyle
j_{*}}}{{\longrightarrow}}U(E)\stackrel{{\scriptstyle\pi_{*}}}{{\longrightarrow}}U(\mathcal{A})\stackrel{{\scriptstyle\eta}}{{\longrightarrow}}K_{0}(\mathcal{B})$
(3.1)
(cf. [12, lemma 2.2]), where $j_{*}$ (resp. $\pi$) is the induced homomorphism
of the inclusion $j\colon\mathcal{B}\rightarrow E$ (resp. $\pi$) on
$U(\mathcal{B})$ (resp. $U(E)$), $\eta=\partial_{0}\circ i_{\mathcal{A}}$ and
$\partial_{0}\colon K_{1}(\mathcal{A})\rightarrow K_{0}(\mathcal{B})$ is the
index map.
Since, in general, we have the exact sequence of groups
$U(\mathcal{B})\stackrel{{\scriptstyle
j_{*}}}{{\longrightarrow}}U(E)\stackrel{{\scriptstyle\pi_{*}}}{{\longrightarrow}}U(\mathcal{A}),$
(for $\pi(\mathfrak{U}_{0}(E))=\mathfrak{U}_{0}(\mathcal{A})$), i.e.,
$U(\cdot)$ is a half–exact and homotopic invariant functor, it follows from
Proposition 21.4.1, Corollary 21.4.2 and Theorem 24.4.3 of [1] that the
sequence of groups
$U(S\mathcal{A})\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}U(\mathcal{B})\stackrel{{\scriptstyle
j_{*}}}{{\longrightarrow}}U(E)\stackrel{{\scriptstyle\pi_{*}}}{{\longrightarrow}}U(\mathcal{A})$
(3.2)
is exact, where $\partial=e_{*}^{-1}\circ i_{*}$ and
$e\colon\mathcal{B}\rightarrow C_{\pi}$ given by $e(b)=(b,0)\in C_{\pi}$,
$e_{*}$ is isomorphic and $i\colon S\mathcal{A}\rightarrow C_{\pi}$ is defined
by $i(g)=(0,g)$, here
$C_{\pi}=\\{(x,f)\in E\oplus C_{0}([0,1),\mathcal{A})|\,\pi(x)=f(0)\\},\quad
S\mathcal{A}=C_{0}((0,1),\mathcal{A}).$
We also have the exact sequence
$K_{1}(S\mathcal{A})\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}K_{1}(\mathcal{B})\stackrel{{\scriptstyle
j_{*}}}{{\longrightarrow}}K_{1}(E)\stackrel{{\scriptstyle\pi_{*}}}{{\longrightarrow}}K_{1}(\mathcal{A}).$
(3.3)
###### Proposition 3.1.
Suppose that $i_{\mathcal{A}}$, $i_{\mathcal{B}}$ are isomorphic and
$i_{S\mathcal{A}}$ is surjective. Assume that $\mathcal{B}$ has
$1$–cancellation. Then $i_{E}$ is an isomorphism.
###### Proof.
Combining (3.1), (3.2) with (3.3), we have following diagram
$\begin{array}[]{ccccccccc}U(S\mathcal{A})&\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}&U(\mathcal{B})\stackrel{{\scriptstyle
j_{*}}}{{\longrightarrow}}&U(E)\stackrel{{\scriptstyle\pi_{*}}}{{\longrightarrow}}&U(\mathcal{A})\stackrel{{\scriptstyle\eta}}{{\longrightarrow}}&K_{0}(\mathcal{B})\\\
\downarrow i_{S\mathcal{A}}&&\downarrow i_{\mathcal{B}}&\downarrow
i_{E}&\downarrow i_{\mathcal{A}}&\parallel\\\
K_{1}(S\mathcal{A})&\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}&K_{1}(\mathcal{B})\stackrel{{\scriptstyle
j_{*}}}{{\longrightarrow}}&K_{1}(E)\stackrel{{\scriptstyle\pi_{*}}}{{\longrightarrow}}&K_{1}(\mathcal{A})\stackrel{{\scriptstyle\partial_{0}}}{{\longrightarrow}}&K_{0}(\mathcal{B})\\\
\end{array},$ (3.4)
in which two rows are exact and
$\eta=\partial_{0}\circ i_{\mathcal{A}},\quad\pi_{*}\circ
i_{E}=i_{\mathcal{A}}\circ\pi_{*},\quad j_{*}\circ i_{\mathcal{B}}=i_{E}\circ
j_{*}.$
Since $e_{*}$ is isomorphic, it follows from the commutative diagram
$\begin{array}[]{ccccc}U(S\mathcal{A})&\stackrel{{\scriptstyle
i_{*}}}{{\longrightarrow}}&U(C_{\pi})\stackrel{{\scriptstyle
e_{*}}}{{\longleftarrow}}&U(\mathcal{B})\\\ \downarrow
i_{S\mathcal{A}}&&\downarrow i_{C_{\pi}}&\downarrow i_{\mathcal{B}}\\\
K_{1}(S\mathcal{A})&\stackrel{{\scriptstyle
i_{*}}}{{\longrightarrow}}&K_{1}(C_{\pi})\stackrel{{\scriptstyle
e_{*}}}{{\longleftarrow}}&K_{1}(\mathcal{B})\\\ \end{array}$
that $\partial\circ i_{S\mathcal{A}}=i_{\mathcal{B}}\circ\partial$. Thus,
(3.4) is a commutative diagram. Using the Five–Lemma to (3.4), we can obtain
the assertion. ∎
For a $C^{*}$–algebra $\mathcal{E}$, let $\mathrm{csr}(\mathcal{E})$ and
$\mathrm{gsr}(\mathcal{E})$ be the connected stable rank and general stable
rank of $\mathcal{E}$, respectively, defined in [6]. We summrize some
properties of these stable ranks as follows:
###### Lemma 3.2.
Let $\mathcal{E}$ be a $C^{*}$–algebra. Then
1. (1)
$\mathrm{gsr}(\mathcal{E})\leq\mathrm{csr}(\mathcal{E})$ (cf. [6]);
2. (2)
$\mathrm{csr}(\mathcal{E})\leq 2$ when $\mathcal{E}$ is a stable
$C^{*}$–algebra (cf. [9, Theorem 3.12]);
3. (3)
$\mathcal{E}$ has $1$–cancellation if $\mathrm{gsr}(\mathcal{E})\leq 2$ (cf.
[12]);
4. (4)
if $\mathrm{csr}(\mathcal{E})\leq 2$ and
$\mathrm{gsr}(C(\mathbf{S}^{1},\mathcal{E}))\leq 2$, then $i_{\mathcal{E}}$ is
isomorphic (cf. [7, Theorem 2.9] or [12, Corollary 2.2]).
Now we present the main result of this section as follows:
###### Theorem 3.3.
Assume that $\mathcal{A}$ is a unital simple purely infinite $C^{*}$–algebra
and $\mathcal{B}$ is a stable $C^{*}$–algebra. Let $X$ be a compact Hausdorff
space. Then $i_{C(X,E)}$ is an isomorphism.
###### Proof.
If $\mathcal{B}$ is stable, then so is $C(Y,\mathcal{B})$ for any compact
Hausdorff space $Y$. Thus,
$\mathrm{gsr}(C(\mathbf{S}^{1},C(X,\mathcal{B})))\leq 2$ and
$\mathrm{csr}(C(X,\mathcal{B}))\leq 2$ by Lemma 3.2 (1) and (2). So we get
that $i_{C(X,\mathcal{B})}$ is isomorphic by Lemma 3.2 (4).
Since $\mathcal{A}$ is unital simple purely infinite, it follows from [12,
Corollary 3.1] that $i_{C(X,\mathcal{A})}$ and $i_{SC(X,\mathcal{A})}$ are all
surjective. Now we prove $i_{C(X,\mathcal{A})}$ is injective by using some
methods appeared in [8].
Let $f\in\mathfrak{U}(C(X,\mathcal{A}))$ with $i_{C(X,\mathcal{A})}([f])=0$ in
$K_{1}(C(X,\mathcal{A}))$. Let $p$ be a non–trivial projection in
$\mathcal{A}$. Then there exists $g\in\mathfrak{U}(C(X,p\mathcal{A}p))$ such
that $f$ is homotopic to $g+1-p$ by [13, Lemma 2.7]. Thus, there is a
continuous path
$f_{t}\colon[0,1]\rightarrow\mathfrak{U}(\mathrm{M}_{n+1}(C(X,\mathcal{A})))$
such that $f_{0}=1_{n+1}$ and $f_{1}=\mathrm{diag}(g+1-p,1_{n})$ for some
$n\geq 2$. Since $\mathrm{M}_{n+1}(\mathcal{A})$ is purely infinite, we can
find a partial isometry $v=(v_{ij})\in\mathrm{M}_{n+1}(\mathcal{A})$ such that
$\mathrm{diag}(1-p,1_{n})=v^{*}v$, $vv^{*}\leq\mathrm{diag}(1-p,0)$.
Consequently, we get that
$v_{11}^{*}v_{11}=1-p,\ v^{*}_{1j}v_{1,j}=1,\ v^{*}_{1j}v_{1,i}=0,\ i\not=j,\
\sum^{n+1}_{i=1}v_{1i}v^{*}_{1i}\leq 1-p.$
Set $v_{1}=p+v_{11}$, $v_{i}=v_{1i}$, $i=2,\cdots n+2$. Then $v_{1},\cdots
v_{n+1}$ are isometries in $\mathcal{A}$ and $v_{i}^{*}v_{j}=0$, $i\not=j$,
$s=\sum\limits^{n+1}_{i=1}v_{i}v_{i}^{*}$ is a projection. Put
$w_{t}(x)=(v_{1},\cdots,v_{n+1})f_{t}(x)\begin{pmatrix}v_{1}^{*}\\\
v_{2}^{*}\\\ \vdots\\\ v^{*}_{n+1}\end{pmatrix}+1-s,\quad t\in[0,1],\ x\in X.$
It is easy to check that $w_{t}$ is a continuous path in
$\mathfrak{U}(\mathrm{M}_{n}(C(X,\mathcal{A})))$ with $w_{0}=1$ and
$w_{1}=g+1-p$. Thus, $i_{C(X,\mathcal{A})}$ is injective.
The final result follows from Proposition 3.1. ∎
Combining Theorem 3.3 with standard argument in Algebraic Topology, we can get
###### Corollary 3.4.
Let $\mathcal{A}$, $\mathcal{B}$ and $E$ be as in Theorem 3.3. Then
$\pi_{n}(\mathfrak{U}(E))=\begin{cases}K_{0}(E)&\ n\ \text{odd}\\\ K_{1}(E)&\
n\ \text{even}\end{cases}.$
## References
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* [2] Cuntz, J., K–theory for certain $C^{*}$–algebras. J. Ann. Math., 113(1981),181–197.
* [3] Brown, L.G. and Pedersen, G.K., $C^{*}$–algebras of real rank zero. J. Funct. Anal., 99(1991), 131–149.
* [4] Higson, H. and Rørdam, M., The Weyl–Von Neumann theorem for multipliers of some AF–algebras, Canadian J. Math., 43(2) (1991),322–330.
* [5] Liu, S and Fang, X., K–theory for extensions of purely infinite simple $C^{*}$–algebras, Chinese Ann. of Math., 29A(2)(2008), 195–202.
* [6] Rieffel, M.A., Dimensionl and stable rank in the $K$–theory of $C^{*}$–Algebras, Proc. London Math. Soc., 46(3) (1983), 301–333.
* [7] Rieffel, M.A., The homotopy groups of the unitary groups of non–commutative tori, J. Operator Theory, 17 (1987), 237–254.
* [8] Rørdam, M., Larsen, F. and Laustsen, N., An introduction to K-theory for $C^{*}$–algebras, London Math. Soc. Student, Text 49, Cambridge University Press, 2000.
* [9] Sheu, A.J.L., A cancellation theorem for modules over the group $C^{*}$–algebras of certain nipotent Lie groups, Canadian J. Math., 39(1987), 365–427.
* [10] Visinescu, B., Topological structure of the unitary group of certain $C^{*}$-algebras. J. Operator Theory, 60 (2008), 113–124.
* [11] Xue, Y., The reduced minimum modulus in $C^{*}$–algebras, Integr. equ. Oper. Theory, 59 (2007), 269–280.
* [12] Xue, Y., The general stable rank in nonstable K–theory, Rocky Mountain J. Math., 30(2)(2000), 761–775.
* [13] Zhang, S., On the homotopy type of the unitary group and the Grassmann space of purely infinite simple $C^{*}$–algebras, K-Theory, 24(2001), 203–225.
|
arxiv-papers
| 2010-06-29T20:49:24 |
2024-09-04T02:49:11.320471
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhihua Li and Yifeng Xue",
"submitter": "Yifeng Xue",
"url": "https://arxiv.org/abs/1006.5725"
}
|
1007.0009
|
789200620051199988
11institutetext: Sydney Institute for Astronomy (SIfA), School of Physics,
University of Sydney, NSW 2006, Australia 22institutetext: Department of
Astronomy, Yale University, P.O. Box 208101, New Haven, CT 06520-8101
33institutetext: Department of Physics and Astronomy, Aarhus University, 8000
Aarhus C, Denmark 44institutetext: LESIA, CNRS, Université Pierre et Marie
Curie, Université Denis Diderot, Observatoire de Paris, 92195 Meudon, France
55institutetext: School of Physics and Astronomy, University of Birmingham,
Edgbaston, Birmingham B15 2TT, UK 66institutetext: Laboratoire AIM, CEA/DSM-
CNRS, Université Paris 7 Diderot, IRFU/SAp, Centre de Saclay, 91191, Gif-sur-
Yvette, France 77institutetext: Space Telescope Science Institute, 3700 San
Martin Drive, Baltimore, Maryland 21218, USA 88institutetext: GEPI,
Observatoire de Paris, CNRS, Université Paris Diderot, 5 Place Jules Janssen,
92195 Meudon, France 99institutetext: High Altitude Observatory, NCAR, P.O.
Box 3000, Boulder, CO 80307, USA 1010institutetext: Harvard-Smithsonian Center
for Astrophysics, 60 Garden Street, Cambridge, MA, 02138, USA
1111institutetext: Instytut Astronomiczny Uniwersytetu Wrocławskiego, ul.
Kopernika 11, 51-622 Wrocław, Poland 1212institutetext: Institut
d’Astrophysique et de Géophysique de l’Université de Liège, 17 Allée du 6
Août, B-4000 Liège, Belgium 1313institutetext: Queen Mary University of
London, Mile End Road, London E1 4NS, UK 1414institutetext: Max Planck
Institute for Astrophysics, Karl Schwarzschild Str. 1, Garching bei München,
D-85741, Germany 1515institutetext: Vrije Universiteit Brussel, Pleinlaan 2,
B-1050 Brussels, Belgium 1616institutetext: Konkoly Observatory, H-1525
Budapest, P.O. Box 67, Hungary
later
# Solar-like oscillations in cluster stars††thanks: Data from Kepler
D. Stello Corresponding author:
11 stello@physics.usyd.edu.au S. Basu 22 T. R. Bedding 11 K. Brogaard 33 H.
Bruntt 44 W. J. Chaplin 55 J. Christensen-Dalsgaard 33 P. Demarque 22 Y. P.
Elsworth 55 R. A. García 66 R. L. Gilliland 77 S. Hekker 55 D. Huber 11 C.
Karoff 55 H. Kjeldsen 33 Y. Lebreton 88 S. Mathur 99 S. Meibom 1010 J.
Molenda-Żakowicz 1111 A. Noels 1212 I. W. Roxburgh 1313 V. S. Aguirre 1414 C.
Sterken 1515 R. Szabó 1616
(25 May 2010; 24 June 2010)
###### Abstract
We present a brief overview of the history of attempts to obtain a clear
detection of solar-like oscillations in cluster stars, and discuss the results
on the first clear detection, which was made by the Kepler Asteroseismic
Science Consortium (KASC) Working Group 2.
###### keywords:
stars: fundamental parameters — stars: oscillations — stars: interiors —
techniques: photometric — open clusters and associations: individual (NGC
6819)
## 1 Introduction
Star clusters are extremely important in stellar astrophysics. Most stars form
in open clusters, many of which disperse into the diversity of field stars in
the interstellar medium. Understanding the formation and evolution of cluster
stars is therefore important for achieving a comprehensive theory of stellar
evolution. Stars in a cluster are thought to be formed coevally, from the same
interstellar cloud of gas and dust. Each cluster member is therefore expected
to have some properties in common (age, composition, distance), which
strengthens our ability to constrain our stellar models when tested against an
ensemble of cluster stars, especially for asteroseismic analyses (Gough &
Novotny 1993). Asteroseismology has the capability to probe the interior of
stars and hence help us understand the fundamental physical process that
govern stellar structure and evolution (e.g., Christensen-Dalsgaard 2002). In
particular, the detection of solar-like oscillations provide many modes, which
each carrying unique information about the stellar interior. Stars that
potentially exhibit solar-like oscillations, covering most stars that we see,
are cooler than the red edge of the classical instability strip, and have a
convection zone near the surface (necessary for the excitation of the modes).
Solar-like oscillations are reasonably well described by current theory,
giving us some confidence that we can use them as tools to understand stellar
physics, and hopefully also to learn more about the more subtle aspects of the
oscillations themselves. Combining asteroseismic analysis of solar-like
oscillations with the study of cluster stars has therefore been a long-sought
goal.
## 2 Previous attempts
Kepler is certainly not the first attempt to detect solar-like oscillations in
cluster stars. A quick (and hence incomplete) perusal of the history of
previous attempts to detect solar-like oscillations in open and globular
clusters shows that several attempts were made to detect oscillations since
the early 1990s. Among the most ambitious was that of Gilliland et al. (1993),
who used 4-m class telescopes to target the stars in the open cluster M67 at
the cluster turn-off in a multi-site campaign that lasted one week. While an
impressively low noise level was obtained, the data did not reveal the clear
detection of stellar oscillations (Figure 1). However, a red giant star that
happened to be in the field did show intriguing evidence of excess power in
the expected frequency range (Figure 2). Unfortunately, the length of the time
series did not allow individual modes to be resolved for such an evolved star
with much smaller frequency separations between modes. A clear detection
remained elusive, as oscillations could not be distinguished from the rising
background towards low frequency.
Figure 1: Amplitude spectrum (high-pass filtered) of one of the stars targeted
by Gilliland et al. (1993). The horizontal line marks the expected location of
the oscillations. Figure 2: Amplitude spectrum of red giant star observed by
Gilliland et al. (1993). Horizontal line marks the expected location of the
oscillations.
Inspired by Gilliand’s results, Stello et al. (2007) targeted specifically the
red giants in M67 during a 6-week long multi-site campaign of 1–2m class
telescopes. Strong evidence for excess power was found in a number of stars,
but no unambiguous detection of the solar-like pattern of equally spaced modes
was claimed by the authors (Figure 3).
Figure 3: Power spectra of three red giant stars observed by Stello et al.
(2007). Black arrow marks the location of the oscillations expected from
scaling the solar value.
In parallel, several attempts to detect oscillations in globular clusters were
carried out. From the ground, Frandsen et al. (2007) aimed at the red giants
in M4, which delivered lower limits on amplitudes, indicating that the low
metallicity of M4 could have the effect of lowering the oscillation
amplitudes. Again, detection was hindered by long-term stability not being
high enough and varying data quality resulting in strong aliasing in the
weighted amplitude spectra. Slightly more successful were the efforts using
the Hubble Space Telescope by Edmonds & Gilliland (1996); Stello & Gilliland
(2009). In the former study, clear variation was found in a large number of
red giants in 47 Tuc, but the low frequency resolution provided by the 40-hour
time series did not allow the authors to establish this as solar-like
oscillations. The later study was aimed at the red giants in the extremely
metal poor NGC 6397, using archival data originally obtained to detect the
cluster’s faint white dwarf population. The far from ideal data of highly
saturated photometry of the red giants meant that only one star showed good
evidence for oscillations, with excess power at the right frequency range and
amplitude. Despite the 27-day long time series, this fell just short for an
unambiguous detection of equally spaced frequencies in this highly evolved
asymptotic giant branch star.
The main conclusion from these previous efforts is that dedicated space-based
missions are required to achieve the ultra-high precision photometry and long-
term stability in order to detect solar-like oscillations in clusters with
such accuracy that they will be useful for asteroseismic analysis.
We note that in addition to the previous marginal detections, these campaigns
resulted in firm detection of oscillations in a number of classical pulsators
that exhibit much large amplitude than solar-like oscillations (see e.g.
Bruntt 2007, and references therein).
## 3 First results from Kepler
Kepler has a unique capability to overcome the shortcomings that have limited
previous efforts aimed at stellar clusters. Both quality and quantity of the
Kepler data outshine that of early explorations by several orders of
magnitude, and it will undoubtedly be the front runner for cluster seismology
in the next 5–10 years.
As reported by Stello et al. (2010), the first month of Kepler data already
revealed clear detection of solar-like oscillations in a large sample of red
giant stars in the open cluster NGC 6819 (see also Gilliland et al. 2010).
Based on the spacecrafts so called long-cadence mode, which provides a time
averaged exposure every 29.4 minutes, detection was reported in 47 red giant
stars that range almost from the bottom to the tip of the red giant branch
(Figure 4). We saw periodicity in the light curves that span about a factor of
100, corresponding to a factor of $\sim 10$ in radius. Two sample light curves
are shown in Figure 5.
Figure 4: HR-diagram of NGC 6819. Empty symbols mark those where a detection
of solar-like oscillations was reported by Stello et al. (2010). Figure 5:
Kepler time series for two red giants in NGC 6819. Numbers refer to the
numbering in Figure 4. Note the different time scale of the variation.
Photometry and isochrone is of Hole et al. (2009) and Marigo et al. (2008),
respectively.
Power spectra of the stars marked with numbers in Figure 4 are shown in Figure
6. Panels are sorted according to apparent magnitude (brightest at the top),
which for a cluster is indicative of luminosity. One noticeable result is that
not all stars with high membership probability from radial velocity surveys
(see Hole et al. 2009) follow the expected monotonic trend of increasing
frequency of the oscillations (and decreasing amplitude) for decreasing
luminosity. We indicate the expected frequency location with an arrow for
stars that seem to behave strangely compared to the classical scaling
relations for the amplitude and the frequency of maximum power (e.g. Kjeldsen
& Bedding 1995).
Figure 6: Power spectra of 11 stars marked in Figure 4, which are
representative for the entire sample. ‘AM’ indicates that the star is an
asteroseismic member (i.e. observation agrees with scaling relations). Dashed
lines show the measured large frequency separation. For stars where the large
separation could not be determined (no dashed lines), we localised the power
excess from the hump of power in the smoothed power spectrum (solid black
curve). The arrows indicate the expected location of the excess power for
stars where observations do not agree with expectation.
Possible explanations for this behaviour are that these “odd” stars are not
members, or that they have unusual evolution histories.
Stello et al. (2010) were further able to measure the amplitudes of the modes
using the method by Kjeldsen et al. (2009), assuming the relative amplitudes
of the modes of different spherical degree was the same as for the Sun. From
this we could test the $L/M$ scaling relation (Kjeldsen & Bedding 1995; Samadi
et al. 2007), and found that $(L/M)^{0.7}/T_{\mathrm{eff}}^{2}$ provided the
best match to the data.
For further details on what is reported here, we refer to the source paper of
Stello et al. (2010).
## 4 Future
There are four open clusters in Kepler’s field of view. They span a range in
metallicity and age, which brackets the solar values, and are therefore ideal
for testing our current models of stellar evolution (Table 1).
Table 1: Open clusters in Kepler field Cluster | Age | [Fe/H] | Mturnoff
---|---|---|---
| Gyr | | M⊙
NGC 6866 | $\sim$0.4 | $\sim-$0.1 | $\sim$1.7
NGC 6811 | $\sim$1.0 | $\sim-$0.07 | $\sim$1.5
NGC 6819 | $\sim$2.5 | $\sim-$0.05 | $\sim$1.3
NGC 6791 | $\sim$8.5 | $\sim+$0.4 | $\sim$1.0
Values are from Grundahl et al. (2008) (NGC 6791),
Hole et al. (2009) (NGC 6819), Loktin & Matkin (1994) (NGC 6866)
and unpublished work by Meibom.
In Figure 7 we show $\log(g$) vs $T_{\mathrm{eff}}$ for a representative
sample of the stars in Kepler’s field of view together with the representative
isochrones for the four open clusters that are targets in our future
asteroseismic analyses.
Figure 7: $\log(g$) vs $T_{\mathrm{eff}}$ for stars in Kepler’s field of view.
We represent the four open clusters by suitable isochrones. The order in which
we have plotted the cluster names corresponds to their turn-off stars, with
NGC 6866 having the hottest (heaviest) turn-off stars and NGC 6791 the coolest
(lightest). The dashed line indicates the red edge of the classical
instability strip.
For NGC 6819 we expect to achieve a signal-to-noise level for the turn-off
stars that after 3.5 years of data matches what we see in the bottom panels of
Figure 6. This will provide detection in up to 100 stars ranging stellar
evolution from the main sequence F stars to the asymptotic giant branch
including M giants, as well as a number of blue stragglers. This will
potentially provide unprecedented tests of state-of-the-art stellar evolution
models.
In NGC 6791 we already see evidence for power in the red giants, and expect
firm detections for all stars on this highly populated red giant branch, with
unique potential for testing intrinsic variation among practically identical
stars.
The two younger clusters NGC 6811 and NGC 6866 are less populated but provide
the opportunity to investigate classical pulsators in great detail. NGC 6811
also contains a few He-core burning red giants.
The combination of results from all four clusters promises great prospects for
testing asteroseismic scaling relations on distinct stellar populations that
span a large range in stellar age and brackets the solar metallicity.
###### Acknowledgements.
Funding of the Discovery mission is provided by NASA’s Science Mission
Directorate. The authors thank the entire Kepler team without whom this
investigation would not have been possible. The authors also thank all funding
councils and agencies that have supported the activities for Working Group 2
of the KASC. In particular, DS would like to thank HELAS for support to attend
the HELAS IV meeting in Lanzarote.
## References
* Bruntt (2007) Bruntt, H. 2007, Communications in Asteroseismology, 150, 326
* Christensen-Dalsgaard (2002) Christensen-Dalsgaard, J. 2002, Reviews of Modern Physics, 74, 1073
* Edmonds & Gilliland (1996) Edmonds, P. D., & Gilliland, R. L. 1996, ApJ, 464, L157
* Frandsen et al. (2007) Frandsen, S., et al. 2007, A&A, 475, 991
* Gilliland et al. (1993) Gilliland, R. L., et al. 1993, AJ, 106, 2441
* Gilliland et al. (2010) —. 2010, PASP, 122, 131
* Gough & Novotny (1993) Gough, D. O., & Novotny, E. 1993, in ASP Conf. Ser. 42: GONG 1992. Seismic Investigation of the Sun and Stars, ed. T. M. Brown, 355
* Grundahl et al. (2008) Grundahl, F., Clausen, J. V., Hardis, S., & Frandsen, S. 2008, A&A, 492, 171
* Hole et al. (2009) Hole, K. T., Geller, A. M., Mathieu, R. D., Platais, I., Meibom, S., & Latham, D. W. 2009, AJ, 138, 159
* Kjeldsen & Bedding (1995) Kjeldsen, H., & Bedding, T. R. 1995, A&A, 293, 87
* Kjeldsen et al. (2009) Kjeldsen, H., Bedding, T. R., & Christensen-Dalsgaard, J. 2009, in IAU Symposium, Vol. 253, IAU Symposium, 309–317
* Loktin & Matkin (1994) Loktin, A. V., & Matkin, N. V. 1994, Astronomical and Astrophysical Transactions, 4, 153
* Marigo et al. (2008) Marigo, P., Girardi, L., Bressan, A., Groenewegen, M. A. T., Silva, L., & Granato, G. L. 2008, A&A, 482, 883
* Samadi et al. (2007) Samadi, R., Georgobiani, D., Trampedach, R., Goupil, M. J., Stein, R. F., & Nordlund, Å. 2007, A&A, 463, 297
* Stello & Gilliland (2009) Stello, D., & Gilliland, R. L. 2009, ApJ, 700, 949
* Stello et al. (2007) Stello, D., et al. 2007, MNRAS, 377, 584
* Stello et al. (2010) —. 2010, ApJ, 713, L182
|
arxiv-papers
| 2010-06-30T20:00:52 |
2024-09-04T02:49:11.335133
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. Stello, S. Basu, T. R. Bedding, K. Brogaard, H. Bruntt, W. J.\n Chaplin, J. Christensen-Dalsgaard, P. Demarque, Y. P. Elsworth, R.A.\n Garc\\'ia, R. L. Gilliland, S. Hekker, D. Huber, C. Karoff, H. Kjeldsen, Y.\n Lebreton, S. Mathur, S. Meibom, J. Molenda-\\.Zakowicz, A. Noels, I. W.\n Roxburgh, V. S. Aguirre, C. Sterken, R. Szab\\'o",
"submitter": "Dennis Stello",
"url": "https://arxiv.org/abs/1007.0009"
}
|
1007.0108
|
# Jacob’s ladders and the $\tilde{Z}^{2}$-transformation of the orthogonal
system of trigonometric functions
Jan Moser Department of Mathematical Analysis and Numerical Mathematics,
Comenius University, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA
jan.mozer@fmph.uniba.sk
###### Abstract.
It is shown in this paper that there is a continuum set of orthogonal systems
relative to the weight function $\tilde{Z}^{2}(t)$. The corresponding
integrals cannot be obtained in known theories of Balasubramanian, Heath-Brown
and Ivic.
###### Key words and phrases:
Riemann zeta-function
## 1\. The first result
### 1.1.
In this paper we obtain some new properties of the signal
(1.1) $Z(t)=e^{i\vartheta(t)}\zeta\left(\frac{1}{2}+it\right)$
that is generated by the Riemann zeta-function, where
(1.2)
$\vartheta(t)=-\frac{t}{2}\ln\pi+\text{Im}\ln\Gamma\left(\frac{1}{4}+i\frac{t}{2}\right)=\frac{t}{2}\ln\frac{t}{2\pi}-\frac{t}{2}-\frac{\pi}{8}+\mathcal{O}\left(\frac{1}{t}\right).$
Let us remind that
(1.3) $\tilde{Z}^{2}(t)=\frac{{\rm d}\varphi_{1}(t)}{{\rm d}t},\
\varphi_{1}(t)=\frac{1}{2}\varphi(t)$
where
(1.4)
$\tilde{Z}^{2}(t)=\frac{Z^{2}(t)}{2\Phi^{\prime}_{\varphi}[\varphi(t)]}=\frac{Z^{2}(t)}{\left\\{1+\mathcal{O}\left(\frac{\ln\ln
t}{\ln t}\right)\right\\}\ln t}$
(see [12], (5.1)-(5.3)) and $\varphi_{1}(T),\ T\geq T_{0}[\varphi_{1}]$ is the
Jacob’s ladder.
### 1.2.
It is known that the system of trigonometric functions
(1.5)
$\left\\{1,\cos\left(\frac{\pi}{l}t\right),\sin\left(\frac{\pi}{l}t\right),\dots,\cos\left(\frac{\pi}{l}nt\right),\sin\left(\frac{\pi}{l}nt\right),\dots\right\\}$
is the orthogonal system on the segment $[0,2l]$. In this direction the
following theorem holds true.
###### Theorem 1.
Let $\mathcal{J}(2l)=\varphi_{1}\\{\mathring{\mathcal{J}}(2l)\\}$, where
Then the system of functions
(1.7)
$\left\\{1,\cos\left(\frac{\pi}{l}\varphi_{1}(t)\right),\sin\left(\frac{\pi}{l}\varphi_{1}(t)\right),\dots,\cos\left(\frac{\pi}{l}n\varphi_{1}(t)\right),\sin\left(\frac{\pi}{l}n\varphi_{1}(t)\right),\dots\right\\}$
is the orthogonal system on $\mathring{\mathcal{J}}(2l)$ with respect to the
weight function $\tilde{Z}^{2}(t)$, i.e. the following new system of integrals
(1.8)
$\begin{split}&\int_{\mathring{\mathcal{J}}(2l)}\cos\left(\frac{\pi}{l}m\varphi_{1}(t)\right)\cos\left(\frac{\pi}{l}n\varphi_{1}(t)\right)\tilde{Z}^{2}(t){\rm
d}t=\left\\{\begin{array}[]{rcl}0&,&m\not=n,\\\ l&,&m=n,\end{array}\right.\\\
&\int_{\mathring{\mathcal{J}}(2l)}\sin\left(\frac{\pi}{l}m\varphi_{1}(t)\right)\sin\left(\frac{\pi}{l}n\varphi_{1}(t)\right)\tilde{Z}^{2}(t){\rm
d}t=\left\\{\begin{array}[]{rcl}0&,&m\not=n,\\\ l&,&m=n,\end{array}\right.\\\
&\int_{\mathring{\mathcal{J}}(2l)}\sin\left(\frac{\pi}{l}m\varphi_{1}(t)\right)\cos\left(\frac{\pi}{l}n\varphi_{1}(t)\right)\tilde{Z}^{2}(t){\rm
d}t=0,\\\
&\int_{\mathring{\mathcal{J}}(2l)}\cos\left(\frac{\pi}{l}n\varphi_{1}(t)\right)\tilde{Z}^{2}(t){\rm
d}t=0,\\\
&\int_{\mathring{\mathcal{J}}(2l)}\sin\left(\frac{\pi}{l}n\varphi_{1}(t)\right)\tilde{Z}^{2}(t){\rm
d}t=0\end{split}$
for all $m,n\in\mathbb{N}$ is obtained, where
(A) $t-\varphi_{1}(t)\sim(1-c)\pi(t),$ (B) $2l(K+1)<\widering{2lK},$ (C)
$\rho\\{\mathcal{J}(2l);\mathring{\mathcal{J}}(2l)\\}\sim(1-c)\pi(t)\to\infty,$
as $K\to\infty$, and $\rho$ denotes the distance of the corresponding
segments, $c$ is the Euler constant and $\pi(t)$ is the prime-counting
function.
###### Remark 1.
Theorem 1 gives the contact point between the functions
$\zeta\left(\frac{1}{2}+it\right),\ \pi(t),\ \varphi_{1}(t)$ and the
orthogonal system of trigonometric functions.
###### Remark 2.
It is clear that the formulae (1.8) - for the modulated function
$\tilde{Z}^{2}(t)$ \- cannot be obtained in the known theories of
Balasubramanian, Heath-Brown and Ivic (comp. [1]).
This paper is a continuation of the series [2]-[15].
## 2\. New method of the quantization of the Hardy-Littlewood integral (a
special case)
### 2.1.
We obtain from the first two formulae in (1.8)
(2.1)
$\begin{split}&\int_{\mathring{\mathcal{J}}(2l)}\cos^{2}\left(\frac{\pi}{l}m\varphi_{1}(t)\right)\tilde{Z}^{2}(t){\rm
d}t=\frac{1}{2}|\mathcal{J}(2l)|,\\\
&\int_{\mathring{\mathcal{J}}(2l)}\sin^{2}\left(\frac{\pi}{l}m\varphi_{1}(t)\right)\tilde{Z}^{2}(t){\rm
d}t=\frac{1}{2}|\mathcal{J}(2l)|\end{split}$
for all $m\in\mathbb{N}$. Next, from (2.1) we obtain
###### Corollary 1.
(2.2) $\int_{\mathring{\mathcal{J}}(2l)}\tilde{Z}^{2}(t){\rm
d}t=|\mathcal{J}(2l)|;\ |\mathcal{J}(2l)|=2l.$
### 2.2.
Let us consider now the problem concerning the solid of revolution
corresponding to the graph of the function (comp. [5])
$\tilde{Z}(t),\ t\in[\widering{2lK},+\infty),\ 2lK>T_{0}[\varphi_{1}].$
###### Problem.
To divide this solid of revolution on parts of equal volumes.
From (2.2) we obtain the resolution of this problem.
###### Corollary 2.
Since
(a)
$[\widering{2lK},+\infty)=\bigcup_{r=1}^{\infty}\mathring{\mathcal{J}}(2l,r),\
\mathring{\mathcal{J}}(2l,r)=[\widering{2l(K+r-1)},\widering{2l(K+r)}],$ (b)
$\pi\int_{\mathring{\mathcal{J}}(2l,r)}\tilde{Z}^{2}(t){\rm d}t=2\pi l,\
r=1,2,3,\dots\ ,$
it follows that the sequence of points
$\\{\widering{2l(K+r-1)}\\}_{r=2}^{+\infty}$
is the resolution to the Problem for arbitrary fixed $2l\in(0,T/\ln T]$.
## 3\. Generalization of the formula (2.2)
### 3.1.
The following theorem holds true.
###### Theorem 2.
Let
$\mathcal{J}(T,U)=[T,T+U],\
J(T,U)=\varphi_{1}\\{\mathring{\mathcal{J}}(T,U)\\};\
\mathring{\mathcal{J}}(T,U)=[\mathring{T},\widering{T+U}].$
Then
(3.1) $\int_{\mathring{\mathcal{J}}(T,U)}\tilde{Z}^{2}(t){\rm
d}t=|\mathcal{J}(T,U)|=U,$
for every $T\geq T_{0}[\varphi_{1}],\ U\in(0,T/\ln T]$.
###### Remark 3.
From (3.1) the general method for quantization of the Hardy-Littlewood
integral follows (comp. Corollary 2: $2lK\to\forall\ T\geq
T_{0}[\varphi_{1}],\ \mathcal{J}(2l)\to\mathcal{J}(T,U)$).
Next, we obtain, using the mean-value theorem in (3.1)
###### Corollary 3.
(3.2)
$\tilde{Z}^{2}(\xi)=\frac{|\mathcal{J}(T,U)|}{|\mathring{\mathcal{J}}(T,U)|},\
\xi\in\xi(\mathring{T},\widering{T+U}),\ \tilde{Z}(\xi)\not=0,$
i.e.
$\tilde{Z}^{2}(\xi):1=|\mathcal{J}(T,U)|:|\mathring{\mathcal{J}}(T,U)|.$
### 3.2.
Let $\\{[T^{\prime},T^{\prime}+1]\\}$ stands for the continuum set of segments
$[T^{\prime},T^{\prime}+1]\subset[T,T+T/\ln T]$. Since
$\frac{1}{|\mathring{\mathcal{J}}(T^{\prime},1)|}=\tilde{Z}^{2}(\xi),\
\xi=\xi(T^{\prime})\in(\mathring{T}^{\prime},\widering{T^{\prime}+1})$
then by the Riemann-Siegel formula
$Z(t)=2\sum_{n\leq\sqrt{\frac{t}{2\pi}}}\frac{1}{\sqrt{n}}\cos\\{\vartheta(t)-t\ln
n\\}+\mathcal{O}(t^{-1/4})$
we obtain (see (1.4))
###### Corollary 4.
(3.3)
$\frac{1}{\sqrt{|\mathring{\mathcal{J}}(T^{\prime},1)|}}\sim\frac{2}{\sqrt{\ln\xi}}\left|\sum_{n\leq\sqrt{\frac{\xi}{2\pi}}}\frac{1}{\sqrt{n}}\cos\\{\vartheta(\xi)-\xi\ln
n\\}+\mathcal{O}(\xi^{-1/4})\right|$
where $\xi=\xi(T^{\prime})$.
###### Remark 4.
The formula (3.3) describes the complicated oscillations of the value
$|\mathring{\mathcal{J}}(T^{\prime},1)|$ generated by the nonlinear
transformation
$\mathcal{J}(T^{\prime},1)=\varphi_{1}\\{\mathring{\mathcal{J}}(T^{\prime},1)\\}$.
## 4\. Proof of Theorems 1 and 2
### 4.1.
Let us remind that the following lemma is true (see [12], (5.1)-(5.3))
###### Lemma.
For every integrable function (in the Lebesgue sense) $f(x),\
x\in[\varphi_{1}(T),\varphi_{1}(T+U)]$ the following is true
(4.1) $\int_{T}^{T+U}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{\varphi_{1}(T)}^{\varphi_{1}(T+U)}f(x){\rm d}x,\ U\in(0,T/\ln T],$
where $t-\varphi_{1}(t)\sim(1-c)\pi(t)$.
###### Remark 5.
The formula (4.1) is true also in the case when the integral on the right-hand
side of eq. (4.1) is convergent but not absolutely (in the Riemann sense).
### 4.2.
If $\varphi_{1}\\{[\mathring{T},\widering{T+U}]\\}=[T,T+U]$ then we obtain
from (4.1) the following formula
(4.2)
$\int_{\mathring{T}}^{\widering{T+U}}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{T}^{T+U}f(x){\rm d}x,\ U\in(0,T/\ln T].$
Next, in the case $[T,T+U]=[2lK,2lK+2l]=\mathcal{J}(2l)$, we have
(4.3) $\int_{\mathcal{J}(2l)}F(t){\rm d}t=\int_{0}^{2l}F(t){\rm d}t$
for every (integrable) $2l$-periodic function $F(t)$. Then from the known
formulae
$\begin{split}&\int_{\mathcal{J}(2l)}\cos\left(\frac{\pi}{l}m\varphi_{1}(t)\right)\cos\left(\frac{\pi}{l}n\varphi_{1}(t)\right){\rm
d}t=\left\\{\begin{array}[]{rcl}0&,&m\not=n,\\\ l&,&m=n,\end{array}\right.\\\
&\int_{\mathcal{J}(2l)}\sin\left(\frac{\pi}{l}m\varphi_{1}(t)\right)\sin\left(\frac{\pi}{l}n\varphi_{1}(t)\right){\rm
d}t=\left\\{\begin{array}[]{rcl}0&,&m\not=n,\\\ l&,&m=n,\end{array}\right.\\\
&\int_{\mathcal{J}(2l)}\sin\left(\frac{\pi}{l}m\varphi_{1}(t)\right)\cos\left(\frac{\pi}{l}n\varphi_{1}(t)\right){\rm
d}t=0,\\\
&\int_{\mathcal{J}(2l)}\cos\left(\frac{\pi}{l}n\varphi_{1}(t)\right){\rm
d}t=0,\quad\int_{\mathcal{J}(2l)}\sin\left(\frac{\pi}{l}n\varphi_{1}(t)\right){\rm
d}t=0,\quad m,n\in\mathbb{N},\end{split}$
by the $\tilde{Z}^{2}$-transformation (see (4.2), (4.3);
$[\mathring{T},\widering{T+U}]=\mathring{\mathcal{J}}(2l)$) the formulae (1.8)
follow. The properties (B), (C) in Theorem 1 are identical with [13], (A1),
(B1).
### 4.3.
The formula (3.1) follows from (4.2) in the case $f(x)\equiv 1$.
## 5\. Another type of the orthogonal systems
It follows from (4.2) that the continuum set $\mathcal{S}(T,2l)$ of the
systems
$\begin{split}&\left\\{|\tilde{Z}(t)|,|\tilde{Z}(t)|\cos\left(\frac{\pi}{l}(\varphi_{1}(t)-T)\right),|\tilde{Z}(t)|\sin\left(\frac{\pi}{l}(\varphi_{1}(t)-T)\right),\dots,\right.\\\
&\left.|\tilde{Z}(t)|\cos\left(\frac{\pi}{l}n(\varphi_{1}(t)-T)\right),|\tilde{Z}(t)|\sin\left(\frac{\pi}{l}n(\varphi_{1}(t)-T)\right),\dots\right\\},\\\
&t\in[\mathring{T},\widering{T+2l}]\end{split}$
for all
$T\geq T_{0}[\varphi_{1}],\ 2l\in(0,T/\ln T]$
is the set of orthogonal systems on $[\mathring{T},\widering{T+2l}]$.
###### Remark 6.
Let us call the elements of the system $\mathcal{S}(T,2l)$ for fixed
$2l\in(0,T/\ln T]$ and for all $T\geq T_{0}[\varphi_{1}]$ as _the clones_ of
the known orthogonal trigonometric system
$\left\\{1,\cos\left(\frac{\pi}{l}t\right),\sin\left(\frac{\pi}{l}t\right),\dots,\cos\left(\frac{\pi}{l}nt\right),\sin\left(\frac{\pi}{l}nt\right),\dots\right\\},\
t\in[0,2l].$
I would like to thank Michal Demetrian for helping me with the electronic
version of this work.
## References
* [1] A. Ivic, ‘The Riemann zeta-function‘, A Willey-Interscience Pub., New York, 1985.
* [2] J. Moser, ‘Jacob’s ladders and the almost exact asymptotic representation of the Hardy-Littlewood integral’, (2008), arXiv:0901.3973.
* [3] J. Moser, ‘Jacob’s ladders and the tangent law for short parts of the Hardy-Littlewood integral’, (2009), arXiv:0906.0659.
* [4] J. Moser, ‘Jacob’s ladders and the multiplicative asymptotic formula for short and microscopic parts of the Hardy-Littlewood integral’, (2009), arXiv:0907.0301.
* [5] J. Moser, ‘Jacob’s ladders and the quantization of the Hardy-Littlewood integral’, (2009), arXiv:0909.3928.
* [6] J. Moser, ‘Jacob’s ladders and the first asymptotic formula for the expression of the sixth order $|\zeta(1/2+i\varphi(t)/2)|^{4}|\zeta(1/2+it)|^{2}$’, (2009), arXiv:0911.1246.
* [7] J. Moser, ‘Jacob’s ladders and the first asymptotic formula for the expression of the fifth order $Z[\varphi(t)/2+\rho_{1}]Z[\varphi(t)/2+\rho_{2}]Z[\varphi(t)/2+\rho_{3}]\hat{Z}^{2}(t)$ for the collection of disconnected sets‘, (2009), arXiv:0912.0130.
* [8] J. Moser, ‘Jacob’s ladders, the iterations of Jacob’s ladder $\varphi_{1}^{k}(t)$ and asymptotic formulae for the integrals of the products $Z^{2}[\varphi^{n}_{1}(t)]Z^{2}[\varphi^{n-1}(t)]\cdots Z^{2}[\varphi^{0}_{1}(t)]$ for arbitrary fixed $n\in\mathbb{N}$‘ (2010), arXiv:1001.1632.
* [9] J. Moser, ‘Jacob’s ladders and the asymptotic formula for the integral of the eight order expression $|\zeta(1/2+i\varphi_{2}(t))|^{4}|\zeta(1/2+it)|^{4}$‘, (2010), arXiv:1001.2114.
* [10] J. Moser, ‘Jacob’s ladders and the asymptotically approximate solutions of a nonlinear diophantine equation‘, (2010), arXiv: 1001.3019.
* [11] J. Moser, ‘Jacob’s ladders and the asymptotic formula for short and microscopic parts of the Hardy-Littlewood integral of the function $|\zeta(1/2+it)|^{4}$‘, (2010), arXiv:1001.4007.
* [12] J. Moser, ‘Jacob’s ladders and the nonlocal interaction of the function $|\zeta(1/2+it)|$ with $\arg\zeta(1/2+it)$ on the distance $\sim(1-c)\pi(t)$‘, (2010), arXiv: 1004.0169.
* [13] J. Moser, ‘Jacob’s ladders and the $\tilde{Z}^{2}$ \- transformation of polynomials in $\ln\varphi_{1}(t)$‘, (2010), arXiv: 1005.2052.
* [14] J. Moser, ‘Jacob’s ladders and the oscillations of the function $|\zeta\left(\frac{1}{2}+it\right)|^{2}$ around the main part of its mean-value; law of the almost exact equality of the corresponding areas‘, (2010), arXiv: 1006.4316
* [15] J. Moser, ‘Jacob’s ladders and the nonlocal interaction of the function $Z(t)$ with the function $\tilde{Z}^{2}(t)$ on the distance $\sim(1-c)\pi(t)$ for a collection of disconneted sets‘, (2010), arXiv: 1006.5158
|
arxiv-papers
| 2010-07-01T09:13:21 |
2024-09-04T02:49:11.348225
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jan Moser",
"submitter": "Michal Demetrian",
"url": "https://arxiv.org/abs/1007.0108"
}
|
1007.0176
|
# Generalized Polya-Szegö inequality
Hichem Hajaiej Ipeit (Institut préparatoire aux études d’ingénieur de Tunis)
2, Rue Jawaher Lel Nahru - 1089 Montfleury - tunis Tunisie
###### Abstract.
We generalize Polya-Szegö inequality to integrands depending on $u$ and its
gradient. Under minimal additional assumptions, we establish equality cases in
this generalized inequality.
###### Key words and phrases:
Generalized Polya-Szegö inequalities, identity results, radial symmetry, non-
compact minimization problems
###### 2000 Mathematics Subject Classification:
46E35, 26B25, 26B99, 47B38
Ipeit (Institut préparatoire aux études d’ingénieur de Tunis) 2, Rue Jawaher
Lel Nahru - 1089 Montfleury - Tunis, Tunisie. E-mail: hichem.hajaiej@gmail.com
###### Contents
1. 1 Introduction
2. 2 Preliminary stuff
3. 3 Generalized Polya-Szegö inequality
## 1\. Introduction
The Polya-Szegö inequality asserts that the $L^{2}$ norm of the gradient of a
positive function $u$ in $W^{1,p}({\mathbb{R}}^{N})$ cannot increase under
Schwarz symmetrization,
(1.1) $\int_{{\mathbb{R}}^{N}}|\nabla
u^{*}|^{2}dx\leq\int_{{\mathbb{R}}^{N}}|\nabla u|^{2}dx.$
The Schwarz rearrangement of $u$ is denoted here by $u^{*}$. Inequality (1.1)
has numerous applications in physics. It was first used in 1945 by G. Polya
and G. Szegö to prove that the capacity of a condenser diminishes or remains
unchanged by applying the process of Schwarz symmetrization (see [30]).
Inequality (1.1) was also the key ingredients to show that, among all bounded
bodies with fixed measure, balls have the minimal capacity (see [26, Theorem
11.17]). Finally (1.1) has also played a crucial role in the solution of the
famous Choquard’s conjecture (see [25]). It is heavily connected to the
isoperimetric inequality and to Riesz-type rearrangement inequalities.
Moreover, it turned out that (1.1) is extremely helpful in establishing the
existence of ground states solutions of the nonlinear Schrödinger equation
(1.2) $\begin{cases}{\rm i}\partial_{t}\Phi+\Delta\Phi+f(|x|,\Phi)=0&\text{in
${\mathbb{R}}^{N}\times(0,\infty)$},\\\ \Phi(x,0)=\Phi_{0}(x)&\text{in
${\mathbb{R}}^{N}$}.\end{cases}$
A ground state solution of equation (1.2) is a positive solution to the
following associated variational problem
(1.3) $\inf\left\\{\frac{1}{2}\int_{{\mathbb{R}}^{N}}|\nabla
u|^{2}dx-\int_{{\mathbb{R}}^{N}}F(|x|,u)dx:\,u\in
H^{1}({\mathbb{R}}^{N}),\,\,\|u\|_{L^{2}}=1\right\\},$
where $F(|x|,s)$ is the primitive of $f(|x|,\cdot)$ with $F(|x|,0)=0$.
Inequality (1.1) together with the generalized Hardy-Littlewood inequality
were crucial to prove that (1.3) admits a radial and radially decreasing
solution. Furthermore, under appropriate regularity assumptions on the
nonlinearity $F$, there exists a Lagrange multiplier $\lambda$ such that any
minimizer of (1.3) is a solution of the following semi-linear elliptic PDE
$-\Delta u+f(|x|,u)+\lambda u=0,\quad\text{in ${\mathbb{R}}^{N}$}.$
We refer the reader to [20] for a detailed analysis. The same approach applies
to the more general quasi-linear PDE
$-\Delta_{p}u+f(|x|,u)+\lambda u=0,\quad\text{in ${\mathbb{R}}^{N}$}.$
where $\Delta_{p}u$ means ${\rm div}(|\nabla u|^{p-2}\nabla u)$, and we can
derive similar properties of ground state solutions since (1.1) extends to
gradients that are in $L^{p}({\mathbb{R}}^{N})$ in place of
$L^{2}({\mathbb{R}}^{N})$, namely
(1.4) $\int_{{\mathbb{R}}^{N}}|\nabla
u^{*}|^{p}dx\leq\int_{{\mathbb{R}}^{N}}|\nabla u|^{p}dx.$
Due to the multitude of applications in physics, rearrangement inequalities
like (1.1) and (1.4) have attracted a huge number of mathematicians from the
middle of the last century. Different approaches were built up to establish
these inequalities such as heat-kernel methods, slicing and cut-off techniques
and two-point rearrangement.
A generalization of inequality (1.4) to suitable convex integrands
$A:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$,
(1.5) $\int_{{\mathbb{R}}^{N}}A(|\nabla
u^{*}|)dx\leq\int_{{\mathbb{R}}^{N}}A(|\nabla u|)dx,$
was first established by Almgren and Lieb (see [1]). Inequality (1.5) is
important in studying the continuity and discontinuity of Schwarz
symmetrization in Sobolev spaces (see e.g. [1, 11]). It also permits us to
study symmetry properties of variational problems involving integrals of type
$\int_{{\mathbb{R}}^{N}}A(|\nabla u|)dx$. Extensions of Polya-Szegö inequality
to more general operators of the form
$j(s,\xi)=b(s)A(|\xi|),\quad s\in{\mathbb{R}},\,\xi\in{\mathbb{R}}^{N},$
on bounded domains have been investigated by Kawohl, Mossino and Bandle. More
precisely, they proved that
(1.6) $\int_{\Omega^{*}}b(u^{*})A(|\nabla
u^{*}|)dx\leq\int_{\Omega}b(u)A(|\nabla u|)dx,$
where $\Omega^{*}$ denotes the ball in ${\mathbb{R}}^{N}$ centered at the
origin having the Lebesgue measure of $\Omega$, under suitably convexity,
monotonicity and growth assumptions (see e.g. [3, 24, 29]). Numerous
applications of (1.6) have been discussed in the above references. In [35],
Tahraoui claimed that a general integrand $j(s,\xi)$ with appropriate
properties can be written in the form
$\sum_{i=1}^{\infty}b_{i}(s)A_{i}(|\xi|)+R_{1}(s)+R_{2}(\xi),\quad
s\in{\mathbb{R}},\,\xi\in{\mathbb{R}}^{N},$
where $b_{i}$ and $A_{i}$ are such that inequality (1.6) holds. However, there
are some mistakes in [35] and we do not believe that this density type result
holds true. Until quite recently there were no results dealing with the
generalized Polya-Szegö inequality, namely
(1.7) $\int_{\Omega^{*}}j(u^{*},|\nabla u^{*}|)dx\leq\int_{\Omega}j(u,|\nabla
u|)dx.$
While writing down this paper we have learned about a very recent survey by F.
Brock [6] who was able to prove (1.7) under continuity, monotonicity,
convexity and growth conditions.
Following a completely different approach, we prove (1.7) without requiring
any growth conditions on $j$. As it can be easily seen it is important to drop
these conditions to the able to cover some relevant applications. Our approach
is based upon a suitable approximation of the Schwarz symmetrized $u^{*}$ of a
function $u$. More precisely, if $(H_{n})_{n\geq 1}$ is a dense sequence in
the set of closed half spaces $H$ containing $0$ and $u\in
L^{p}_{+}({\mathbb{R}}^{N})$, there exists a sequence $(u_{n})$ consisting of
iterated polarizations of the $H_{n}$s which converges to $u^{*}$ in
$L^{p}({\mathbb{R}}^{N})$ (see [17, 38]). On the other hand, a straightforward
computation shows that
$\|\nabla u\|_{L^{p}({\mathbb{R}}^{N})}=\|\nabla
u_{0}\|_{L^{p}({\mathbb{R}}^{N})}=\cdots=\|\nabla
u_{n}\|_{L^{p}({\mathbb{R}}^{N})},\quad\text{for all $n\in{\mathbb{N}}$}.$
By combining these properties with the weak lower semicontinuity of the
functional $J(u)=\int j(u,|\nabla u|)dx$ enable us to conclude (see Theorem
3.1). Note that (1.5) was proved using coarea formula; however this approach
does not apply to integrands depending both on $u$ and its gradient since one
has to apply simultaneously the coarea formula to $|\nabla u|$ and to
decompose $u$ with the Layer-Cake principle.
Notice that Brock’s method is based on an intermediate maximization problem
and cannot yield to the establishment of equality cases. Our approximation
approach was also fruitful in determining the relationship between $u$ and
$u^{*}$ such that
(1.8) $\int_{{\mathbb{R}}^{N}}j(u^{*},|\nabla
u^{*}|)dx=\int_{{\mathbb{R}}^{N}}j(u,|\nabla u|)dx.$
Indeed, under very general conditions on $j$, we prove that (1.8) is
equivalent to
$\int_{{\mathbb{R}}^{N}}|\nabla u^{*}|^{p}dx=\int_{{\mathbb{R}}^{N}}|\nabla
u|^{p}dx.$
For $j(\xi)=|\xi|^{p}$, identity cases were completely studied in the
breakthrough paper of Brothers and Ziemer [10].
The paper is organized as follows. Section 2 is dedicated to some preliminary
stuff, especially the ones concerning the invariance of a class of functionals
under polarization. These observations are crucial, in Section 3, to establish
in a simple way the generalized Polya-Szegö inequality.
Notations.
1. (1)
For $N\in{\mathbb{N}}$, $N\geq 1$, we denote by $|\cdot|$ the euclidean norm
in ${\mathbb{R}}^{N}$.
2. (2)
${\mathbb{R}}_{+}$ (resp. ${\mathbb{R}}_{-}$) is the set of positive (resp.
negative) real values.
3. (3)
$\mu$ denotes the Lebesgue measure in ${\mathbb{R}}^{N}$.
4. (4)
$M({\mathbb{R}}^{N})$ is the set of measurable functions in
${\mathbb{R}}^{N}$.
5. (5)
For $p>1$ we denote by $L^{p}({\mathbb{R}}^{N})$ the space of $f$ in
$M({\mathbb{R}}^{N})$ with $\int_{{\mathbb{R}}^{N}}|f|^{p}dx<\infty$.
6. (6)
The norm $(\int_{{\mathbb{R}}^{N}}|f|^{p}dx)^{1/p}$ in
$L^{p}({\mathbb{R}}^{N})$ is denoted by $\|\cdot\|_{p}$.
7. (7)
For $p>1$ we denote by $W^{1,p}({\mathbb{R}}^{N})$ the Sobolev space of
functions $f$ in $L^{p}({\mathbb{R}}^{N})$ having generalized partial
derivatives $D_{i}f$ in $L^{p}({\mathbb{R}}^{N})$, for $i=1,\dots,N$.
8. (8)
$D^{1,p}({\mathbb{R}}^{N})$ is the space of measurable functions whose
gradient is in $L^{p}({\mathbb{R}}^{N})$.
9. (9)
$L^{p}_{+}({\mathbb{R}}^{N})$ is the cone of positive functions of
$L^{p}({\mathbb{R}}^{N})$.
10. (10)
$W^{1,p}_{+}({\mathbb{R}}^{N})$ is the cone of positive functions of
$W^{1,p}({\mathbb{R}}^{N})$.
11. (11)
For $R>0$, $B(0,R)$ is the ball in ${\mathbb{R}}^{N}$ centered at zero with
radius $R$.
## 2\. Preliminary stuff
In the following $H$ will design a closed half-space of ${\mathbb{R}}^{N}$
containing the origin, $0_{{\mathbb{R}}^{N}}\in H$. We denote by
${\mathcal{H}}$ the set of closed half-spaces of ${\mathbb{R}}^{N}$ containing
the origin. We shall equip ${\mathcal{H}}$ with a topology ensuring that
$H_{n}\to H$ as $n\to\infty$ if there is a sequence of isometries
$i_{n}:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ such that $H_{n}=i_{n}(H)$ and
$i_{n}$ converges to the identity as $n\to\infty$. We first recall some basic
notions. For more details, we refer the reader to [12].
###### Definition 2.1.
A reflection $\sigma:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ with respect to $H$
is an isometry such that the following properties hold
1. (1)
$\sigma\circ\sigma(x)=x$, for all $x\in{\mathbb{R}}^{N}$;
2. (2)
the fixed point set of $\sigma$ separates ${\mathbb{R}}^{N}$ in $H$ and
${\mathbb{R}}^{N}\setminus H$ (interchanged by $\sigma$);
3. (3)
$|x-y|<|x-\sigma(y)|$, for all $x,y\in H$.
Given $x\in{\mathbb{R}}^{N}$, the reflected point $\sigma_{H}(x)$ will also be
denoted by $x^{H}$.
###### Definition 2.2.
Let $H$ be a given half-space in ${\mathbb{R}}^{N}$. The two-point
rearrangement (or polarization) of a nonnegative real valued function
$u:{\mathbb{R}}^{N}\to{\mathbb{R}}_{+}$ with respect to a given reflection
$\sigma_{H}$ (with respect to $H$) is defined as
$u^{H}(x):=\begin{cases}\max\\{u(x),u(\sigma_{H}(x))\\},&\text{for $x\in
H$},\\\ \min\\{u(x),u(\sigma_{H}(x))\\},&\text{for
$x\in{\mathbb{R}}^{N}\setminus H$}.\end{cases}$
###### Definition 2.3.
We say that a nonnegative measurable function $u$ is symmetrizable if
$\mu(\\{x\in{\mathbb{R}}^{N}:u(x)>t\\})<\infty$ for all $t>0$. The space of
symmetrizable functions is denoted by $F_{N}$ and, of course,
$L^{p}_{+}({\mathbb{R}}^{N})\subset F_{N}$. Also, two functions $u,v$ are said
to be equimeasurable (and we shall write $u\sim v$) when
$\mu(\\{x\in{\mathbb{R}}^{N}:u(x)>t\\})=\mu(\\{x\in{\mathbb{R}}^{N}:v(x)>t\\}),$
for all $t>0$.
###### Definition 2.4.
For a given $u$ in $F_{N}$, the Schwarz symmetrization $u^{*}$ of $u$ is the
unique function with the following properties (see e.g. [19])
1. (1)
$u$ and $u^{*}$ are equimeasurable;
2. (2)
$u^{*}(x)=h(|x|)$, where $h:(0,\infty)\to{\mathbb{R}}_{+}$ is a continuous and
decreasing function.
In particular, $u$, $u^{H}$ and $u^{*}$ are all equimeasurable functions (see
e.g. [2]).
###### Lemma 2.5.
Let $u\in W^{1,p}_{+}({\mathbb{R}}^{N})$ and let $H$ be a given half-space.
Then $u^{H}\in W^{1,p}_{+}({\mathbb{R}}^{N})$ and, setting
$v(x):=u(x^{H}),\quad w(x):=u^{H}(x^{H}),\qquad x\in{\mathbb{R}}^{N},$
the following facts hold:
1. (1)
We have
$\displaystyle\nabla u^{H}(x)$ $\displaystyle=\begin{cases}\nabla
u(x)&\text{for $x\in\\{u>v\\}\cap H$},\\\ \nabla v(x)&\text{for $x\in\\{u\leq
v\\}\cap H$},\\\ \end{cases}$ $\displaystyle\nabla w(x)$
$\displaystyle=\begin{cases}\nabla v(x)&\text{for $x\in\\{u>v\\}\cap H$},\\\
\nabla u(x)&\text{for $x\in\\{u\leq v\\}\cap H$}.\\\ \end{cases}$
2. (2)
For all $i=1,\dots,N$ and $p\in(1,\infty)$, we have
(2.1)
$\|D_{i}u^{H}\|_{L^{p}({\mathbb{R}}^{N})}=\|D_{i}u\|_{L^{p}({\mathbb{R}}^{N})}.$
3. (3)
Let $j:[0,\infty)\times[0,\infty)\to{\mathbb{R}}$ be a Borel measurable
function. Then
(2.2) $\int_{{\mathbb{R}}^{N}}j(u,|\nabla
u|)dx=\int_{{\mathbb{R}}^{N}}j(u^{H},|\nabla u^{H}|)dx,$
provided that $0\in H$ and that both integrals are finite.
###### Proof.
Observing that, for all $x\in H$, we have
$u^{H}(x)=v(x)+(u(x)-v(x))^{+},\qquad w(x)=u(x)-(u(x)-v(x))^{+},$
in light of [26, Corollary 6.18] it follows that $v,w$ belong to
$W^{1,p}_{+}({\mathbb{R}}^{N})$. Assertion (1) follows by a simple direct
computation. Assertion (2) follows as a consequence of assertion (1).
Concerning (3), writing $\sigma_{H}$ as $\sigma_{H}(x)=x_{0}+Rx$, where $R$ is
an orthogonal linear transformation, taking into account that $|{\rm
det}\,R|=1$ and
$|\nabla v(x)|=|\nabla(u(\sigma_{H}(x)))|=|R(\nabla
u(\sigma_{H}(x)))|=|(\nabla u)(\sigma_{H}(x))|,$
we have
$\displaystyle\int_{{\mathbb{R}}^{N}}j(u,|\nabla u|)dx$
$\displaystyle=\int_{H}j(u,|\nabla u|)dx+\int_{{\mathbb{R}}^{N}\setminus
H}j(u,|\nabla u|)dx$ $\displaystyle=\int_{H}j(u,|\nabla
u|)dx+\int_{H}j(u(\sigma_{H}(x)),|(\nabla u)(\sigma_{H}(x))|)dx$
$\displaystyle=\int_{H}j(u,|\nabla u|)dx+\int_{H}j(v,|\nabla v|)dx.$
In a similar fashion, we have
$\displaystyle\int_{{\mathbb{R}}^{N}}j(u^{H},|\nabla u^{H}|)dx$
$\displaystyle=\int_{H}j(u^{H},|\nabla
u^{H}|)dx+\int_{H}j(u^{H}(\sigma_{H}(x)),|(\nabla u^{H})(\sigma_{H}(x))|)dx$
$\displaystyle=\int_{H}j(u^{H},|\nabla u^{H}|)dx+\int_{H}j(w,|\nabla w|)dx$
$\displaystyle=\int_{\\{u>v\\}\cap H}j(u,|\nabla u|)dx+\int_{\\{u>v\\}\cap
H}j(v,|\nabla v|)dx$ $\displaystyle+\int_{\\{u\leq v\\}\cap H}j(v,|\nabla
v|)dx+\int_{\\{u\leq v\\}\cap H}j(u,|\nabla u|)dx$
$\displaystyle=\int_{H}j(u,|\nabla u|)dx+\int_{H}j(v,|\nabla v|)dx,$
which concludes the proof ∎
## 3\. Generalized Polya-Szegö inequality
The first main result of the paper is the following
###### Theorem 3.1.
Let $\varrho:[0,\infty)\times{\mathbb{R}}^{N}\to{\mathbb{R}}$ be a Borel
measurable function. For any function $u\in W^{1,p}_{+}({\mathbb{R}}^{N})$,
let us set
$J(u)=\int_{{\mathbb{R}}^{N}}\varrho(u,\nabla u)dx.$
Moreover, let $(H_{n})_{n\geq 1}$ be a dense sequence in the set of closed
half spaces containing $0_{{\mathbb{R}}^{N}}$. For $u\in
W^{1,p}_{+}({\mathbb{R}}^{N})$, define a sequence $(u_{n})$ by setting
$\begin{cases}u_{0}=u&\\\ u_{n+1}=u_{n}^{H_{1}\ldots H_{n+1}}.&\end{cases}$
Assume that the following conditions hold:
1. (1)
$-\infty<J(u)<+\infty;$
2. (2)
(3.1) $\liminf_{n}J(u_{n})\leq J(u);$
3. (3)
if $(u_{n})$ converges weakly to some $v$ in $W^{1,p}_{+}({\mathbb{R}}^{N})$,
then
$J(v)\leq\liminf_{n}J(u_{n}).$
Then
$J(u^{*})\leq J(u).$
###### Proof.
By the (explicit) approximation results contained in [17, 38], we know that
$u_{n}\to u^{*}$ in $L^{p}({\mathbb{R}}^{N})$ as $n\to\infty$. Moreover, by
Lemma 2.5 applied with $j(s,|\xi|)=|\xi|^{p}$, we have
(3.2) $\|\nabla u\|_{L^{p}({\mathbb{R}}^{N})}=\|\nabla
u_{0}\|_{L^{p}({\mathbb{R}}^{N})}=\cdots=\|\nabla
u_{n}\|_{L^{p}({\mathbb{R}}^{N})},\quad\text{for all $n\in{\mathbb{N}}$}.$
In particular, up to a subsequence, $(u_{n})$ is weakly convergent to some
function $v$ in $W^{1,p}({\mathbb{R}}^{N})$. By uniqueness of the weak limit
in $L^{p}({\mathbb{R}}^{N})$ one can easily check that $v=u^{*}$, namely
$u_{n}\rightharpoonup u^{*}$ in $W^{1,p}({\mathbb{R}}^{N})$. Hence, using
assumption (3) and (3.1), we have
(3.3) $J(u^{*})\leq\liminf_{n}J(u_{n})\leq J(u),$
concluding the proof. ∎
###### Remark 3.2.
A quite large class of functionals $J$ which satisfy assumption (3.1) of the
previous Theorem is provided by Lemma 2.5.
###### Corollary 3.3.
Let $j:[0,\infty)\times[0,\infty)\to{\mathbb{R}}$ be a function satisfying the
following assumptions:
1. (1)
$j(\cdot,t)$ is continuous for all $t\in[0,\infty)$;
2. (2)
$j(s,\cdot)$ is convex for all $s\in[0,\infty)$ and continuous at zero;
3. (3)
$j(s,\cdot)$ is nondecreasing for all $s\in[0,\infty)$.
Then, for all function $u\in W^{1,p}_{+}({\mathbb{R}}^{N})$ such that
$\int_{{\mathbb{R}}^{N}}j(u,|\nabla u|)dx<\infty,$
we have
$\int_{{\mathbb{R}}^{N}}j(u^{*},|\nabla
u^{*}|)dx\leq\int_{{\mathbb{R}}^{N}}j(u,|\nabla u|)dx.$
###### Proof.
The assumptions on $j$ imply that $\\{\xi\mapsto j(s,|\xi|)\\}$ is convex so
that the weak lower semicontinuity assumption of Theorem 3.1 holds (we refer
the reader e.g. to the papers [21, 22] by A. Ioffe). Also, assumption (3.1) of
Theorem 3.1 is provided by means of Lemma 2.5. ∎
###### Remark 3.4.
In [6, Theorem 4.3], F. Brock proved Corollary 3.3 for Lipschitz functions
having compact support. In order to prove the most interesting cases in the
applications, the inequality has to hold for functions $u$ in
$W^{1,p}_{+}({\mathbb{R}}^{N})$. This forces him to assume some growth
conditions of the Lagrangian $j$, for instance to assume that there exists a
positive constant $K$ and $q\in[p,p^{*}]$ such that
$|j(s,|\xi|)|\leq K(s^{q}+|\xi|^{p}),\quad\text{for all $s\in{\mathbb{R}}_{+}$
and $\xi\in{\mathbb{R}}^{N}$}.$
By our approach, instead, can include integrands such as
$j(s,|\xi|)=\frac{1}{2}(1+s^{2\alpha})|\xi|^{p},\quad\text{for all
$s\in{\mathbb{R}}_{+}$ and $\xi\in{\mathbb{R}}^{N}$},$
for some $\alpha>0$, which have meaningful physical applications (for instance
quasi-linear Schrödinger equations, see [27] and references therein). We also
stress that the approach of [6] cannot yield the establishment of equality
cases (see Theorem 3.6).
###### Corollary 3.5.
Let $m\geq 1$ and $p_{1},\dots,p_{m}\in(1,\infty)$. Then
$\sum_{i=1}^{m}\int_{{\mathbb{R}}^{N}}|D_{i}u^{*}|^{p_{i}}dx\leq\sum_{i=1}^{m}\int_{{\mathbb{R}}^{N}}|D_{i}u|^{p_{i}}dx,$
for all $u\in\bigcap_{i=1}^{m}W^{1,p_{i}}_{+}({\mathbb{R}}^{N})$.
###### Proof.
The assertion follows by a simple combination of Theorem 3.1 with inequality
(2.1) of Lemma 2.5. ∎
###### Theorem 3.6.
In addition to the assumptions of Theorem 3.1, assume that
(3.4) $\text{$J(u_{n})\to J(u^{*})$ as $n\to\infty$ implies that $u_{n}\to
u^{*}$ in $D^{1,p}({\mathbb{R}}^{N})$ as $n\to\infty$}.$
Then
$J(u)=J(u^{*})\,\,\Longrightarrow\,\,\|\nabla
u\|_{L^{p}({\mathbb{R}}^{N})}=\|\nabla u^{*}\|_{L^{p}({\mathbb{R}}^{N})}.$
###### Proof.
Assume that $J(u)=J(u^{*})$. Then, by assumption (3.1), we obtain
$J(u^{*})=\lim_{n}J(u_{n})=J(u).$
In turn, by assumption, $u_{n}\to u^{*}$ in $D^{1,p}({\mathbb{R}}^{N})$ as
$n\to\infty$. Then, taking the limit inside equalities (3.2), we conclude the
assertion. ∎
###### Remark 3.7.
Assume that $\\{\xi\mapsto j(s,|\xi|)\\}$ is strictly convex for any
$s\in{\mathbb{R}}_{+}$ and there exists $\nu^{\prime}>0$ such that
$j(s,|\xi|)\geq\nu^{\prime}|\xi|^{p}$ for all $s\in{\mathbb{R}}_{+}$ and
$\xi\in{\mathbb{R}}^{N}$. Then assumption (3.4) is fulfilled for
$J(u)=\int_{{\mathbb{R}}^{N}}j(u,|\nabla u|)dx$. We refer to [39, Section 3].
###### Remark 3.8.
Equality cases of the type $\|\nabla u\|_{L^{p}({\mathbb{R}}^{N})}=\|\nabla
u^{*}\|_{L^{p}({\mathbb{R}}^{N})}$ have been completely characterized in the
breakthrough paper by Brothers and Ziemer [10].
Let us now set
$\displaystyle{M={\rm esssup}_{{\mathbb{R}}^{N}}u={\rm
esssup}_{{\mathbb{R}}^{N}}u^{*}},\qquad C^{*}=\\{x\in{\mathbb{R}}^{N}:\nabla
u^{*}(x)=0\\}.$
###### Corollary 3.9.
Assume that $\\{\xi\mapsto j(s,|\xi|)\\}$ is strictly convex and there exists
a positive constant $\nu^{\prime}$ such that
$j(s,|\xi|)\geq\nu^{\prime}|\xi|^{p},\quad\text{for all $s\in{\mathbb{R}}$ and
$\xi\in{\mathbb{R}}^{N}$}.$
Moreover, assume that
$\int_{{\mathbb{R}}^{N}}j(u,|\nabla
u|)dx=\int_{{\mathbb{R}}^{N}}j(u^{*},|\nabla
u^{*}|)dx,\quad\mu(C^{*}\cap(u^{*})^{-1}(0,M))=0.$
Then there exists $x_{0}\in{\mathbb{R}}^{N}$ such that
$u(x)=u^{*}(x-x_{0}),\quad\text{for all $x\in{\mathbb{R}}^{N}$},$
namely $u$ is radially symmetric after a translation in ${\mathbb{R}}^{N}$.
###### Proof.
It is sufficient to combine Theorem 3.6 with [10, Theorem 1.1]. ∎
## References
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|
arxiv-papers
| 2010-07-01T14:18:02 |
2024-09-04T02:49:11.354210
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H.Hajaiej",
"submitter": "Hichem Hajaiej",
"url": "https://arxiv.org/abs/1007.0176"
}
|
1007.0226
|
# On the motion of high energy wave packets and the transition radiation by
“half-bare” electron
N.F. Shul’ga shulga@kipt.kharkov.ua V.V. Syshchenko syshch@bsu.edu.ru S.N.
Shul’ga Akhiezer Institute for Theoretical Physics, National Science Center
“Kharkov Institute of Physics and Technology”, Akademicheskaya st., 1, Kharkov
61108, Ukraine Belgorod State University, Pobedy st., 85, Belgorod 308015,
Russian Federation
###### Abstract
The problem of the motion of high-energy wave packets combined of free
electromagnetic waves is considered. It is demonstrated that the
transformation of such packets to the packet of spherically diverging waves
happens on long distances along the packet’s motion direction, that
substantially exceed the radiated wavelength. The transition radiation by the
“half-bare” ultrarelativistic electron is considered. It is demonstrated that
the transition radiation by such an electron on the targets located inside and
outside the coherence length of the radiation process would be substantially
different.
###### keywords:
equivalent photons method , wave packet , half-bare electron
###### PACS:
41.20.-q , 41.60.-m
††journal: Physics Letters A
## 1 Introduction
Moving electron is the charge and the eigenfield (Coulomb field) moving
together with it. Changing the electron’s trajectory disturbs that field. The
disturbance of the field could be treated as a packet of free plane
electromagnetic waves. On large distances from the region where the
acceleration had happened the packet transforms to the packet of diverging
waves (the radiation field). For non-relativistic particles that happens on
the distances of order of the length $\lambda$ of radiated wave [1]. High
energies make the stabilizing influence to wave packets that leads to a
substantial increase of the length on which the packet’s transformation takes
place. This length could have macroscopic size, exceeding not only interatomic
distances in matter, but also the size of the target and just the size of the
experimental installation (detector). Hence it is important to know the
behavior of such high-energy packets of electromagnetic waves in the region
where that transformation happens. The present article is devoted to the
examination of this problem.
Primarily, we consider the motion of Gaussian packet combined of plane waves
with directions of the wave vector $\mathbf{k}$ close to each other. It is
demonstrated that the shape of such a packet changes on the lengths that
substantially exceed the wavelength $\lambda=1/|\mathbf{k}|$ corresponding to
the given absolute value of the wave vector $|\mathbf{k}|$.
Then we consider the motion of the wave packet that coincides in the time
moment $t=0$ with the eigenfield of the ultrarelativistic electron. It is
demonstrated that the last packet also conserves its shape for a long time
interval. Fourier component of this packet with the wavelength $\lambda$
changes only on the distances $z$ along the packet’s direction of motion that
exceed the length $2\gamma^{2}\lambda$, where $\gamma$ is the electron’s
Lorentz factor. This length coincides with the coherence length of the
radiation process of the relativistic electron $l=2\gamma^{2}\lambda$ [2, 3].
The problem of special interest is the radiation under sharp (at the time
moment $t=0$) changing of the ultrarelativistic electron’s velocity [3 - 5].
We demonstrate that the packets of electromagnetic waves arising in this case
are close in their structure to the packets considered above. However, their
manifestations in the direction of the initial and final motion of the
electron are substantially different. Namely, on the distances
$z<2\gamma^{2}\lambda$, Fourier components with the wavelength $\lambda$ of
the packet moving along the direction of the initial electron motion will
practically coincide with the Fourier components of the initial packet and,
consequently, to the Fourier components of the Coulomb field of the electron
moving in the initial direction without scattering. Oppositely, in the final
electron’s direction of motion, the field of the packet of free waves will
screen the particle’s eigenfield. The electron under such conditions was
called in [4] as “half-bare particle”, that is the particle whose specific
Fourier components of the surrounding field are practically absent for a long
time. We put attention to that the transition radiation by such particles and
wave packets on the targets placed on the distances from the point of
scattering larger and smaller than $2\gamma^{2}\lambda$ would be substantially
different. The corresponding experiment would permit to observe direct
manifestation of the “half-bare” electron and the process of its dressing.
Let us note that for the charged particle the Gauss theorem is applicable,
according to which the number of force lines of the electromagnetic field
surrounding the electron does not change with time [1]. Under this the
radiation process by electron can be presented as bending of these force lines
[6 - 10]. Such a concept of radiation process relates to the complete
electromagnetic field surrounding the electron. However, it does not contain
such characteristics of the radiation process as coherence length and wave
zone which are connected with determined Fourier components of this field. The
term “half-bare electron” relates also to a determined Fourier component of
the field surrounding the electron which is defined by the wavelength
$\lambda$. So, the analysis of a space-time evolution of these Fourier
components (wave packets) gives us a supplement for the picture of evolution
of complete field surrounding the electron which is in accelerated motion.
We use the system of units in which the speed of light in vacuum is taken
equal to the unit: $c=1$.
## 2 Gaussian packet
The scalar potential of the packet of free electromagnetic waves could be
expressed in the form of the following Fourier decomposition:
$\varphi(\mathbf{r},t)=\int\frac{d^{3}q}{(2\pi)^{3}}e^{i(\mathbf{q}\mathbf{r}-qt)}C_{q},$
(1)
where $C_{q}$ are the coefficients of the decomposition, $q=|\mathbf{q}|$.
Consider at first the behavior of the packet combined at $t=0$ of plane waves
with the wave vectors $\mathbf{k}$ directed closely to some given direction
(the $z$-axis). Supposing for simplicity that the distribution of the waves
over directions of the vector $\mathbf{k}$ is Gaussian at $t=0$, let us write
the potential (1) in the form
$\varphi_{k}(\mathbf{r},0)=\frac{1}{\pi\Delta^{2}}\int d^{2}\vartheta
e^{-\vartheta^{2}/\Delta^{2}}e^{i\mathbf{k}\mathbf{r}},$ (2)
where $\vartheta$ is the angle between $\mathbf{k}$ and the $z$-axis, and
$\Delta^{2}$ is the mean square value of the angle $\vartheta$, $\Delta^{2}\ll
1$. Coefficients $C_{\mathbf{q}}$ of a such packet have the form
$C_{\mathbf{q}}=(2\pi)^{3}\int\frac{d^{2}\vartheta}{\pi\Delta^{2}}e^{-\vartheta^{2}/\Delta^{2}}\delta(\mathbf{k}-\mathbf{q}),$
(3)
where $\delta(\mathbf{k}-\mathbf{q})$ is the delta-function. In this case,
according to (1),
$\varphi_{k}(\mathbf{r},t)=\frac{1}{1+ikz\Delta^{2}/2}\exp\left\\{ik(z-t)-\frac{(k\rho\Delta/2)^{2}}{1+i(kz\Delta^{2}/2)}\right\\},$
(4)
where $\rho$ is the transverse (in relation to the $z$-axis) component of
$\mathbf{r}$.
Eq. (4) demonstrates that under $kz\Delta^{2}/2\ll 1$
$\varphi_{k}(\mathbf{r},t)\approx\exp\left\\{ik(z-t)-(k\rho\Delta/2)^{2}\right\\},$
(5)
and under the condition $kz\Delta^{2}/2\gg 1$
$\varphi_{k}(\mathbf{r},t)\approx-\frac{2i}{kz\Delta^{2}}\exp\left\\{ik(z-t)+ik\frac{\rho^{2}}{2z}-\frac{\rho^{2}}{z^{2}\Delta^{2}}\right\\}.$
(6)
In the case $z\gg\rho$ the last formula could be written in the form of
diverging wave:
$\varphi_{k}(\mathbf{r},t)\approx-\frac{2i}{kr\Delta^{2}}\exp\left\\{ik(r-t)-\frac{\rho^{2}}{z^{2}\Delta^{2}}\right\\},$
(7)
where $r=\sqrt{\rho^{2}+z^{2}}\approx z+\rho^{2}/2z$.
So, on the distances $z$ from the center of the packet that satisfy the
condition
$kz\Delta^{2}/2\ll 1,$ (8)
the shape of the packet (4) coincides with the packet’s shape at $t=0$. Only
on the distances $z$ that satisfy the condition
$kz\Delta^{2}/2>1,$ (9)
the transformation of the packet of plane waves (4) into the packet of
diverging spherical waves happens.
In the theory of radiation of electromagnetic waves, the spatial region where
the field of moving charges acquires the form of spherically diverging waves,
is called as wave zone (see, e.g. [1, 11]). Particularly, for non-relativistic
charged particles the wave zone begins just on the distances from the
radiating system that exceed the radiated wavelength (see [1]). Condition (9)
demonstrates, however, that under $\Delta^{2}\ll 1$ the formation of the wave
zone takes place not on the distances $z>\lambda$, like in the problem of
radiation of the non-relativistic particle, but on the distances
$z>2\lambda/\Delta^{2},$ (10)
which are much larger than the wavelengths $\lambda=1/k$, of which the packet
is composed (4). For small values of $\Delta^{2}$ the length
$z=2\lambda/\Delta^{2}$ could reach macroscopic sizes.
## 3 Approximation of Coulomb field by the packet of plane waves
Such problem arises in the equivalent photons method (or the method of virtual
quanta) when the Coulomb field of relativistic electron is replaced at some
specific time moment ($t=0$) by the packet of free electromagnetic waves.
Indeed, the Fourier decomposition of the electron’s Coulomb field could be
written in the form
$\varphi_{c}(\mathbf{r},t)=\mathop{\mathrm{Re}}\nolimits\int\frac{d^{3}k}{(2\pi)^{3}}e^{i(\mathbf{k}\mathbf{r}-\mathbf{k}\mathbf{v}t)}C_{k}^{c},$
(11)
where $\mathbf{v}$ is the electron’s velocity directed along the $z$-axis, and
$C_{k}^{c}=\frac{8\pi e\Theta(k_{z})}{k_{\perp}^{2}+k_{z}^{2}/\gamma^{2}}.$
(12)
Here $\gamma$ is the electron’s Lorentz factor, $k_{z}$ and
$\mathbf{k}_{\perp}$ are the components of the vector $\mathbf{k}$, parallel
and orthogonal to the $z$-axis, $\Theta(k_{z})$ is the Heaviside’s step
function.
It is supposed in the equivalent photons method that at $t=0$ the packet (1)
composed of free electromagnetic waves coincides with the electron’s Coulomb
field moving with the velocity $\mathbf{v}$ [11 - 13]. That corresponds to
Fourier decomposition (1) with the coefficients $C_{q}=C_{k}^{c}$.
For $\gamma\gg 1$ the main contribution to (1) would be made by the values
$\mathbf{q}=\mathbf{k}$ which directions are close to the direction of the
electron’s velocity $\mathbf{v}$. Taking this into account, the packet (1)
could be written in the form
$\varphi(\mathbf{r},t)=\mathop{\mathrm{Re}}\nolimits\int_{0}^{\infty}dk\,\varphi_{k}(\mathbf{r},t),$
(13)
where
$\varphi_{k}(\mathbf{r},t)=\frac{2}{\pi}\exp\left[ik(z-t)\right]\int_{0}^{\infty}\frac{\vartheta
d\vartheta}{\vartheta^{2}+\gamma^{-2}}J_{0}(k\rho\vartheta)\,e^{-ikz\,\vartheta^{2}/2}.$
(14)
Here $\vartheta$ is the angle between $\mathbf{k}$ and $\mathbf{v}$
($\vartheta\ll 1$), and $J_{0}(x)$ is the Bessel function.
The function $\varphi_{k}(\mathbf{r},t)$ has the same structure as the
function (4) corresponding to Gaussian distribution of the vectors
$\mathbf{k}$ over the angles $\vartheta$. Namely, if $kz\vartheta^{2}/2\ll 1$,
the main contribution to the integral (14) is made by the values
$\vartheta\sim\gamma^{-1}$ and
$\varphi_{k}(\mathbf{r},t)\approx\frac{2}{\pi}K_{0}(k\rho/\gamma)\,e^{ik(z-t)},$
(15)
where $K_{0}(x)$ is the modified Hankel function. In this case after
integration over $k$ in (13) we find that
$\varphi(\mathbf{r},t)\approx\frac{e}{\sqrt{(z-t)^{2}+\rho^{2}/\gamma^{2}}}.$
(16)
The main contribution to (13) is made by the values $k\sim\gamma/\rho$, hence
Eq. (16) is valid in the range of coordinates $\rho$ and $z$ that satisfy the
condition $z<\gamma\rho$. In this range of coordinates the packet under
consideration moves with the velocity of light in the $z$-axis direction.
So, on the distances $z\lesssim 2\gamma^{2}\lambda$ the considered wave packet
practically coincides with the initial one (at $t=0$). Substantial
transformation of the packet would happen only on the distances
$z>2\gamma^{2}\lambda.$ (17)
In this case for the evaluation of the integral in (14) over $\vartheta$ one
could apply the method of stationary phase. As a result of using of this
method we find that
$\varphi_{k}(\mathbf{r},t)=-\frac{2i}{\pi}\frac{1}{\vartheta_{0}^{2}+\gamma^{-2}}\frac{1}{kr}e^{ik(r-t)},$
(18)
where $r\approx z+\rho^{2}/2z$ and $\vartheta_{0}=\rho/z$ is the point of
stationary phase of the integral (14). We see that the components (18) of our
packet have in the case under consideration the form of diverging spherical
waves. Under this condition the angle $\vartheta_{0}$ corresponds to the
direction of radiation, and the function before the diverging wave describes
the angular distribution of the radiation. So, the condition (17) draws out
the wave zone in application to given problem.
The value $2\gamma^{2}\lambda$ presenting in the condition (17) is known in
the theory of radiation by ultrarelativistic particles as the formation length
or the coherence length [2, 3].
## 4 Transition radiation by a “half-bare electron”
High-energy packets of electromagnetic waves considered above manifest
themselves in many problems connected with bremsstrahlung and diffraction
radiation (see, e.g., [5, 14, 15]). Let us pay attention to some
manifestations of such packets in the problem of transition radiation arising
after sharp scattering of the high-energy electron on large angle.
The retarded solution for the potential of the electromagnetic field after the
scattering of the electron at the time moment $t=0$ on large angle could be
expressed in the following form [3]:
$\varphi(\mathbf{r},t)=\Theta(r-t)\varphi_{\mathbf{v}}(\mathbf{r},t)+\Theta(t-r)\varphi_{\mathbf{v}^{\prime}}(\mathbf{r},t),$
(19)
where $\varphi_{\mathbf{v}}(\mathbf{r},t)$ and
$\varphi_{\mathbf{v}^{\prime}}(\mathbf{r},t)$ are potentials of the Coulomb
field of the electrons moving all the time with the velocity $\mathbf{v}$
along the $z$-axis and with the velocity $\mathbf{v}^{\prime}$ along the
$z^{\prime}$-axis, respectively. Eq. (19) demonstrates that after scattering
of the electron at $t=0$ its eigenfield strips out and after that transforms
into the radiation field. In the direction of the final particle’s motion the
electron’s eigenfield arises only in the region $r<t$ which is achieved by the
signal about the scattering act at $t=0$ (see Fig. 1, where the isolines of
the scalar potential (19) are presented).
Consider the Fourier decomposition of (19):
$\displaystyle\varphi(\mathbf{r},t)={e\over
2\pi^{2}}\mathop{\mathrm{Re}}\nolimits\int{d^{3}k\over
k}e^{i\mathbf{k}\mathbf{r}}\left\\{{1\over
k-\mathbf{k}\mathbf{v}}e^{-ikt}\right.$ $\displaystyle+\left.{1\over
k-\mathbf{k}\mathbf{v}^{\prime}}\left[1-e^{-i(k-\mathbf{k}\mathbf{v}^{\prime})t}\right]e^{-i\mathbf{k}\mathbf{v}^{\prime}t}\right\\}.$
(20)
The first term in this formula has the form of the packet of free waves moving
along initial direction of the electron’s velocity $\mathbf{v}$. This packet
coincides with the electron’s eigenfield at $t=0$. According to (17), (18),
the transformation of the Fourier components of this packet with the
wavelength $\lambda$ to the packet of diverging waves would happen on the
distances $z>2\gamma^{2}\lambda$. On smaller distances the packet of waves
with the given value of $|\mathbf{k}|$ would be close to the initial one.
The length $l=2\gamma^{2}\lambda$ on which the formation of the wave zone
takes place could have macroscopic size. For example, for the electrons of
energy 50 MeV in the range of wavelengths $\lambda\sim 10^{-1}$ cm this length
is about 20 m (the measuring technique in such conditions is developed today —
see, e.g. [15, 16] ). So in the frames of that length one could arrange a thin
target (see the target in Fig.1 which is arranged along the z-axis at
$z<2\gamma^{2}\lambda$) and examine the “transition radiation” of the
considered packet (reflection of the waves, their passage through target
etc.). The characteristics of such “transition radiation” practically would
not differ from the characteristics of the transition radiation of the
electron moving in the same direction (however, the electron in the packet
under consideration is absent). But if the target would be located on the
distance $z>2\gamma^{2}\lambda$ (see dashed-line box in Fig.1), the features
of the considered “transition radiation” would change due to the changing of
the packet’s shape (formation of the diverging waves).
The second term in (20) describes the field surrounding the electron after its
scattering at $t=0$, when its velocity became equal to $\mathbf{v}^{\prime}$.
This field consists of the electron’s eigenfield moving with the velocity
$\mathbf{v}^{\prime}$ (the first term in square brackets in (20)) and the
packet of free waves moving in the direction of $\mathbf{v}^{\prime}$
coinciding at $t=0$ with the opposite sign with Coulomb field of the electron
(the second term in square brackets).
As it was demonstrated above, transformation of the packet of plane waves to
the packet of diverging waves takes place on the distances $z^{\prime}\sim
2\gamma^{2}\lambda$, where the axis $z^{\prime}$ is directed along
$\mathbf{v}^{\prime}$. During the time interval $t$ over which the electron
passes that distance, the substantial cancellation of the terms in the square
brackets in (20) takes place. This mean that the electron stays on that
distance in a “half-bare” state: the Fourier components with the wave vector
$\mathbf{k}$ of its surrounding field would be suppressed comparing to the
case $z^{\prime}>2\gamma^{2}\lambda$. Transition radiation of the electron
with such field (“half-bare” electron) on the target located on the distance
$z^{\prime}<2\gamma^{2}\lambda$ from the point of scattering (see the target
on Fig.1 which is arranged along the $z^{\prime}$-axis) would be suppressed in
comparison to the case $z^{\prime}>2\gamma^{2}\lambda$.
Figure 1: Equipotential surfaces of (19) and possible positions of targets for
producing of the transition radiation.
The results obtained are correct for sharp scattering of an electron at a
large angle. Sharp scattering means that it takes place on the length which is
much smaller than the coherent radiation length. At macroscopical values of
the coherent length $l=2\gamma^{2}\lambda$ it can occur not only under
scattering of an electron by an atom but also under its scattering by a
magnet. The only condition required for this is that the size of a scatterer
was small as compared with the coherent radiation length.
Note that bremsstrahlung arising under collisions of the “half-bare” electron
with the atoms of the medium located in the frames of the radiation formation
length is suppressed comparing to the case when the collisions happen out of
that length [5, 17, 18]. That lead, particularly, to such effects as Landau-
Pomeranchuk-Migdal effect of suppression of the radiation by ultrarelativistic
electrons in amorphous medium, the effect of suppression of the coherent
bremsstrahlung in crystals and the effect of suppression of the radiation in
thin layers of substance (see recent reviews and monographs [19 - 22] devoted
to this topic, and the references therein). Experimental studies of these
effects were carried out during last years and are made at present time on the
accelerators of ultra high energies (see, e.g., [21 - 23]). Examination of the
process of transition radiation by “half-bare” electron creates one more
opportunity for study of manifestations of such an electron under its
interaction with matter.
## Acknowledgements
This work is supported in part by the internal grant of Belgorod State
University.
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* [14] N.F. Shul’ga, S.N. Dobrovol’sky, JETP 90 (2000) 579; Nucl. Instrum. and Methods B 201 (2003) 123.
* [15] G. Naumenko, X. Artru, A. Potylitsyn et al., arXiv:0901.2630 [physics.acc-ph]
* [16] Y. Shibata, K. Ishi, T. Takahashi et al., Phys. Rev. E 49 (1994) 785.
* [17] A.I. Akhiezer, N.F. Shul’ga, Sov. Phys. Usp. 30 (1987) 197.
* [18] E.L. Feinberg, Sov. Phys. Usp. 22 (1979) 479.
* [19] R. Blankenbecler, S. Drell, Phys. Rev. D 53 (1996) 6265.
* [20] A.I. Akhiezer, N.F. Shul’ga, S.P. Fomin, Landau-Pomeranchuk-Migdal Effect, Cambridge Sci. Pub., 2005\.
* [21] S. Klein, Rev. of Mod. Phys. 71 (1999) 1501.
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|
arxiv-papers
| 2010-07-01T17:56:00 |
2024-09-04T02:49:11.360882
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "N. F. Shul'ga, V. V. Syshchenko, S. N. Shul'ga",
"submitter": "Vladislav Syshchenko",
"url": "https://arxiv.org/abs/1007.0226"
}
|
1007.0291
|
# Landau level states on a topological insulator thin film
Zhihua Yang Department of Physics and BK21 Physics Research Division,
Sungkyunkwan University, Suwon 440-746, Korea Xinjiang Technical Institute of
Physics $\&$ Chemistry, Chinese Academy of Sciences, Urumqi 830011, China
Jung Hoon Han hanjh@skku.edu Department of Physics and BK21 Physics Research
Division, Sungkyunkwan University, Suwon 440-746, Korea
###### Abstract
We analyze the four-dimensional Hamiltonian proposed to describe the band
structure of the single-Dirac-cone family of topological insulators in the
presence of a uniform perpendicular magnetic field. Surface Landau level(LL)
states appear, decoupled from the bulk levels and following the quantized
energy dispersion of a purely two-dimensional surface Dirac Hamiltonian. A
small hybridization gap splits the degeneracy of the central $n=0$ LL with
dependence on the film thickness and the field strength that can be obtained
analytically. Explicit calculation of the spin and charge densities show that
surface LL states are localized within approximately one quintuple layer from
the surface termination. Some new surface-bound LLs are shown to exist at a
higher Landau level index.
###### pacs:
73.20.-r,73.43.-f,85.75.-d
## I Introduction
Insulating materials with topologically protected surface states known as
topological insulators (TIs) are a matter of great current interestkane-hasan
; qi-zhang ; moore . The surface metallic states in this new class of
materials is characterized by Dirac-like quasiparticle dispersion, and a one-
to-one correspondence between momentum and spin quantum numbers of the single-
particle states thus representing an extreme form of spin-orbit coupling. Both
these aspects have been confirmed for the first time in BixSb1-x familykane-fu
of topological insulators by ARPESARPES-on-BiSb and STMSTM-on-BiSb studies.
More recently, a lot of experimental efforts has been given to the synthesis
and characterization of Bi2Se3, Bi2Te3, and Sb2Te3ARPES-on-Bi2Te3 ; STM-on-
Bi2Se3 ; ong ; Bi2Se3-exp ; thin-film-TI-exp-Japan ; thin-film-TI-exp-China ;
STM-B-thin-film ; hanaguri following the prediction of their topological
behaviorBi2Se3-theory ; Bi2Se3-theory2 , due to their simple surface band
structure consisting of a single-cone Dirac spectrum centered at the
$\Gamma$-point and a relatively large band gap. Topological insulators of the
single-Dirac-cone family in the thin-film form has been synthesized by a
number of groupsthin-film-TI-exp-Japan ; thin-film-TI-exp-China ; STM-B-thin-
film . Theoretically, the thin-film TIs bear close analogy to another heavily
studied topological material, i.e. graphenecastro-neto . For instance, the
well-known pair of valley-degenerate Dirac bands of graphene becomes the top
and bottom surface Dirac bands of TIs with finite thickness. Perpendicular
magnetic field quantizes the surface Landau levels (LLs) with the energies
that scale with the LL index $n$ as $\pm\sqrt{n}$ in TIs as well as in
graphene.
Previous treatments of the magnetic field effect on TI surface started from
the two-dimensional (2D) Dirac Hamiltonian focusing only on the surface
electronic states and ignoring the bulk states altogetherqi ; SQShen-LL .
These methods relied on first projecting the bulk Hamiltonian to the surface,
obtaining the 2D Dirac model, then including the field effect by way of
Peierls substitution. In another vein, several recent papers theoretically
examined the properties of a thin slab of TI in which the bulk and surface
electronic states are treated on an equal footinglinder ; liu ; shen in the
absence of the magnetic field. It is thus natural to consider how the magnetic
field effect plays out for a thin film geometry of TI, following the spirit of
solving the bulk Hamiltonian adopted in Refs. linder, ; liu, ; shen, . In
fact, an attempt of precisely this sort has been made in a recent paper by Liu
et al.Bi2Se3-theory2 Here, the authors solved the $4\times 4$ tight-binding
Hamiltonian with the Peierls substitution for the magnetic field and even
including the Zeeman field coupling. We point out in this paper that the
method adopted in Ref. Bi2Se3-theory2, does not treat the surface and bulk
electronic states simultaneously, and as a result the bands arising from
surface LLs penetrate into the bulk LL states, while physically such
overlapping of energy levels will not occur.
Our approach follows closely the spirit of zero-field case studied in Refs.
linder, ; liu, ; shen, and takes care of the boundary conditions properly.
Some parts of our report are technical, dealing with the characteristic
equation resulting from the boundary conditions and the methods of solving
them. Several physically meaningful results follow from our analysis. First,
hybridization of the zeroth-LL states localized on the top and the bottom
surfaces for a sufficiently thin sample is shown to manifest itself as the
splitting of the degeneracy of zeroth-LL states with the gap magnitude that
can be calculated analytically. The zeroth-LL gap size oscillates with the
film thickness. Our finding naturally extrapolates a similar observation of
the gap oscillation observed previouslylinder to finite magnetic field.
Interestingly, we find that a new kind of surface-bound LL states appear for
higher-LL indices where the conventional surface LL band of $\sim\sqrt{n}$
variety has merged into the bulk continuum. Justification of the new surface
LLs is made on the basis of careful numerical study and an approximate
analytic solution of the characteristic equation. Properties of the bulk
single-particle states for higher-LL indices are examined in detail. Finally,
both charge and spin density profiles of the surface LLs at low LL indices
along the thickness of the sample are explicitly worked out.
In Sec. II we formulate the LL problem based on the $4\times 4$ Hamiltonian
proposed previously for Bi2Se3-family of topological insulators. Boundary
conditions are imposed on the two surface layers for a thin-film geometry and
characteristic equations are derived in Sec. III. In Sec. IV several physical
results are shown and its relevance to recent STM are discussed. Summary of
results and an outlook is given in Sec. V. Technical discussion for the new
surface-bound LLs can be found in the Appendix.
## II Formulation
The 3D tight-binding Hamiltonian proposed as a minimal model for single-Dirac-
cone family of TIs first in Ref. Bi2Se3-theory, and detailed in Ref.
Bi2Se3-theory2, is
$\displaystyle H({\bf p})=\varepsilon({\bf p})\\!+\\!\begin{pmatrix}M({\bf
p})\tau_{z}\\!+\\!A_{1}p_{z}\tau_{x}&A_{2}p_{-}\tau_{x}\\\
A_{2}p_{+}\tau_{x}&M({\bf p})\tau_{z}\\!-\\!A_{1}p_{z}\tau_{x}\end{pmatrix}$
(1)
in the basis spanned by
$(\mathrm{Bi}^{+}_{\uparrow},\mathrm{Se}^{-}_{\uparrow},\mathrm{Bi}^{+}_{\downarrow},\mathrm{Se}^{-}_{\downarrow})$.
Pauli matrices $\bm{\tau}$ are introduced and $p_{\pm}=p_{x}\pm ip_{y}$ are
momentum operators. The upper and lower indices in the basis set refer to the
parity and spin quantum numbers for the $p_{z}$ orbitals of Bi or Se atoms,
respectively. It was shownBi2Se3-theory that $\varepsilon({\bf p})$ and
$M({\bf p})$ depend on the momentum ${\bf p}$ as
$\displaystyle\varepsilon({\bf p})$ $\displaystyle=$ $\displaystyle
C+D_{1}p_{z}^{2}+D_{2}(p_{x}^{2}+p_{y}^{2}),$ $\displaystyle M({\bf p})$
$\displaystyle=$ $\displaystyle
M_{0}-B_{1}p_{z}^{2}-B_{2}(p_{x}^{2}+p_{y}^{2}).$ (2)
Values of the various constants can be found in Refs. Bi2Se3-theory, ; linder,
; liu, ; shen, ; Bi2Se3-theory2, . In our paper all the material parameters
are re-scaled in terms of the one mass scale $M_{0}$. Two length parameters
emerge as a result, $l_{z}=A_{1}/M_{0}$ and $l_{\perp}=A_{2}/M_{0}$, each
characterizing the length scale within the plane and perpendicular to it. With
the material parameters given in Ref. Bi2Se3-theory, they read
$l_{\perp}=14.64$Å and $l_{z}=7.9$Å. We use them as the measure of length in
each direction. All equations can be cast in dimensionless form as well as the
two functions $\varepsilon({\bf p})$ and $M({\bf p})$ which now become
(following the parameterization of Ref. Bi2Se3-theory, )
$\displaystyle\varepsilon({\bf p})$ $\displaystyle=$
$\displaystyle-0.024+0.075p_{z}^{2}+0.3265p_{\perp}^{2},$ $\displaystyle
M({\bf p})$ $\displaystyle=$ $\displaystyle
1-0.58p_{z}^{2}-0.94p_{\perp}^{2}.$ (3)
Coefficient-by-coefficient, expressions in $\varepsilon({\bf p})$ are smaller
than the ones in $M({\bf p})$. In this study, we will ignore $\varepsilon({\bf
p})$ for calculational simplicity and restore particle-hole symmetry of the
spectrum as a consequence.
The four-dimensional single-particle eigenstates can be constructed in terms
of two, two-dimensional spinors $u$ and $v$. For an infinite medium one can
write the eigenstate as $\psi=e^{i{\bf k}\cdot{\bf r}}\chi$, where
$\chi=\begin{pmatrix}u\\\ v\end{pmatrix}$ is a 4-component constant spinor to
be determined by solving
$\displaystyle k_{+}u$ $\displaystyle=$
$\displaystyle\Bigl{(}E\tau_{x}+k_{z}+iM\tau_{y}\Bigr{)}v,$ $\displaystyle
k_{-}v$ $\displaystyle=$
$\displaystyle\Bigl{(}E\tau_{x}-k_{z}+iM\tau_{y}\Bigr{)}u,$ $\displaystyle M$
$\displaystyle=$ $\displaystyle
1-\alpha_{z}k_{z}^{2}-\alpha_{\perp}k_{\perp}^{2},$ (4)
with $E$ as the energy, $k_{\pm}=k_{x}\pm ik_{y}$ as the momentum, and
$\alpha_{z}$ and $\alpha_{\perp}$ are two material constants. They read
$\alpha_{z}=0.58$ and $\alpha_{\perp}=0.94$ in the parametrization of Ref.
Bi2Se3-theory, .
Figure 1: (color online) (a) Surface energy spectra without magnetic field
when $\varepsilon_{k}=0$ (red) and $\varepsilon_{k}\neq 0$ (blue). (b) Bulk
and surface energy dispersions in the absence of magnetic field and
$\varepsilon_{k}=0$. $L_{z}/l_{z}$=3000 was used.
As our interest lies in the case of finite thickness $L_{z}$ for the
$z$-direction the above equation will be deformed as
$ik_{z}\rightarrow\lambda_{z}$linder ; liu ; shen . Due to the boundary
conditions at $z=\pm L_{z}/2$, surface state solutions appearlinder ; liu ;
shen . Figure 1(a) shows the difference in the surface energy spectra when the
diagonal energy $\varepsilon_{k}$ is turned on/off. As stressed earlier we
will suppress the diagonal energy $\varepsilon_{k}$ and work with the
particle-hole symmetric model in the following section where we consider the
magnetic field effect. Figure 1(b) shows the surface state energy together
with the bulk energy as a function of the transverse momentum $k_{\perp}$. The
edge state energy dispersion is precisely linear in $|k_{\perp}|$ for large
$|k_{\perp}|$ but opens an exponentially small hybridization gap at
$k_{\perp}=0$. The gap at the $\Gamma$-point is given byshen
$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle{8\alpha\over\beta}\
e^{-\alpha L_{z}}|\sin(\beta L_{z})|,$ (5)
where $\alpha$, $\beta$ are
$\displaystyle\alpha={1\over
2\alpha_{z}},~{}~{}\beta={\sqrt{4\alpha_{z}-1}\over 2\alpha_{z}}.$ (6)
This result will be generalized in the following section to the nonzero
magnetic field, with the revised meaning for the gap as the energy difference
of symmetric and anti-symmetric combinations of zeroth-Landau levels localized
to top and bottom surface layers.
## III Landau Levels
Magnetic field ${\bf H}=(0,0,H)$ perpendicular to the slab modifies the
momentum operator ${\bf p}\rightarrow{\bf p}+{\bf A}=(p_{x}-Hy,p_{y},p_{z})$
in the Hamiltonian. A pair of canonical operators
$\displaystyle\mathcal{A}={1\over\sqrt{2}}\Bigl{(}{y\\!-\\!y_{0}\over
l_{H}}\\!+\\!l_{H}\partial_{y}\Bigr{)},~{}\mathcal{A}^{\dagger}={1\over\sqrt{2}}\Bigl{(}{y\\!-\\!y_{0}\over
l_{H}}\\!-\\!l_{H}\partial_{y}\Bigr{)},$ (7)
are introduced such that $[\mathcal{A},\mathcal{A}^{\dagger}]=1$. The magnetic
length (measured in units of $l_{\perp}$) appears as $l_{H}=1/\sqrt{H}$, as
well as the guiding center $y_{0}=l_{H}^{2}k_{x}$. Relation to the physical
field strength $H_{\mathrm{phys}}$ in Tesla is
$\displaystyle H={l_{\perp}^{2}eH_{\mathrm{phys}}/\hbar}\simeq 3.25\times
10^{-3}H_{\mathrm{phys}}/[\mathrm{T}].$ (8)
Taking $k_{+}=-(\sqrt{2}/l_{H})\mathcal{A}^{\dagger}$ and
$k_{-}=-(\sqrt{2}/l_{H})\mathcal{A}$, the eigenvalue equation for a slab with
perpendicular magnetic field becomes
$\displaystyle\mathcal{A}^{\dagger}u$ $\displaystyle=$
$\displaystyle-{l_{H}\over\sqrt{2}}\Bigl{(}E\tau_{x}-i\lambda_{z}+iM_{\hat{N}}\tau_{y}\Bigr{)}v,$
$\displaystyle\mathcal{A}v$ $\displaystyle=$
$\displaystyle-{l_{H}\over\sqrt{2}}\Bigl{(}E\tau_{x}+i\lambda_{z}+iM_{\hat{N}}\tau_{y}\Bigr{)}u,$
(9)
with several new definitions ($\hat{N}=\mathcal{A}^{\dagger}\mathcal{A}$)
$\displaystyle\alpha_{H}={2\alpha_{\perp}\over
l_{H}^{2}},~{}~{}M_{\hat{N}}=1+\alpha_{z}\lambda_{z}^{2}-\alpha_{H}\Bigl{(}\hat{N}+{1\over
2}\Bigr{)}.$ (10)
The rest of this section is concerned with the solution of this equation,
together with the boundary conditions at the two terminations $z=\pm L_{z}/2$.
The structure of the equation invites for a solution of the form
$u=\phi_{n-1}\begin{pmatrix}a_{n}\\\ b_{n}\end{pmatrix}$, and
$v=\phi_{n}\begin{pmatrix}c_{n}\\\ d_{n}\end{pmatrix}$, where $\phi_{n}$ is
the $n$-th Landau level (LL) oscillator wave function centered at $y=y_{0}$.
By substituting the ansatz to Eq. (9) we getcomment
$\displaystyle\sqrt{n}\begin{pmatrix}a_{n}\\\
b_{n}\end{pmatrix}=-{l_{H}\over\sqrt{2}}\Bigl{(}E\tau_{x}\\!-\\!i\lambda_{z}\\!+\\!iM_{n}\tau_{y}\Bigr{)}\begin{pmatrix}c_{n}\\\
d_{n}\end{pmatrix},$ $\displaystyle\sqrt{n}\begin{pmatrix}c_{n}\\\
d_{n}\end{pmatrix}=-{l_{H}\over\sqrt{2}}\Bigl{(}E\tau_{x}\\!+\\!i\lambda_{z}\\!+\\!iM_{n-1}\tau_{y}\Bigr{)}\begin{pmatrix}a_{n}\\\
b_{n}\end{pmatrix},$ $\displaystyle
M_{n}=1+\alpha_{z}\lambda_{z}^{2}-\alpha_{H}\Bigl{(}n+{1\over 2}\Bigr{)}.$
(11)
We can parameterize the spinor solution $u$ and $v$ satisfying Eq. (11) in the
following form
$\displaystyle u=\phi_{n-1}\cos\varphi\begin{pmatrix}-i\sin{\theta}\\\
\cos{\theta}\end{pmatrix},~{}~{}v=\phi_{n}\sin\varphi\begin{pmatrix}\cos{\theta}\\\
i\sin{\theta}\end{pmatrix}$ (12)
with the two complex angles $(\theta,\varphi)$ fixed by
$\displaystyle\tan{\theta}={\lambda_{z}\over
E+\mu},~{}~{}~{}\tan\varphi=-{{M_{n-1}\\!+\\!\mu}\over{\sqrt{2n}/l_{H}}}.$
(13)
Here $\mu$ means
$\displaystyle\mu\alpha_{H}=M_{n}^{2}-E^{2}-\lambda_{z}^{2}+\alpha_{H}M_{n}+{2n\over
l_{H}^{2}}.$ (14)
The eigenvalues are fixed up by the relation $\mu^{2}=E^{2}+\lambda_{z}^{2}$
which reads when $\mu$ is explicitly written out
$\displaystyle\left(M_{n}^{2}\\!-\\!E^{2}\\!-\\!\lambda_{z}^{2}+\alpha_{H}M_{n}\\!+\\!{2n\over
l_{H}^{2}}\right)^{2}=\alpha_{H}^{2}(E^{2}\\!+\\!\lambda_{z}^{2}).$ (15)
This is the desired characteristic equation for the energy $E$.
Being eighth-power in $\lambda_{z}$, one can find eight different
$\lambda_{z}$’s for a given energy. We call them $a\lambda_{b}$ as in the non-
magnetic caselinder ; liu ; shen , with $a=\pm$ and $b=1,2,3,4$. There are
thus eight independent solutions of the same energy $E$ for a given LL index
$n$ and the guiding center $y_{0}$,
$\displaystyle\chi_{nab}(y\\!-\\!y_{0})$ $\displaystyle=$
$\displaystyle\begin{pmatrix}-ia\sin{\theta_{b}}\cos\varphi_{b}\phi_{n-1}\\\
\cos{\theta_{b}}\cos\varphi_{b}\phi_{n-1}\\\
\cos{\theta_{b}}\sin\varphi_{b}\phi_{n}\\\
ia\sin{\theta_{b}}\sin\varphi_{b}\phi_{n}\end{pmatrix}.$ (16)
Taking the linear combination among the eight states gives out the most
general eigenstate before the boundary condition is imposed as
$\displaystyle\psi_{nk_{x}}(x,y-y_{0},z)=e^{ik_{x}x}\sum_{ab}A_{ab}e^{a\lambda_{b}z}\chi_{nab}(y-y_{0}).$
(17)
To facilitate the further solution, the coefficients $A_{ab}$ can be
classified into symmetric ($A_{ab}=A_{b}$) and anti-symmetric
($A_{ab}=aA_{b}$) types. For the symmetric case the boundary conditions at
$z=\pm L_{z}/2$,
$\psi_{nk_{x}}(x,y-y_{0},L_{z}/2)=\psi_{nk_{x}}(x,y-y_{0},-L_{z}/2)=0$, can be
satisfied if we require that $A_{b}$ obey
$\displaystyle\sum_{b}A_{b}\sinh(\lambda_{b}L_{z}/2)\sin{\theta_{b}}\sin\varphi_{b}=0,$
$\displaystyle\sum_{b}A_{b}\sinh(\lambda_{b}L_{z}/2)\sin{\theta_{b}}\cos\varphi_{b}=0,$
$\displaystyle\sum_{b}A_{b}\cosh(\lambda_{b}L_{z}/2)\cos{\theta_{b}}\sin\varphi_{b}=0,$
$\displaystyle\sum_{b}A_{b}\cosh(\lambda_{b}L_{z}/2)\cos{\theta_{b}}\cos\varphi_{b}=0.$
(18)
A nontrivial solution exists provided the characteristic equation of the above
4$\times$4 matrix is zero. With the aid of Eq. (13) this condition can be
expressed as
$\displaystyle\Bigl{(}{\lambda_{1}\lambda_{2}\tanh{\lambda_{1}L_{z}\over
2}\tanh{\lambda_{2}L_{z}\over
2}\over(E_{n}\\!+\\!\mu_{n,1})(E_{n}\\!+\\!\mu_{n,2})}+{\lambda_{3}\lambda_{4}\tanh{\lambda_{3}L_{z}\over
2}\tanh{\lambda_{4}L_{z}\over
2}\over(E_{n}\\!+\\!\mu_{n,3})(E_{n}\\!+\\!\mu_{n,4})}\Bigr{)}(\mu_{n,1}\\!-\\!\mu_{n,2}\\!+\alpha_{z}(\lambda_{1}^{2}-\lambda_{2}^{2}))(\mu_{n,3}\\!-\\!\mu_{n,4}\\!+\alpha_{z}(\lambda_{3}^{2}-\lambda_{4}^{2}))$
$\displaystyle+$
$\displaystyle\Bigl{(}{\lambda_{1}\lambda_{4}\tanh{\lambda_{1}L_{z}\over
2}\tanh{\lambda_{4}L_{z}\over
2}\over(E_{n}+\mu_{n,1})(E_{n}+\mu_{n,4})}+{\lambda_{2}\lambda_{3}\tanh{\lambda_{2}L_{z}\over
2}\tanh{\lambda_{3}L_{z}\over
2}\over(E_{n}\\!+\\!\mu_{n,2})(E_{n}\\!+\\!\mu_{n,3})}\Bigr{)}(\mu_{n,1}\\!-\\!\mu_{n,4}\\!+\alpha_{z}(\lambda_{1}^{2}-\lambda_{4}^{2}))(\mu_{n,2}\\!-\\!\mu_{n,3}\\!+\alpha_{z}(\lambda_{2}^{2}-\lambda_{3}^{2}))$
$\displaystyle=$
$\displaystyle\Bigl{(}{\lambda_{1}\lambda_{3}\tanh{\lambda_{1}L_{z}\over
2}\tanh{\lambda_{3}L_{z}\over
2}\over(E_{n}+\mu_{n,1})(E_{n}\\!+\\!\mu_{n,3})}+{\lambda_{2}\lambda_{4}\tanh{\lambda_{2}L_{z}\over
2}\tanh{\lambda_{4}L_{z}\over
2}\over(E_{n}\\!+\\!\mu_{n,2})(E_{n}\\!+\\!\mu_{n,4})}\Bigr{)}(\mu_{n,1}\\!-\\!\mu_{n,3}\\!+\alpha_{z}(\lambda_{1}^{2}-\lambda_{3}^{2}))(\mu_{n,2}\\!-\\!\mu_{n,4}\\!+\alpha_{z}(\lambda_{2}^{2}-\lambda_{4}^{2})),$
where $M_{n,b}=1+\alpha_{z}\lambda_{b}^{2}-\alpha_{H}(n+1/2)$, and
$\mu_{n,b}\alpha_{H}=M_{n,b}^{2}-E_{n}^{2}-\lambda_{b}^{2}+\alpha_{H}M_{n,b}+2n/l_{H}^{2}$.
The case of anti-symmetric coefficients $A_{ab}=aA_{b}$ can be handled by
interchanging $\sinh(\lambda_{b}L_{z}/2)$ and $\cosh(\lambda_{b}L_{z}/2)$ in
Eq. (18), and replacing $\tanh(\lambda_{b}L_{z}/2)$ by
$\coth(\lambda_{b}L_{z}/2)$ in Eq. (LABEL:eq:char-eq-for-LL). Equation
(LABEL:eq:char-eq-for-LL) and its anti-symmetric counterpart can be solved
numerically for given $n$, giving out simultaneously surface and bulk energy
solutions in the presence of the field $H$. When $L_{z}$ becomes large both
$\tanh(\lambda_{b}L_{z}/2)$ and $\coth(\lambda_{b}L_{z}/2)$ tend to the same
value and we will have a pair of degenerate states for each energy, each state
being localized either at the top or the bottom surface and not coupled to the
opposite layer.
The zeroth-LL $n=0$ requires a separate treatment. In this case $u$ is
identically zero, and $v^{T}=(c_{0},d_{0})$ is found from solving
$\displaystyle-{l_{H}\over\sqrt{2}}\Bigl{(}E\tau_{x}\\!-\\!i\lambda_{z}\\!+\\!iM_{0}\tau_{y}\Bigr{)}\begin{pmatrix}c_{0}\\\
d_{0}\end{pmatrix}=0,$ (20)
with $M_{0}=1+\alpha_{z}\lambda_{z}^{2}-\alpha_{H}/2$. For a given energy
$E_{0}$, $E_{0}^{2}=M_{0}^{2}-\lambda_{z}^{2}$ results in four different
$\lambda_{z}$’s, $a\lambda_{b}$ with $a=\pm$ and $b=1,2$. Similar to Eq. (16)
one can assume the spinor solution for $n=0$
$\displaystyle\chi_{0ab}(y-y_{0})=\phi_{0}(y-y_{0})\begin{pmatrix}0\\\ 0\\\
\cos\theta_{b}\\\ ia\sin\theta_{b}\end{pmatrix}$ (21)
where $\theta_{b}$ is given by
$\displaystyle\tan\theta_{b}={\lambda_{b}\over E+M_{0,b}},$ (22)
and
$\displaystyle M_{0,b}$ $\displaystyle=$ $\displaystyle
1+\alpha_{z}\lambda_{b}^{2}-\alpha_{H}/2,$ $\displaystyle\lambda_{b}$
$\displaystyle=$
$\displaystyle{1\over\sqrt{2}\alpha_{z}}\Bigl{(}1-2\alpha_{z}+\alpha_{\perp}^{\prime}\alpha_{z}$
(23)
$\displaystyle-(-1)^{b}\sqrt{1-4\alpha_{z}+2\alpha_{H}\alpha_{z}+4E^{2}\alpha_{z}^{2}}\Bigr{)}^{{1\over
2}}.$
A linear combination
$\displaystyle\psi_{0k_{x}}(y-y_{0})=\sum_{ab}A_{ab}e^{a\lambda_{b}z}\chi_{0ab}(y-y_{0})$
(24)
can be formed with the boundary conditions at $z=\pm L_{z}/2$. Again assuming
symmetric ($A_{ab}=A_{b}$) and anti-symmetric ($A_{ab}=aA_{b}$) coefficients
separately and denoting the corresponding energies by $E_{0}^{S}$ and
$E_{0}^{A}$, we have
$\displaystyle{\lambda_{2}\over\lambda_{1}}{E_{0}^{S}\\!+\\!M_{0,1}\over
E_{0}^{S}\\!+\\!M_{0,2}}$ $\displaystyle=$
$\displaystyle{\tanh{\lambda_{1}L_{z}\over 2}\over\tanh{\lambda_{2}L_{z}\over
2}},$ (25)
and
$\displaystyle{\lambda_{2}\over\lambda_{1}}{E_{0}^{A}\\!+\\!M_{0,1}\over
E_{0}^{A}\\!+\\!M_{0,2}}$ $\displaystyle=$
$\displaystyle{\tanh{\lambda_{2}L_{z}\over 2}\over\tanh{\lambda_{1}L_{z}\over
2}}.$ (26)
One can easily show from Eqs. (25) and (26) that $E_{0}^{A}=-E_{0}^{S}$.
This completes the derivation of the full energy spectra and eigenstates for a
thin-slab geometry of TI model with the perpendicular magnetic field. In the
following section we discuss several physical results obtained from the
analysis of the solution.
## IV Physical Results
Figure 2 shows the dependence of surface and bulk energies on the LL index
$n$, for a sufficiently large thickness $L_{z}$. The numerical results remain
consistently similar for $L_{z}$ larger than about ten times $l_{z}$. A most
surprising aspect of the numerical analysis is the existence of three distinct
branches of surface-localized states, labeled as (I), (II), and (III) in Fig.
2.
The behavior of the first surface branch $E^{(1)}_{n}$ is remarkably close to
the formula:
$\displaystyle E^{(s)}_{n}\simeq\pm\sqrt{2n}/l_{H}=\pm\sqrt{2nH}.$ (27)
This is exactly what is expected of the purely two-dimensional Dirac
Hamiltonian with the Fermi velocity $v_{F}=1$ (equal to
$M_{0}l_{\perp}/\hbar=A_{2}/\hbar\approx 6.2\times 10^{5}$ m/s in physical
units)Bi2Se3-theory . Restoring all physical units, the surface LLs occur at
$\displaystyle E^{(s)}_{n}=\pm{A_{2}\over l_{H_{\mathrm{phys}}}}\sqrt{2n}.$
(28)
Physical magnetic field $H_{\mathrm{phys}}=11$T results in the magnetic length
$l_{H_{\mathrm{phys}}}=\sqrt{\hbar/eH_{\mathrm{phys}}}\sim 100$Å and the
energy levels $\pm 58\sqrt{n}$ mV. Indeed the spacing in the $n=0$ and $n=1$
LL peaks were found to be about $40\sim 50$ mVSTM-B-thin-film .
Using the physical magnetic field $H_{\mathrm{phys}}=10$T we find that at
$n>n_{c}\approx 12$ the first branch of surface LL begins to merge with the
bulk spectrum (Fig. 2). Here $n_{c}$ corresponds to the Landau level index for
which the surface LL begins to touch the bottom of the bulk band. A sharper
criterion to determine $n_{c}$ can be drawn by keeping track of the
eigenvalues $\lambda_{b}$ for the surface-bound LLs. With increasing $n$, one
of the four $\lambda_{b}$’s forming the surface LL eigenstate has its real
part decrease and eventually touch zero at $n=n_{c}$. This signals the mixture
of an extended state in the wave function just as the surface LL merges with
the bulk continuum. Recently, the number of surface LLs that can be resolved
in the tunneling spectra of STMSTM-B-thin-film was shown to be about 12,
consistent with our estimate of $n_{c}$. For $n>n_{c}$, the surface branch no
longer exists independently of the bulk LL, but rather seems to form the
bottom of the bulk band as depicted in Fig. 2.
Figure 2: (color online) Landau level energies for $L_{z}/l_{z}=3\times
30^{3}$ and Hphys=10T, showing both surface(sky blue square) and bulk
states(black square). Three surface branches are labeled (I) through (III)
with analytic fits shown as red solid curves to $\sqrt{2n}/l_{H}$ (branch I)
and $\sqrt{A\pm\delta-Bn}$ (branches II and III). $A$ and $B$ values are
derived in the Appendix and a small offset $\delta$ is used to fit the
numerical results. When Hphys=10T, $A=0.918,~{}\delta=0.068,~{}B=0.037$,
respectively.
A recent paper by Liu et al. also computed the bulk and surface LLs based on
the model Hamiltonian for Bi2Se3. In their Fig. 7 it appeared as though the
surface LLs can exist well inside the bulk spectra as an independent branch.
We believe this is an artifact of their calculation not taking care of the
boundary conditions precisely. Once the boundary conditions at $z=\pm L_{z}/2$
are handled properly, the correct energy profile for $n>n_{c}$ is the one in
which the surface-localized wave functions are hybridized with the extended
states to form a “hybrid” state. To confirm this assertion, we have made a
careful analysis of all the $\lambda_{b}$ values for eigenstates with energies
both at the bottom of, and deep inside the bulk for $n>n_{c}$. While the
details are too tedious to report here, we can say with certainty that states
forming the bulk LL are typically a linear combination of solutions with real
$\lambda_{b}$ (localized to surface) and some with purely imaginary
$\lambda_{b}$ (extended). See Eq. (17) for a general definition of the
eigenstate. Only for the three surface branches (I) through (III) is it
possible to get all $\lambda_{b}$’s of the eigenstate being real and the wave
function completely localized.
The existence of extra two surface branches, labeled (II) and (III) in Fig. 2,
is unexpected. They begin to appear at $n\approx 4$ and $n\approx 8$
respectively for $H_{\mathrm{phys}}=10$T. We have confirmed their existence
for $L_{z}/l_{z}$ as small as 10 and as large as 3000. Due to the
insensitivity of their features to surface thickness, we can first of all
conclude that the extra surface modes are bound to one particular surface and
not hybridized with the other one. To further confirm that these branches are
genuine, we have carried out an approximate analytic treatment valid at large
LL index $n$ and infinite thickness $L_{z}$ and indeed found that two extra
branches exist. Details of this analysis are given in the Appendix.
Figure 3: (color online) Hybridization gap energies for
$H_{\mathrm{phys}}=0$T, 10T, and 50T with varying thickness $L_{z}$.
Hybridization effect mixes the two degenerate $n=0$ LLs previously associated
with each surface layer and opens a gap. We have derived the $n=0$ surface LL
energies analytically for symmetric $(E^{S}_{0})$ and anti-symmetric
($E^{A}_{0}$) combinations as
$\displaystyle E_{0}^{S}$ $\displaystyle\simeq$
$\displaystyle-{4\alpha(H)\over\beta(H)}(1\\!-\\!\alpha_{\perp}H)\sin[\beta(H)L_{z}]e^{-\alpha(H)L_{z}},$
(29)
and $E_{0}^{A}=-E_{0}^{S}$. The gap is defined as
$\Delta_{0}=|E_{0}^{S}-E_{0}^{A}|=2|E_{0}^{S}|$. For practically available
field strengths where $H\ll 1$, $\alpha(H)$ and $\beta(H)$ in Eq. (29) are
$\displaystyle\alpha(H)\simeq{1\over
2\alpha_{z}},~{}~{}\beta(H)\simeq{\sqrt{4\alpha_{z}\\!-\\!1\\!-\\!4\alpha_{z}\alpha_{\perp}H}\over
2\alpha_{z}}.$ (30)
They reduce exactly to $\alpha$ and $\beta$ coefficients obtained in Eq. (6)
as $H\rightarrow 0$. The gap still exhibits an oscillatory decay similar to
the gap at the $\Gamma$-point without magnetic field. In Fig. 3 we compare the
energy gaps for zero-field and for $H_{\mathrm{phys}}=10$T and 50T. The
similarity of their $L_{z}$-dependence is a strong clue that the origins of
the gaps are the same. Ignoring the small field-induced shift, the gap can be
$\displaystyle\Delta_{0}\approx 7M_{0}e^{-L_{z}/[9.2\AA]}\approx
2e^{-L_{z}/[9.2\AA]}\mathrm{eV}.$ (31)
It gives a value $\approx 10$ meV for a seven quintuple-layer thin film and
may well be resolved as two split $n=0$ LLs in a careful STM spectroscopy
study. Currently available thin-film STM study was done on 50 quintuple-layer
sampleSTM-B-thin-film . In Ref. liu, it was argued that the oscillation in
the sign of the hybridization gap under zero magnetic field marks the
transition between topologically trivial and non-trivial insulator phases. If
this is so, our calculation seems to reveal that well-defined Dirac-like LLs
exists regardless of the thickness and the sign of the gap, implying that
changes in the topological character of the thin-film TI will not be revealed
by examination of the surface LLs alone.
Figure 4: (color online) $n$th-LL wave function density $\rho_{n,c}(z)$ and
the spin density $\rho_{n,s}(z)$for $L_{z}=60$Å.
The charge $(c)$ and spin $(s)$ densities of each $n$-th surface LL wave
function can be defined as
$\displaystyle\rho_{n,s(c)}(z)$ $\displaystyle=$ $\displaystyle{\int
dxdy~{}\psi^{\dagger}_{n}\Gamma_{s(c)}\psi_{n}\over\int
dxdydz~{}\psi^{\dagger}_{n}\psi_{n}},$ $\displaystyle\Gamma_{s}$
$\displaystyle=$ $\displaystyle\mathrm{diag}(1,1,-1,-1),$
$\displaystyle\Gamma_{c}$ $\displaystyle=$
$\displaystyle\mathrm{diag}(1,1,1,1).$ (32)
Figure 4 shows results for a few surface LLs with small LL index $n$. All
surface LLs are localized to within one $l_{z}$ of the termination, or within
about one quintuple layer. As one can see from Fig. 4(b), the zeroth-LL is
completely spin-polarized, $\int\rho_{0,s}(z)dz=-1$, while other higher
surface LLs are nearly spin-quenched, $\int\rho_{n>0,s}(z)dz\approx 0$. The
zeroth-LL has only the lower two elements of the four-component spinor $\chi$
take nonzero values, which refer to the amplitudes for Bi and Se states of
spin-$\downarrow$ (See text following Eq. 1). There are two $n=0$ LL in the
solution, and both of them are fully spin-$\downarrow$-polarized. The origin
of the spin polarization is the analogue of the sublattice polarization of the
$n=0$ LL in graphenecastro-neto . The difference is that the two valley $n=0$
Landau levels occupy the opposite sublattices, so that the overall sublattice
symmetry is restored.
Here, by contrast, both top and bottom surface LLs give the same spin
polarization. The reason is that for the top surface Dirac states the magnetic
field is pointing out of the bulk but the bottom surface states experience the
field pointing into the bulk, so that effectively the sense of the field
direction is also reversed between the two surface layers. By reversing the
field direction from $+\hat{z}$ to $-\hat{z}$ one will generate $n=0$ of
spin-$\uparrow$ polarization. As a result a thin slab of TI subject to
quantizing magnetic field creates two $n=0$ LLs which are completely spin
polarized. Such spin-polarized surface layers are detectable by Faraday or
Kerr rotation experimentsBi2Se3-theory .
## V Conclusion
We showed how to derive the Landau level solution for a slab geometry of the
topological insulator based on the four-band modelBi2Se3-theory ;
Bi2Se3-theory2 . Previoius approaches were to first project the zero-field
bulk Hamiltonian to the surface, then using the Peierls substitution to
address the magnetic field effectSQShen-LL ; Bi2Se3-theory2 . Our strategy by
contrast is to introduce the Peierls substitution directly into the bulk
Hamiltonian and use the boundary conditions appropriate for a slab geometry.
The obtained surface Landau level energies are in good accord with those
obtained from the surface Dirac Hamiltonian, and we conclude that surface
projection and the Peierls substitution can be implemented in any order with
the same physical spectrum.
A dramatic departure of the present Dirac LL problem with an analogous one
posed by the graphene systemcastro-neto is that the surface LLs are
eventually bounded by the bulk spectra, and one has to face the issue what
will happen to the surface LLs as they begin to merge with the bulk continuum.
We addressed such a question numerically and analytically in this paper, with
a prediction for the existence of new surface-bound LLs appearing at higher-LL
indices. Detection of the predicted new surface modes presents an interesting
challenge for the future surface-sensitive measurements on TI materials.
###### Acknowledgements.
H. J. H. is supported by Mid-career Researcher Program through NRF grant
funded by the MEST (No. R01-2008-000-20586-0).
## Appendix A Analysis of New Surface Modes
After some trial and error, we find that the following ansatz describe the
numerically found surface modes (2) and (3) with good accuracy.
$\displaystyle\lambda_{b}^{2}={\alpha_{H}\over\alpha_{z}}(n-m_{b}\sqrt{n}).$
(33)
Here $m_{b}$ is a constant, to be determined later. This gives for $M_{n}$,
$\displaystyle M_{n}=1-\alpha_{H}m_{b}\sqrt{n}-{1\over 2}\alpha_{H}.$ (34)
We can also make an ansatz for the surface energy mode of the form
$\displaystyle E_{n}^{2}=A-Bn,$ (35)
with two undetermined positive coefficients $A$ and $B$. Inserting Eqs. (33)
through (35) into Eq. (15) gives
$\displaystyle\left[\left({\alpha^{2}_{H}}m_{b}^{2}+B-{\alpha_{H}\over\alpha_{z}}+{2\over
l_{H}^{2}}\right)n+\left({1\over\alpha_{z}}-2\right)\alpha_{H}m_{b}\sqrt{n}+\cdots\right]^{2}=\alpha_{H}^{2}\left({\alpha_{H}\over\alpha_{z}}-B\right)n+\cdots.$
(36)
The terms in $\cdots$ have subleading order in $n$ than the ones shown.
Assuming a sufficiently large $n$ we require that the two sides of the
equation cancel out at each order in $n$. From the equality of $n^{2}$, $n$,
and $\sqrt{n}$-order terms we obtain the three following equations:
$\displaystyle(\alpha_{H}m_{b})^{2}$ $\displaystyle=$
$\displaystyle{\alpha_{H}\over\alpha_{z}}-{2\over l_{H}^{2}}-B,$
$\displaystyle\left({1\over\alpha_{z}}-2\right)^{2}(\alpha_{H}m_{b})^{2}$
$\displaystyle=$
$\displaystyle\alpha_{H}^{2}\left({\alpha_{H}\over\alpha_{z}}-B\right),$
$\displaystyle 2(1-{\alpha_{H}^{2}\over
4}-A)({1\over\alpha_{z}}-2)\alpha_{H}^{2}m_{b}$ $\displaystyle=$
$\displaystyle-{\alpha_{H}\over\alpha_{z}}\alpha_{H}^{2}m_{b}.$ (37)
Upon solving them we obtain
$\displaystyle m_{b}^{2}$ $\displaystyle=$
$\displaystyle{2/l_{H}^{2}\over(2-1/\alpha_{z})^{2}-\alpha_{H}^{2}},$
$\displaystyle B$ $\displaystyle=$ $\displaystyle{2\over
l_{H}^{2}}-{\alpha_{H}\over\alpha_{z}}-{2\alpha_{H}^{2}/l_{H}^{2}\over(2-1/\alpha_{z})^{2}-\alpha_{H}^{2}},$
$\displaystyle A$ $\displaystyle=$ $\displaystyle 1-{\alpha_{H}^{2}\over
4}+{2\alpha_{H}^{2}/\alpha_{z}\over 1/\alpha_{z}-2}.$ (38)
Further consideration of sub-leading corrections finally yield a splitting of
$A$ into two branches responsible for (II) and (III) in Fig. 2. Rather than
going into the complicated sub-leading order analysis, we can simply split $A$
into two branches by writing $A\pm\delta$ with $\delta$ chosen to fit the two
branches in Fig. 2 while $A$ itself is completely determined from the
parameters such as $\alpha_{z}$ and $\alpha_{\perp}$. It is shown that both
branches (II) and (III) match quite well the ansatz for energy, Eq. (35).
We can also discuss the stability of the new surface branches by recalling the
numerical value $\alpha_{z}=0.58$, and
$\alpha_{H}=2\alpha_{\perp}/l_{H}^{2}=1.84/l_{H}^{2}$. It follows that
positive $(\alpha_{H}m_{b})^{2}$ is possible if
$2-(0.58)^{-1}-1.84/l_{H}^{2}=0.276-1.84/l_{H}^{2}>0$, or if $l_{H}^{2}>6.66$.
Returning to physical length scales, this implies the magnetic length greater
than $\sim$38Å, or the magnetic field strength less than 46T. We then expect
that surface modes (II) and (III) should co-exist with the more familiar mode
(I) inside the bulk gap for typical laboratory magnetic field ranges.
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* (23) It follows immediately that an eigenstate with energy $-E$ is obtained from the state with energy $E$ in Eq. (9) by the operation $\tau_{y}$.
|
arxiv-papers
| 2010-07-02T04:45:44 |
2024-09-04T02:49:11.369529
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhihua Yang and Jung Hoon Han",
"submitter": "Zhihua Yang",
"url": "https://arxiv.org/abs/1007.0291"
}
|
1007.0319
|
# Possible quantum gravity effects on the gravitational deflection of light
Xin Li1,3 lixin@itp.ac.cn Zhe Chang2,3 changz@ihep.ac.cn 1Institute of
Theoretical Physics, Chinese Academy of Sciences, 100190 Beijing, China
2Institute of High Energy Physics, Chinese Academy of Sciences, 100049
Beijing, China
3Theoretical Physics Center for Science Facilities, Chinese Academy of
Sciences
###### Abstract
We investigate possible quantum gravity (QG) effects on the gravitational
deflection of light. Two forms of deformation of the Schwarzschild spacetime
are proposed. The first ansatz is a given Finslerian line element, it could be
regarded as a weak QG effect on the Schwarzschild spacetime. Starting from
this ansatz, we deduce the deflection angle of the light ray which passes a
weak gravitational source. The second ansatz could be regarded as a strong QG
effect on the Schwarzschild spacetime. The deflection angle of the light ray
which passes a weak gravitational source is deduced in this Riemannian
spacetime. This QG effect may distinguish the mixed light rays in the absence
of gravitational source by a “spectroscope” (the gravitational source). The
solutions of gravitational field equation in this Riemannian spacetime
indicate that the QG effect could be regarded as the vacuum energy and the
energy density of vacuum is related to the spacetime deformation parameter.
###### pacs:
04.60.Bc
## I Introduction
The quantum theory of gravity (QG) has been studied for more than 70 years
Stachel . The most famous theories of quantum gravity, such as string theory
string and loop quantum gravity loop , provide us some key features of QG.
Amelino-Camelia Amelino listed possible candidates of QG effects: the
violation of Lorentz symmetry of Standard Model Garay as well as discrete
symmetry (CPT symmetry) Ellis ; Kostelecky , Planck-scale fuzziness of
spacetime Amelino1 . Other possible QG effects include: deviations from Newton
s law at very short distances Hoyle , possible production of mini-black holes
Dimopoulos . The researches of such possible consequence of QG are refered as
quantum gravity phenomenology (QGP). The rapid progress in technology makes
experiments have the opportunities to test the sub-Planckian consequences of
QG scenarios.
The QGP covers a wide range of subjects. One of the most important QG effects
is the violation of the Lorentz invariance (LI)Mattingly . A feature of QG,
which is the most debated possibility for a quantum spacetime, manifests that
the spacetime in Planck-scale may be noncommutative Connes ; Majid . A huge
numbers of investigations of noncommutative spacetime manifest that the Lie-
algebra Poincare symmetries are either broken to a smaller symmetry (Lie
algebra or deformed into Hopf-algebra Majid1 symmetries). The LI violation
implies that the dispersion relations for elementary particles should be
modified. Moreover, studying on the dispersion relation is a convenient way
for physicists to test the departure from LI. In the past few years, Amelino-
Camelia and Smolin as well as their collaborators have developed the Doubly
Special Relativity (DSR) Amelino2 ; Amelino3 ; Amelino4 ; Smolin1 ; Smolin2
to take Planck-scale effects into account by introducing an invariant Planckin
parameter in the theory of Special Relativity. The general form of dispersion
relation for free particles in the DSR is of the form
$E^{2}=m^{2}+p^{2}+\sum_{n=1}^{\infty}\alpha_{n}(\mu,M_{p})p^{n}~{},$ (1)
where $p=\sqrt{\parallel\vec{p}\parallel^{2}}$, $\mu$ denotes a parameter of
the theory with mass scale and $M_{p}$ is the Planck mass. The modified
dispersion relations (MDR) have been tested through observations on gamma-ray
bursts and ultra-high energy cosmic raysJacobson . Girelli et al.Girelli
showed that the MDR can be incorporated into the framework of Finsler
geometry. The symmetry of the MDR was described in the Hamiltonian formalism.
The generators of symmetry commute with ${\cal M}(p)$ (here ${\cal
M}(p)=m^{2}$ gives the mass shell condition). The mass shell condition is
invariant under the deformed Lorentz transformations.
The research of Girelli et al. Girelli gives a possible origin of MDR, which
means the quantum spacetime may have a Finslerian form. Randers space, as a
special kind of Finsler space, was first proposed by G. Randers Randers .
Within the framework of Randers space, modified dispersion relation has been
discussed RF , and the threshold of ultra high energy cosmic rays was
investigated UHC . The investigation of the isometric group of Finsler space
indicates that the Lorentz symmetry is broken in Finsler space Li .
Regardless of any particular theories of QG, we generally agree that the
spacetime in QG scenario should be deformed. In this paper, we start from a
giving deformed spacetime to investigate possible QG effects on the
gravitational deflection of light. This paper is organized as follows. In Sec.
II, we introduce a Finslerian line element which could be regarded as a weak
deformation from the Schwarzschild spacetime. Starting from such line element,
we investigate the trajectory of the light ray. In Sec. III, we introduce a
Riemannian line element which could be regarded as a strong deformation from
the Schwarzschild spacetime. The trajectory of the light ray in this
Riemannian spacetime is studied. The Einstein’s gravitational field equation
is presented in this Riemannian spacetime. The light ray deflected by a weak
gravitational source is studied. In Sec. IV, we give conclusions on the
possible QG effects on the gravitational deflection of light.
## II Weak quantum gravity effect on the gravitational deflection of light
Instead of defining an inner product structure over the tangent bundle in
Riemann geometry, Finsler geometry is based 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
$g_{\mu\nu}\equiv\frac{\partial}{\partial y^{\mu}}\frac{\partial}{\partial
y^{\nu}}\left(\frac{1}{2}F^{2}\right).$ (2)
Finsler geometry has its genesis in integrals of the form
$\int^{r}_{s}F(x^{1},\cdots,x^{n};\frac{dx^{1}}{d\tau},\cdots,\frac{dx^{n}}{d\tau})d\tau~{}.$
(3)
So that the Finsler structure represents the length element of Finsler space.
Following the calculus of variations, one get the geodesic equation of Finsler
spaceBook by Bao
$\frac{d^{2}\sigma^{\lambda}}{d\tau^{2}}+\gamma^{\lambda}_{\mu\nu}\frac{d\sigma^{\mu}}{d\tau}\frac{d\sigma^{\nu}}{d\tau}=\frac{d\sigma^{\mu}}{d\tau}\frac{d}{d\tau}\left(\log
F(\sigma,\frac{d\sigma}{d\tau})\right),$ (4)
where
$\gamma^{\lambda}_{\mu\nu}=\frac{g^{\lambda\alpha}}{2}\left(\frac{\partial
g_{\mu\alpha}}{\partial x^{\nu}}+\frac{\partial g_{\nu\alpha}}{\partial
x^{\mu}}-\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}\right)$ (5)
is the formal Christoffel symbols of the second kind with the same form of
Riemannian connection. The parallel transport has been studied in the
framework of Cartan connectionMatsumoto ; Antonelli ; Szabo . The notation of
parallel transport in Finsler manifold means that the length
$F\left(\frac{d\sigma}{d\tau}\right)$ is constant. Thus, the autoparallel
equation can be got from the equation (4)
$\frac{d^{2}\sigma^{\lambda}}{d\tau^{2}}+\gamma^{\lambda}_{\mu\nu}\frac{d\sigma^{\mu}}{d\tau}\frac{d\sigma^{\nu}}{d\tau}=0.$
(6)
Since the geodesic equation (4) is directly derived from the integral length
of $\sigma$
$L=\int F\left(\frac{d\sigma}{d\tau}\right)d\tau,$ (7)
the inner product
$\left(\sqrt{g_{\mu\nu}\frac{d\sigma^{\mu}}{d\tau}\frac{d\sigma^{\nu}}{d\tau}}=F\left(\frac{d\sigma}{d\tau}\right)\right)$
of two parallel transported vectors is preserved.
In a phenomenological approach, it is reasonable to suppose that the line
element of spacetime is given as
$\displaystyle F^{2}d\tau^{2}=$ $\displaystyle
adt^{2}-a^{-1}dr^{2}-r^{2}(\sin^{2}\theta d\theta^{2}+d\phi^{2})$ (8)
$\displaystyle+$
$\displaystyle\kappa\sqrt{a}\frac{GM}{r}\left(1-a\frac{J^{2}}{E^{2}r^{2}}\right)^{\frac{3}{4}}dt\sqrt{dtdr},$
where $a=1-\frac{2GM}{r}$, $\frac{GM}{r}\ll 1$, $M$ is the mass of
gravitational source, $E$ is the energy per unit mass of the particle, $J$ is
the angular momentum per unit mass of the particle and $\kappa$ is
dimensionless and constancy parameter which is function of the new physics
scale. The dimensionless parameter $\kappa$ just plays the role of the
measurement of QG effect. While $\kappa$ vanishes, the line element (8)
returns to the famous Schwarzschild metric. The line element (8) could be
regarded as the deformation of the Schwarzschild metric. Since the coefficient
of the non Riemannian term in (8) is isotropic, it is convenient to consider
that the motion of particle is confined in the plane of
$\theta=\frac{\pi}{2}$. Therefore, the trajectory of the particle is described
by the $r$ and $\phi$ coordinates, and the line element reduces to
$\displaystyle F^{2}d\tau^{2}=$ $\displaystyle
adt^{2}-a^{-1}dr^{2}-r^{2}d\phi^{2}$ (9) $\displaystyle+$
$\displaystyle\kappa\sqrt{a}\frac{GM}{r}\left(1-a\frac{J^{2}}{E^{2}r^{2}}\right)^{\frac{3}{4}}dt\sqrt{dtdr}.$
It should be noticed that the coefficients of the line element (9) do not
depend either on $r$ or $\phi$. By making use of the autoparallel geodesic
equation (6), we obtain two integrals of motion
$\displaystyle
a\dot{t}+\kappa\sqrt{a}\frac{3GM}{4r}\left(1-a\frac{J^{2}}{E^{2}r^{2}}\right)^{\frac{3}{4}}\sqrt{\dot{t}\dot{r}}$
$\displaystyle=$ $\displaystyle E,$ (10) $\displaystyle r^{2}\dot{\phi}$
$\displaystyle=$ $\displaystyle J,$ (11)
where a dot denotes the derivatives with respect to trajectory parameter
$\tau$. The deformation term in (9) is very small, here we can approximately
choose the two integral constants to be $E$ and $J$, respectively. In this
paper, we mainly consider the motion of light ray. There is an additional
constraint arising from $F=0$ for the motion of light ray
$a\dot{t}^{2}-a^{-1}\dot{r}^{2}-r^{2}\dot{\phi}^{2}+\kappa\sqrt{a}\frac{GM}{r}\left(1-a\frac{J^{2}}{E^{2}r^{2}}\right)^{\frac{3}{4}}\dot{t}^{\frac{3}{2}}\dot{r}^{\frac{1}{2}}=0.$
(12)
By making use of the equations (10), (11) and (12), we obtain an approximate
equation for $\dot{r}$, which is valid for $\frac{GM}{r}\ll 1$ (it was assumed
in the beginning of this section)
$\dot{r}=\sqrt{E^{2}-a\frac{J^{2}}{r^{2}}}\left(1-\kappa\frac{GM}{4r}\right).$
(13)
Combining (13) with equation (11), we obtain
$\left(\frac{1}{r^{2}}\frac{dr}{d\phi}\right)^{2}=\left(\frac{E^{2}}{J^{2}}-\frac{a}{r^{2}}\right)\left(1-\kappa\frac{GM}{2r}\right).$
(14)
In terms of the variable $u=\frac{GM}{r}$, the equation (14) changes as
$\left(\frac{du}{d\phi}\right)^{2}=\left(\left(\frac{EGM}{J}\right)^{2}-u^{2}(1-2u)\right)\left(1-\frac{\kappa
u}{2}\right).$ (15)
The derivatives $\frac{d}{d\phi}$ of the equation (15) gives
$\frac{d^{2}u}{d\phi}^{2}+u=3u^{2}-\frac{\kappa}{4}\left(\left(\frac{EGM}{J}\right)^{2}-3u^{2}\right)+O(u^{3}).$
(16)
Solving the equation (16) to order $u$, we get that
$u=u_{0}\cos\phi.$ (17)
Substituting the above solution into (15), we obtain the approximate solution
for $u_{0}$
$u_{0}=\frac{EGM}{J}=\frac{GM}{\xi},$ (18)
where $\xi$ is the minimum distance of the light ray to the gravitational
source with mass $M$. The closest approach of the light ray to the
gravitational source implies $\frac{dr}{d\phi}=0$. Then, deducing from the
equation (14), we obtain approximate solution for $\xi$$(=\frac{J}{E}$). This
is the reason for the second equation in (18). By making use of the first
order approximation (17), one may get the second order approximation of (16)
$u=u_{0}^{2}\left((1+\kappa/4)(1+\sin^{2}\phi)-\kappa/4\right).$ (19)
Thus, the solution of equation (16) is
$u=u_{0}\cos\phi+u_{0}^{2}\left((1+\kappa/4)(1+\sin^{2}\phi)-\kappa/4\right).$
(20)
At infinity ($u=0$), the solution (20) shows that the angle
$\phi=\pm(\frac{\pi}{2}+\alpha)$, and the small angle $\alpha$ satisfies the
constraint
$-u_{0}\sin\alpha+u_{0}^{2}\left((1+\kappa/4)(1+\cos^{2}\alpha)-\kappa/4\right)=0.$
(21)
Since the angle $\alpha$ is very small, the solution of the above equation
(21) is
$\alpha=u_{0}(2+\kappa/4)=(2+\kappa/4)\frac{GM}{\xi}.$ (22)
The two asymptotic directions differs from $\pi$ by the deflection angle
$\hat{\alpha}=2\alpha=(4+\kappa/2)\frac{GM}{\xi}$ (23)
In general relativity, the light passing a massive object $M$ at a minimum
distance $\xi$ suffers deflection, and the deflection angle (“Einstein angle”)
is $\hat{\alpha}_{E}=\frac{4GM}{\xi}$ Weinberg . The spacetime (8) deformed
from Schwarzschild spacetime implies a deformed deflection angle. The formula
for the deformed deflection angle (23) differs from $\hat{\alpha}_{E}$ by
$\kappa\frac{GM}{2\xi}$, which is proportional to the deformed parameter
$\kappa$.
In astronomical observations, the gravitational lensing surveys is used to
calculate the mass distribution that projected onto the sky. A large amount of
observations manifest that the expected gravitational lensing effects deducing
by the Einstein angle are not in accord with the experimental data. An example
is the full-sky data product for the Bullet Cluster 1E0657-558 Clowe . We wish
the deformed deflection angle (23) may account to these observations.
## III Strong quantum gravity effect on the gravitational deflection of light
In Sec. II, we got a deformed deflection angle for the light ray. The
Finslerian metric (8) could be regarded as a weak deformation from the
Schwarzschild metric. In this section, we discuss the strong quantum gravity
effect on the gravitational deflection of light. Again, in a phenomenological
approach, we propose that the line element of spacetime is given as
$ds^{2}=adt^{2}-(4n+1)^{2}a^{-1}dr^{2}-r^{2}d\phi^{2},$ (24)
where the deformation parameter $n=0,1,2\cdots$. In ansatz (24), we already
confined the motion of particle in the plane of $\theta=\frac{\pi}{2}$. The
reason is the same within Sec. II. The ansatz (24) manifests a large
deformation from the Schwarzschild metric, it returns to the Schwarzschild
metric while the deformation parameter vanishes.
By making use of the autoparallel geodesic equation of Riemannian space
Weinberg , we obtain two integrals of motion
$\displaystyle a\dot{t}$ $\displaystyle=$ $\displaystyle E,$ (25)
$\displaystyle r^{2}\dot{\phi}$ $\displaystyle=$ $\displaystyle J.$ (26)
The additional constraint arising from $\frac{ds}{d\tau}=0$ for the motion of
light is
$a\dot{t}^{2}-(4n+1)^{2}a^{-1}\dot{r}^{2}-r^{2}\dot{\phi}^{2}=0$ (27)
By making use of the equations (25), (26) and (27), we get an approximate
equation for $\dot{r}$
$(4n+1)^{2}\dot{r}^{2}=E^{2}-a\frac{J^{2}}{r^{2}}.$ (28)
Combining this equation (28) with (26), we obtain that
$\left(\frac{4n+1}{r^{2}}\frac{dr}{d\phi}\right)^{2}=\frac{E^{2}}{J^{2}}-\frac{a}{r^{2}}.$
(29)
The closest approach of the light ray to the gravitational source $M$ implies
$\frac{dr}{d\phi}=0$. Then, deducing from the equation (29), we obtain
approximate solution for the distance of closest approach
$\xi$($=\frac{J}{E}$). Changing variable to $u=\frac{GM}{r}$, we obtain from
equation (14) that
$(4n+1)^{2}\left(\frac{du}{d\phi}\right)^{2}=\left(\frac{EGM}{J}\right)^{2}-u^{2}(1-2u).$
(30)
Calculating the derivatives $\frac{d}{d\phi}$ of the equation (30) gives
$\frac{d^{2}u}{d\phi^{2}}+\frac{u}{(4n+1)^{2}}=\frac{3u^{2}}{(4n+1)^{2}}.$
(31)
Noticing that $u=\frac{GM}{r}\ll 1$, the equation (31) has solution
$u=u_{0}\cos\frac{\phi}{4n+1}+u_{0}^{2}\left(1+\sin^{2}\frac{\phi}{4n+1}\right).$
(32)
Substituting the above solution into (30), we get the approximate solution for
$u_{0}$
$u_{0}=\frac{EGM}{J}=\frac{GM}{\xi}.$ (33)
At infinity ($u=0$), to first order in $u_{0}$, the solution (32) shows
$\phi=\pm(4n+1)\frac{\pi}{2}.$ (34)
The angle $\phi$ in (34) for different $n$ only differ from an integer times
of $2\pi$. The formula (34) means each light ray which corresponds to a given
parameter $n$ is mixed at infinity in the absence of the gravitational source
$M$, and the light rays all move with the same direction. When they suffer
from a week gravitational source, to second order in $u_{0}$, the solution
(32) implies
$\phi=\pm\left((4n+1)\frac{\pi}{2}+\alpha\right),$ (35)
and the small angle $\alpha$ satisfies the constraint
$-u_{0}\sin\frac{\alpha}{4n+1}+u_{0}^{2}\left(1+\cos^{2}\frac{\alpha}{4n+1}\right)=0.$
(36)
Since the angle $\alpha$ is very small, the solution of the above equation
(36) is
$\alpha=2(4n+1)u_{0}=2(4n+1)\frac{GM}{\xi}.$ (37)
The difference of two asymptotic directions
$\pm\left((4n+1)\frac{\pi}{2}+\alpha\right)$ differs from $(4n+1)\pi$ by the
deflection angle
$\hat{\alpha}=2\alpha=4(4n+1)\frac{GM}{\xi}.$ (38)
This formula (38) means the deflection angles for different $n$ are different.
Therefore, the weak gravitational source $M$ just plays the role of the
“spectroscope”. The mixed light rays come from infinity, refracted by the
“spectroscope” $M$ at the same distance of closest approach $\xi$, go into
different directions. And the deflection angle is in proportion to the
spacetime deformation parameter $n$.
One should notice from the formula (28) that the radial momentum of light rays
is modified. The ansatz (24) is a Riemannian line element. Therefore, the
motion of particle in this spacetime satisfies the Einstein’s gravitational
field equation. The Einstein’s gravitational field equation is taken the from
$R_{\mu\nu}=-8\pi
G\left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T^{\lambda}_{~{}\lambda}\right),$ (39)
where $R_{\mu\nu}$ is the Ricci tensor. The energy-momentum tensor of a
perfect fluid is taken the from as
$T^{\mu\nu}=-pg^{\mu\nu}+(p+\rho)U^{\mu}U^{\nu},$ (40)
here $U^{\mu}$ is the fluid four-velocity and satisfies
$g_{\mu\nu}U^{\mu}U^{\nu}=1$ , $p$ and $\rho$ are pressure and energy density
respectively. By making use of the line element (24), we show that the
combination of the $tt$ and $rr$ component of the field equation (39) gives
$p=-\rho,$ (41)
and the $\phi\phi$ component of the field equation (39) gives
$-1+\frac{1}{(4n+1)^{2}}=-4\pi Gr^{2}(\rho-p).$ (42)
The equation (41) implies that there is a quantum vacuum with energy density
$\rho$ outside the gravitational source. Combining the equation (41) with
(42), we obtain
$\rho=\frac{1}{8\pi Gr^{2}}\left(1-\frac{1}{(4n+1)^{2}}\right).$ (43)
The equation (43) indicates that the energy density of vacuum is the function
of spacetime deformation parameter $n$. From the point of view of particle
physics, the cosmological constant, a popular candidate of the dark energy,
naturally arises as an energy density of the vacuum Copeland . It implies that
this kind of QG effect (9) may arise from the influence of the dark energy.
## IV Conclusion
In this paper, we investigated the possible quantum gravity effects on the
gravitational deflection of light. One of the most expected QG effect is the
deformation of spacetime geometry. In a phenomenological approach, we proposed
two deformations of the Schwarzschild spacetime. The first ansatz (8) is a
given Finslerian line element, it could be regarded as a weak QG effect on the
Schwarzschild spacetime. The deformation term (the non Riemannian term) is
very small. Starting from the ansatz (8), we deduced the deflection angle
(23). This deformed deflection angle may account to the observations of
gravitational lensing which can not be explained in the framework of general
relativity.
The second ansatz (24) could be regarded as a strong QG effect on the
Schwarzschild spacetime, for the deformation term $(4n+1)^{2}\geq 1$. Starting
from the ansatz (24), we deduced the deflection angle (38). This QG effect may
distinguish the mixed light rays in the absence of gravitational source by a
“spectroscope” $M$. The solutions of gravitational field equation in spacetime
(24) indicate that the QG effect could be regarded as the vacuum effect and
the energy density of vacuum is related to the spacetime deformation parameter
$n$. While $n$ vanishes, the spacetime with quantum effect (24) returns to the
Schwarzschild spacetime and the vacuum energy vanishes, it means that there is
nothing exist outside the gravitational source $M$.
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 and 10875129.
## References
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|
arxiv-papers
| 2010-07-02T09:01:53 |
2024-09-04T02:49:11.378617
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin Li and Zhe Chang",
"submitter": "Xin Li",
"url": "https://arxiv.org/abs/1007.0319"
}
|
1007.0647
|
# The distance and internal composition of the neutron star in EXO 0748$-$676
with XMM-Newton
Guobao Zhang1, Mariano Méndez1 , Peter Jonker2,3,4 and Beike Hiemstra1.
1Kapteyn Astronomical Institute, University of Groningen, P.O. BOX 800, 9700
AV Groningen, The Netherlands
2SRON, Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA,
Utrecht, The Netherlands
3Harvard–Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138, U.S.A.
4Department of Astrophysics, IMAPP, Radboud University Nijmegen, PO Box 9010,
NL-6500 GL Nijmegen, the Netherlands
E-mail: zhang@astro.rug.nl
(Accepted. Received; in original form)
###### Abstract
Recently, the neutron star X-ray binary EXO 0748-676 underwent a transition to
quiescence. We analyzed an XMM-Newton observation of this source in
quiescence, where we fitted the spectrum with two different neutron-star
atmosphere models. From the fits we constrained the allowed parameter space in
the mass-radius diagram for this source for an assumed range of distances to
the system. Comparing the results with different neutron-star equations of
state, we constrained the distance to EXO 0748-676. We found that the EOS
model ’SQM1’ is rejected by the atmosphere model fits for the known distance,
and the ’AP3’ and ’MS1’ is fully consistent with the known distance.
###### keywords:
stars: neutron — X-rays: binaries — dense matter: equation of state — stars:
individual: EXO 0748$-$676
Accepted. Received; in original form
## 1 Introduction
The low-mass X-ray binary (LMXB) EXO 0748–676 was discovered as a transient
source with the European X-ray Observatory Satellite (EXOSAT) in 1985 (Parmar
et al., 1986). The source exhibits simultaneous X-ray and optical eclipses
from which the orbital period of P = 3.82 hr was deduced (Crampton et al.,
1986). EXO 0748$-$676 also exhibited irregular X-ray dipping activity (Parmar
et al., 1986), and type-I X-ray bursts (Gottwald et al., 1986). Burst
oscillations in EXO 0748$-$676 were first reported by Villarreal & Strohmayer
(2004) at 45 Hz in the average Fourier Power Spectrum of 38 type-I X-ray
bursts; the 45-Hz signal was then interpreted as the spin frequency of the
neutron star. Recently, Galloway et al. (2009) detected millisecond
oscillations in the rising phase of two type-I X-ray bursts in EXO 0748-676 at
a frequency of 552 Hz. They concluded that the spin frequency of EXO 0748-676
is close to 522 Hz, rather than 45 Hz as suggested by Villarreal & Strohmayer
(2004). The 45 Hz oscillation may arise in the boundary layer between the disk
and the neutron star (Balman, 2009) or it could be a statistical fluctuation
(Galloway et al., 2009). Cottam, Paerels & Méndez (2002) reported a
measurement of the gravitational redshift from iron and oxygen X-ray
absorption lines arising from the atmosphere of the neutron star in EXO
0748$-$676 during type-I X-ray bursts, but subsequent observations failed to
confirm these features (Cottam et al., 2008). Based on the gravitational
redshift, Özel (2006) suggested that the mass, radius and distance of EXO
0748$-$676 are $2.10\pm 0.28$ ${\rm M}_{\odot}$, $13.8\pm 1.8$ km and $9.2\pm
1.0$ kpc, respectively, which would rule out many neutron-star equations of
state.
Measuring the distance to LMXBs is difficult, except for the case of sources
in globular clusters. A way to get the distance is using type-I X-ray bursts.
The peak flux for some very bright bursts can reach the Eddington luminosity
at the surface of the neutron star. From a strong X-ray burst, Wolff et at.
(2005) derived a distance to EXO 0748$-$676 of 7.7 kpc for a helium-dominated
burst photosphere, and 5.9 kpc for a hydrogen-dominated burst photosphere.
Galloway et al. (2008a) analyzed several type-I X-ray bursts from EXO
0748$-$676 and estimated a distance of 7.4 kpc, different from the value of
9.2 kpc reported by Özel (2006). Taking into account the touchdown flux and
high-inclination in EXO 0748$-$676 , recently Galloway et al. (2008b) gave a
distance of 7.1 $\pm$ 1.2 kpc.
Another way to get the distance to an LMXB is through observations of
quiescent X-ray emission from the neutron-star surface. During the quiescent
state, X-ray emission originates from the atmosphere of the neutron star. By
fitting the X-ray spectrum of the neutron-star system with hydrogen atmosphere
models, one can estimate the mass, radius and distance of the neutron star.
Recently, the neutron-star X-ray transient EXO 0748$-$676 underwent a
transition into quiescence (Degenaar et al., 2009; Bassa et al., 2009).
In this paper, we report on the distance to EXO 0748$-$676 that we constrained
from XMM-Newton data. We use two different neutron-star atmophere models to
fit the X-ray spectrum, and compare the results of the spectral fitting with
different neutron-star equation of state (EOS). In the next section, we
describe the observation and data analysis. We show the fitting results in §3,
and we discuss our findings in §4.
## 2 OBSERVATIONS AND DATA ANALYSIS
EXO 0748$-$676 was observed with the European Photon Imaging Camera (EPIC PN
and MOS) on board the XMM-Newton on 2008 November 6 at 08:30:03 UTC (obsID
0560180701). The PN and the two MOS cameras were operated in Full-Window mode.
We reduced the XMM-Newton Observation Data Files (ODF) using version 8.0.0 of
the science analysis software (SAS). We used the epproc and emproc tasks to
extract the event files for the PN and the two MOS cameras, respectively.
Source light curves and spectra were extracted in the 0.2 $-$ 12.0 keV band
using a circular extraction region with a radius of 30 arcsec centered on the
position of the source. Background light curves and spectra were extracted
from a circular source-free region of 35 arcsec source-free on the same CCD.
We applied standard filtering and examined the light curves for background
flares. No flares were present and we used the whole exposure for our
analysis. The exposure time for the PN camera was 24.2 ks, and for each MOS
camera was 29.03 ks. The source count rate was $0.496\pm 0.005$ cts/s for PN,
and $0.135\pm 0.002$ cts/s and $0.127\pm 0.002$ cts/s for MOS1 and MOS2,
respectively. We checked the filtered event files for photon pile-up by
running the task epatplot. No pile-up was apparent in the PN, MOS1 and MOS2
data. The photon redistribution matrices and ancillary files for the source
spectra were created using the SAS tools rmfgen and arfgen, respectively. We
rebinned the source spectra using the tool pharbn111M. Guainazzi, private
communication, such that the number of bins per resolution element of the PN
and MOS spectra was 3 and the minimum number of counts per channel was 20.
We fitted the PN and MOS spectra simultaneously in the 0.5$-$10.0 keV range
with XSPEC 12.50 (Arnaud, 1996), using either of two neutron-star hydrogen-
atmosphere models: NSAGRAV (Zavlin et al., 1996) and NSATMOS (Heinke et al.,
2006). The NSAGRAV model provides the spectra emitted from a nonmagnetic
hydrogen atmosphere of a neutron star with surface gravitational acceleration,
$g$, ranging from $10^{13}$ to $10^{15}$ cm s-2. This model uses the mass
($M_{\rm NS}$) and radius ($R_{\rm NS}$) of the neutron star and the
unredshifted effective temperature of the surface of the star ($kT_{\rm eff}$)
as parameters. The normalization of the model is defined as $1/D^{2}$, where
$D$ is the distance to the source in pc. The second model that we used,
NSATMOS, includes a range of surface gravities and effective temperatures, and
incorporates thermal electron conduction and self-irradiation by photons from
the compact object. This model assumes negligible magnetic fields (less than
$10^{9}$ G) and a pure hydrogen atmosphere. NSATMOS parameters are $M_{\rm
NS}$, $R_{\rm NS}$, log$T_{\rm eff}$ (the same as for NSAGRAV), distance in
kpc, and a separate normalization $K$, which corresponds to the fraction of
the neutron-star surface that is emitting. We fixed $K$ to be 1 in all our
fits with NSATMOS.
Figure 1: XMM-Newton PN (black), MOS1 (red) and MOS2 (green) spectrum of EXO
0748$-$676 in the 0.5$-$ 10.0 keV energy band. The spectrum was fitted with a
neutron-star hydrogen atmosphere model (NSATMOS) and a power-law model with
$\Gamma$ fixed to 1. The lower panel shows the residuals to the best-fit
model.
We included the effect of interstellar absorption using PHABS assuming cross-
sections of Balucinska-Church & McCammon (1992) and solar abundances from
Anders & Grevesse (1989), and we let $N_{\rm H}$, column density along the
line of sight free to vary during the fitting. In order to account for
differences in effective area between the different cameras, we introduced a
multiplicative factor in our model. First, this factor was fixed to unity for
PN and free for MOS1 and MOS2. Then, we set the scaling factor to unity for
MOS1 and MOS2, respectively, and set the factor free for the other cameras. We
found that, fixing the scaling factor for different cameras gives similar
best-fit results. Therefore in the rest of the paper we fixed the factor to be
1 for PN and free to vary for the other cameras. None of the atmosphere models
alone fitted the spectrum above $\sim 2-3$ keV properly. Adding a power-law
component improved the fits significantly, however, all parameters were less
constrained than when fitting the data with the neutron-star atmosphere model
only. We first fixed the power-law index to 0.5, 1.0 and 1.5 to get better
constraints on the parameters of the neutron-star atmosphere model (Degenaar
et al. 2009). Further, we initially fixed the distance to the NS at 7.1 kpc,
which is the value inferred from the touchdown flux of Galloway et al. (2008b)
.
## 3 RESULTS
### 3.1 Results from the spectral fits
Table 1: Best-fit parameters of neutron-star atmosphere models fit to the XMM-
Newton data of EXO 0748$-$676 .
model | NH | $T^{\infty}_{\rm eff}$ | $M_{\rm NS}$ | $R_{\rm NS}$ | $\Gamma$ | $F_{pow}$ | $F_{X}$ | $\chi^{2}$/d.o.f.
---|---|---|---|---|---|---|---|---
| ($10^{20}cm^{-2}$) | (eV) | (${\rm M}_{\odot}$) | (km) | | 10-13 ergs cm-2 s-1 | 10-12 ergs cm-2 s-1 |
NSAGRAV | $5.6\pm 1.8$ | $113^{+14}_{-8}$ | $1.55\pm 0.18$ | $15.2\pm 1.8$ | 0.5 | $1.15\pm 0.21$ | $1.18\pm 0.15$ | 0.986/219
NSATMOS | $5.4\pm 1.5$ | $113\pm 4$ | $1.29\pm 0.20$ | $16.1^{+0.9}_{-1.2}$ | 0.5 | $1.17\pm 0.20$ | $1.23\pm 0.16$ | 0.985/219
NSAGRAV | $6.2^{+1.3}_{-1.8}$ | $114^{+24}_{-3}$ | $1.62\pm 0.11$ | $15.8^{+0.25}_{-3.5}$ | 1.0 | $1.10\pm 0.15$ | $1.14\pm 0.13$ | 0.977/219
NSATMOS | $6.1\pm 1.5$ | $114\pm 4$ | $1.55\pm 0.12$ | $16.0^{+0.7}_{-1.3}$ | 1.0 | $1.11\pm 0.15$ | $1.13\pm 0.06$ | 0.977/219
NSAGRAV | $6.7\pm 1.5$ | $110\pm 8$ | $1.71\pm 0.30$ | $16.5\pm 0.5$ | 1.5 | $1.00\pm 0.19$ | $1.01\pm 0.15$ | 0.987/219
NSATMOS | $6.7\pm 1.4$ | $110\pm 5$ | $1.77\pm 0.45$ | $16.6^{+1.8}_{-7.5}$ | 1.5 | $1.03\pm 0.22$ | $1.03\pm 0.10$ | 0.985/219
* Note. – NH is the equivalent hydrogen column density, $T^{\infty}_{\rm eff}$ the effective temperature of the neutron-star surface as seen at infinity, $M_{\rm NS}$ and $R_{\rm NS}$ are the mass and radius of the neutron star, respectively. $F_{pow}$ is the unabsorbed flux of the power-law component in the 0.5$-$10 keV energy band, and $F_{X}$ is the total unabsorbed X-ray flux in the same energy band. The last column gives the reduced $\chi^{2}$ for 219 degrees of freedom. The quoted errors represent the 90% confidence levels.
Figure 1 shows the XMM-Newton spectra of EXO 0748-676 fitted with the model
“phabs (NSATMOS + powerlaw) ”. The power-law index is fixed at 1.0. The best
fit of this model gives $N_{\rm H}=6.1\pm 1.5$ $\times 10^{20}$ cm-2, neutron-
star mass $M_{\rm NS}=1.55\pm 0.12{\rm M}_{\odot}$, neutron-star radius
$R_{\rm NS}=16.0^{+0.7}_{-1.3}$ km, and effective temperature log$T_{\rm
eff}=6.20\pm 0.02$ (in $K$). According to the same formula $T^{\infty}_{\rm
eff}=T_{\rm eff}\sqrt{1-(2GM_{\rm NS})/(R_{\rm NS}c^{2})}$ used by Degenaar et
al. (2009), we converted $T_{\rm eff}$ to the effective temperature as seen by
an observer at infinity, $T^{\infty}_{\rm eff}=114\pm 4$ eV. In the formula,
$G$ is the gravitational constant and $c$ is the speed of light. The model
predicts 0.5$-$10 keV an unabsorbed X-ray flux $F_{X}=1.13\pm 0.06\times
10^{-12}$ ergs cm-2 s-1. The flux of the power-law component in the same
energy band is $F_{pow}=1.11\pm 0.15\times 10^{-13}$ ergs cm-2 s-1, which
corresponds to $\sim 10\%$ of the total unabsorbed flux. The reduced
$\chi^{2}$ is 0.977 for 219 degrees of freedom. The best-fit results of the
models NSAGRAV and NSATMOS for the three different power-law index are given
in Table-LABEL:tab:model. Errors are given at the 90% confidence level for one
fit parameter.
We note from Table LABEL:tab:model that both atmosphere models, regardless of
the value of $\Gamma$, yield a good fit with similar $\chi^{2}$. In the rest
of the analysis, we used a power-law index fixed to 1. Further, $N_{\rm H}$
and $T_{\rm eff}$ are well constrained and are consistent for the different
fits. Both NSAGRAV and NSATMOS models also give consistent results on $M_{\rm
NS}$ and $R_{\rm NS}$. The NSATMOS model is more accurate in constraining
$T_{\rm eff}$ than the NSAGRAV model.
### 3.2 Equation of state
Fitting the quiescence XMM-Newton spectrum of EXO 0748$-$676 with two
different atmosphere models and comparing the results allows us to test the
reliability and accuracy of both models. From the fits we get a mass and
radius of the neutron star at a specified distance, and then by comparing the
inferred mass and radius with the different neutron-star EOS we can give upper
limits to the source distance for the different EOS.
We used the steppar command in xspec to vary the mass, radius and distance
parameters simultaneously, allowing other parameters to be free to find the
best fit at each step. For the mass we go from 0.5 to 2.5 $M_{\odot}$ with
steps of 0.1 $M_{\odot}$, and for the distance we go from 5 to 10 kpc with
steps of 0.25 kpc. The minimum and maximum radius allowed with these models
are 5.0 km and 25.0 km, respectively. In Fig 2 we show the contour plots
obtained from the STEPPAR procedure for the NSATMOS model. Each plot is for a
different distance, ranging from 5 to 10 kpc. The contour lines (red) are for
the confidence levels of 90% (solid) and 99% (dashed). Further, in Fig 2 we
give different neutron-star EOS (black) taken from Lattimer & Prakash (2007).
We did the same analysis for the NSAGRAV model as well. Both two models give
consistent result, in accordance with the findings of Webb & Barret (2007).
Using the optical data from the Very Large Telescope (VLT), moderate-
resolution spectroscopy of the optical counterpart and Doppler tomography,
Muñoz et al. (2009) provided the first dynamical constraints on the stellar
mass of LMXB EXO 0748$-$676 . The mass range of the neutron star that they
derived is $1{\rm M}_{\odot}\leq{\rm M_{\rm NS}}\leq 2.4{\rm M}_{\odot}$.
Subsequently, Bassa et al. (2009) analyzed optical spectra of EXO 0748$-$676
when the source was in the quiescent state, and they gave a lower limit to the
neutron-star mass of ${\rm M_{\rm NS}}\geq$ 1.27 ${\rm M}_{\odot}$. As upper
limit we used the value reported by Muñoz et al. (2009), but since at the time
of their observation the source was still in outburst, we used the lower limit
reported in Bassa et al. (2009). In Figure 2 we also give the lower
(pink/dotted) and upper (green/dashed) limits to the neutron-star mass.
(a) 5.0 kpc
(b) 6.0 kpc
(c) 7.0 kpc
(d) 8.0 kpc
(e) 9.0 kpc
(f) 10.0 kpc
Figure 2: Contour plots showing the results of modeling the neutron-star in
EXO 0748$-$676 with the xspec model NSATMOS and power-law. The power-law index
is fixed to 1. The plots show two confidence levels in the mass-radius diagram
obtained from our fit; the contour lines (red) are for the confidence levels
of 90% and 99%, respectively. The pink line “A” is for the lower limit of
$M_{\rm NS}$ given by Bassa et al.(2009), and the green line “B” is for the
upper limit given by Muñoz et al. (2009).
In order to test the EOS and identify the upper limit to the source distance,
we assume three different EOS models: normal nucleonic matter (AP3), boson
condensates matter (MS1) and strange quark matter (SQM1). By varying the
source distance from 5 to 10 kpc, the contour lines for the fitted model move
on the NS mass-radius diagram. We can estimate the probability of the distance
for each EOS when the contour lines pass through the EOS curves. Note that as
the distance increases (see Figure 2), the satisfied area of the model moves
from bottom left to top right in the plot. The results using NSAGRAV are
similar to those shown in Figure 2. For a certain distance we found that not
all the EOSs are consistent with the two neutron-star atmosphere models that
we used.
If the neutron star in EXO 0748$-$676 follows the EOS model ‘AP3’, the
probability that the source has a distance of 10.0 kpc is $1\times 10^{-4}$
and $1\times 10^{-6}$ for NSAGRAV and NSATMOS, respectively. If we want to get
a probability for the distance larger than $1\times 10^{-2}$ (99% confidence),
the distance for NSAGRAV and NSATMOS should be smaller than 8.9 kpc and 8.5
kpc, respectively. The distance at 90% confidence for NSAGRAV and NSATMOS is
less than 8.3 kpc and 8.2 kpc, respectively. Both models are consistent with
the distance of 7.1 kpc given by type-I X-ray bursts (Galloway et al., 2008b).
For the EOS model ’MS1’, the probabilities that EXO 0748$-$676 is at a
distance of 10 kpc is $10^{-5}$ and $10^{-6}$ for NSAGRAV and NSATMOS,
respectively. For both models, respectively, the distance at 99% confidence
level is less than 7.3 kpc and 7.1 kpc, and the distance at 90% confidence
level is less than 6.9 kpc and 6.8 kpc. Both neutron-star atmosphere models
with the ’MS1’ model have an upper limit for the distance smaller than 7.1
kpc.
For a EOS model ‘SQM1’, the distance at 99% confidence level is less than 5.2
kpc, and the distance at 90% confidence level is less than 5.0 kpc for both
atmosphere models. The upper limits on the distance to EXO 0748$-$676 for
different EOS are shown in Table LABEL:tab:upper_limit. The ’SQM1’ model is
rejected at a 99% confidence level for this neutron star, unless the source is
closer than 5.2 kpc.
Table 2: Upper limits on the distance to EXO 0748$-$676 for different EOS
models.
EOS | AP3 | AP3 | MS1 | MS1 | SQM1 | SQM1
---|---|---|---|---|---|---
confidence | 90% | 99% | 90% | 99% | 90% | 99%
NSAGRAV | $<$ 8.3 | $<$ 8.9 | $<$ 6.9 | $<$ 7.3 | $<$ 5.0 | $<$ 5.2
NSATMOS | $<$ 8.2 | $<$ 8.5 | $<$ 6.8 | $<$ 7.1 | $<$ 5.0 | $<$ 5.2
* Note. –The 90% and 99% confidence levels upper limit for the two NS atmosphere models NSAGRAV and NSATMOS for the EOS models: ‘AP3’, ‘MS1’ and ’SQM1’. The distance is in kpc.
## 4 DISCUSSION
We analyzed an XMM-Newton observation of the neutron star EXO 0748$-$676 in
the quiescent state. The unabsorbed X-ray flux in the 0.5$-$10.0 keV energy
band was $\sim 1.1\times$ 10-12 ergs cm-2 s-1. We found that the non-thermal
(power-law) component only contributes $\sim 10\pm 2\%$ of the 0.5$-$10 keV
X-ray flux, which is lower than what Degenaar et al. (2009) found from Chandra
data ($F_{pow}$ was $\sim$ 16$-$17$\%$ of the 0.5$-$10 keV X-ray flux from the
fit with $\Gamma=1$) about a month earlier than our observation. The total
unabsorbed flux (0.5$-$10.0 keV) decreased from $1.3\times 10^{-12}$ ergs cm-2
s-1 in the Chandra observation to $1.1\times 10^{-12}$ ergs cm-2 s-1 in our
observation, whereas $N_{\rm H}$ changed from $\sim 1.2$ $\times 10^{21}$ $\rm
cm^{-2}$ to $\sim 0.6$ $\times 10^{21}$ $\rm cm^{-2}$. The effective
temperature, however, did not show large variations in one month time.
According to the above comparisons, the reduction of the total flux is due to
a lower contribution of the power-law component.
Because the X-ray spectrum in the quiescent state is dominated by thermal
emission originating from the NS surface, our data allow us to constrain the
mass and radius of the neutron star. From the two different NS atmosphere
models (NSAGRAV and NSATMOS) that we used to fit the X-ray spectrum, we found
that both models show similar results and set good constraints on the neutron-
star radius. Even taking into account the $M_{\rm NS}$ lower limit (from Bassa
et al., 2009), upper limit (from Muñoz et al., 2009) and our best fit
$\Delta\chi^{2}$ contour, we still have a large area on the mass-radius
diagram, and many EOSs are still possible (see Figure 2). In order to
constrain the allowed space of mass and radius at a specified distance, we
choose three typical neutron-star EOS, ‘AP3’, ‘MS1’ and ’SQM1’. We found that
the smaller the distance to the NS the more EOSs are consistent with the data.
For any specific EOS, as the upper limit of the distance we took the value of
the distance where the 99% confidence contour just intersects the curve of
that EOS. We found that the upper limits on the distance as derived from the
NSAGRAV model are slightly higher than those for the NSATMOS model. The EOS
model ‘MS1’ can be just satisfied at a distance of 7.1 kpc. If we assume that
the neutron star in EXO 0748$-$676 is a normal neutron star, following the EOS
‘AP3’, the source should be closer than 8.9 kpc for the NSAGRAV model, or 8.5
kpc for the NSATMOS model. Both the ’MS1’ and ’AP3’ EOS are fully consistent
with the measured distance of 7.1 kpc (Galloway et al., 2008b; Wolff et at.,
2005). For larger distances more EOS are ruled out. The EOS ’SQM1’ is rejected
by the atmosphere model fits for a distance of 7.1 kpc measured from the X-ray
bursts Galloway et al. (2008b). We note, however, the neutron-star atmosphere
models may not appropriate for ’bare’ quark matter stars, but only for those
normal quark star where a crust is present.
## Acknowledgments
This work is based on the observations obtained from XMM-Newton. PGJ
acknowledges support from a VIDI grant from the Netherlands Organisation for
Scientific Research.
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* (24)
|
arxiv-papers
| 2010-07-05T09:44:35 |
2024-09-04T02:49:11.399336
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guobao Zhang (1), Mariano Mendez (1), Peter Jonker (2), and Beike\n Hiemstra (1) ((1) Groningen, (2) SRON)",
"submitter": "Guobao Zhang",
"url": "https://arxiv.org/abs/1007.0647"
}
|
1007.0655
|
# Primitivity and Independent Sets in Direct Products of Vertex-Transitive
Graphs ††thanks: Supported by the National Natural Science Foundation of China
(No.10826084) and Zhejiang Innovation Project (Grant No. T200905).
Huajun Zhang
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
and
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, P.R.
China
E-mail: huajunzhang@zjnu.cn
Abstract. We introduce the concept of the primitivity of independent set in
vertex-transitive graphs, and investigate the relationship between the
primitivity and the structure of maximum independent sets in direct products
of vertex-transitive graphs. As a consequence of our main results, we
positively solve an open problem related to the structure of independent sets
in powers of vertex-transitive graphs.
## 1 Introduction
The direct product $G\times H$ of two graphs $G$ and $H$ is defined by
$V(G\times H)=V(G)\times V(H)$
and
$E(G\times H)=\\{[(u_{1},u_{2}),(v_{1},v_{2})]:[u_{1},v_{1}]\in E(G)\mbox{\
and \ }[u_{2},v_{2}]\in E(H)\\}.$
For a graph $G$, let $G^{n}=G\times\cdots\times G$ denote the $n$-th power of
$G$.
It is clear that if $I$ is an independent set of $G$ (or $H$), then $I\times
H$ (or $G\times I$) is an independent set of $G\times H$. We say that $G\times
H$ is MIS-normal (maximum-independent-set-normal) if each of its maximum
independent sets is of this form. Then the independence number
$\displaystyle\alpha(G\times H)=\max\\{\alpha(G)|H|,\alpha(H)|G|\\}$ (1)
if $G\times H$ is MIS-normal. A product $G_{1}\times G_{2}\times\cdots\times
G_{n}$ is said to be MIS-normal if all of its maximum independent sets are
preimages of projections of maximum independent sets of one of its factors.
This poses two immediate problems: whether (1) holds for all graphs $G$ and
$H$, and whether $G\times H$ is MIS-normal when (1) holds. In general,
however, (1) does not hold for some non-vertex-transitive graphs (see [7]).
So, Tardif [3] asked whether (1) holds for all vertex-transitive graphs $G$
and $H$. Larose and Tardif [2] investigated the relationship between the
projectivity and the structure of maximal independent sets in powers of a
circular graph, Kneser graph, or truncated simplex. Recently, Mario and Vera
[5] proved that (1) holds for some special vertex-transitive graphs, e.g.,
circular graphs and Kneser graphs. In fact, Frankl [6] proved in 1996, one
year before Tardif’s question was posed, that (1) holds for Kneser graphs.
Subsequently, Ahlswede, Aydinian and Khachatrian [8] generalized Frankl’s
result.
In the context of vertex-transitive graphs, the “No-Homomorphism” lemma of
Albertson and Collins [1] is useful to get bounds on the size of independent
sets.
###### Lemma 1.1
([1]) Let $G$ and $H$ be two graphs such that $G$ is vertex-transitive and
there exists a homomorphism $\phi:H\mapsto G$. Then
$\frac{\alpha(G)}{|V(G)|}\leq\frac{\alpha(H)}{|V(H)|},$ and the equality holds
if and only if for any independent set $I$ of cardinality $\alpha(G)$ in $G$,
$\phi^{-1}(I)$ is an independent set of cardinality $\alpha(H)$ in $H$.
By this lemma, it is easy to deduce that $\alpha(G^{n})=\alpha(G)|V(G)|^{n-1}$
for any vertex-transitive graph $G$ and positive integer $n$ (see [2]). So it
is natural to ask whether $G^{n}$ is MIS-normal. Evidently, if $G^{n}$ is MIS-
normal for some $n>2$, so is $G^{2}$. Conversely, Larose and Tardif [2] posed
the following problem.
###### Problem 1.2
(see [2] )Let $G$ be a non-bipartite vertex-transitive graph. If $G^{2}$ is
MIS-normal, is the same for all powers of $G$?
This paper is organized as follows. In the next section, we introduce a
concept of the primitivity of independent sets in a vertex-transitive graph,
and prove that the primitivity can be preserved in direct products under
certain conditions. Based on these results we establish in section $3$ a
direct product theorem on the MIS-normality. As a consequence, Problem 1.2 is
solved.
## 2 Primitivity of independent sets
In the sequel of this paper, let $G$ and $H$ be vertex-transitive graphs. By
$I(G)$ we denote the set of all maximum independent sets of $G$. For any
subset $A$ of $V(G)$, let $\alpha(A)$ denote the independence number of the
induced subgraph of $G$ by $A$, and we define
$N_{G}(A)=\\{b\in G:\mbox{$(a,b)\in E(G)$ for some $a\in A$}\\},$
$N_{G}[A]=N_{G}(A)\cup A\ \mbox{and}\ \overline{N}_{G}[A]=G-N_{G}[A].$
In Lemma 1.1, by taking $H$ as an induced subgraph of $G$ and $\phi$ as the
embedding mapping, we obtain the following lemma (cf. [4]).
###### Lemma 2.1
$\frac{\alpha(G)}{|V(G)|}\leq\frac{\alpha(B)}{|B|}$ holds for all $B\subseteq
V(G)$. Equality implies that $|S\cap B|=\alpha(B)$ for every $S\in I(G)$.
A graph $G$ is said to be non-empty if $E(G)\neq\emptyset$. Lemma 2.1 implies
that $\alpha(G)\leq|V(G)|/2$ for all non-empty vertex-transitive graphs.
Equality holds if and only if $G$ is bipartite, which we state as a corollary
for reference.
###### Corollary 2.2
Let $G$ be a non-empty vertex-transitive graph. Then
$\frac{\alpha(G)}{|G|}\leq\frac{1}{2}$, and equality holds if and only if $G$
is bipartite.
###### Proposition 2.3
Let $A$ be an independent set of $G$. Then
$\frac{|A|}{|N_{G}[A]|}\leq\frac{\alpha(G)}{|V(G)|}$. Equality implies that
$|S\cap N_{G}[A]|=|A|$ for every $S\in I(G)$, and in particularly $A\subseteq
S$ for some $S\in I(G)$.
Proof. Since $A$ is an independent set, clearly
$\frac{|A|+\alpha(\overline{N}_{G}[A])}{|N_{G}[A]|+|\overline{N}_{G}[A]|}\leq\frac{\alpha(G)}{|V(G)|}.$
By Lemma 2.1 we see that
$\frac{\alpha(\overline{N}_{G}[A])}{|\overline{N}_{G}[A]|}\geq\frac{\alpha(G)}{|V(G)|}$,
so $\frac{|A|}{|N_{G}[A]|}\leq\frac{\alpha(G)}{|V(G)|}$. Equality in the
latter implies equality in the former. In this case any $S\in I(G)$ must be
the union of a maximum independent set in $\overline{N}_{G}[A]$ and an
independent set of size $|A|$ in $N_{G}[A]$, and thus $|S\cap N_{G}[A]|=|A|$.
$\Box$
An independent set $A$ in $G$ is said to be imprimitive if $|A|<\alpha(G)$ and
$\frac{|A|}{|N_{G}[A]|}=\frac{\alpha(G)}{|V(G)|}$. We say that $G$ is IS-
imprimitive if $G$ has an imprimitive independent set. In the other case, $G$
is _IS-primitive_.
###### Proposition 2.4
Let $A$ be a maximum imprimitive independent set of $G$. Set
$B=\overline{N}_{G}[A]$. Then $\frac{\alpha(B)}{|B|}=\frac{\alpha(G)}{|V(G)|}$
and $\\{\sigma(B)|\sigma\in\mbox{Aut}(G)\\}$ forms a nontrivial partition of
$V(G)$, i.e., $\sigma(B)\cap B=\emptyset$ or $B$ for each
$\sigma\in\mbox{Aut}(G)$.
Proof. Clearly
$\frac{|A|+\alpha(B)}{|N_{G}[A]|+|B|}\leq\frac{\alpha(G)}{|V(G)|}$. Combining
the condition of $A$ and Lemma 2.1, we have
$\frac{\alpha(B)}{|B|}=\frac{\alpha(G)}{|V(G)|}$. By definition,
$N_{G}[\sigma(A)]=\sigma(N_{G}[A])$ for all $\sigma\in\mbox{Aut}(G)$. Suppose
that there exists a $\sigma\in\mbox{Aut}(G)$ such that $\sigma(B)\neq B$ and
$\sigma(B)\cap B\neq\emptyset$. Then $\sigma(N_{G}[A])\neq N_{G}[A]$ and
$\displaystyle|V(G)|>|N_{G}[A]\cup\sigma\big{(}N_{G}[A]\big{)}|>|N_{G}[A]|.$
(2)
Let $C=\sigma(A)\cup(A-N_{G}[\sigma(A)])$. Then $C$ is also an independent set
and
$N_{G}[C]\subseteq N_{G}[A]\cup\sigma(N_{G}[A]).$
By Proposition 2.3, $|S\cap N_{G}[A]|=|A|$ for all $S\in I(G)$, which implies
that $(S-N_{G}[A])\cup A\in I(G)$ for all $S\in I(G)$. Similarly,
$\displaystyle((S-N_{G}[A])\cup A)-N_{G}[\sigma(A)])\cup\sigma(A)$
$\displaystyle=$ $\displaystyle(S-N_{G}[A]\cup
N_{G}[\sigma(A)])\cup(A-N_{G}[\sigma(A)])\cup\sigma(A)$ $\displaystyle=$
$\displaystyle(S-N_{G}[A]\cup N_{G}[\sigma(A)])\cup C$
is also a maximum independent set of $G$, which implies $|S\cap(N_{G}[A]\cup
N_{G}[\sigma(A)])|=|C|$ for all $S\in I(G)$.
Given a $u\in V(G)$, suppose that there are $r$ $S$’s in $I(G)$ such that
$u\in S$. Since $G$ is vertex-transitive, the number $r$ is independent of the
choice of $u$. Thus $r|V(G)|=\alpha(G)|I(G)|$. On the other hand, since
$|S\cap(N_{G}[A]\cup N_{G}[\sigma(A)])|=|C|$ for all $S\in I(G)$,
$|C||I(G)|=r|N_{G}[A]\cup N_{G}[\sigma(A)]|$. Combining the above two
equalities, we have $\frac{|C|}{|N_{G}[A]\cup
N_{G}[\sigma(A)]|}=\frac{\alpha(G)}{|V(G)|}$. Thus, by Proposition 2.3 we have
$\frac{\alpha(G)}{|V(G)|}\geq\frac{|C|}{N_{G}[C]}\geq\frac{|C|}{|N_{G}[A]\cup
N_{G}[\sigma(A)]|}=\frac{\alpha(G)}{|V(G)|},$
which implies $N_{G}[C]=N_{G}[A]\cup N_{G}[\sigma(A)]$ and
$\frac{|C|}{|N_{G}[C]|}=\frac{\alpha(G)}{|V(G)|}$. By (2), we have
$|A|<|C|<\alpha(G)$, contradicting the maximality of $|A|$. This completes the
proof. $\Box$
The concept of primitivity comes from permutation groups: A permutation group
$\Gamma$ acting on a set $X$ is called primitive if $\Gamma$ preserves no
nontrivial partition of $X$. In the other case, $\Gamma$ is imprimitive. As
usual (see e.g. [2]), a vertex-transitive graph $G$ is called primitive if its
automorphism group, as a permutation group on $V(G)$, is primitive. By
Proposition 2.4 we see that if $G$ is primitive, then $G$ is IS-primitive. But
the converse is not true.
For any $S\subseteq V(G)\times V(H)$, $a\in G$ and $u\in H$, define
$\partial_{G}(u,S)=\\{b\in G:(b,u)\in S\\},\ \ \partial_{H}(a,S)=\\{v\in
H:(a,v)\in S\\},$
and
$\partial_{G}(S)=\\{b\in G:\partial_{H}(b,S)\neq\emptyset\\},\ \
\partial_{H}(S)=\\{v\in H:\partial_{G}(v,S)\neq\emptyset\\}.$
By definition we see that $\partial_{G}(S)$ and $\partial_{H}(S)$ are in fact
the projections of $S$ on $G$ and $H$, respectively.
###### Lemma 2.5
Suppose $G\times H$ is MIS-normal and
$\frac{\alpha(H)}{|H|}\leq\frac{\alpha(G)}{|G|}$. If $G\times H$ is IS-
imprimitive, then one of the following two possible cases holds:
* (i)
$\frac{\alpha(H)}{|H|}=\frac{\alpha(G)}{|G|}$, and one of them is IS-
imprimitive or both $G$ and $H$ are bipartite;
* (ii)
$\frac{\alpha(H)}{|H|}<\frac{\alpha(G)}{|G|}$, and $G$ is IS-imprimitive or
$H$ is disconnected.
Proof. Throughout this proof, we denote $N_{G\times H}[A]$ by $N[A]$ for
brevity. Suppose that $G\times H$ is IS-imprimitive and let $A$ be a maximum
imprimitive independent set of $G\times H$. Clearly, $\alpha(G\times
H)=\alpha(G)|V(H)|$, and thus $\frac{|A|}{|N[A]|}=\frac{\alpha(G\times
H)}{|V(G\times H)|}=\frac{\alpha(G)}{|V(G)|}$. If $E(G)=\emptyset$, the result
is trivial, so we suppose $E(G)\neq\emptyset$, then Corollary 2.2 implies that
$\frac{\alpha(H)}{|V(H)|}\leq\frac{\alpha(G)}{|V(G)|}\leq\frac{1}{2}$. By
Proposition 2.3, there exists some $S\in I(G\times H)$ such that $A=S\cap
N[A]$. Since $G\times H$ is MIS-normal, we may assume that $S=S^{\prime}\times
H$ for some $S^{\prime}\in I(G)$. Thus $A=(S^{\prime}\times H)\cap N[A]$. Set
$B=\overline{N}[A]$. Then, by Proposition 2.4, $\sigma(B)\cap B=\emptyset$ or
$B$ for every $\sigma\in\mbox{Aut}(G\times H)$.
Set $C=\partial_{G}(B)$. For every pair $a$ and $b$ of $C$, select
$u\in\partial_{H}(a,B)$ and $v\in\partial_{H}(b,B)$. Since $G$ and $H$ are
vertex-transitive, there exist $\gamma\in\mbox{Aut}(G)$ and
$\tau\in\mbox{Aut}(H)$ such that $a=\gamma(b)$ and $u=\tau(v)$. It is clear
that $\sigma=(\gamma,\tau)\in\mbox{Aut}(G\times H)$ and
$(a,u)=\sigma(b,v)\in\sigma(B)\cap B$. By Proposition 2.4, we conclude that
$\sigma(B)=B$. Thus, we have $\partial_{H}(a,B)=\tau(\partial_{H}(b,B))$.
Therefore, $|\partial_{H}(a,B)|=|\partial_{H}(b,B))|$ for any $a,b\in C$. In
the following, we will complete the proof by two cases.
Case $1$: $C\neq V(G)$. Set $\overline{C}=(V(G)-C)$. Then $(\overline{C}\times
H)\cap B=\emptyset$, and thus $\overline{C}\times H\subseteq N[A]$. For every
$S^{\prime\prime}\in I(G)$, it is clear that $S^{\prime\prime}\times H$ is a
maximum independent set of $G\times H$. Since
$\frac{\alpha(B)}{|B|}=\frac{\alpha(G\times H)}{|G\times
H|}=\frac{\alpha(G)}{|V(G)|}$ and $|\partial_{H}(a,B)|=|\partial_{H}(b,B)|$
for all $a,b\in\partial_{G}(B)$, from Lemma 2.1 and the MIS-normality of
$G\times H$ it follows that
$\frac{|(S^{\prime\prime}\times H)\cap B|}{|B|}=\frac{|S^{\prime\prime}\cap
C|}{|C|}=\frac{\alpha(G)}{|V(G)|}.$
Thus for every $S^{\prime\prime}\in I(G)$,
$\displaystyle\frac{\alpha(G)}{|V(G)|}=\frac{|S^{\prime\prime}|}{|V(G)|}=\frac{|S^{\prime\prime}\cap
C|+|S^{\prime\prime}\cap\overline{C}|}{|C|+|\overline{C}|}=\frac{|S^{\prime\prime}\cap\overline{C}|}{|\overline{C}|}=\frac{|S^{\prime\prime}\cap
C|}{|C|}.$ (3)
Recall that $\overline{C}\times H\subseteq N[A]$ and $A\subseteq
S^{\prime}\times H$, it is easy to see that $A=N[A]\cap(S^{\prime}\times H)$
and $\partial_{G}(A\cap(\overline{C}\times H))=S^{\prime}\cap\overline{C}$.
Setting $F=S^{\prime}\cap\overline{C}$, we have that $a\times H\subseteq A$
for every $a\in F$. If $N_{G}[F]\cap C\neq\emptyset$, then there exist $a\in
F$ and $b\in{C}$ such that $(a,b)\in E(G)$. Since $B=\overline{N}[A]$ and
$a\times H\subseteq A$, by definition, $(b,u)\subseteq\overline{N}[a\times H]$
for every $u\in\partial_{H}(b,B)$. Hence $N_{H}[H]\neq\emptyset$ and
$E(H)=\emptyset$, which contradicts that
$\frac{\alpha(H)}{|H|}\leq\frac{1}{2}$. Thus $N_{G}[F]\cap C=\emptyset$, i.e.,
$N_{G}[F]\subseteq\overline{C}$. By Proposition 2.3 and (3),
$\frac{\alpha(G)}{|V(G)|}\geq\frac{|F|}{|N_{G}[F]|}=\frac{|{S^{\prime}}\cap\overline{C}|}{|N_{G}[F]|}\geq\frac{|{S^{\prime}}\cap\overline{C}|}{|\overline{C}|}=\frac{\alpha(G)}{|V(G)|}.$
Therefore $\frac{|F|}{|N_{G}[F]|}=\frac{\alpha(G)}{|V(G)|}$, so $G$ is IS-
imprimitive and (i) holds.
Case $2$: $C=V(G)$. Since $|\partial_{H}(a,B)|=|\partial_{H}(b,B))|$ for all
$a,b\in V(G)$, we have $\partial_{G}(N[A])=V(G)$ and
$|\partial_{H}(a,N[A])|=|\partial_{H}(b,N[A])|<|H|$ for all $a,b\in V(G)$.
Since $A=(S^{\prime}\times H)\cap N[A]$,
$\partial_{H}(a,N[A])\subseteq\partial_{H}(a,S^{\prime}\times H)$ for all
$a\in\partial_{G}(A)$. Thus $\partial_{H}(a,A)=\partial_{H}(a,N[A])$ for all
$a\in\partial_{G}(A)$. Select two vertices $a$ and $b$ of $V(G)$ such that
$a\in\partial_{G}(A)$ and $(a,b)\in E(G)$. Then, for every
$u\in[V(H)-\partial_{H}(b,N[A])]$ and $v\in\partial_{H}(a,N[A])$, it is clear
that $[(b,u),(a,v)]\not\in E(G\times H)$, so $(u,v)\not\in E(H)$. This means
$u\not\in N_{H}(\partial_{H}(a,N[A]))$, that is,
$\displaystyle V(H)-\partial_{H}(b,N[A])\subseteq
V(H)-N_{H}(\partial_{H}(a,N[A])).$ (4)
If $\partial_{H}(b,N[A])=\partial_{H}(a,N[A])$, it follows from (4) that $H$
is disconnected, and so either (i) or (ii) holds.
Suppose that $\partial_{H}(b,N[A])\neq\partial_{H}(a,N[A])$ and set
$D=\partial_{H}(a,N[A])-\partial_{H}(b,N[A])$. It is easy to check that
$\displaystyle
2|D|=|\partial_{H}(a,N[A])\cup\partial_{H}(b,N[A])-\partial_{H}(a,N[A])\cap\partial_{H}(b,N[A])|.$
Since $D\subseteq H-\partial_{H}(b,N[A])$ and
$D\subseteq\partial_{G}(a,N[A])$, by (4), we have
$D\subseteq V(H)-\partial_{H}(b,N[A])\subseteq
V(H)-{N}_{H}(\partial_{H}(a,N[A]))\subseteq V(H)-{N}_{H}(D).$
So $D$ is an independent set of $H$ and
$\displaystyle N_{H}[D]$ $\displaystyle\subseteq$ $\displaystyle
D\cup[\partial_{H}(b,N[A])-\partial_{H}(a,N[A])]$ $\displaystyle=$
$\displaystyle\partial_{H}(a,N[A])\cup\partial_{H}(b,N[A])-\partial_{H}(a,N[A])\cap\partial_{H}(b,N[A]),$
which implies that
$\frac{1}{2}\geq\frac{\alpha(H)}{|V(H)|}\geq\frac{|D|}{|N_{H}[D]|}\geq\frac{1}{2}$.
Thus $\frac{\alpha(G)}{|V(G)|}=\frac{\alpha(H)}{|V(H)|}=\frac{1}{2}$. By
Corollary 2.2, $G$ and $H$ are both bipartite, so (i) holds and the proof
completed. $\Box$
###### Theorem 2.6
Let $G$ and $H$ be two non-bipartite vertex-transitive graph such that
$\frac{\alpha(H)}{|V(H)|}=\frac{\alpha(G)}{|V(G)|}$. If $G\times H$ is MIS-
normal, then $G$, $H$ and $G\times H$ are all IS-primitive.
Proof. First, suppose that $G$ is IS-imprimitive and let $A$ be an imprimitive
independent set in $G$. For any $S\in I(H)$, let
$S^{\prime}=(\overline{N}_{G}[A]\times S)\cup(A\times H)$. It is clear that
$S^{\prime}$ is an independent set of $G\times H$ and
$\displaystyle|S^{\prime}|$ $\displaystyle=$
$\displaystyle|\overline{N}_{G}[A]\alpha(H)|+|A||V(H)|=(|\overline{N}_{G}[A]|+|N_{G}[A]|)\alpha(H)$
$\displaystyle=$ $\displaystyle|V(G)|\alpha(H)=\alpha(G\times H),$
i.e., $S^{\prime}$ is a maximum independent set of $G\times H$, contradicting
the MIS-normality of $G$. Therefore, $G$ is IS-primitive. Similarly, $H$ is
also IS-primitive. By Lemma 2.5, $G\times H$ is IS-primitive. $\Box$
## 3 MIS-normality of the Products of Graphs
The following theorem is the main result on the MIS-normality of products of
vertex-transitive graphs in this paper.
###### Theorem 3.1
Let $G$ and $H$ be two vertex-transitive graphs. Suppose that there exists an
induced subgraph $G^{\prime}$ of $G$ such that $G^{\prime}\times H$ is MIS-
normal and
$\frac{\alpha(G^{\prime})}{|V(G^{\prime})|}=\frac{\alpha(G)}{|V(G)|}$. Then
either: (i) $G\times H$ is MIS-normal, or (ii)
$\frac{\alpha(G)}{|V(G)|}=\frac{\alpha(H)}{|V(H)|}$ and $G$ is IS-imprimitive,
or (iii) $\frac{\alpha(G)}{|V(G)|}<\frac{\alpha(H)}{|V(H)|}$ and $G$ is
disconnected.
Proof. If $E(H)=\emptyset$, the result is obvious, so we assume that
$E(H)\neq\emptyset$. By Lemma 2.1 and the MIS-normality of $G^{\prime}\times
H$, we have the following inequality
$\frac{\alpha(G\times H)}{|V(G)||V(H)|}\leq\frac{\alpha(G^{\prime}\times
H)}{|V(G^{\prime})||V(H)|}=\max\left\\{\frac{\alpha(G)}{|V(G)|},\frac{\alpha(H)}{|V(H)|}\right\\}\leq\frac{\alpha(G\times
H)}{|V(G)||V(H)|},$
yielding
$\displaystyle\frac{\alpha(G\times
H)}{|V(G)||V(H)|}=\frac{\alpha(G^{\prime}\times
H)}{|V(G^{\prime})||V(H)|}=\max\left\\{\frac{\alpha(G)}{|V(G)|},\frac{\alpha(H)}{|V(H)|}\right\\}.$
(5)
For every $\sigma\in\mbox{Aut}(G)$, it is clear that $\sigma(G^{\prime})\times
H$ is MIS-normal. Let $S$ be a maximum independent set of $G\times H$. By
Lemma 2.1 and (5), $S\cap(\sigma(G^{\prime})\times H)$ is a maximum
independent set of $\sigma(G^{\prime})\times H$. Clearly, for each
$a\in\partial_{G}(S)$, there is a $\sigma\in\mbox{Aut}(G)$ such that
$a\in\sigma(G^{\prime})$. We therefore have that $|\partial_{H}(a,S)|=|H|$ or
$\alpha(H)$ for each $a\in\partial_{G}(S)$. In the following we distinguish
three cases to complete the proof.
Case $1$: $|\partial_{H}(a,S)|=|V(H)|$ for every $a\in\partial_{G}(S)$. By
(5), we obtain that $|\partial_{G}(S)|=\alpha(G)$. Since $E(H)\neq\emptyset$,
$\partial_{G}(S)$ is an independent set of $G$. This implies that
$S=\partial_{G}(S)\times H$.
Case $2$: $|\partial_{H}(a,S)|=\alpha(H)$ for every $a\in\partial_{G}(S)$. By
(5), we have that $\partial_{G}(S)=G$,
$\frac{\alpha(H)}{|V(H)|}\geq\frac{\alpha(G)}{|V(G)|}$ and $\partial_{H}(a,S)$
is a maximum independent set of $H$ for every $a\in G$. Let $a$ be a fixed
vertex of $G$, and set
$C=\\{c\in G:\partial_{H}(c,S)=\partial_{H}(a,S)\\}.$
If $C=G$, then $S=G\times\partial_{H}(a,S)$. If $C\neq G$, then choose $d\in
G-C$ and $c\in C$. Since $\partial_{H}(c,S)\neq\partial_{H}(d,S)$, there
exists $u\in\partial_{H}(c,S)$ and $v\in\partial_{H}(d,S)$ such that $(u,v)\in
E(H)$ and $[(c,u),(d,v)]\not\in E(G\times H)$. This implies that $(c,d)\in
E(G)$ and thus G is disconnected.
Case $3$: $|\partial_{H}(a,S)|=|V(H)|$ and $|\partial_{H}(b,S)|=\alpha(H)$ for
some $a,b\in\partial_{G}(S)$. By (5),
$\frac{\alpha(H)}{|V(H)|}=\frac{\alpha(G)}{|V(G)|}$ and $\alpha(G\times
H)=\alpha(G)|V(H)|=\alpha(H)|V(G)|$. Set
$C=\\{c\in G:|\partial_{H}(c,S)|=|V(H)|\\}\mbox{\ and \ }D=\\{d\in
G:|\partial_{H}(d,S)|=\alpha(H)\\}.$
Since $E(H)\neq\emptyset$, it is clear that $C$ is an independent set of $G$
and $(c,d)\not\in E(G)$ for every $c\in C$ and $d\in D$. So $N_{G}[C]\subseteq
V(G)-D$. Moreover,
$|S|=\alpha(H)|V(G)|=|C||V(H)|+|D|\alpha(H).$
Thus
$\frac{|C|}{|N_{G}[C]|}\geq\frac{|C|}{|V(G)|-|D|}=\frac{\alpha(H)}{|V(H)|}=\frac{\alpha(G)}{|V(G)|}$.
By Proposition 2.3, $\frac{|C|}{|N_{G}[C]|}=\frac{\alpha(G)}{|V(G)|}$, that
is, $G$ is IS-imprimitive.
This completes the proof. $\Box$
The following Corollary solves Problem 1.2 in a bit more general setting.
###### Corollary 3.2
Let $G$ be a vertex-transitive, non-bipartite graph. If $G^{2}$ is MIS-normal,
then $G^{n}$ is also MIS-normal and IS-primitive for all $n\geq 3$.
Proof. We prove by induction on $n$. Since $G^{2}$ is MIS-normal, by Theorem
2.6, $G$ and $G^{2}$ are both IS-primitive. Assume that $G^{d}$ is MIS-normal
and IS-primitive for all $d=2,\ldots,n-1$. We now prove that $G^{n}$ is MIS-
normal and IS-primitive. Note that $G^{n}=G^{2}\times G^{n-2}$. Let
$G^{\prime}$ be some subgraph of $G^{2}$ that is isomorphic to $G$, for
instance, the subgraph induced by the set of vertices $\\{(u,u):u\in V(G)\\}$.
It is clear that
$\frac{\alpha(G^{\prime})}{|V(G^{\prime})|}=\frac{\alpha(G)}{|V(G)|}=\frac{\alpha(G^{2})}{|V(G^{2})|}$
and $G^{\prime}\times G^{n-2}$ is isomorphic to $G^{n-1}$. Thus by assumption,
$G^{\prime}\times G^{n-2}$ is MIS-normal. By Theorem 3.1 and Theorem 2.6, it
is easy to see that $G^{n}$ is MIS-normal and IS-primitive. This completes the
proof. $\Box$
Acknowledgement The author is greatly indebted to the anonymous referees for
giving useful comments and suggestions that have considerably improved the
manuscript. He is grateful also for many valuable discussions with Professor
J. Wang and Professor C.J. Zhou.
## References
* [1] M.O. Albertson and K.L. Collins, Homomorphisms of $3$-chromatic graphs, Discrete Math., 54 (1985) 127-132.
* [2] B. Larose and C. Tardif, Projectivity and independent sets in powers of graphs, J. Graph Theory, 40 (2002) 162-171.
* [3] C. Tardif, Graph products and the chromatic difference sequence of vertex-transitive graphs, Discrete Math., 185 (1998) 193-200.
* [4] P.J. Cameron and C.Y. Ku, Intersecting families of permutations, European J. Comb., 24 (2003) 881-890.
* [5] V.P. Mario and J. Vera, Independent and coloring properties of direct products of some vertex-transitive graphs, Discrete Math., 306 (2006) 2275-2281.
* [6] P. Frankl, An Erdős-Ko-Rado Theorem for direct products, European J. Combin., 17 (1996) 727-730.
* [7] P.K. Jha and S. Klavz̆ar, Independence in direct-product graphs, Ars Combin., 50 (1998) 53-60.
* [8] R. Ahlswede, H. Aydinian and L.H. Khachatrian, The intersection theorem for direct products, European J. Combin., 19 (1998) 649-661.
|
arxiv-papers
| 2010-07-05T10:42:45 |
2024-09-04T02:49:11.405131
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhang Huajun",
"submitter": "Zhang Huajun",
"url": "https://arxiv.org/abs/1007.0655"
}
|
1007.0795
|
and
††thanks: Corresponding author.
# Cross-intersecting families and primitivity of symmetric systems
Jun Wang jwang@shnu.edu.cn Huajun Zhang huajunzhang@zjnu.cn Department of
Mathematics, Shanghai Normal University, Shanghai 200234, P.R. China
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, P.R.
China
###### Abstract
Let $X$ be a finite set and $\mathfrak{p}\subseteq 2^{X}$, the power set of
$X$, satisfying three conditions: (a) $\mathfrak{p}$ is an ideal in $2^{X}$,
that is, if $A\in\mathfrak{p}$ and $B\subset A$, then $B\in\mathfrak{p}$; (b)
For $A\in 2^{X}$ with $|A|\geq 2$, $A\in\mathfrak{p}$ if
$\\{x,y\\}\in\mathfrak{p}$ for any $x,y\in A$ with $x\neq y$; (c)
$\\{x\\}\in\mathfrak{p}$ for every $x\in X$. The pair $(X,\mathfrak{p})$ is
called a symmetric system if there is a group $\Gamma$ transitively acting on
$X$ and preserving the ideal $\mathfrak{p}$. A family
$\\{A_{1},A_{2},\ldots,A_{m}\\}\subseteq 2^{X}$ is said to be a
cross-$\mathfrak{p}$-family of $X$ if $\\{a,b\\}\in\mathfrak{p}$ for any $a\in
A_{i}$ and $b\in A_{j}$ with $i\neq j$. We prove that if $(X,\mathfrak{p})$ is
a symmetric system and $\\{A_{1},A_{2},\ldots,A_{m}\\}\subseteq 2^{X}$ is a
cross-$\mathfrak{p}$-family of $X$, then
$\sum_{i=1}^{m}|{A}_{i}|\leq\left\\{\begin{array}[]{cl}|X|&\hbox{if
$m\leq\frac{|X|}{\alpha(X,\,\mathfrak{p})}$,}\\\
m\,\alpha(X,\,\mathfrak{p})&\hbox{if
$m\geq\frac{|X|}{\alpha{(X,\,\mathfrak{p})}}$,}\end{array}\right.$
where $\alpha(X,\,\mathfrak{p})=\max\\{|A|:A\in\mathfrak{p}\\}$. This
generalizes Hilton’s theorem on cross-intersecting families of finite sets,
and provides analogs for cross-$t$-intersecting families of finite sets,
finite vector spaces and permutations, etc. Moreover, the primitivity of
symmetric systems is introduced to characterize the optimal families.
###### keywords:
intersecting family, cross-intersecting family, symmetric system, Erdős-Ko-
Rado theorem
MSC: 05D05, 06A07
## 1 Introduction
A family $\mathcal{A}$ of sets is said to be intersecting if $A\cap
B\neq\emptyset$ for any $A,B\in\mathcal{A}$. A classical result on
intersecting families is due to Erdős, Ko and Rado, which says that if
$\mathcal{A}$ is an intersecting family consisting of $k$-element subsets of
an $n$-element set with $n\geq 2k$, then $|\mathcal{A}|\leq{n-1\choose k-1}$,
and if $n>2k$, equality holds if and only if every subset in $\mathcal{A}$
contains a fixed element.
The Erdős-Ko-Rado theorem has many generalizations, analogs and variations.
First, the notion of intersection is generalized to $t$-intersection, and
finite sets are analogous to finite vector spaces, permutations and other
mathematical objects. Second, intersecting families are generalized to cross-
intersecting families:
$\mathcal{A}_{1},\mathcal{A}_{2},\ldots,\mathcal{A}_{m}$ are said to be cross-
intersecting if $A\cap B\neq\emptyset$ for any $A\in\mathcal{A}_{i}$ and
$B\in\mathcal{A}_{j}$, $i\neq j$. Clearly, if
$\mathcal{A}_{1}=\mathcal{A}_{2}=\ldots=\mathcal{A}_{m}=\mathcal{A}$, then
$\mathcal{A}$ is an intersecting family. Combining the two points of view, we
may consider the cross-$t$-intersecting families over finite vector spaces,
permutations, etc.
A nice result on cross-intersecting families is given by Hilton [19] as
follows.
###### Theorem 1.1
(Hilton [19]) Let $\mathcal{A}_{1},\mathcal{A}_{2},\ldots,\mathcal{A}_{m}$ be
cross-intersecting families of $k$-element subsets of an $n$-element set $X$
with $\mathcal{A}_{1}\neq\emptyset$. If $k\leq n/2$, then
$\displaystyle\sum_{i=1}^{m}|\mathcal{A}_{i}|\leq\left\\{\begin{array}[]{cl}\binom{n}{k},&\hbox{if
$m\leq\frac{n}{k}$;}\\\ m\binom{n-1}{k-1},&\hbox{if
$m\geq\frac{n}{k}$.}\end{array}\right.$ (3)
Unless $m=2=n/k$, the bound is attained if and only if one of the following
holds:
1. (i)
$m<n/k$ and $\mathcal{A}_{1}=\\{A\subset X:|A|=k\\}$, and
$\mathcal{A}_{2}=\cdots=\mathcal{A}_{m}=\emptyset$;
2. (ii)
$m>n/k$ and
$|\mathcal{A}_{1}|=|\mathcal{A}_{2}|=\ldots=|\mathcal{A}_{m}|=\binom{n-1}{k-1}$;
3. (iii)
$m=n/k$ and $\mathcal{A}_{1},\mathcal{A}_{2},\ldots,\mathcal{A}_{m}$ are as in
(i) or (ii).
Recently, Borg gives a simple proof of the above theorem [7], and generalizes
it to labeled sets [4] and permutations [8]. Inspired by his proofs we shall
present a general result on cross-intersecting, or cross-$t$-intersecting
families of finite sets, finite vector spaces, permutations, etc. To do this,
we introduce a general definition.
Let $X$ be a finite set and $\mathfrak{p}\subseteq 2^{X}$, the power set of
$X$, satisfying three conditions as follows:
1. * (a)
$\mathfrak{p}$ is an ideal in $2^{X}$, that is, if $A\in\mathfrak{p}$ and
$B\subset A$, then $B\in\mathfrak{p}$;
* (b)
For $A\in 2^{X}$ with $|A|\geq 2$, $A\in\mathfrak{p}$ if
$\\{x,y\\}\in\mathfrak{p}$ for any $x,y\in A$ with $x\neq y$;
* (c)
$\\{x\\}\in\mathfrak{p}$ for every $x\in X$.
Note that condition (a) is essential and (c) is to avoid trivial cases. If
ignore conditions (b) and (c), the pair $(X,\mathfrak{p})$ is an (abstract)
simplicial complex in topology, or a hereditary family in extremal set theory
(see e.g. [12, p.86] or [6]). If ignore (b), $\mathfrak{p}$ is called a full
hereditary family in [12, p.86]. Condition (b) is not redundant in most
discussions on extremal combinatorics, and is necessary in our argument.
Clearly, $\mathfrak{p}$ defines a binary relation “$\sim_{\mathfrak{p}}$” on
$X$: $x\sim_{\mathfrak{p}}y$ if and only if $\\{x,y\\}\in\mathfrak{p}$ for any
$x,y\in X$. This relation is reflexive and symmetric, i.e.,
$x\sim_{\mathfrak{p}}x$ for every $x\in X$, and $x\sim_{\mathfrak{p}}y$
implies $y\sim_{\mathfrak{p}}x$. Conversely, given a reflexive and symmetric
binary relation “$\sim$” on $X$, we can get an ideal $\mathfrak{p}$ in
$2^{X}$: $A\subset X$ is in $\mathfrak{p}$ if $a\sim b$ for any $a,b\in A$.
Moreover, $\mathfrak{p}$ also defines a property on $2^{X}$: a subset $A$ of
$X$ has the property $\mathfrak{p}$ if $A\in\mathfrak{p}$. Therefore, we call
the pair $(X,\mathfrak{p})$ a $\mathfrak{p}$-system, or a system, for short.
An element of $\mathfrak{p}$ is also called a $\mathfrak{p}$-subset of $X$. A
family $\\{A_{1},A_{2},\ldots,A_{m}\\}\subseteq 2^{X}$ is said to be a
cross-$\mathfrak{p}$-family of $X$ if $\\{a,b\\}\in\mathfrak{p}$ for any $a\in
A_{i}$ and $b\in A_{j}$ with $i\neq j$. By definition we see that if
$\\{A_{1},A_{2},\ldots,A_{m}\\}$ is a cross-$\mathfrak{p}$-family and
$A_{1}=A_{2}=\cdots=A_{m}=A$, then $A$ is a $\mathfrak{p}$-subset. Write
$\alpha(X,\mathfrak{p}):=\max\\{|A|:A\in\mathfrak{p}\\}$
and
$\alpha_{m}(X,\mathfrak{p}):=\max\left\\{\sum_{i=1}^{m}|A_{i}|:\\{A_{1},A_{2},\ldots,A_{m}\\}\
\mbox{is a cross-$\mathfrak{p}$-family}\right\\}.$
A cross-$\mathfrak{p}$-family $\\{A_{1},A_{2},\ldots,A_{m}\\}$ is said to be
optimal if $\sum_{i=1}^{m}|A_{i}|=\alpha_{m}(X,\mathfrak{p})$.
We call a system $(X,\mathfrak{p})$ symmetric if there is a group $\Gamma$
transitively acting on $X$ and preserving the property $\mathfrak{p}$, i.e.,
for every pair $a,b\in X$ there is a $\gamma\in\Gamma$ such that
$b=\gamma(a)$, and $A\in\mathfrak{p}$ implies $\delta(A)\in\mathfrak{p}$ for
every $\delta\in\Gamma$. In this case we say that the group $\Gamma$
transitively acts on $(X,\mathfrak{p})$.
Two typical examples of symmetric systems are as follows.
###### Example 1.2
For a positive integer $n$, let $[n]$ denote the set $\\{1,2,\ldots,n\\}$. By
$\mathcal{C}_{n}^{k}$ we denote the set of all $k$-element subsets of $[n]$,
as known for $\binom{[n]}{k}$ in many literatures. Then
$|\mathcal{C}_{n}^{k}|={n\choose k}$. A subset $\mathcal{A}$ of
$\mathcal{C}_{n}^{k}$ is said to be a $t$-intersecting family if $|A\cap
B|\geq t$ for any $A,B\in\mathcal{A}$, where $1\leq t\leq k$. For convenience,
we regard the empty set as a $t$-intersecting family. Let $\mathfrak{i}_{t}$
be the collection of all $t$-intersecting families in $\mathcal{C}_{n}^{k}$.
Then, it is clear that $\mathfrak{i}_{t}$ is an ideal of the power set of
$\mathcal{C}_{n}^{k}$, and satisfies condition (b). When $t=1$,
$\mathfrak{i}_{t}$ is abbreviated as $\mathfrak{i}$. The Erdős-Ko-Rado theorem
and Theorem 1.1 say that $\alpha(\mathcal{C}_{n}^{k},\mathfrak{i})={n-1\choose
k-1}$ and $\alpha_{m}(\mathcal{C}_{n}^{k},\mathfrak{i})=\max\left\\{{n\choose
k},m{n-1\choose k-1}\right\\}$ for $n\geq 2k$, respectively. In fact, Erdős,
Ko and Rado [13] also proved
$\alpha(\mathcal{C}_{n}^{k},\mathfrak{i}_{t})=\binom{n-t}{k-t}$ for $t>1$ and
$n\geq n_{0}(k,t)$, a sufficiently large positive integer depending on $k$ and
$t$. The smallest $n_{0}(k,t)=(k-t+1)(t+1)$ was determined by Frankl [14] for
$t\geq 15$ and subsequently determined by Wilson [27] for all $t$. It is well
known that the symmetric group $S_{n}$ transitively acts on
$\mathcal{C}_{n}^{k}$ in a natural way, and preserves $\mathfrak{i}_{t}$.
Therefore, $(\mathcal{C}_{n}^{k},\mathfrak{i}_{t})$ is symmetric.
###### Example 1.3
Let $\mathcal{L}_{n,k}(q)$ denote the set of all $k$-dimensional subspaces of
an $n$-dimensional vector space over a $q$-element field. Then
$|\mathcal{L}_{n,k}(q)|={n\atopwithdelims[
]k}=\frac{\\{n\\}!}{\\{k\\}!\\{n-k\\}!}$ where $\\{k\\}=1+q+\cdots+q^{k-1}$
and $\\{k\\}!=\\{k\\}\\{k-1\\}\cdots\\{1\\}$. A subset $\mathcal{A}$ of
$\mathcal{L}_{n,k}(q)$ is said to be a $t$-intersecting family if $\dim(A\cap
B)\geq t$ for any $A,B\in\mathcal{A}$, where $1\leq t\leq k$. We still use
$\mathfrak{i}_{t}$ to denote the collection of all $t$-intersecting families
in $\mathcal{L}_{n,k}(q)$, and abbreviate $\mathfrak{i}_{1}$ as
$\mathfrak{i}$. That
$\alpha(\mathcal{L}_{n,k}(q),\mathfrak{i})={n-1\atopwithdelims[ ]k-1}$ was
first established by Hsieh [18] for $k<n/2$, and by Greene and Kleitman [16]
for $k|n$. For $t\geq 2$, Frankl and Wilson [15] proved that
$\alpha(\mathcal{L}_{n,k}(q),\mathfrak{i}_{t})=\max\left\\{{n-t\atopwithdelims[
]k-t},{2k-t\atopwithdelims[ ]k}\right\\}$ for $n\geq 2k-t$. Analogously to
$(\mathcal{C}_{n}^{k},\mathfrak{i}_{t})$, the general linear group $GL(n,q)$
transitively acts on $\mathcal{L}_{n,k}(q)$ and preserves $\mathfrak{i}_{t}$.
Therefore, $(\mathcal{L}_{n,k}(q),\mathfrak{i}_{t})$ is also symmetric.
To our knowledge, there is no information on
$\alpha_{m}(\mathcal{C}_{n}^{k},\mathfrak{i}_{t})$ for $t>1$ and
$\alpha_{m}(\mathcal{L}_{n,k}(q),\mathfrak{i}_{t})$ for $t\geq 1$.
In this paper we shall generalize Theorem 1.1 to all symmetric systems
$(X,\mathfrak{p})$ up to $\alpha(X,\mathfrak{p})$. The main result will be
presented in the next section. To characterize the optimal
cross-$\mathfrak{p}$-families we introduce the primitivity of the symmetric
systems, and give its main characters in Section 3. As applications of results
in Section 3, we prove in Section 4 that the symmetric systems defined on
finite sets, finite vector spaces and symmetric groups are all primitive
except a few trivial cases.
## 2 Cross-intersecting families of symmetric systems
Given a system $(X,\mathfrak{p})$, we can construct a simple graph, written as
$G(X,\mathfrak{p})$, whose vertex set is $X$, and $\\{a,b\\}$ is an edge if
$\\{a,b\\}\not\in\mathfrak{p}$. Then every subset of $X$ in $\mathfrak{p}$
corresponds to an independent set of $G(X,\mathfrak{p})$. Conversely, given a
simple graph $G$, we obtain a system $(X(G),\mathfrak{p}(G))$, where $X(G)$ is
the vertex set $V(G)$ of $G$ and $\mathfrak{p}(G)$ consists of all independent
sets of $G$. It is clear that $\alpha(X(G),\mathfrak{p}(G))=\alpha(G)$, the
independence number of $G$.
By $I(X,\mathfrak{p})$ we denote the set of all maximal-sized
$\mathfrak{p}$-subsets of $X$. Similarly, for a graph $G$, let $I(G)$ denote
the set of all maximal-sized independent sets of $G$. For $B\subseteq V(G)$,
let $G[B]$ denote the induced subgraph of $G$ by $B$.
The notations introduced below have graph-theoretic intuition.
Let $(X,\mathfrak{p})$ be a $\mathfrak{p}$-system. For $B\subseteq X$, we
abbreviate $\alpha(B,\mathfrak{p}\cap 2^{B})$ as $\alpha(B,\mathfrak{p})$.
Clearly, $\alpha(B,\mathfrak{p})$ equals $\alpha(G[B])$, where
$G=G(X,\mathfrak{p})$. For $A\subseteq X$, set
$N_{X,\mathfrak{p}}[A]=A\cup\\{b\in X:\mbox{ $\\{a,b\\}\not\in\mathfrak{p}$
for some $a\in A$ }\\}$
and
$\bar{N}_{X,\mathfrak{p}}[A]=X-N_{X,\mathfrak{p}}[A].$
If there is no possibility of confusion, we abbreviate $N_{X,\mathfrak{p}}[A]$
as $N[A]$. From definition we see that $N[\emptyset]=\emptyset$; $N[A]=X$ if
$A\in I(X,\mathfrak{p})$; if both $B\subseteq A$ and $C\subseteq\bar{N}[A]$
are in $\mathfrak{p}$, then $B\cup C\in\mathfrak{p}$.
We call $(X,\mathfrak{p})$ connected (disconnected) if the graph
$G(X,\mathfrak{p})$ is connected (disconnected). By definition we see that
$(X,\mathfrak{p})$ is disconnected if and only if there is a proper subset
$A\subset X$ such that $\bar{N}[A]=X-A$, and, $(X,\mathfrak{p})$ is symmetric
if and only if $G(X,\mathfrak{p})$ is vertex-transitive.
In the context of vertex-transitive graphs, the “No- Homomorphism” lemma is
useful to get bounds on the size of independent sets.
###### Lemma 2.1
( Albertson and Collins [1]) Let $G$ and $H$ be two graphs such that $G$ is
vertex-transitive and there exists a homomorphism $\phi:H\mapsto G$. Then
$\frac{\alpha(G)}{|V(G)|}\leq\frac{\alpha(H)}{|V(H)|}$, and equality holds if
and only if for each $I\in I(G)$, $\phi^{-1}(I)\in I(H)$.
In the above lemma, by taking $H$ as an induced subgraph of $G$ and $\phi$ as
the embedding mapping, we obtain the following theorem, which is more
convenient in our argument.
###### Theorem 2.2
(Cameron and Ku [10]) Let $G$ be a vertex-transitive graph and $B$ a subset of
$V(G)$. Then any independent set $S$ in $G$ satisfies that
$\frac{|S|}{|V(G)|}\leq\frac{\alpha(G[B])}{|B|}$, equality implies that
$|S\cap B|=\alpha(G[B])$.
In [28], the second author of this paper proved Lemma 2.3 and Theorem 3.2
below in terms of graph theory. He also introduced the concept of imprimitive
independent sets of a vertex-transitive graph. For completeness we restate
them in terms of symmetric systems and provide proofs for them.
###### Lemma 2.3
Let $(X,\mathfrak{p})$ be a symmetric system. Then
$\frac{|A|}{|N[A]|}\leq\frac{\alpha(X,\mathfrak{p})}{|X|}$ for an arbitrary
$\mathfrak{p}$-subset $A$ of $X$. Equality implies that $|S\cap N[A]|=|A|$ for
every $S\in I(X,\mathfrak{p})$, and
$\frac{\alpha(\bar{N}[A]\\!,\,\mathfrak{p})}{|\bar{N}[A]|}=\frac{\alpha(X\\!,\,\mathfrak{p})}{|X|}$.
Proof. Let $C$ be a maximal-sized $\mathfrak{p}$-subset of $\bar{N}[A]$.
Clearly, $A\cup C$ is a $\mathfrak{p}$-subset of $X$ and
$\frac{|A\cup
C|}{|X|}=\frac{|A|+\alpha(\bar{N}[A],\mathfrak{p})}{|N[A]|+|\bar{N}[A]|}\leq\frac{\alpha(X,\mathfrak{p})}{|X|}.$
Since
$\frac{\alpha(\bar{N}[A],\,\mathfrak{p})}{|\bar{N}[A]|}\geq\frac{\alpha(X,\,\mathfrak{p})}{|X|}$
by Theorem 2.2, $\frac{|A|}{|N[A]|}\leq\frac{\alpha(X,\,\mathfrak{p})}{|X|}$.
Equality implies that
$\frac{\alpha(\bar{N}[A],\,\mathfrak{p})}{|\bar{N}[A]|}=\frac{\alpha(X,\,\mathfrak{p})}{|X|}$
and $\alpha(X,\mathfrak{p})=\alpha(\bar{N}[A],\mathfrak{p})+|A|$. Again by
Theorem 2.2, we have that
$|S\cap\bar{N}[A]|=|\alpha(\bar{N}[A],\mathfrak{p})|$ and $|S|=|S\cap
N[A]|+|S\cap\bar{N}[A]|$ for every $S\in I(X,\mathfrak{p})$. Therefore,
$|S\cap N[A]|=|A|$ for every $S\in I(X,\mathfrak{p})$, completing the proof. ∎
In [28], a graph $G$ is called IS-imprimitive (independent-set-imprimitive) if
there is an independent set $A$ of $G$ such that $|A|<\alpha(G)$ and
$\frac{|A|}{|N[A]|}=\frac{\alpha(G)}{|V(G)|}$, and $A$ is called an
imprimitive independent set of $G$. In any other case, $G$ is called IS-
primitive. In this paper, we say a system $(X,\mathfrak{p})$ is
$\mathfrak{p}$-imprimitive ($\mathfrak{p}$-primitive) if the graph
$G(X,\mathfrak{p})$ is IS-imprimitive (IS-primitive); a $\mathfrak{p}$-subset
$A$ is called imprimitive if $A$ is an imprimitive independent set of
$G(X,\mathfrak{p})$. From definition we see that a disconnected symmetric
system $(X,\mathfrak{p})$ is $\mathfrak{p}$-imprimitive and hence a
${\mathfrak{p}}$-primitive symmetric system $(X,\mathfrak{p})$ is connected.
We now contribute to $\alpha_{m}(X,\mathfrak{p})$. Note that in a series of
papers [4, 7, 8, 9] Borg determined this value for various cross-intersecting
families. An important step in his proofs was inequality (4) below he
established for some special intersecting families. We find that the
inequality for $\mathfrak{p}$-subsets in symmetric systems is a consequence of
Theorem 2.2, stated as follows.
###### Corollary 2.4
Let $(X,\mathfrak{p})$ be a symmetric system, and let $A$ be a
$\mathfrak{p}$-subset of $X$. Then
$|A|+\frac{\alpha(X,\mathfrak{p})}{|X|}|\bar{N}[A]|\leq\alpha(X,\mathfrak{p}).$
(4)
Equality holds if and only if $A=\emptyset$ or $|A|=\alpha(X,\mathfrak{p})$ or
$A$ is an imprimitive $\mathfrak{p}$-subset.
Proof. If $A=\emptyset$ or $|A|=\alpha(X,\mathfrak{p})$, equality trivially
holds. Suppose that $0<|A|<\alpha(X,\mathfrak{p})$ and $B$ is a maximal-sized
$\mathfrak{p}$-subset in $\bar{N}[A]$, that is,
$|B|=\alpha(\bar{N}[A],\mathfrak{p})$. Then $A\cup B$ is also a
$\mathfrak{p}$-subset of $X$, so $|A|+|B|\leq\alpha(X,\mathfrak{p})$, and
Theorem 2.2 implies that
$\frac{|B|}{|\bar{N}[A]|}\geq\frac{\alpha(X,\mathfrak{p})}{|X|}$. Therefore,
$|A|+\frac{\alpha(X,\mathfrak{p})}{|X|}|\bar{N}[A]|\leq|A|+|B|\leq\alpha(X,\mathfrak{p}).$
If
$\alpha(X,\mathfrak{p})=|A|+\frac{\alpha(X,\mathfrak{p})}{|X|}|\bar{N}[A]|=|A|+\frac{\alpha(X,\mathfrak{p})}{|X|}(|X|-|N[A]|)$,
then $\frac{|A|}{|N[A]|}=\frac{\alpha(X,\mathfrak{p})}{|X|}$, i.e., $A$ is an
imprimitive $\mathfrak{p}$-subset. ∎
The following theorem is the main result of this paper.
###### Theorem 2.5
Let $(X,\mathfrak{p})$ be a connected symmetric system, and let
$\\{{A}_{1},{A}_{2},\ldots,{A}_{m}\\}$ be a cross-$\mathfrak{p}$-family over
$X$ with $A_{1}\neq\emptyset$. Then
$\sum_{i=1}^{m}|{A}_{i}|\leq\left\\{\begin{array}[]{cl}|X|&\hbox{if
$m\leq\frac{|X|}{\alpha(X,\,\mathfrak{p})}$;}\\\
m\,\alpha(X,\,\mathfrak{p})&\hbox{if
$m\geq\frac{|X|}{\alpha{(X,\,\mathfrak{p})}}$,}\end{array}\right.$
and the bound is attained if and only if one of the following holds:
1. (i)
$m<\frac{|X|}{\alpha(X,\,\mathfrak{p})}$ and $A_{1}=X$,
$A_{2}=\ldots=A_{m}=\emptyset$,
2. (ii)
$m>\frac{|X|}{\alpha(X,\,\mathfrak{p})}$ and ${A}_{1}=\ldots={A}_{m}=I\in
I(X,\mathfrak{p})$,
3. (iii)
$m=\frac{|X|}{\alpha(X,\,\mathfrak{p})}$ and either
${A}_{1},{A}_{2},\ldots,{A}_{m}$ are as in (i) or (ii), or there is an
imprimitive $\mathfrak{p}$-subset $A$ such that $A\subseteq{A}_{i}$,
$i=1,2,\ldots,m$, and
$\\{A_{1}^{\prime},A_{2}^{\prime},\ldots,A_{m}^{\prime}\\}$ is a
cross-$\mathfrak{p}$-family and a partition of $\bar{N}[A]$, where
$A_{i}^{\prime}=A_{i}-A$, $i=1,2\ldots,m$.
Proof. Following Borg’s notation in [7, 8, 9], write $A_{i}^{*}=\\{a\in
A_{i}:\\{a,b\\}\in\mathfrak{p}\ \mbox{for every $b\in A_{i}$}\\}$,
$A_{i}^{\prime}=A_{i}-A_{i}^{*}$,
$A^{*}=\displaystyle{\cup_{i=1}^{m}}{A}_{i}^{*}$ and
$A^{\prime}=\displaystyle{\cup_{i=1}^{m}}{A}_{i}^{\prime}$. It is clear that
$A^{*}$ is a $\mathfrak{p}$-subset and ${A}^{\prime}\subseteq\bar{N}[A^{*}]$.
From definition it follows that $A_{i}\cap A_{j}\subseteq A_{i}^{*}\cap
A_{j}^{*}$, therefore $A_{i}^{\prime}\cap A_{j}^{\prime}=\emptyset$ for $i\neq
j$, thus $|{A}^{\prime}|=\sum_{i=1}^{m}|{A}^{\prime}_{i}|$. By Corollary 2.4
we have that
$\displaystyle\sum_{i=1}^{m}|{A}_{i}|$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{m}|{A}^{\prime}_{i}|+\sum_{i=1}^{m}|{A}_{i}^{*}|\leq|{A}^{\prime}|+m|A^{*}|\leq|\bar{N}[A^{*}]|+m|A^{*}|$
$\displaystyle=$
$\displaystyle\frac{|X|}{\alpha(X,\mathfrak{p})}\left(\frac{\alpha(X,\mathfrak{p})}{|X|}|\bar{N}[A^{*}]|+|A^{*}|\right)+\left(m-\frac{|X|}{\alpha(X,\mathfrak{p})}\right)|A^{*}|$
$\displaystyle\leq$
$\displaystyle|X|+\left(m-\frac{|X|}{\alpha(X,\mathfrak{p})}\right)|A^{*}|.$
If $m<\frac{|X|}{\alpha(X,\mathfrak{p})}$, then
$\sum_{i=1}^{m}|{A}_{i}|\leq|X|$, and equality implies $A^{*}=\emptyset$,
hence ${A_{i}}={A_{i}}^{\prime}$ for every $i\in[m]$, and we thus have that
the corresponding graph $G(X,\mathfrak{p})$ is a union of the induced
subgraphs $G(X,\mathfrak{p})[A_{i}^{\prime}]$’s. Then, the connectivity of
$(X,\mathfrak{p})$ yields that one of them is $X$ and the others are empty, as
(i).
If $m>\frac{|X|}{\alpha(X,\mathfrak{p})}$, then $\sum_{i=1}^{m}|{A}_{i}|\leq
m\,\alpha(X,\mathfrak{p})$ and equality implies that
${A}_{1}^{*}=\cdots=A_{m}^{*}=A^{*}$ and $|A^{*}|=\alpha(X,\mathfrak{p})$, as
(ii).
If $m=\frac{|X|}{\alpha(X,\mathfrak{p})}$, then
$\sum_{i=1}^{m}|{A}_{i}|\leq|X|$, and equality implies that
${A}_{1}^{*}=\cdots=A_{m}^{*}=A^{*}$ and
$\frac{\alpha(X,\mathfrak{p})}{|X|}|\bar{N}[A^{*}]|+|A^{*}|=\alpha(X,\mathfrak{p})$.
Then Corollary 2.4 implies that $|A^{*}|=0$ or $|A|=\alpha(X,\mathfrak{p})$ or
$A^{*}$ is an imprimitive $\mathfrak{p}$-subset. In the last case,
$\\{A_{1}^{\prime},A_{2}^{\prime},\dots,A_{m}^{\prime}\\}$ is a
cross-$\mathfrak{p}$-family, and a partition of $\bar{N}[A^{*}]$. ∎
From the above theorem we see that if $(X,\mathfrak{p})$ is symmetric and
$\mathfrak{p}$-primitive (hence connected), then $\alpha_{m}(X,\mathfrak{p})$
is uniquely determined by $\alpha(X,\mathfrak{p})$, i.e.,
$\alpha_{m}(X,\mathfrak{p})=\max\left\\{|X|,m\,\alpha(X,\mathfrak{p})\right\\},$
and an optimal cross-$\mathfrak{p}$-family is one of the forms
$\\{X,\emptyset,\ldots,\emptyset\\}$ and $\\{A,A,\ldots,A\\}$ where
$A\in\mathfrak{p}$ with $|A|=\alpha(X,\mathfrak{p})$.
For the $(X,\mathfrak{p})$ dealt with in this field, however,
$\alpha(X,\mathfrak{p})$ is usually well known, and the symmetric property of
$(X,\mathfrak{p})$ is easy to verify. So we concentrate on the primitivity of
symmetric systems in the next two sections.
## 3 Primitivity of symmetric systems
This concept comes from permutation groups. Let $X$ be a set, and $\Gamma$ a
group transitively acting on $X$. Then $\Gamma$ is said to be imprimitive on
$X$ if it preserves a nontrivial partition of $X$, called a block system, each
element of which is called a block. In any other case $\Gamma$ is primitive on
$X$. More precisely, $\Gamma$ is imprimitive on $X$ if there is nontrivial
partition $X=\cup_{i=1}^{k}X_{i}$ such that $\gamma(X_{i})$ is a block of the
partition for every $\gamma\in\Gamma$ and $i=1,2,\ldots,k$. Here
$\gamma(X_{i})$ denotes the set $\\{\gamma(x):x\in X_{i}\\}$.
A classical result on the primitivity of group actions is the following
theorem (cf. [20, Theorem 1.12]).
###### Theorem 3.1
Suppose that a group $\Gamma$ transitively acts on $X$. Then $\Gamma$ is
primitive on $X$ if and only if for each $a\in X$, $\Gamma_{a}$ is a maximal
subgroup of $\Gamma$. Here $\Gamma_{a}=\\{\gamma\in\Gamma:\gamma(a)=a\\}$, the
stabilizer of $a\in X$.
The following theorem explains why a symmetric system is called primitive or
imprimitive.
###### Theorem 3.2
Let $(X,\mathfrak{p})$ be an imprimitive symmetric system, $A$ a maximal-sized
imprimitive $\mathfrak{p}$-subset of $X$, $D=X-N[A]$, and let $\Gamma$ be the
group transitively acting on $(X,\mathfrak{p})$. Then
$\frac{\alpha(D\\!,\,\mathfrak{p})}{|D|}=\frac{\alpha(X\\!,\,\mathfrak{p})}{|X|}$
and $\\{\sigma(D):\sigma\in\Gamma\\}$ forms a partition of $X$.
Proof. First, suppose that $A$ and $B$ are two imprimitive
$\mathfrak{p}$-subsets of $X$, and write $C=A\cup(B-N[A])$. We claim that $C$
is a $\mathfrak{p}$-subset satisfying $N[C]=N[A]\cup N[B]$ and
$\frac{|C|}{|N[C]|}=\frac{\alpha(X,\mathfrak{p})}{|X|}$.
To prove this claim we write $N[A]\cup N[B]=M$. From definition it is easily
seen that $C$ is also a $\mathfrak{p}$-subset and $N[C]\subseteq M$. Since
$\frac{|B|}{|N[B]|}=\frac{\alpha(X,\mathfrak{p})}{|X|}$, by Lemma 2.3 we have
that $|S\cap N[B]|=|B|$ for all $S\in I(X,\mathfrak{p})$. So, $B\cup(S-N[B])$
is also a maximal-sized $\mathfrak{p}$-subset of $X$ for every $S\in
I(X,\mathfrak{p})$. By repeating this process for the maximal-sized
$\mathfrak{p}$-subset $B\cup(S-N[B])$ and the imprimitive
$\mathfrak{p}$-subset $A$ we have that
$\displaystyle A\cup((B\cup(S-N[B]))-N[A])$ $\displaystyle=$ $\displaystyle
A\cup(B-N[A])\cup((S-N[B])-N[A])=C\cup(S-M)$
is also a maximal-sized $\mathfrak{p}$-subset of $X$, which implies that
$|S\cap M|=|C|$ for every $S\in I(X,\mathfrak{p})$. Given a $u\in X$, suppose
there are $r$ maximal-sized $\mathfrak{p}$-subsets containing $u$. Since
$(X,\mathfrak{p})$ is symmetric, it is easily seen that the number $r$ is
independent on the choice of $u$. Let us count pairs $(x,S)$ with $x\in M\cap
S,\ S\in I(X,\mathfrak{p})$, in two ways. Since $|M\cap S|=|C|$ for every
$S\in I(X,\mathfrak{p})$, the number of the pairs is clearly equal to
$|C||I(X,\mathfrak{p})|$. On the other hand, for each $x\in M$ there are $r$
$S$’s in $I(X,\mathfrak{p})$ with $x\in S$. So the number is also equal to
$r|M|$, proving $r|M|=|C||I(X,\mathfrak{p})|$. Similarly, by counting pairs
$(x,S)$ with $x\in S\in I(X,\mathfrak{p})$ in two ways we obtain
$r|X|=\alpha(X,\mathfrak{p})|I(X,\mathfrak{p})|$. Combining the above two
equalities gives $\frac{|C|}{|M|}=\frac{\alpha(X,\mathfrak{p})}{|X|}$. Thus,
by Lemma 2.3 we have that
$\frac{\alpha(X,\mathfrak{p})}{|X|}\geq\frac{|C|}{|N[C]|}\geq\frac{|C|}{|M|}=\frac{\alpha(X,\mathfrak{p})}{|X|}.$
Hence $N[C]=M$ and $\frac{|C|}{|N[C]|}=\frac{\alpha(X,\mathfrak{p})}{|X|}$,
proving our claim.
We now close the proof of the theorem. Let $A$ be a maximal-sized imprimitive
$\mathfrak{p}$-subset of $X$. From definition it follows that
$N[\sigma(A)]=\sigma(N[A])$ for all $\sigma\in\Gamma$. Suppose that there
exists a $\sigma\in\Gamma$ such that $\sigma(D)\neq D$ and $\sigma(D)\cap
D\neq\emptyset$. Then $\sigma(N[A])\neq N[A]$, hence
$|N[A]\cup\sigma\big{(}N[A]\big{)}|>|N[A]|$. Set
$A^{\prime}=A\cup(\sigma(A)-N[A])$. Then $A^{\prime}$ is also a
$\mathfrak{p}$-subset of $X$. By the above claim we have that
$N[A^{\prime}]=N[A]\cup\sigma\big{(}N[A]\big{)}$ and
$\frac{|A^{\prime}|}{|N[A^{\prime}]|}=\frac{\alpha(X,\mathfrak{p})}{|X|}=\frac{|A|}{|N[A]|}$,
which implies $|A^{\prime}|>|A|$. On the other hand, from definition it
follows that each element of $\sigma(D)\cap D$ does not belong to
$N[A]\cup\sigma\big{(}N[A]\big{)}$, so $N[A^{\prime}]\neq X$, yielding
$|A^{\prime}|<\alpha(X,\mathfrak{p})$. It contradicts the maximality of $A$,
thus proving that $\sigma(D)=D$ or $\sigma(D)\cap D=\emptyset$ for each
$\sigma\in\Gamma$. The transitivity of $\Gamma$ on $X$ implies that
$X=\cup_{\sigma\in\Gamma}\sigma(D)$. Furthermore, for any
$\sigma,\gamma\in\Gamma$, if $\sigma(D)\cap\gamma(D)\neq\emptyset$, then
$(\gamma^{-1}\sigma)(D)\cap D\neq\emptyset$, implying
$(\gamma^{-1}\sigma)(D)=D$, i.e., $\sigma(D)=\gamma(D)$. Therefore,
$\\{\sigma(D):\sigma\in\Gamma\\}$ is a partition of $X$. ∎
By Theorem 3.2 and Theorem 3.1 we obtain the following consequences.
###### Corollary 3.3
Suppose that a group $\Gamma$ transitively acts on $(X,\mathfrak{p})$. Then
$(X,\mathfrak{p})$ is $\mathfrak{p}$-primitive if one of the following
conditions holds.
1. * (i)
$\Gamma$ is primitive on $X$, or equivalently, $\Gamma_{a}$ is a maximal
subgroup of $\Gamma$ for each $a\in X$.
* (ii)
$\Gamma$ is imprimitive on $X$, but each block $D$ satisfies
$\frac{\alpha(D,\mathfrak{p})}{|D|}>\frac{\alpha(X,\mathfrak{p})}{|X|}$.
## 4 Primitivity of some classical symmetric systems
Finite sets, finite vector spaces and permutations are among the most
important finite structures in combinatorics, especially in extremal
combinatorics. In what follows we prove the primitivity of three symmetric
systems defined on them.
###### Proposition 4.1
$(\mathcal{C}_{n}^{k},\mathfrak{i}_{t})$ is $\mathfrak{i}_{t}$-primitive for
$n\geq(k-t+1)(t+1)$ unless $n=2k\geq 4$ and $t=1$.
Proof. Since the case $n\leq 3$ is trivial, we assume that $n\geq 4$. From
Example 1.2 we know that $(\mathcal{C}_{n}^{k},\mathfrak{i}_{t})$ is symmetric
and $\alpha(\mathcal{C}_{n}^{k},\mathfrak{i}_{t})={n-t\choose k-t}$ for
$n\geq(k-t+1)(t+1)$. Consider the action of the symmetric group $S_{n}$ on
$\mathcal{C}_{n}^{k}$. It is well known that for each
$A\in\mathcal{C}_{n}^{k}$, the stabilizer $S_{n,A}$ of $A$ is isomorphic to
$S_{k}\times S_{n-k}$, which is a maximal subgroup of $S_{n}$ if $n\neq 2k$
(See e.g [3]). Therefore, $(\mathcal{C}_{n}^{k},\mathfrak{i}_{t})$ is
$\mathfrak{i}_{t}$-primitive when $n\neq 2k$. It is easily seen that
$\\{A,[2k]-A\\}$ is a block in $\mathcal{C}_{2k}^{k}$ under the action of
$S_{2k}$, and every block is of this form. On the other hand,
$\frac{\alpha(\\{A,\bar{A}\\},\mathfrak{i}_{t})}{2}=\frac{1}{2}\geq\frac{{2k-t\choose
k-t}}{{2k\choose
k}}=\frac{\alpha(\mathcal{C}_{n}^{k},\mathfrak{i}_{t})}{|\mathcal{C}_{n}^{k}|}$
for all $1\leq t\leq k$, and equality holds if and only if $t=1$. By Corollary
3.3, $(\mathcal{C}_{2k}^{k},\mathfrak{i}_{t})$ is $\mathfrak{i}_{t}$-primitive
for $t>1$. It is clear that $(\mathcal{C}_{2k}^{k},\mathfrak{i})$ is
disconnected, hence $\mathfrak{i}$-imprimitive. ∎
###### Proposition 4.2
$(\mathcal{L}_{n,k}(q),\mathfrak{i}_{t})$ is $\mathfrak{i}_{t}$-primitive for
all $n\geq 2k-t$.
Proof. It is well known [2] that for each $A\in\mathcal{L}_{n,k}(q)$, the
stabilizer of $A$ is a maximal subgroup of $GL(n,q)$. By Corollary 3.3
$(\mathcal{L}_{n,k}(q),\mathfrak{i}_{t})$ is $\mathfrak{i}_{t}$-primitive.∎
In the foregoing two examples, the primitivity of systems follows directly
from the primitivity of groups acting on them. However, it is not always the
case, as we shall see.
Let us consider the set $S_{n}$. A subset $A$ of $S_{n}$ is said to be
$t$-intersecting if any two permutations in $A$ agree in at least $t$ points,
i.e. for any $\sigma,\tau\in A$, $|\\{i\in[n]:\sigma(i)=\tau(i)\\}|\geq t$. We
still denote this property by $\mathfrak{i}_{t}$. When $t=1$, Deza and Frankl
[11] showed that a $1$-intersecting subset $A\subseteq S_{n}$ has size at most
$(n-1)!$ and conjectured that for $t$ fixed, and $n$ sufficiently large
depending on $t$, a $t$-intersecting subset $A\subseteq S_{n}$ has size at
most $(n-t)!$. Cameron and Ku [10] proved a $1$-intersecting subset of size
$(n-1)!$ is a coset of the stabilizer of a point. A few alternative proofs of
Cameron and Ku’s result are given in [23], [17] and [26]. To show the
transitivity of $(S_{n},\mathfrak{i}_{t})$ we consider the action of $S_{n}$
on itself by the multiplication on the left. It is evident that the action is
transitive, but is far from primitive because the stabilizer of a point is the
identity.
###### Proposition 4.3
$(S_{n},\mathfrak{i}_{t})$ is $\mathfrak{i}_{t}$-primitive unless $n=3$ and
$t=1$.
Proof. The case $n=2$ is trivial. If $n=3$, it is easy to verify that the
graph $G(S_{3},\mathfrak{i})$ is disconnected and hence
$\mathfrak{i}$-imprimitive, while $(S_{3},\mathfrak{i}_{t})$ for $t=2,3$ is
$\mathfrak{i}_{t}$-primitive. We now assume that $n\geq 4$.
We first prove that $(S_{n},\mathfrak{i}_{t})$ is connected, i.e, the
corresponding graph $G(S_{n},\mathfrak{i}_{t})$ is connected. Since
$\mathfrak{i}_{t}\subseteq\mathfrak{i}_{1}$ for $t\geq 2$, it suffices to
prove that $G(S_{n},\mathfrak{i})$ is connected. For any pair $\gamma,\eta\in
S_{n}$, let $A_{j}=\\{i\in[n]:\eta(j)\neq i\neq\gamma(j)\\}$ for $1\leq j\leq
n$. Clearly, $|A_{j}|\geq n-2$. For every $J\subseteq[n]$, if $|J|=2$, then
$|\cup_{j\in J}A_{j}|\geq|A_{j}|=n-2\geq 2$. Suppose that $|J|\geq 3$. Then,
for each $k\in[n]$, since there are at most two points $i_{1},i_{2}\in[n]$
such that $\gamma(i_{1})=\eta(i_{2})=k$, we can find a $j\in J$ such that
$k\in A_{j}$, so $\cup_{j\in J}A_{j}=[n]$. Therefore $|\cup_{j\in
J}A_{j}|\geq|J|$ for all $J\subseteq[n]$. By the well-known Hall theorem [24]
on distinct representatives of subsets, there is a system of distinct
representatives $i_{1},i_{2},\ldots,i_{n}$ for $A_{1},A_{2},\ldots,A_{n}$.
Define a permutation $\tau$ by $\tau(j)=i_{j}$ for $1\leq j\leq n$. It is
clear that both $\\{\eta,\tau\\}$ and $\\{\tau,\gamma\\}$ belong to
$E(G(S_{n},\mathfrak{i}))$, proving that $G(S_{n},\mathfrak{i})$ is connected.
Suppose that $(S_{n},\mathfrak{i}_{t})$ is $\mathfrak{i}_{t}$-imprimitive for
some $n\geq 4$ and $t\geq 1$. Let $A$ be a maximal-sized imprimitive
$\mathfrak{i}_{t}$-subset of $S_{n}$, and $D=\bar{N}[A]=S_{n}-N[A]$. From
Theorem 3.2, it follows that
$\frac{\alpha(D,\mathfrak{i}_{t})}{|D|}=\frac{\alpha(S_{n},\mathfrak{i}_{t})}{|S_{n}|}$,
and $\tau D\cap D=\emptyset$ or $D$ for all $\tau\in S_{n}$, and Theorem 2.2
implies that $|S\cap D|=\alpha(D,\mathfrak{i}_{t})$ for every $S\in
I(S_{n},\mathfrak{i}_{t})$. Let $\sigma$ be a fixed $n$-cycle permutation in
$S_{n}$, and $H=\\{\sigma,\sigma^{2},\ldots,\sigma^{n}=1\\}$, the cyclic group
generated by $\sigma$. Then any two distinct elements of a right coset of $H$
disagree at every point. Therefore $H\rho\subset N[\\{\rho\\}]$ for every
$\rho\in S_{n}$, so $HA\subseteq N[A]$. Set $B=\\{\rho\in S_{n}:H\rho\subset
D\\}$ and $C=\\{\rho\in S_{n}:\mbox{$H\rho\cap N[A]\neq\emptyset$ and
$H\rho\cap D\neq\emptyset$}\\}$. We now complete the proof by two cases.
Case 1: $t\geq 2$. For any $\tau,\rho\in S_{n}$, set
$F_{i}=F_{i}(\tau,\rho)=\\{j:\tau(j)=\sigma^{i}\rho(j)\\}$, $i=1,2,\ldots,n$.
It is easily seen that for every $j\in[n]$ there is a unique $i\in[n]$ such
that $j\in F_{i}$, which yields $\sum_{i=1}^{n}|F_{i}|=n$. From this we see
that there are at least half $F_{i}$’s with at most one point, meaning that
there are at least $\lceil n/2\rceil$ $i$’s such that $\tau$ and
$\sigma^{i}\rho$ do not agree on $t$ points. In other words, $|H\rho\cap
N[\\{\tau\\}]|\geq\lceil\frac{n}{2}\rceil\geq 2$, which implies that
$B=\emptyset$ and $D\subset\cup_{\rho\in C}H\rho$. If $\sigma D\cap
D\neq\emptyset$, then $\sigma D=D$, hence $HD=D$, contradicting $B=\emptyset$.
We therefore obtain that $\sigma D\cap D=\emptyset$. Moreover, since
$\frac{\alpha(\sigma D,\mathfrak{i}_{t})}{|\sigma
D|}=\frac{\alpha(D,\mathfrak{i}_{t})}{|D|}=\frac{\alpha(S_{n},\mathfrak{i}_{t})}{|S_{n}|}$,
from Theorem 2.2 it follows that $|S\cap\sigma D|=\alpha(\sigma
D,\mathfrak{i}_{t})=\alpha(D,\mathfrak{i}_{t})$ for every $S\in
I(S_{n},\mathfrak{i}_{t})$. Note that for each $S_{D}\in
I(D,\mathfrak{i}_{t})$, we have $A\cup S_{D}\in I(S_{n},\mathfrak{i}_{t})$, so
$|(A\cup S_{D})\cap\sigma D|=\alpha(D,\mathfrak{i}_{t})$. Recalling that
$HA\subseteq N[A]$, we have
$(A\cup S_{D})\cap\sigma D=A\cap\sigma D\subseteq HA\cap\sigma D=\sigma(HA\cap
D)\subseteq\sigma(N[A]\cap D)=\emptyset,$
yielding a contradiction. Thus $(S_{n},\mathfrak{i}_{t})$ is
$\mathfrak{i}_{t}$-primitive for $t\geq 2$.
Case 2: $t=1$. By definition we see that $|A\cap H|\leq 1$. On the other hand,
from $HA\subseteq N[A]$ and
$\frac{|A|}{|N[A]|}=\frac{\alpha(S_{n},\mathfrak{i})}{|S_{n}|}=\frac{1}{n}$ it
follows that $N[A]=HA$, that is, $N[A]$ is a union of some right cosets of
$H$, so $D$ is a union of other right cosets of $H$, i.e., $D=HB$. By
definition we also have that $A\subseteq\bar{N}[D]\subseteq\bar{N}[H\rho]$ for
every $\rho\in B$. However, if $\tau\in\bar{N}[H\rho]$, i.e.
$F_{i}(\tau,\rho)=\\{j:\tau(j)=\sigma^{i}\rho(j)\\}\neq\emptyset$ for every
$i\in[n]$, then
$F_{i}(\sigma^{k}\tau,\rho\\}=\\{j:\sigma^{k}\tau(j)=\sigma^{i}\rho(j)\\}=\\{j:\tau(j)=\sigma^{i-k}\rho(j)\\}=F_{i-k}(\tau,\rho)\neq\emptyset$
for all $i,k\in[n]$ (here $i-k$ is taken to be the least positive residue
modulo $n$), therefore $H\tau\subseteq\bar{N}[H\rho]$. From this it follows
that $N[A]=HA\subseteq\bigcap_{\rho\in B}\bar{N}[H\rho]=\bar{N}[D]$, which
implies that $(S_{n},\mathfrak{i})$ is disconnected, yielding a contradiction.
Thus $(S_{n},\mathfrak{i})$ is $\mathfrak{i}$-primitive for $n\geq 4$.∎
Analogously, we may consider the primitivity of symmetric systems defined on
labeled sets [4] (or signed sets [5], colored sets [25] etc) and some other
permutations (see [21], [22] and [26]).
## Acknowledgements
The authors are greatly indebted to the anonymous referees for giving useful
comments and suggestions that have considerably improved the manuscript. This
work was partially supported by the National Natural Science Foundation of
China (No. 10826084 and No.10731040), Ph.D. Programs Foundation of Ministry of
Education of China (No. 20093127110001), and Innovation Program of Shanghai
Municipal Education Commission (No. 09zz134).
## References
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* [2] M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984) 469-514.
* [3] B. Newton, B. Benesh, A classification of certain maximal subgroups of symmetric groups, J. Algebra 304 (2006) 1108-1113.
* [4] P. Borg, Intersecting and cross-intersecting families of labeled sets, Electron. J. Combin. 15 (2008) N9.
* [5] P. Borg, On $t$-intersecting families of signed sets and permutations, Discrete Math. 309 (2009) 3310-3317.
* [6] P. Borg, Extremal $t$-intersecting sub-families of hereditary families, J. London Math. Soc. 79 (2009) 167-185
* [7] P. Borg, A short proof of a cross-intersection theorem of Hilton, Discrete Math. 309 (2009) 4750-4753.
* [8] P. Borg, Cross-intersecting families of permutations, J. Combin. Theory Ser. A 117 (2010) 483-487.
* [9] P. Borg and I. Leader, Multiple cross-intersecting families of signed sets, J. Combin. Theory Ser. A 117 (2010) 583-588.
* [10] P.J. Cameron, C.Y. Ku, Intersecting families of permutations, European J. Combin. 24 (2003) 881-890.
* [11] M. Deza, P. Frankl, On the maximum number of permutations with given maximal or minimal distance, J. Combin. Theory Ser. A 22 (1977) 352-360.
* [12] K. Engel, Sperner Theory, Cambridge University Press, Cambridge, 1997.
* [13] P. Erdős, C. Ko and R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. 2 (1961) 313-318.
* [14] P. Frankl, The Erdős-Ko-Rado theorem is true for $n=ckt$, Col. Soc. Math. J. Bolyai 18 (1978) 365-75.
* [15] P. Frankl, R.M. Wilson, The Erdős-Ko-Rado theorem for vector spaces, J. Combin. Theory Ser. A 43 (1986) 228-236.
* [16] C. Greene, D. J. Kleitman, Proof techniques in the ordered sets, in: G.-C. Rota, ed., “Studies in Combinatorics” (Math. Assn. America, Washington DC, 1978) 22-79.
* [17] C. Godsil, K. Meagher, A new proof of the Erdős-Ko-Rado theorem for intersecting families of permutations, European J. Combin. 30 (2009) 404-414.
* [18] W. N. Hsieh, Intersection theorems for systems of finite vector spaces, Discrete Math. 12 (1975) 1-16.
* [19] A.J.W. Hilton, An intersection theorem for a collection of families of subsets of a finite set, J. London Math. Soc. 2 (1977) 369-384.
* [20] N. Jacobson, Basic algebra. I, Second edition. W. H. Freeman and Company, New York, 1985.
* [21] C.Y. Ku, I. Leader, An Erdős-Ko-Rado theorem for partial permutations, Discrete Math. 306 (2006) 74-86.
* [22] C.Y. Ku, T.W.H. Wong, Intersecting families in the alternating group and direct product of symmetric groups, Electron. J. Combin. 14 (2007) R25.
* [23] B. Larose, C. Malvenuto, Stable sets of maximal size in Kneser-type graphs, European J. Combin. 25 (2004) 657-673.
* [24] P. Hall, On representatives of subsets, J. London Math. Soc. 10 (1935) 26-30.
* [25] Y.S. Li, J. Wang, Erdős-Ko-Rado-Type Theorems for Colored Sets, Elecron. J. Combin. 14 (2007) R1.
* [26] J. Wang, S.J. Zhang, An Erdős-Ko-Rado-type theorem in Coxeter groups, European J. Combin. 29 (2008) 1112-1115.
* [27] R.M. Wilson, The exact bound in the Erdős-Ko-Rado theorem, Combinatorica 4 (1984) 247-57.
* [28] H.J. Zhang, Primitivity and independent sets in direct products of vertex-transitive graphs, J. Graph Theory, to appear.
|
arxiv-papers
| 2010-07-06T02:41:23 |
2024-09-04T02:49:11.413220
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jun Wang and Huajun Zhang",
"submitter": "Zhang Huajun",
"url": "https://arxiv.org/abs/1007.0795"
}
|
1007.0797
|
# Independent Sets in Direct Products of Vertex-transitive Graphs
Huajun Zhang huajunzhang@zjnu.cn Department of Mathematics, Zhejiang Normal
University, Jinhua 321004, P.R. China Department of Mathematics, Shanghai
Normal University, Shanghai 200234, P.R. China
###### Abstract
The direct product $G\times H$ of graphs $G$ and $H$ is defined by:
$V(G\times H)=V(G)\times V(H)$
and
$E(G\times H)=\left\\{[(u_{1},v_{1}),(u_{2},v_{2})]:(u_{1},u_{2})\in
E(G)\mbox{\ and\ }(v_{1},v_{2})\in E(H)\right\\}.$
In this paper, we will prove that the equality
$\alpha(G\times H)=\max\\{\alpha(G)|H|,\alpha(H)|G|\\}$
holds for all vertex-transitive graphs $G$ and $H$, which provides an
affirmative answer to a problem posed by Tardif (Discrete Math. 185 (1998)
193-200). Furthermore, the structure of all maximum independent sets of
$G\times H$ are determined.
###### keywords:
direct product; primitivity; independence number; vertex-transitive
MSC: 05D05, 06A07
## 1 Introduction
Let $G$ and $H$ be two graphs. The direct product $G\times H$ of $G$ and $H$
is defined by
$\displaystyle V(G\times H)=V(G)\times V(H)$
and
$\displaystyle E(G\times
H)=\left\\{[(u_{1},v_{1}),(u_{2},v_{2})]:(u_{1},u_{2})\in E(G)\mbox{\ and\
}(v_{1},v_{2})\in E(H)\right\\}.$
It is easy to see this product is commutative and associative, and the product
of more than two graphs is well-defined. For a graph $G$, the products
$G^{n}=G\times G\times\cdots\times G$ is called the $n$-th powers of $G$.
An interesting problem is the independence number of $G\times H$. It is clear
that if $I$ is an independent set of $G$ or $H$, then the preimage of $I$
under projections is an independent set of $G\times H$, and so $\alpha(G\times
H)\geq\max\\{\alpha(G)|H|,\alpha(H)|G|\\}.$ It is natural to ask whether the
equality holds or not. In general, the equality does not hold for non-vertex-
transitive graphs (see [13]). So Tardif [17] posed the following problem.
###### Problem 1.1
(Tardif [17]) Does the equality
$\alpha(G\times H)=\max\\{\alpha(G)|H|,\alpha(H)|G|\\}$
hold for all vertex-transitive graphs $G$ and $H$?
Furthermore, it immediately raises another interesting problem:
###### Problem 1.2
When $\alpha(G\times H)=\max\\{\alpha(G)|H|,\alpha(H)|G|\\}$, is every maximum
independent set of $G\times H$ the preimage of an independent set of one
factor under projections?
If the answer is yes, we then say the direct product $G\times H$ is MIS-normal
(maximum-independent-set-normal). Furthermore, the direct products
$G_{1}\times G_{2}\times\cdots\times G_{n}$ is said to be MIS-normal if every
maximum independent set of it is the preimage of an independent set of one
factor under projections.
About these two problems, there are some progresses have been made for some
very special vertex-transitive graphs.
Let $n,r$ and $t$ be three integers with $n\geq r\geq t\geq 1$. The graph
$K(t,r,n)$ is defined by: whose vertices set is the set of all $r$-element
subsets of $[n]=\\{1,2,\ldots,n\\}$, and $A$ and $B$ of which are adjacent if
and only if $|A\cap B|<t$. If $n\geq 2r$, then $K(1,r,n)$ is the well-known
Kneser graph. The classical Erdős-Ko-Rado Theorem [8] states that
$\alpha(K(1,r,n))=\binom{n-1}{r-1}$ (where $n\geq 2r$), and Frankl [9] first
investigated the independence number of the direct products of Kneser graphs.
Subsequently, Ahlswede, Aydinian and Khachatrian investigated the general case
[2].
###### Theorem 1.3
Let $n_{i}\geq r_{i}\geq t_{i}$ for $i=1,2,\ldots,k$.
(i) (Frankl [9]) if $t_{1}=\cdots=t_{k}=1$ and
$\frac{r_{i}}{n_{i}}\geq\frac{1}{2}$ for $i=1,2,\ldots,k$, then
$\alpha\left(\prod_{1\leq i\leq
k}K(1,r_{i},n_{i})\right)=\max\left\\{\frac{r_{1}}{n_{1}},\frac{r_{2}}{n_{2}},\ldots,\frac{r_{k}}{n_{k}}\right\\}\prod_{1\leq
i\leq k}|K(1,r_{i},n_{i})|.$
(ii) (Ahlswede, Aydinian and Khachatrian [2])
$\alpha\left(\prod_{1\leq i\leq
k}K(t_{i},r_{i},n_{i})\right)=\max\left\\{\frac{\alpha(K(t_{i},r_{i},n_{i}))}{|K(t_{i},r_{i},n_{i})|}:1\leq
i\leq k\right\\}\prod_{1\leq i\leq k}|K(t_{i},r_{i},n_{i})|.$
The circular graph $Circ(r,n)$ ($n\geq 2r$) is defined by:
$V(Circ(r,n))=\mathbb{Z}_{n}=\\{0,1,\ldots,n-1\\}$
and
$E(Circ(r,n))=\left\\{(i,j):|i-j|\in\\{r,r+1,\ldots,n-r\\}\right\\}.$
It is well known that $\alpha(Circ(r,n))=r$. Mario and Juan [16] determined
the independence number of the direct products of circular graphs.
###### Theorem 1.4
(Mario and Juan [16]) Let $n_{i}\geq 2r_{i}$ for $i=1,2\ldots,k$. Then
$\alpha\left(\prod_{1\leq i\leq
k}Circ(r_{i},n_{i})\right)=\max\left\\{\frac{r_{1}}{n_{1}},\frac{r_{2}}{n_{2}},\ldots,\frac{r_{k}}{n_{k}}\right\\}\prod_{1\leq
i\leq k}n_{i}.$
For positive integers $n$, let $S_{n}$ denote the permutation group on $[n]$.
Two permutations $f$ and $g$ are said to be intersecting if there exists an
$i\in[n]$ such that $f(i)=g(i)$. We define a graph on $S_{n}$ as that two
permutations are adjacent if and only if they are not intersecting. For
brevity, this graph is also denoted by $S_{n}$. Deza and Frankl [7] first
obtained that $\alpha(S_{n})=(n-1)!$. Cameron and Ku [6] proved that each
maximum independent set of $S_{n}$ is a coset of the stabilizer of a point, to
which Larose and Malvenuto [14], Wang and Zhang [18] and Godsil and Meagher
[10] gave alternative proofs, respectively. Recently, Cheng and Wong [11]
further investigated the independence number and the MIS-normality of the
direct products of $S_{n}$.
###### Theorem 1.5
(Cheng and Wong[11]) Let $2\leq n_{1}=\cdots=n_{p}<n_{p+1}\leq\ldots,n_{q}$,
$1\leq p\leq q$. Then
$\alpha\big{(}S_{n_{1}}\times S_{n_{2}}\times\cdots\times
S_{n_{q}}\big{)}=(n_{1}-1)!\prod_{2\leq i\leq q}n_{i}!,$
and the direct products $S_{n_{1}}\times S_{n_{2}}\times\cdots\times
S_{n_{q}}$ is MIS-normal except for the following cases:
1. (i)
$n_{1}=\cdots=n_{p}<n_{p+1}=3\leq n_{p+2}\leq\cdots\leq n_{q}$;
2. (ii)
$n_{1}=n_{2}=3\leq n_{3}\leq\cdots\leq n_{q}$;
3. (iii)
$n_{1}=n_{2}=n_{3}\leq n_{4}\leq\cdots\leq n_{q}$.
In [15], Larose and Tardif investigated the relationship between projectivity
and the structure of maximum independent sets in powers of some vertex-
transitive graphs, and obtained the MIS-normality of the powers of Kneser
graphs and circular graphs.
###### Theorem 1.6
(Larose and Tardif [15]) Let $n$ and $r$ be two positive integers. If $n>2r$,
then both $K^{k}(1,r,n)$ and $Circ^{k}(r,n)$ are MIS-normal for all positive
integer $k$.
Besides the above results, Larose and Tardif [15] prove that if $G$ is vertex-
transitive, then $\alpha(G^{n})=\alpha(G)|V(G)|^{n-1}$ for all $n>1$. They
also ask whether or not $G^{n}$ is MIS-normal if $G^{2}$ is MIS-normal.
Recently, Ku and Mcmillan [12] gave an affirmative answer to this problem, and
we solved this problem in a more general setting [20].
In this paper we shall solve both Problem 1.1 and Problem 1.2. To state our
results we need to introduce some notations and notions.
For a graph $G$, let $I(G)$ denote the set of all maximum independent sets of
$G$. Given a subset $A$ of $V(G)$, we define
$N_{G}(A)=\\{b\in V(G):\mbox{$(a,b)\in E(G)$ for some $a\in A$}\\}$
$N_{G}[A]=N_{G}(A)\cup A\mbox{ and }\bar{N}_{G}[A]=V(G)-N_{G}[A].$
If $G$ is clear from the context, for simplicity, we will omit the index $G$.
In [20], by the so-called “No-Homomorphism” lemma of Albertson and Collins [1]
we proved the following result.
###### Proposition 1.7
([20]) Let $G$ be a vertex-transitive graph. Then, for every independent set
$A$ of $G$, $\frac{|A|}{|N_{G}[A]|}\leq\frac{\alpha(G)}{|V(G)|}$. Equality
implies that $|S\cap N_{G}[A]|=|A|$ for every $S\in I(G)$, and in particularly
$A\subseteq S$ for some $S\in I(G)$.
An independent set $A$ in $G$ is said to be imprimitive if $|A|<\alpha(G)$ and
$\frac{|A|}{|N[A]|}=\frac{\alpha(G)}{|V(G)|}$. And $G$ is called IS-
imprimitive if $G$ has an imprimitive independent set. In any other cases, $G$
is called _IS-primitive_. From definition we see that a disconnected vertex-
transitive graph $G$ is IS-imprimitive and hence an IS-primitive vertex-
transitive graph $G$ is connected.
The following Theorem is the main result of this paper.
###### Theorem 1.8
Let $G$ and $H$ be two vertex-transitive graphs with
$\frac{\alpha(G)}{|G|}\geq\frac{\alpha(H)}{|H|}$. Then
$\alpha(G\times H)=\alpha(G)|H|,$
and either:
1. (i)
$G\times H$ is MIS-normal, or
2. (ii)
$\frac{\alpha(G)}{|G|}=\frac{\alpha(H)}{|H|}$ and one of them is IS-
imprimitive, or
3. (iii)
$\frac{\alpha(G)}{|G|}>\frac{\alpha(H)}{|H|}$ and $H$ is disconnected.
We leave the proof of Theorem 1.8 to the next section, while in Section 3, we
discuss the MIS-normality of the direct products of more than two vertex-
transitive graphs.
## 2 Proof of Theorem 1.8
Let $S$ be a maximum independent set of $G\times H$. Then
$|S|\geq\alpha(G)|H|\geq|G|\alpha(H)$. We now prove $\alpha(G\times
H)\leq\alpha(G)|H|$.
For every $a\in G$, define
$X_{a}=\\{x\in H:(a,x)\in S\\}.$
Since $S$ is an independent set of $G\times H$, for each $x\in X_{a}$ and
$y\in X_{b}$, $(x,y)\not\in E(H)$ whenever $(a,b)\in E(G)$. In this case, we
say that $X_{a}$ and $X_{b}$ are cross-independent. This concept is equivalent
to cross-intersecting families in extremal set theory. We refer [19] for
details.
In the language of cross-intersecting families, Borg [3, 4, 5] introduce a
decomposition of $X_{a}$ as follows.
$X^{*}_{a}=\\{x\in X_{a}:N_{H}(x)\cap X_{a}=\emptyset\\},$
$X^{\prime}_{a}=\\{x\in X_{a}:N_{H}(x)\cap X_{a}\neq\emptyset\\}$
and
$X^{\prime}=\bigcup_{a\in V(G)}X^{\prime}_{a}.$
Clearly, $X_{a}^{*}$ is an independent set of $H$ for every $a\in V(G)$, and
$|S|=\sum_{a\in V(G)}|X_{a}|$. Here, the empty set is regarded as an
independent set.
We list all distinct $X^{*}_{a}$’s as $Y_{1},Y_{2},\ldots,Y_{k}$, and define
$B_{i}=\\{a\in V(G):X^{*}_{a}=Y_{i}\\},\ i=1,2,\ldots,k.$
We then obtain a partition of $V(G)$ as $V(G)=B_{1}\cup B_{2}\cup\cdots\cup
B_{k}$. Then
$\displaystyle|S|$ $\displaystyle=$ $\displaystyle\sum_{a\in
V(G)}|X_{a}|=\sum_{a\in V(G)}(|X_{a}^{*}|+|X^{\prime}_{a}|)=\sum_{i=1}^{k}\
\sum_{a\in B_{i}}|X_{a}^{*}|+\sum_{a\in V(G)}|X_{a}^{\prime}|$ (1)
$\displaystyle=$ $\displaystyle\sum_{i=1}^{k}|Y_{i}||B_{i}|+\sum_{x\in
X^{\prime}}|A_{x}|,$
where
$A_{x}=\\{a\in V(G):x\in X^{\prime}_{a}\\}.$
For every pair $a,b\in V(G)$, it is easy to verify that $(a,b)\not\in E(G)$ if
$X^{\prime}_{a}\cap X^{\prime}_{b}\neq\emptyset$. Therefore, $A_{x}$ is an
independent set of $G$. By Proposition 1.7 we have that
$|A_{x}|\leq\frac{\alpha(G)}{|V(G)|}|N_{G}[A_{x}]|,$ (2)
and equality holds if and only if $|A_{x}|=0$, or $|A_{x}|=\alpha(G)$, or
$A_{x}$ is an imprimitive independent set of $G$.
Suppose $x\in N_{H}[Y_{i}]=N_{H}(Y_{i})\cup Y_{i}$. If $x\in N_{H}(Y_{i})$,
then there exists $y\in Y_{i}$ such that $(x,y)\in E(H)$ and
$\\{(a,x),(b,y)\\}\subset S$ for any $b\in B_{i}$ and $a\in A_{x}$, hence
$(a,b)\not\in E(G)$ since $S$ is an independent set; if $x\in Y_{i}$, then for
each $a\in A_{x}$, there is a $z\in X_{a}$ with $(x,z)\in E(H)$ and
$\\{(a,z),(b,x)\\}\subset S$, yielding $(a,b)\not\in E(G)$. Thus proving that
$B_{i}\subseteq\bar{N}_{G}[A_{x}]$ if $x\in N_{H}[Y_{i}]$. From this it
follows that
$\sum_{i:x\in
N_{H}[Y_{i}]}|B_{i}|\leq|\bar{N}_{G}[A_{x}]|=|V(G)|-|N_{G}[A_{x}]|,$
i.e.,
$|N_{G}[A_{x}]|\leq|V(G)|-\sum_{i:x\in
N_{H}[Y_{i}]}|B_{i}|=\sum_{i:x\in\bar{N}_{H}[Y_{i}]}|B_{i}|.$ (3)
Note that
$X^{\prime}\subseteq\bigcup_{i=1}^{k}\bar{N}_{H}[Y_{i}].$ (4)
Together with (2), (3)and (4), we then obtain that
$\displaystyle\sum_{x\in
X^{\prime}}|A_{x}|\leq\frac{\alpha(G)}{|V(G)|}\sum_{x\in
X^{\prime}}\sum_{i:x\in\bar{N}_{H}[Y_{i}]}|B_{i}|$ (5) $\displaystyle\leq$
$\displaystyle\frac{\alpha(G)}{|V(G)|}\sum_{i=1}^{k}\sum_{x\in\bar{N}_{H}[Y_{i}]}|B_{i}|=\frac{\alpha(G)}{|V(G)|}\sum_{i=1}^{k}|B_{i}||\bar{N}_{H}[Y_{i}]|.$
Combining (1) and (5) gives that
$\displaystyle|S|$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{k}|Y_{i}||B_{i}|+\sum_{x\in X^{\prime}}|A_{x}|$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{k}|Y_{i}||B_{i}|+\frac{\alpha(G)}{|V(G)|}\sum_{i=1}^{k}|B_{i}||\bar{N}_{H}[Y_{i}]|$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k}|B_{i}|\left(\frac{\alpha(G)}{|V(G)|}|H|+|Y_{i}|-\frac{\alpha(G)}{|V(G)|}|N_{H}[Y_{i}]|\right)$
$\displaystyle=$
$\displaystyle\alpha(G)|H|+\sum_{i=1}^{k}|B_{i}|\left(|Y_{i}|-\frac{\alpha(G)}{|V(G)|}|N_{H}[Y_{i}]|\right)$
$\displaystyle\leq$ $\displaystyle\alpha(G)|H|.$
The last inequality follows from that
$\displaystyle|Y_{i}|-\frac{\alpha(G)}{|G|}|N_{H}[Y_{i}]|\leq|Y_{i}|-\frac{\alpha(H)}{|V(H)|}|N_{H}[Y_{i}]|\leq
0,$ (6)
by Proposition 1.7.
The maximum of $|S|$ implies that $|S|=\alpha(G)|H|$, from which it follows
that equalities (2), (3), (4) and (6) hold. Also, from Proposition 1.7,
equality (6) means that either $Y_{i}=\emptyset$, or
$\frac{\alpha(G)}{|G|}=\frac{\alpha(H)}{|V(H)|}$ and $Y_{i}$ is either
imprimitive or a maximum independent set of $H$ for $i=1,2,\ldots,k$.
We now prove that either $S$ is the preimages of projections of a maximum
independent set of $G$ or $H$, or (ii) or (iii) holds. There are two cases to
be considered.
Case 1: $\frac{\alpha(G)}{|G|}>\frac{\alpha(H)}{|H|}$. Then, equality (6)
means that $Y_{i}=\emptyset$ for all $i$, and so $X^{\prime}=V(H)$ by equality
(4). Hence, from equality (2) it follows that $A_{x}$ is a maximum independent
set of $G$ for all $x\in V(H)$. With this assumption we have that for any
$x,y\in V(H)$ with $(x,y)\in E(H)$, if $A_{x}\neq A_{y}$, there must exist
$a\in A_{x}$ and $b\in A_{y}$ with $(a,b)\in E(G)$ since both $A_{x}$ and
$A_{y}$ are maximum independent set, so $[(a,x),(b,y)]\in E(G\times H)$,
contradicting $\\{(a,x),(b,y)\\}\subset S$. Therefore, $A_{x}=A_{y}$ whenever
$(x,y)\in E(H)$, which implies that $S$ is the preimage of a maximum
independent set of $G$ under projections if $H$ is connected.
Case 2: $\frac{\alpha(G)}{|G|}=\frac{\alpha(H)}{|H|}$. Then, equality (6)
means that either $|Y_{i}|=0$ or $\alpha(H)$, or $Y_{i}$ is an imprimitive
independent set of $H$ for each index $i$. If $Y_{i}$ is an imprimitive
independent set of $H$ for some $i$, then $H$ is IS-imprimitive. If
$|Y_{i}|=\alpha(H)$ for all $i$, then $X_{a}=X_{a}^{*}$ is a maximum
independent set of $H$ for all $a\in V(G)$, and we can prove in the similar
way as in Case 1 that $S$ is the preimage of a maximum independent set of $H$
under projections if $G$ is connected. We now suppose that $|Y_{i}|=0$ for
some $i$. With this assumption, then equality (4) implies $X^{\prime}=V(H)$,
and then equality (3) means that either $A_{x}$ is either imprimitive or a
maximum independent set of $G$ for all $x\in V(H)$. If the former holds for
some $x\in V(H)$, we have that $H$ is IS-imprimitive; otherwise, the latter
holds for all $x\in V(H)$, and then we can prove in the similar way as in Case
1 that $S$ is the preimage of a maximum independent set of $G$ under
projections if $H$ is connected.
## 3 Concluding Remark.
Let $G_{1},G_{2},\ldots,G_{n}$ be $n$ non-empty vertex-transitive graphs, and
set $G=G_{1}\times G_{2}\times\cdots\times G_{n}$. From Theorem 1.8 it
immediately follows that
$\alpha(G)=\alpha(G_{1})\prod_{2\leq i\leq n}|G_{i}|.$
We now discuss the MIS-normality of $G$. For convenience, we say $G$ is MIS-
normal if $n=1$.
A graph $H$ is said to be non-empty if $E(H)\neq\emptyset$. It is well known
that if $H$ is a non-empty vertex-transitive graph, then
$\frac{\alpha(H)}{|H|}\leq\frac{1}{2}$, and equality holds if and only if $H$
is a bipartite graph.
Without loss of generality we may assume that
$\frac{1}{2}\geq\frac{\alpha(G_{1})}{|G_{1}|}=\cdots=\frac{\alpha(G_{\ell})}{|G_{\ell}|}>\frac{\alpha(G_{\ell+1})}{|G_{\ell+1}|}\geq\cdots\geq\frac{\alpha(G_{n})}{|G_{n}|}$,
and write $H_{0}=G_{1}\times\cdots\times G_{\ell}$ and $H_{i}=H_{i-1}\times
G_{\ell+i}$ for $i=1,\ldots,n-\ell$ subject to $n>\ell$. Then $G=H_{n-\ell}$
and with
$\frac{\alpha(H_{i-1})}{|H_{i-1}|}>\frac{\alpha(G_{\ell+i})}{|G_{\ell+i}|}$
for $i\geq 1$.
###### Proposition 3.1
Suppose $n>\ell$. Then $G$ is MIS-normal if and only if $H_{0}$ is MIS-normal
and $G_{\ell+1},\ldots,G_{n}$ are all connected.
Proof. Since $\alpha(G)=\alpha(H_{0})\prod_{i=\ell+1}^{n}|G_{i}|$, we have
that if $H_{0}$ is not MIS-normal, then $G$ is not MIS-normal. Furthermore, if
$G_{i}$ is not connected for for some $i\geq 1$, writing
$G_{i}=G_{i}^{\prime}\cup G_{i}^{\prime\prime}$, a union of disjoint
subgraphs, then, for all $I_{1},I_{2}\in I(H_{i-1})$ with $I_{1}\neq I_{2}$,
it is clear that $S=(I_{1}\times G_{i}^{\prime})\cup(I_{2}\times
G_{i}^{\prime\prime})\in I(H_{i})$, which is not a preimage of any independent
set of one factor under projections, i.e., $H_{i}$ is not MIS-normal, hence
$G$ is not MIS-normal.
Conversely, suppose $H_{0}$ is MIS-normal, and $G_{\ell+i}$ is connected for
$i\geq 1$. Since
$\frac{\alpha(H_{i-1})}{|H_{i-1}|}>\frac{\alpha(G_{\ell+i})}{|G_{\ell+i}|}$,
Theorem 1.8 implies that each maximal-sized independent set is of the form
$S\times G_{\ell+i}$, where $S\in I(H_{i-1})$, which means that $H_{i}$ is
MIS-normal for $i\geq 1$. We thus prove that $G$ is MIS-normal. ∎
We now discuss the case $n=\ell$, that is, each $G_{i}$ has the identical
independence ratio. To deal with this case we need a lemma as follows.
###### Lemma 3.2
Suppose that G is a vertex-transitive bipartite graph. Then $G$ is imprimitive
if and only if $G$ is disconnected.
Proof. It is clear that $G$ is imprimitive if $G$ is disconnected. On the
converse, if $G$ is imprimitive, then there is an imprimitive independent set
$A$ such that $\frac{|A|}{|N_{G}[A]|}=\frac{\alpha(G)}{|G|}=\frac{1}{2}.$ Set
$B=N_{G}(A)$. $|B|=|A|$ and $A\subseteq N_{G}(B)$ is clearly. If $N_{G}(B)\neq
A$, then we obtain that $\sum_{u\in A}d(u)\leq\sum_{v\in B}d(v)$, which
induces a contradiction. Hence $N_{G}(B)=A$, that is to say $G$ is
disconnected.
###### Proposition 3.3
Suppose that
$\frac{\alpha(G_{1})}{|G_{1}|}=\cdots=\frac{\alpha(G_{n})}{|G_{n}|}=\frac{\alpha(G)}{|G|}$.
Then $G$ is MIS-normal if and only if one of the following holds.
(i) $\frac{\alpha(G)}{|G|}<\frac{1}{2}$ and every $G_{i}$ is IS-primitive.
(ii) $\frac{\alpha(G_{1})}{|G_{1}|}=\frac{1}{2}$, $n=2$ and both $G_{1}$ and
$G_{2}$ are connected.
Proof. For $1\leq i\leq n$, set $\hat{G}_{i}=G_{1}\times\cdots\times
G_{i-1}\times G_{i+1}\times\cdots\times G_{n}$. Then $G=\hat{G}_{i}\times
G_{i}$ for $i=1,2,\ldots,n$. If $G_{i}$ is imprimitive, letting $A_{i}$ be an
imprimitive independent set of $G_{i}$, for every $I\in I(\hat{G}_{i})$, it is
easy to see that $S=(\hat{G}_{i}\times
A_{i})\cup(I\times\bar{N}_{G_{i}}[A_{i}])\in I(G)$, which is not a preimage of
any independent set of $\hat{G}_{i}$ or $G_{i}$ under projections, therefore,
$G$ is not MIS-normal. Conversely, if both $\hat{G}_{i}$ and $G_{i}$ are IS-
primitive, Theorem 1.8 implies that $G$ is MIS-normal. It remains to check
when $\hat{G}_{i}$ is IS-primitive. Summing up the above, $G$ is MIS-normal if
and only if both $\hat{G}_{i}$ and $G_{i}$ are IS-primitive. To complete the
proof, it remains to check when $\hat{G}_{i}$ is IS-primitive. We distinguish
two cases.
Case (i):$\frac{\alpha(G)}{|G|}<\frac{1}{2}$. In this case, Theorem 2.6 in
[20] says that if $G$ is MIS-normal, then both $\hat{G}_{i}$ and $G_{i}$ are
IS-primitive. The induction implies (i).
Case (ii): $\frac{\alpha(G_{1})}{|G_{1}|}=\frac{1}{2}$, i.e., every $G_{i}$ is
bipartite. From Lemma 3.2 it follows that $\hat{G}_{i}$ and $G_{i}$ is IS-
primitive if and only if both $\hat{G}_{i}$ and $G_{i}$ are connected.
However, it is well known that $\hat{G}_{i}$ is disconnected if $n>2$, thus
proving (ii). ∎
Combining Proposition 3.1 and Proposition 3.3 gives the following theorem.
###### Theorem 3.4
Let $G_{1},G_{2},\ldots,G_{n}$ be connected vertex-transitive graphs with
$\frac{1}{2}\geq\frac{\alpha(G_{1})}{|G_{1}|}=\cdots=\frac{\alpha(G_{\ell})}{|G_{\ell}|}>\frac{\alpha(G_{\ell+1})}{|G_{\ell+1}|}\geq\cdots\geq\frac{\alpha(G_{n})}{|G_{n}|}$,
where $n\geq 2$ and $1\leq\ell\leq n$. Then $G_{1}\times
G_{2}\times\cdots\times G_{n}$ is MIS-normal if and only if one of the
following holds:
(i) $\frac{\alpha(G_{1})}{|G_{1}|}<\frac{1}{2}$ and
$G_{1},G_{2},\ldots,G_{\ell}$ are all IS-primitive whenever $\ell>1$.
(ii) $\frac{\alpha(G_{1})}{|G_{1}|}=\frac{1}{2}$ and $\ell\leq 2$.
Acknowledgement The author is greatly indebted to Professor J. Wang for giving
useful comments, suggestions and helps that have considerably improved the
manuscript.
## References
* [1] M.O. Albertson and K.L. Collins, Homomorphisms of $3$-chromatic graphs, Discrete Math., 54 (1985) 127-132.
* [2] R. Ahlswede, H. Aydinian and L.H. Khachatrian, The Intersection Theorem for Direct Products, European J. Combin., 19 (1998) 649-661.
* [3] P. Borg, A short proof of a cross-intersection theorem of Hilton, Discrete Math., 309 (2009) 4750-4753.
* [4] P. Borg, Cross-intersecting families of permutations, J. Combin. Theory Ser. A, 117 (2010) 483-487.
* [5] P. Borg and I. Leader, Multiple cross-intersecting families of signed sets, J. Combin. Theory Ser. A, 117 (2010) 583-588.
* [6] P.J. Cameron and C.Y. Ku, Intersecting families of permutations, European J. Combin., 24 (2003) 881-890.
* [7] M. Deza and P. Frankl, On the maximum number of permutations with given maximal or minimal distance, J. Combin. Theory Ser. A, 22 (1977) 352-362.
* [8] P. Erdős, C. Ko and R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser., 2 (12) (1961) 313-318.
* [9] P. Frankl, An Erdős-Ko-Rado Theorem for direct products, European J. Combin., 17 (1996) 727-730.
* [10] C. Godsil and K. Meagher, A new proof of the Erdős-Ko-Rado theorem for intersecting families of permutations, Eurpean J. Combin., 30 (2008) 404-414.
* [11] C.Y. Ku and T.W.H. Wong, Intersecting families in the alternating group and direct product of symmetric groups, Electron. J. Combin., 14 (2007).
* [12] C.Y. Ku and B.B. Mcmillan, Independent sets of maximal size in tensor powers of vertex-transitive graphs, J. Graph Theory, 60 (2009) 295-301.
* [13] P.K. Jha and S. Klavz̆ar, Independence in direct-product graphs, Ars Combin., 50 (1998) 53-60.
* [14] B. Larose and C. Malvenuto, Stable sets of maximal size in Kneser-type graphs, European J. Combin., 25 (2004) 657-673.
* [15] B. Larose and C. Tardif, Projectivity and independent sets in powers of graph, J. Graph Theory, 40 (2002) 162-171.
* [16] V.P. Mario and V. Juan, Independence and coloring properties of direct products of some vertex-transitive graphs, Discrete Math., 306 (2006) 2275-2281.
* [17] C. Tardif, Graph products and the chromatic difference sequence of vertex-transitive graphs, Discrete Math., 185 (1998) 193-200.
* [18] J. Wang and S.J. Zhang, An Erdős-Ko-Rado-Type Theorem in Coxeter Groups, Eurpean J. Combin., 29 (2008) 1112-1115.
* [19] J. Wang and H.J. Zhang, Cross-intersecting families and primitivity of symmetric systems, submitted.
* [20] H.J. Zhang, Primitivity and independent sets in direct products of vertex-transitive graphs, J. Graph Theory, to appear.
|
arxiv-papers
| 2010-07-06T02:44:14 |
2024-09-04T02:49:11.420603
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Huajun Zhang",
"submitter": "Zhang Huajun",
"url": "https://arxiv.org/abs/1007.0797"
}
|
1007.0920
|
# End-Host Distribution in Application-Layer Multicast: Main Issues and
Solutions
Genge Béla and Haller Piroska Department of Electrical Engineering
“Petru Maior” University of Târgu Mureş
Târgu Mureş, Mureş, Romania, 540088
{bgenge,phaller}@engineering.upm.ro
###### Abstract
Application-layer multicast implements the multicast functionality at the
application layer. The main goal of application-layer multicast is to
construct and maintain efficient distribution structures between end-hosts. In
this paper we focus on the implementation of an application-layer multicast
distribution algorithm. We observe that the total time required to measure
network latency over TCP is influenced dramatically by the TCP connection
time. We argue that end-host distribution is not only influenced by the
quality of network links but also by the time required to make connections
between nodes. We provide several solutions to decrease the total end-host
distribution time.
_Keywords— Multicast; Overlay networks; PlanetLab_
## I Introduction
For several years now group communications have been receiving significant
attention from both the industry and scientific communities [1, 2]. The main
goal of group communication is to enable the exchange of information between
group members that can be located across the entire globe.
One of the main application of group communications is in the field of
_multicast_. Historically speaking, the first multicast applications were
implemented over the IP layer, also known as _IP multicast_ [3]. However,
after nearly a decade of research in the field of IP multicast, it was never
fully adopted because of several technical and administrative issues [4].
Later, there have been several proposals for other multicast implementations
that would be easier to deploy over the already existing and well-established
Internet protocols and would require little or no modifications in existing
routers. Such a survey of existing solutions was provided by El-Sayed et al
[5].
One of the directions that has been clearly adopted over the last few years is
_application-layer multicast_ , which implements the multicast functionality
at the application layer. The main goal of application-layer multicast is to
construct and maintain efficient distribution structures between _end-hosts_.
These structures are constructed using an _overlay_ network providing the
necessary infrastructure for data transfer between end-hosts.
Today’s research focuses on the many aspects of application-layer multicast,
including construction of overlay networks [6, 7], optimization issues [8] or
security [9]. In our previous work [10] we have addressed the problem of
optimally distributing end-hosts (i.e. EH) to overlay network hosts (i.e. OH)
in order to minimize network latency and to distribute the load of OH. Based
on a heuristic algorithm we proved that the algorithm ensures a local optimal
distribution of EH in real time and thus can be used to provide a feasible
solution to the distribution problem.
In this paper we focus on the actual deployment of the algorithm proposed in
our previous work in a real and globally-scaled distributed system:
_PlanetLab_ [11]. _PlanetLab_ is a “geographically distributed overlay network
designed to support the deployment and evaluation of planetary-scale network
services” [11]. Using PlanetLab, researchers can test their algorithms and
systems in a real environment where nodes can become unreachable, network
bandwidth can fluctuate and node processing capabilities can drop
dramatically.
In order to test the real applicability of our previously proposed algorithm
we have developed an overlay network in PlanetLab where nodes are connected in
a complete graph model. There are several advantages for using such a graph
model. First, there is no need for implementing complex routing algorithms
[12], which greatly simplifies the implementation and functionality of the
overlay. Second, maintaining routing tables is not more complex than
maintaining connections with all the other nodes. As a downside of this
topology, there is a large number of connections that must be maintained,
which grows exponentially with the number of OH. However, the simplicity of
the routing algorithms between OH makes this topology a great candidate for
using it as a leaf component in hierarchical topologies [13, 14].
Existing research [6, 7, 15] focuses on measuring the delay between nodes
after the overlay has been constructed or measuring the overlay construction
time after TCP connections are done. In deploying our algorithm we have
observed that the total time required to measure network latency over TCP is
influenced dramatically by the TCP connection time. In this paper we also
argue that end-host distribution is not only influenced by the quality of
network links but also by the time required to make connections between nodes.
The paper is structured as follows. In Section II we provide an overall
presentation of the overlay network, we discuss our previous work and we
identify the main problems for deploying the previously proposed algorithm. In
Section III we present the measurement results that were done with nodes
spread across 23 countries and we provide 3 solutions for improving the
performance of the measurements. Finally, we conclude with an overview of the
proposed solutions and we mention some future solutions that could also be
implemented.
## II Problem Statement
The measurements that follow in the next sections are based on a complete
graph overlay topology where EH are distributed using an heuristic algorithm.
An example of such a topology is given in figure Fig. 1, where we have
illustrated the presence of 3 host types:
* •
End-hosts (i.e. EH);
* •
Overlay-hosts (i.e. OH);
* •
Monitor-hosts (i.e. MH ).
EH are the producers and consumers of data transferred by the overlay
containing the OH. MH are used to monitor the load of each OH and to
distribute the connection of EH. The heuristic algorithm we proposed in our
previous work is used to distribute EH to OH in order to minimize latency and
to distribute the load of OH.
Figure 1: Multicast topology
The distribution algorithm uses the measured latency between all OH pairs, the
load of each OH and the measured latency between each EH and OH pairs. The
algorithm is run by the MH each time a new EH must be connected. At this time,
the EH must provide the MH its measurement results on the network latency it
recorded to each OH. Based on this data and the reported load received from
each OH, the MH runs the distribution algorithm.
As mentioned in our previous work, after all data is available, the algorithm
executes very fast. For instance, from the simulations we run, for 100 OH the
algorithm execution time for distributing a single EH is about 3.7 ms. This
execution time provides a real-time applicability of the proposed algorithm.
We have chosen to deploy the proposed multicast in PlanetLab because it
provides globally-available network services that can be used to run any
application type that can run on a Linux OS. From the beginning of the
implementation process we had to deal with several problems. First of all,
network connections between PlanetLab nodes or even node CPUs can be heavily
loaded, sometimes even leading to SYN_ACK timeouts for TCP connections.
Second, nodes can be rebooted at anytime by PlanetLab Central coordinators in
order to ensure a software update, for software maintenance or simply because
of some hardware problems. These problems must be handled by the MH in order
to ensure that EH are not distributed to such nodes and that already
distributed EH nodes are redistributed if necessary (i.e. on OH failure).
We also encountered several problems on the EH side. The proposed algorithm
heavily relies on the measurement data provided by EH. This means that when
joining the network, all EH must first measure the latency with all OH and
then send this data to MH. The problem with this approach is that in some
cases the response time from OH is very long, in the order of seconds as shown
in the next sections. This leads to an overall distribution time in the order
of seconds or even minutes, which is unacceptable.
## III Measurement Issues and Solutions
### III-A Overlay Construction Time
Although the construction of the overlay is done only once, we consider that
measuring the construction time can provide useful perspective of the time
required to re-construct the overlay in possible future developments. The
constructing of the overlay network is not made instantly. In order to
evaluate the performance and the general usability of the proposed overlay, we
have measured the time needed to construct the complete graph between the
overlay nodes.
TABLE I: Country and OH node count Country | Node count | Country | Node count
---|---|---|---
Austria | 1 | Italy | 6
Canada | 2 | Korea | 2
France | 4 | Poland | 3
Germany | 9 | Romania | 2
Greece | 1 | Spain | 2
Hungary | 1 | Switzerland | 1
Israel | 1 | US | 5
Deploying and starting applications on PlanetLab nodes can be done
automatically using applications such as _multicopy_ or _multiquery_ that are
part of the CoDeploy project [16]. These allow a parallel deployment and
execution of commands on a set of nodes. We have considered 5 settings with a
different number of OH nodes. The OH applications were deployed on nodes from
14 countries (for the maximum number of 40 OH nodes), as shown in Table I.
After starting the OH applications, each OH connects to all other OH according
to Alg. 1, where OH corresponds to the set of OH, Cout is the set of outgoing
connections and Cin is the set of incoming connections.
Algorithm 1 Complete connections for one OH
Let $t_{1}$ = @Get_curr_time()
Let $\textsf{Cout}=\phi$
{Start connection sequences}
for all $oh\in\textsf{OH}$ do
$c$ = @Start_conn_sequence($oh$)
$\textsf{Cout}=\textsf{Cout}\cup\\{c\\}$
end for
{Wait for completion}
@Wait_for_completion( Cout )
{Now eliminate duplicate connections}
Let Cin = @Get_incoming_connections()
for all $c\in\textsf{Cout}$ do
if $\exists
c^{\prime}\in\textsf{Cin}:$@Src_address($c^{\prime}$)=@Dest_address($c$) then
$(Meas_{out},Meas_{in})$=@Run_measurements($c,c^{\prime}$)
if $Meas_{out}<Meas_{in}$ then
@End_connection($c$)
$\textsf{Cout}=\textsf{Cout}\setminus\\{c\\}$
end if
end if
end for
{Calculate complete connection time}
Let $t_{2}$ = @Get_curr_time()
Let $G_{Time}=t_{2}-t_{1}$
Figure 2: Complete graph construction time
At first, each OH starts the connection process to other OH nodes. Then, it
waits for the connection process to complete. This process leads to duplicate
connections between each OH node pair. In order to eliminate duplicate
connections we measure the connection latency in each direction by sending a
single package of 1500Bytes and we eliminate the connection with the maximum
latency.
According to Alg. 1, each OH calculates a complete connection time $G_{Time}$.
The complete graph construction time is the maximum of these values, as shown
in Fig. 2. As we can see in Fig. 2 the construction of the overlay is greatly
influenced by the number of nodes. However, the variation is not linear
because the overlay also depends on other factors such as the quality of
network connections and the load of nodes. The result shown in Fig. 2 has the
following explanation. In the first OH set (i.e. 3 nodes), all 3 nodes are
located in European countries, with a minimum load. In the next OH set (i.e.
10 nodes) we have added additional nodes from Europe, one node from the US and
one node from Asia. This almost doubled the graph construction time because
the node from Asia was heavily loaded, with the CPU running at over 80% almost
all the time. In the next set (i.e. 20 nodes) we have added additional nodes
from Asia, Canada and Europe which, because of network connection latencies
and heavily loaded nodes (i.e. from Israel and Germany) has led to a quadruple
time. In the next two sets (i.e. 30 and 40 nodes) we have added additional
nodes from Europe and US, leading to the results shown in Fig. 2.
### III-B EH Connection Measurement Issues
When EH nodes are started, each node first connects to all OH nodes in order
to measure the network latency. The measured values are then sent to the MH
that applies the heuristic algorithm developed in our previous work [10] to
determine the OH node where each EH must connect. We have identified two
components that significantly influence the measured values: connection time
and network latency. Let EH be the set of EH. Then, the total measurement time
$M_{i}$ needed to be executed by an EH is:
$M_{i}=\max_{oh_{j}}\\{Conn(eh_{i},oh_{j})+CummLat(eh_{i},oh_{j})\\}$ (1)
where $eh_{i}\in\textsf{EH}$, $i=\overline{1,|\textsf{EH}|}$ and
$oh_{j}\in\textsf{OH}$, $j=\overline{1,|\textsf{OH}|}$. $Conn$ denotes the
time needed to establish a connection between $eh_{i}$ and $oh_{j}$. $CummLat$
denotes the cumulated round-trip latency calculated by measuring the time
difference between sent and received packages:
$\displaystyle CummLat(eh_{i},oh_{j})$ $\displaystyle=$ $\displaystyle
Lat_{1}(eh_{i},oh_{j})+$ (2) $\displaystyle Lat_{2}(eh_{i},oh_{j})+$
$\displaystyle Lat_{3}(eh_{i},oh_{j})$
where $Lat_{1}$, $Lat_{2}$ and $Lat_{3}$ denote the round-trip latency of 3
packages.
We have considered several scenarios, with EH count ranging from 10 to 1000.
EH nodes were deployed on nodes from 23 countries (for the maximum number of
1000 EH nodes), as shown in Table II.
TABLE II: Country and EH node count Country | Node count | Country | Node count
---|---|---|---
Argentina | 10 | Japan | 10
Australia | 10 | Korea | 20
Austria | 40 | Netherlands | 20
Belgium | 20 | Poland | 40
Canada | 100 | Portugal | 10
China | 20 | Romania | 20
Finland | 10 | Russia | 20
France | 110 | Spain | 40
Germany | 160 | Switzerland | 10
Greece | 10 | Taiwan | 10
Hungary | 20 | US | 240
Italy | 60 | |
Figure 3: Average EH measurement time
Each EH calculates its own $M_{i}$ value that is sent to the MH that
calculates an average measurement time, illustrated in Fig. 3. We can see that
the number of OH nodes clearly influences the overall measurement time. There
are several values that break the linear trajectory. For instance, in the case
of 40 OH nodes, when running 50 EH nodes the average time is 39382ms and when
running 100 EH nodes the average time is reduced to 21571ms. The explanation
for this behavior lies in the way that the measurements were done. Because
PlanetLab offers a set of resources over the Internet that are shared among
researchers, time measurements can change dramatically from one execution to
another. Moreover, the measurements we made span across 10 days. We have
actually seen that in one day a given node can be extremely loaded because
other researchers may also be running experiments, and the next day the node
can show a minimum load. This is in fact the expected behavior of nodes
running in a real networking environment that greatly differs from the
controlled laboratory environments.
The values shown in Fig. 3 include both the connection time and the network
latency. However, as we can see from Fig. 4, the latency is only a small part
of the measurement time, with average values ranging from 68.59ms to 925.86ms.
Figure 4: Average EH-OH measured latency
The values shown in Fig. 3 clearly show that we should improve the performance
of the measuring algorithm. At this stage, the average time needed to measure
the network latency for 1000 EH nodes in the 40 OH node setting is 89000ms,
which corresponds to almost 1.5 minutes. However, this is the average time,
which is much smaller than the maximum time needed for an EH to make the
measurements. The maximum measurement time is shown in Fig. 5, where we can
see that the maximum time needed to make the measurements is in fact 561192ms,
which is almost 9.5 minutes. The values from Fig. 5 show that the time needed
for all nodes to make the measurements are influenced by the number of OH and
by the number of EH, leading to the value of 9.5 minutes, which is
unacceptable.
Figure 5: Maximum EH measurement time
The total accessing distribution time of EH is also influenced by the response
time from the MH. In all our measurements the MH resides on a single node from
Romania. In Fig. 6 we can see the average response time from the MH.
Interestingly, the response time is not influenced by the number of OH or by
the number of EH, but by the number of simultaneous requests that are
received. EH nodes connect to MH only after completing the measurements, this
is why when a large number of EH connect simultaneously to the MH we get the
peaks from the figure. From the measurements we have also seen that after
receiving the measurement data the distribution algorithm is running under 1ms
for each request, thus the values shown in Fig. 6 are given by message
processing and network delay.
Figure 6: Average MH response time
After an EH successfully connects to the OH, it can stay connected for an
unlimited time. However, if the connection is interrupted, it will reconnect
to the designated OH. If the designated OH is no longer available, it must
execute the measurement and distribution all over again. In case of new EH
nodes, these are distributed by the MH without redistributing the already
connected EH nodes.
As mentioned earlier, in case of OH failure, disconnected EH nodes initiate a
new measurement and distribution process. However, in case of network failures
between OH nodes, a reconnect mechanism is activated for each OH node that
tries to re-establish connection with all other OH nodes, effectively trying
to reconstruct the overlay.
### III-C EH Connection Measurement Solutions
As illustrated in the previous section, making network measurements at the
application layer is mainly influenced by the connection time between nodes.
The network latency factor, as opposed to the connection time, has a minimum
impact on the total time.
When EH use the proposed overlay, their main goal is not to make measurements
but to actually use it to effectively distribute data. The time needed to make
the measurements should thus be reduced to a minimum possible.
In this section we propose 3 solutions to the measurement problem. After
implementing them, we have repeated the measurements for the 1000 EH setup,
where the modifications would have a greater impact.
The first solution involves reducing the reconnect process count to 0, meaning
that if a connect attempt fails, the EH removes the OH from its list. EH nodes
usually try to connect over and over again to OH nodes until successful. This
process dramatically increases the overall measurement time, as shown in the
previous section. By eliminating the reconnections, we are in fact eliminating
OH that are overloaded or to which we have a poor connection. The improvements
can be immediately seen, as shown in Fig. 7. In this case, for the maximum
setting, with 40 OH nodes, the average measurement time drops from 89000ms to
22027ms, improving the overall measurement 4 times.
Figure 7: Average improved EH measurement time for 1000 EH
The problem with the first solution is that a connection must be timed out by
the OS to eliminate the OH from the solution. As a second solution we propose
an application-controlled connection timeout, opposed to network OS timeout.
In this case we timed out connections that exceeded 10 seconds, decreasing the
average measurement time from 89000ms to 12284ms and improving the overall
measurement 7 times, as shown in Fig. 7. The 10 seconds were chosen based on
the observation that a lower timeout leads to an increased number of OH nodes
eliminated from the solution. This problem is discussed in detail later in
this section.
The third solution involves partitioning the OH and EH nodes into sub-groups,
thus reducing the total number of OH/EH and the total number of EH/OH. The
partitioning can be seen in Table III. As we can see from Fig. 7, the average
time required for measurements is reduced to 6459ms for 40 OH nodes, improving
the overall measurement time over 13 times.
TABLE III: Sub-group partitioning Sub-Group | 3 OH | 10 OH | 20 OH | 30 OH | 40 OH
---|---|---|---|---|---
| 1OH/EH | 2OH/EH | 4OH/EH | 6OH/EH | 8OH/EH
Grp1 | 333 EH | 200 EH | 200 EH | 200 EH | 200 EH
Grp2 | 333 EH | 200 EH | 200 EH | 200 EH | 200 EH
Grp3 | 333 EH | 200 EH | 200 EH | 200 EH | 200 EH
Grp4 | - | 200 EH | 200 EH | 200 EH | 200 EH
Grp5 | - | 200 EH | 200 EH | 200 EH | 200 EH
The direct effect of the first two solutions is that the number of OH nodes
for which EH nodes test the connection reduces significantly with the
reduction of the timeout. For instance, by using the OS timeout, which can
range from a few seconds to a few minutes we have less eliminated OH nodes
than using a fixed timeout of 10 seconds, as shown in Fig. 8. In case of only
one connection (i.e. OS timeout) the tested percentage is 100% for 3 OH nodes,
however, this drops to 95% for 10 and 20 nodes and then rises to 96.66% for 30
nodes and to 97.43% for 40 nodes. In case of application-layer timeout we have
a 98.1% for 3 OH nodes which drops to 71.79% for 40 OH nodes.
Although the partitioning-based solution provides the best timings, it can
limit sub-groups to a set of OH nodes that may not provide the optimal
solution for the entire group. While the application-layer timeout mechanism
seems to be the next best approach, care must be taken in choosing the timeout
value because a larger connection-time does not necessarily mean that the
specific node is heavily loaded, but several other factors can also influence
this value, such as a momentarily busy OS, or a momentarily busy application.
Other solutions could also be applied, such as using UDP for determining the
network latency between EH and OH. Such a solution would eliminate the
overhead given by TCP connection. However, because the overlay uses TCP for
forwarding data, making measurements by connecting to OH nodes via TCP
provides a more precise view on the future behavior of OH nodes.
Figure 8: Average percentage of measured connections
## IV Conclusions and Future Work
We presented several issues and solutions for deploying application-layer
overlay networks. Based on our measurements conducted over PlanetLab, a real
network testing platform, we have concluded that distributing EH nodes can not
be based only on the measured network latency, but must also include other
elements such as connection time or EH geographical location to reduce the
time required to make the actual latency measurements.
The identified problems have several solutions. In this paper we have proposed
3 such solutions: a first one that eliminates reconnections, a second one that
uses application-layer timeouts and a third one that constructs sub-groups for
reducing the number of OH/EH and EH/OH. By using these solutions we have shown
that the measurement time can be reduced up to 13 times for 1000 EH and 40 OH.
As future work, we intend to use UDP for the initial measurements. However,
special care must be taken because a lower timing for UDP packages does not
necessarily imply lower timings for TCP packages. A study must be made to
determine the correspondence between UDP and TCP timings and how could UDP-
based measurements be used to forecast the overhead introduced by TCP
connections. This study must also take into consideration UDP packet losses
that may also influence the total measurement time.
## References
* [1] F. Bacelli, A. Chaintreau, Z. Liu, A. Riabov, and S. Sahu, “Scalability of Reliable Group Communication Using Overlays”, Proceedings of INFOCOMM, 2004.
* [2] S. M. Venilla, and V. Sankaranarayanan, “Threat Analysis for P-LeaSel, a Multicast Group Communication Model”, Asian Journa of Information Technology, Vol. 7, 2008, pp. 64–68.
* [3] S. Deering, and D. Cheriton, “Multicast Routing in Datagram Internetworks and Extended LANS”, ACM Transactions on Computer Systems, Vol. 8, No. 2, 1990, pp. 85–111.
* [4] C. Diot, B.N. Levine, B. Lyles, H. Kassem, and D. Balensiefen, “Deployment issues for the IP multicast service and architecture”, IEEE Network Magazine, Vol. 14, No. 1, 2000, pp. 78–88.
* [5] A. El-Sayed, and V. Roca, “A Survey of Proposals for an Alternative Group Communication Service”, IEEE Network, Vol. 17, No. 1, 2003, pp. 46–51.
* [6] V. Roca, and A. El-Sayed, “A Host-Based Multicast (HBM) Solution for Group Communications”, Proceedings of the First International Conference on Networking, LNCS, Vol. 2093, 2001, pp. 610–619.
* [7] K. Ragab, and A. Yonezawa, “A Self-organized Clustering-based Overlay Network for Application Level Multicast”, Journal of Networks, Vol. 4, No. 2, 2009, pp. 85–91.
* [8] S. Jaggi, P. Sanders, P. A. Chou, M. Effros, S. Egner, K. Jain, and L. Tolhuizen, “Polynomial Time Algorithms for Multicast Network Code Construction”, IEEE Transactions on Information Theory, Vol. 51, No. 6, 2005, pp. 1973–1982.
* [9] N. Shanthi, and L. Ganesan, “Security In Multicast Mobile Ad-Hoc Networks”, International Journal of Computer Science and Network Security, Vol. 8, No. 7, 2008, pp. 326–330.
* [10] H. Piroska, and R. Balint, “Optimal server distribution in multimedia communication”, IN the Proc. of the 4th International Conference on RoEduNet, 2005, pp. 142–147.
* [11] A. Bavier, M. Bowman, B. Chun, D. Culler, S. Karlin, S. Muir, L. Peterson, T. Roscoe, T. Spalink, and M. Wawrzoniak, “Operating System Support for Planetary-Scale Network Services”, Networked Systems Design and Implementation, 2004.
* [12] T. L. Huang, and D. T. Lee, “A distributed multicast routing algorithm for real-time applications in wide area networks”, Journal of Parallel and Distributed Computing, Vol. 67, Issue 5, 2007, pp. 516–530.
* [13] W. Jia, W. Tu, and J. Wu, “Hierarchical Multicast Tree Algorithms for Application Layer Mesh Networks”, Networking and Mobile Computing, LNCS, Vol. 3619, 2005, pp. 549–559.
* [14] W. Yong, W. Seng, and H. Xianying, “A new Hierarchical Application Layer Multicast algorithm for large-scale video broadcasting”, In the Proc. of the 2nd IEEE International Conference on Computer Science and Information Technology, 2009, pp.610–613.
* [15] S. Ratnasamy, M. Handley, R. Karp, and S. Shenker, “Application-Level Multicast Using Content-Addressable Networks”, Networked Group Communication, LNCS, Vol. 2233, 2001, pp. 14–29.
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|
arxiv-papers
| 2010-07-06T15:16:38 |
2024-09-04T02:49:11.431772
|
{
"license": "Public Domain",
"authors": "Bela Genge and Piroska Haller",
"submitter": "Bela Genge",
"url": "https://arxiv.org/abs/1007.0920"
}
|
1007.0935
|
# Magnetic Response of Interacting Electrons in a Fractal Network: A Mean
Field Approach
Santanu K. Maiti Department of Physics, Narasinha Dutt College, 129 Belilious
Road, Howrah-711 101, India Arunava Chakrabarti Department of Physics,
University of Kalyani, Kalyani, West Bengal-741 235, India.
###### Abstract
The Hubbard model on a Sierpinski gasket fractal is carefully examined within
a Hartree-Fock mean field approach. We examine the influence of a magnetic
flux threading the gasket on its ground state energy, persistent current and
the Drude weight. Both an isotropic gasket and its anisotropic counterpart
have been examined. The variance in the patterns of the calculated physical
quantities are discussed for two situations, viz, at half-filling and when the
‘band’ is less than half-filled. The phase reversal of the persistent currents
and the change of the Drude weight as a function of the Hubbard interaction
are found to exhibit interesting patterns that have so far remained
unaddressed.
###### pacs:
71.27.+a, 73.23.-b, 73.23.Ra
## I Introduction
Deterministic fractals have been known to bridge the gap between systems
possessing perfect periodic order and the completely random ones. The spectrum
of non interacting electrons on such lattices has been exhaustively
investigated in the past domany ; rammal ; banavar ; ghez ; maritan ; gordon1
; gordon2 ; gordon3 ; schwalm1 ; andrade1 ; schwalm2 ; andrade2 ; schwalm3 ;
kappertz ; lin ; wang1 ; andrade3 ; hu ; andrade4 ; macia ; korshu ; meyer ;
new . The principal characteristic features of a deterministic fractal may be
summarized as follows: First, the energy spectrum is a Cantor set, and its
degenerate domany . Second, the density of states displays a variety of
singularities and a magnetic field is shown to broaden up the spectrum banavar
; ghez , and third, the electronic conductance exhibits scaling with a multi-
fractal distribution of the exponents schwalm2 . Apart from these, in certain
cases, isolated extended eigenstates also appear in deterministic, finitely
ramified fractal lattices wang2 ; arun1 ; arun2 ; arun3 , and extensive
numerical work has recently proposed a possible existence of even a continuum
of such extended states schwalm4 .
However, the typical properties exhibited by the deterministic fractals are
obtained within the picture of non-interacting spinless Fermions. The very
fundamental questions such as whether the spectral peculiarities exist even in
the presence of say, electron-electron interaction, or whether the response of
a fractal lattice to an externally applied magnetic (or electric) field brings
out any new features when one looks beyond the non-interacting picture, are
still to be addressed. The effect of electron-electron interaction on the
spectral properties are, to our mind, is of great importance, particularly
because of several experiments done on fractal networks that studied the
magnetoresistance, the superconductor-normal phase boundaries on Sierpinski
gasket wire networks gordon1 ; gordon2 ; gordon3 ; korshu ; meyer . These
experiments, together with the earlier ones on regular square or honeycomb
networks pannet1 ; pannet2 to study the flux quantization effects pioneered
the actual observational studies of spectral properties of planar networks and
the Aharonov-Bohm effect in systems with or without translational invariance.
Although in an early paper the problem of interacting electrons on a
percolating cluster that displays a fractal geometry nedellec , has been
addressed, to the best of our knowledge, no rigorous effort has been made so
far to unravel the effect of an interplay of electron-electron interaction and
an external magnetic field on deterministic networks such as a Sierpinski
gasket (SPG), even at a mean field level.
This inspires us to undertake a detailed study of the ground state energy and
the magnetic response of a Sierpinski gasket (SPG) fractal domany ; rammal ;
banavar that stands out to be a classic example of such lattices, and has
been the subject of the experiments cited above. We examine the persistent
current chung1 ; chung2 in such a fractal in the presence of on-site Hubbard
interaction within an unrestricted Hartree-Fock mean field scheme. Persistent
current in normal metal loops chung1 ; chung2 ; georges ; santanu is an
important effect in mesoscopic dimensions. Here, an SPG network offers a
unique opportunity to study the persistent current in a self-similar
distribution of loops, and with correlated electrons it is likely to give rise
to new observations. This is a major motivation of the present work.
Apart from this, the magnetoconductance (Drude weight) has also been
calculated and the variation of the response of the lattice to the external
magnetic field has been carefully studied as the fractal grows in size. To the
best of our knowledge the interplay of a fractal geometry and electron-
electron correlation in the form of persistent currents and the Drude weight
has not been studied before. With the metallic SPG networks already
synthesized, the present study may motivate experiments for a direct
observation of the effects presented here. In particular, based on the success
of the lithographic techniques it may not be too wild an idea to suggest an
SPG kind of fractal network built by carbon nanotubes that are connected at
the vertices.
As mentioned before, we examine both the isotropic and the anisotropic SPG
fractal networks. The anisotropy is introduced only in the values of the
nearest-neighbor hopping integrals. The response of the lattice is found to
differ grossly for an anisotropic system compared to the isotropic one. This
is of course, dependent on the relative values of the parameters in the
Hamiltonian, through which the anisotropy enters the system. For example, the
anisotropic SPG fractal is found to be more conducting than the isotropic one
in the sense that, the lattice remains conducting over a wide range of values
of the Hubbard interaction. The magnitude of the conductivity however, is
sensitive to the strength of the hopping parameters. This fact has also been
reported recently for non-interacting electrons jana .
In what follows, we present the results. In section II, the model Hamiltonian
is presented. Section III briefly describes the mean field approach, while the
results and the discussion are included in section IV. In section V we draw
our conclusions.
## II The Model
We start by referring to Fig. 1 where a $3$-rd generation SPG in which each
elementary plaquette is threaded by a magnetic flux $\phi$ (measured in unit
of the elementary flux quantum $\phi_{0}=ch/e$)
Figure 1: A $3$-rd generation Sierpinski gasket in which each elementary
plaquette is penetrated by a magnetic flux $\phi$. The filled black circles
correspond to the positions of the atomic sites.
is shown. The filled black circles correspond to the positions of the atomic
sites in the SPG. To describe the system we use a tight-binding framework. In
a Wannier basis the Hamiltonian reads,
$\displaystyle H_{\mbox{SPG}}$ $\displaystyle=$
$\displaystyle\sum_{i,\sigma}\epsilon_{i\sigma}c_{i\sigma}^{\dagger}c_{i\sigma}+\sum_{\langle
ij\rangle,\sigma}t\left[e^{i\theta}c_{i\sigma}^{\dagger}c_{j\sigma}+h.c.\right]$
(1)
$\displaystyle+\sum_{i}Uc_{i\uparrow}^{\dagger}c_{i\uparrow}c_{i\downarrow}^{\dagger}c_{i\downarrow}$
where, $\epsilon_{i\sigma}$ is the on-site energy of an electron at the site
$i$ of spin $\sigma$ ($\uparrow,\downarrow$) and $t$ is the nearest-neighbor
hopping strength. In the case of an anisotropic SPG, the anisotropy is
introduced only in the nearest-neighbor hopping integral $t$ which takes on
values $t_{x}$ and $t_{y}$ for hopping along the horizontal and the angular
bonds, respectively. Due to the presence of magnetic flux $\phi$, a phase
factor $\theta=2\pi\phi/3$ appears in the Hamiltonian when an electron hops
from one site to another site, and accordingly, a negative sign comes when the
electron hops in the reverse direction. As the magnetic filed associated with
the flux $\phi$ does not penetrate any part of the circumference of the
elementary triangle, we ignore the Zeeman term in the above tight-binding
Hamiltonian (Eq. 1). $c_{i\sigma}^{\dagger}$ and $c_{i\sigma}$ are the
creation and annihilation operators, respectively, of an electron at the site
$i$ with spin $\sigma$. $U$ is the strength of on-site Coulomb interaction.
## III The mean field approach
### III.1 Decoupling of the interacting Hamiltonian
To determine the energy eigenvalues of the interacting model of the SPG
described by the tight-binding Hamiltonian given in Eq. 1, first we decouple
the interacting Hamiltonian using the generalized Hartree-Fock approach kato ;
kam . The full Hamiltonian is completely decoupled into two parts. One is
associated with the up-spin electrons, while the other is with the down-spin
electrons. The on-site potentials get modified appropriately, and are given
by,
$\epsilon_{i\uparrow}^{\prime}=\epsilon_{i\uparrow}+U\langle
n_{i\downarrow}\rangle$ (2)
$\epsilon_{i\downarrow}^{\prime}=\epsilon_{i\downarrow}+U\langle
n_{i\uparrow}\rangle$ (3)
where, $n_{i\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma}$ is the number operator.
With these site energies, the full Hamiltonian (Eq. 1) can be written in the
decoupled form (in the mean field approximation) as,
$\displaystyle H_{\mbox{mean field}}$ $\displaystyle=$
$\displaystyle\sum_{i}\epsilon_{i\uparrow}^{\prime}n_{i\uparrow}+\sum_{\langle
ij\rangle}t\left[e^{i\theta}c_{i\uparrow}^{\dagger}c_{j\uparrow}+e^{-i\theta}c_{j\uparrow}^{\dagger}c_{i\uparrow}\right]$
(4) $\displaystyle+$
$\displaystyle\sum_{i}\epsilon_{i\downarrow}^{\prime}n_{i\downarrow}+\sum_{\langle
ij\rangle}t\left[e^{i\theta}c_{i\downarrow}^{\dagger}c_{j\downarrow}+e^{-i\theta}c_{j\downarrow}^{\dagger}c_{i\downarrow}\right]$
$\displaystyle-$ $\displaystyle\sum_{i}U\langle n_{i\uparrow}\rangle\langle
n_{i\downarrow}\rangle$ $\displaystyle=$ $\displaystyle
H_{\uparrow}+H_{\downarrow}-\sum_{i}U\langle n_{i\uparrow}\rangle\langle
n_{i\downarrow}\rangle$
where, $H_{\uparrow}$ and $H_{\downarrow}$ correspond to the effective tight-
binding Hamiltonians for the up and down spin electrons, respectively. The
last term is a constant term which provides a shift in the total energy.
### III.2 Self consistent procedure
With these decoupled Hamiltonians ($H_{\uparrow}$ and $H_{\downarrow}$) of up
and down spin electrons, now we start our self consistent procedure
considering initial guess values of $\langle n_{i\uparrow}\rangle$ and
$\langle n_{i\downarrow}\rangle$. For these initial set of values of $\langle
n_{i\uparrow}\rangle$ and $\langle n_{i\downarrow}\rangle$, we numerically
diagonalize the up and down spin Hamiltonians. Then we calculate a new set of
values of $\langle n_{i\uparrow}\rangle$ and $\langle n_{i\downarrow}\rangle$.
These steps are repeated until a self consistent solution is achieved.
### III.3 The ground state energy
After achieving the self consistent solution, the ground state energy $E_{0}$
for a particular filling at absolute zero temperature ($T=0$K) can be
determined by taking the sum of individual states up to the Fermi energy
($E_{F}$) for both the up and down spins. The final expression of the ground
state energy is written,
$E_{0}=\sum_{n}E_{n\uparrow}+\sum_{n}E_{n\downarrow}-\sum_{i}U\langle
n_{i\uparrow}\rangle\langle n_{i\downarrow}\rangle$ (5)
where, the index $n$ runs over the states up to the Fermi level.
$E_{n\uparrow}$ ($E_{n\downarrow}$) is the single particle energy eigenvalue
for $n$-th eigenstate obtained by diagonalizing the Hamiltonian $H_{\uparrow}$
($H_{\downarrow}$).
### III.4 Calculation of persistent current
At absolute zero temperature, total persistent current of the system is
obtained from the expression chung1 ; chung2
$I(\phi)=-c\frac{\partial E_{0}(\phi)}{\partial\phi}$ (6)
where, $E_{0}(\phi)$ is the ground state energy for a particular filling.
### III.5 Calculation of Drude weight
The conductance can be obtained by calculating the Drude weight $D$ as
originally noted by Kohn kohn . The Drude weight for the SPG is obtained
through the relation,
$D=\left.\frac{N}{4\pi^{2}}\left(\frac{\partial{{}^{2}E_{0}(\phi)}}{\partial{\phi}^{2}}\right)\right|_{\phi\rightarrow
0}$ (7)
where, $N$ gives total number of atomic sites in the gasket. Kohn has shown
that for an insulating system $D$ decays exponentially to zero, while it
becomes finite for a conducting system.
In the present work we inspect all the essential features of magnetic response
of an SPG network at absolute zero temperature and use the units where
$c=h=e=1$. Throughout our numerical work we set
$\epsilon_{i\uparrow}=\epsilon_{i\downarrow}=0$ for all $i$ and choose the
nearest-neighbor hopping strength $t=-1$. In the anisotropic case we select
$t_{x}=-1$ and $t_{y}=-2$ throughout. Energy scale is measured in unit of $t$.
Results are obtained both for an isotropic gasket and its anisotropic
counterpart.
## IV Numerical results and discussion
In Fig. 2 we present the variation of the ground state energy of a $3$-rd
generation isotropic SPG containing $15$ atomic sites as a function of the
magnetic flux through each elementary triangle. Two cases, viz, when the
‘band’ is less that half-filled, and half-filled,
Figure 2: (Color online). Ground state energy levels as a function of flux
$\phi$ for a $3$-rd generation isotropic ($t_{x}=t_{y}=-1$) Sierpinski gasket
($N=15$). The red, green and blue curves correspond to $U=0$, $1$ and $2$,
respectively. (a) $N_{e}=10$ and (b) $N_{e}=15$.
Figure 3: (Color online). Ground state energy levels as a function of flux
$\phi$ for a $3$-rd generation anisotropic ($t_{x}=-1$ and $t_{y}=-2$)
Sierpinski gasket ($N=15$). The red, green and blue curves correspond to
$U=0$, $1$ and $2$, respectively. (a) $N_{e}=10$ and (b) $N_{e}=15$.
are presented as the on-site Coulomb repulsion $U$ is varied. The ground state
energy exhibits a periodicity equal to one flux quantum in all the non-half-
filled cases, while the period changes to half flux quantum at precisely half-
filling. With increasing $U$, the ground state energy increases in both these
cases. In the half-filled case, each site is occupied by at least one
electron, and the placing of a second electron will increase the energy of the
system (the effect of $U$). This is
Figure 4: (Color online). Persistent current as a function of flux $\phi$ for
a $3$-rd generation isotropic ($t_{x}=t_{y}=-1$) Sierpinski gasket ($N=15$).
The red, green and blue curves correspond to $U=0$, $2$ and $4$, respectively.
(a) $N_{e}=10$ and (b) $N_{e}=15$.
Figure 5: (Color online). Persistent current as a function of flux $\phi$ for
a $3$-rd generation anisotropic ($t_{x}=-1$ and $t_{y}=-2$) Sierpinski gasket
($N=15$). The red, green and blue curves correspond to $U=0$, $2$ and $4$,
respectively. (a) $N_{e}=10$ and (b) $N_{e}=15$.
reflected in Fig. 2(b). Also the values of the ground state energy in the
half-filled case turns out to be well separated from each other for $U=0$, $1$
and $2$ compared to the non-half-filled case in (a). This feature remains true
irrespective of the size of the system.
As anisotropy is introduced, the overall features remain unaltered, including
the periodicities. However, as is evident from Fig. 3, the anisotropy lowers
the ground state energy of an SPG, both in the non-half-filled and the half-
filled cases. This will be reflected in the conductance, as will be shown
later.
The variation of the persistent current against the magnetic flux is shown
separately for the isotropic (Fig. 4) and the anisotropic (Fig. 5) SPG for
different values of the Hubbard interaction $U$. Two typical results, when
$N_{e}=10$ (less than half-filled case) and $N_{e}=15$ (half-filling), are
presented for a third generation SPG with $N=15$ sites. In Fig. 4(a) and in
Fig. 5(a) results for the ‘less than half-filled’ case are presented. In Fig.
4(a) the $I(\phi)$-$\phi$ curves exhibit multiple kinks which follows from the
numerous
Figure 6: (Color online). Drude weight as a function of Hubbard interaction
strength $U$ for a $3$-rd generation isotropic ($t_{x}=t_{y}=-1$) Sierpinski
gasket ($N=15$). (a) Non-half-filled case ($N_{e}=10$). (b) Half-filled case
($N_{e}=15$).
band-crossings that are typical of such hierarchical networks banavar ; jana .
Such crossings become less in number, and global gaps open up in the spectrum,
clustering the spectrum into sub-band structures in the case of an anisotropic
SPG, as has recently been reported in the literature even in the case of non-
interacting electrons jana . Kinks are now expected to smooth out. That it
happens, is evident from the anisotropic case, as depicted in Fig. 5(a). So,
anisotropy turns out to be the predominant factor in reducing the band-
crossings here.
On the other hand, in the half-filled case, the isotropic version of the SPG
display (Fig. 4(b)) non-trivial characteristics compared to its anisotropic
counterpart (Fig. 5(b)). In the former case the increasing value of $U$ is
seen to result into a complete reversal of the phase of the persistent
current, converting a diamagnetic response to a paramagnetic one. This however
is not seen to happen (in the half-filled case) in an anisotropic gasket (Fig.
5(b)).
We now present the results of the calculation of Drude weight $D$ both in the
cases of an isotropic and an anisotropic SPG, and observe its variation as $U$
increases. Results are presented in Fig. 6 and Fig. 7, respectively, for a
$3$-rd generation gasket. It is apparent that, the anisotropic gasket turns
out to be more conducting than its isotropic counterpart in the sense that, in
the anisotropic case the Drude weight displays finite values over a wider
range of $U$. The magnitude of $D$ at any specific $U$ of course, depends on
the numerical values of the hopping strength. Interestingly, this fact is also
observed jana for non-interacting electrons on an SPG. In the half-filled
band case, the Drude weight exhibits a much sharper drop in its value compared
to the non-half-filled situation. It is true
Figure 7: (Color online). Drude weight as a function of Hubbard interaction
strength $U$ for a $3$-rd generation anisotropic ($t_{x}=-1$ and $t_{y}=-2$)
Sierpinski gasket ($N=15$). (a) Non-half-filled case ($N_{e}=10$). (b) Half-
filled case ($N_{e}=15$).
for both the isotropic as well as the anisotropic case. The reason can easily
be traced back again to the fact that at half-filling, every site of the SPG
network has one electron occupying it already. So, conduction becomes
difficult as one needs more energy when an electron tries to leave its own
site and occupy a neighboring site. At less than half-filling there are
‘empty’ lattice points and conduction becomes easier. However, we find that in
the anisotropic case, we have to make the on-site Hubbard interaction much
stronger compared to the isotropic case to lower the value of the conductance
close to zero.
Before we end this section, it is pertinent to raise the question as to
whether the features discussed above really represent the characteristics of a
fractal. To get a definite answer to this, we have extended our analysis to
higher generation SPG networks, both in the isotropic and the anisotropic
limits. In each case, the overall features of the ground state energy, the
persistent current or the Drude weight turn out to be the same as in the cases
of lower generations. The effect of a variation of the Hubbard interaction
essentially plays the same role. The difference in the numerical values of the
quantities are of course, obvious. To clarify, we provide the results of our
calculation on a fourth generation SPG network comprising of $42$ sites in the
anisotropic limit, and in the half-filled
Figure 8: (Color online). Magnetic response for a $4$-th generation
anisotropic ($t_{x}=-1$ and $t_{y}=-2$) Sierpinski gasket ($N=42$) in Half-
filled case ($N_{e}=42$). (a) Energy-flux characteristics where the red, green
and blue curves correspond to $U=0$, $1$ and $2$, respectively. (b) Current-
flux characteristics where the red, green and blue curves correspond to $U=0$,
$2$ and $4$, respectively. (c) Drude weight as a function of Hubbard
interaction strength.
band case. This is in Fig. 8. The ground state energy in this case, as in the
previous generations, exhibits the same qualitative variation against the
magnetic flux, and it is the derivative of the ground state energy that
generates the current. So, a qualitative similarity between the curves at
various generations is not unexpected. A direct comparison with Fig. 5 reveals
that, the persistent current for $U=0$ in the present case is a bit rounded
off at the peak compared to the sharp discontinuity exhibited in the
corresponding case in the third generation fractal. This is not un-natural, as
the current depends on the band crossings exhibited by the eigenvalue spectrum
of the finite generation fractals, and the nature of band crossings will
change in every generation. But, the important point to note is that, the
periodicity of the persistent current is not affected, and the gradual phase
shift shown by the $I(\phi)$ curves in every generation, as the Hubbard
interaction is increased, is consistent. The observations remain the same when
we go beyond the fourth generation. This attempts us to believe that the
features are likely to persist for SPG networks of arbitrarily large finite
generations.
## V Closing Remarks
In conclusion, we have performed a thorough mean field analysis of the
response of a Sierpinski gasket fractal to an external magnetic field. We have
examined both the isotropic and the anisotropic limits of the system, where
the anisotropy is introduced only in the values of the nearest-neighbor
hopping integrals along two directions. Within the framework of the
unrestricted Hartree-Fock theory we decouple the Hubbard Hamiltonian and
obtain the ground state energy, the persistent current and the Drude weight.
The persistent current exhibits non trivial patterns in each case, and even
reveals a change in response, from diamagnetic to paramagnetic in the
isotropic case as a function of the interaction $U$. So, the Hubbard
interaction is seen to play its part in the magnetic response. The band
crossing is diminished by the anisotropy. The network remains diamagnetic in
the isotropic case, as far as we have examined. The conductance is obtained
through the Drude weight and, depending on the values of the nearest-neighbor
hopping integrals, the anisotropic gasket may remain conducting than its
isotropic counterpart for a wider range of the Hubbard correlation.
ACKNOWLEDGMENTS
First author thanks Prof. S. N. Karmakar and Prof. Shreekantha Sil for
illuminating comments and suggestions during the calculations.
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|
arxiv-papers
| 2010-07-06T16:08:26 |
2024-09-04T02:49:11.438298
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Santanu K. Maiti and Arunava Chakrabarti",
"submitter": "Santanu Maiti K.",
"url": "https://arxiv.org/abs/1007.0935"
}
|
1007.0943
|
# Spin transport through a quantum network: Effects of Rashba spin-orbit
interaction and Aharonov-Bohm flux
Moumita Dey Theoretical Condensed Matter Physics Division, Saha Institute of
Nuclear Physics, Sector-I, Block-AF, Bidhannagar, Kolkata-700 064, India
Santanu K. Maiti santanu.maiti@saha.ac.in Theoretical Condensed Matter
Physics Division, Saha Institute of Nuclear Physics, Sector-I, Block-AF,
Bidhannagar, Kolkata-700 064, India Department of Physics, Narasinha Dutt
College, 129 Belilious Road, Howrah-711 101, India S. N. Karmakar
Theoretical Condensed Matter Physics Division, Saha Institute of Nuclear
Physics, Sector-I, Block-AF, Bidhannagar, Kolkata-700 064, India
###### Abstract
We address spin dependent transport through an array of diamonds in the
presence of Rashba spin-orbit (SO) interaction where each diamond plaquette is
penetrated by an Aharonov-Bohm (AB) flux $\phi$. The diamond chain is attached
symmetrically to two semi-infinite one-dimensional non-magnetic metallic
leads. We adopt a single particle tight-binding Hamiltonian to describe the
system and study spin transport using Green’s function formalism. After
presenting an analytical method for the energy dispersion relation of an
infinite diamond chain in the presence of Rashba SO interaction, we study
numerically the conductance-energy characteristics together with the density
of states of a finite sized diamond network. At the typical flux
$\phi=\phi_{0}/2$, a delocalizing effect is observed in the presence of Rashba
SO interaction, and, depending on the specific choices of SO interaction
strength and AB flux the quantum network can be used as a spin filter. Our
analysis may be inspiring in designing spintronic devices.
###### pacs:
73.23.-b, 72.25.-b
## I Introduction
In recent times spin transport in low dimensional systems has drawn much
attention both from theoretical as well as experimental point of view, due to
its promising applications in the field of ‘spintronics’ spintronics . It is a
newly developed sub-discipline in condensed matter physics, that deals with
the idea of manipulating spin of the electrons in transport phenomena in
addition to their charge, and holds future promises to integrate memory and
logic into a single device. Since the discovery of Giant Magnetoresistance
(GMR) effect gmr in Fe-Cr magnetic multilayers revolutionary advancement has
taken place in data processing, device making and quantum computation
techniques. Today generation of pure spin current is a major challenge to us
for further development in quantum computation. A more or less usual way of
realization trend1 ; trend2 of spin filters is by using ferromagnetic leads
or by external magnetic field. But in the first case, spin injection from
ferromagnetic leads is difficult due to large resistivity mismatch and for the
second one the difficulty is to confine a very strong magnetic field into a
small region, like, a quantum dot (QD). Therefore, attention is now being paid
for modeling of spin filters using the intrinsic properties intrinsic1 ;
intrinsic2 ; intrinsic3 ; intrinsic4 ; intrinsic5 ; intrinsic6 of the
mesoscopic systems such as spin-orbit interaction or voltage bias. Studies on
Rashba spin orbit interaction rashba1 ; rashba2 ; rashba3 ; rashba4 , which is
present in asymmetric heterostructures has made a significant impact in
semiconductor spintronics as far as the control of spin dynamics is concerned.
It is generally important in narrow gap semiconductors and its strength can be
tuned by electrostatic means, e.g., applying external gate voltages gate1 ;
gate2 ; gate3 . Rashba SO interaction induces spin flipping through a
mechanism known as D’yakonov-Perel’ Dyakonov mechanism, which is a slow spin
scattering process in which spin precession takes place around the Rashba
field during transmission.
Over the last few years quantum networks are becoming prospective candidates
for studying transport phenomena because of the manifestation of several
interesting features, like, quantum interference, interplay of AB flux and
network geometry on electron localization, spin-orbit interaction induced
delocalization, effect of disorder, electron-electron interaction, etc. In
2000, Vidal et al. have shown Hubbard interaction can destroy the localization
induced by magnetic field in a diamond network vidal1 . In some other works
vidal2 ; vidal3 they studied the general formalism to obtain conductance of
any quantum networks and the effect of disorder and interaction. Latter in
2002, they considered Josephson-junction chain of diamonds in a magnetic field
to show a local $Z_{2}$ symmetry at half flux-quantum vidal4 . It may be
interesting to study the effects of Rashba SO coupling and AB flux in such
quantum networks. Depending on their topology these geometries exhibit various
striking spectral properties, and the interplay between AB flux and Rashba SO
strength can also be explored. In 2005, Bercioux et al. bercioux considered
the effect of AB flux and Rashba SO interaction on the energy averaged
conductances of a finite sized diamond chain. They observed that in such a
network spin-orbit interaction or AB flux can induce complete localization,
while the presence of both of them can lead to the effect of weak anti-
localization. The possibility to use such a diamond network as a spin filter
was explored by Aharony et al. in 2008 aharony . In 2009, there was another
work by Chakrabarti et al. sil , where they have shown how such a diamond
network can be implemented as a p-type or n-type semiconductor depending on
the suitable choice of the on-site potentials of the atoms at the vertices of
the network and the strength of magnetic flux penetrating each diamond
plaquette. But the effect of spin-orbit interaction was not considered.
Several other interesting theoretical works have been done considering this
kind of geometry. Gulacsi et al. gul1 ; gul2 in 2007 shown the exact ground
state of diamond Hubbard chain in magnetic field exhibits a wide range of
striking properties, those are tunable by magnetic flux, electron density,
etc. Peeters et al. considered quantum rings in presence of Rashba SO
interaction and magnetic field to obtain various features of
magnetoconductance peeters1 ; peeters2 ; peeters3 . In our present work, we
wish to explore the spectral and transport properties of a diamond network in
the presence of both AB flux and Rashba SO interaction. We calculate spin
conserved and spin flip conductances using single-particle Green’s function
formalism green1 within a tight-binding framework for a finite sized diamond
chain, which is compatible with the analytical dispersion relation obtained by
renormalization group method for an infinite diamond network. Analysis of the
spin-dependent conductances, dispersion relation and the density of states
(DOS) provides an insight about the effect of Rashba SO interaction and AB
flux on the localization behavior of the electrons. Finally, we show that, for
some specific choices of the external parameters this finite sized diamond
network can achieve a high degree of spin polarization.
Our organization of the paper is as follows. Following a brief introduction
(Section I), in Section II, we present the model and the theoretical
formulation. Section III is on our work comprising an analytical form for the
energy dispersion relation for an infinite diamond network, the numerical
calculations of two-terminal conductance, DOS, discussion on delocalizing
effect in presence of SO interaction and demonstration of spin filtering
action for a finite sized diamond array. At the end, the summary of our work
will be available in Section IV.
## II Model and theoretical formulation
At the beginning of our theoretical formulation we start by describing the
geometry of the quasi one-dimensional nanostructure through which spin
transport properties are being investigated. In Fig. 1 we illustrate
schematically the quantum network, in which the square loops are connected at
the vertices (termed as Diamond Network (DN) or
Figure 1: (Color online). A finite sized diamond network (central region)
connected to two semi-infinite one-dimensional non-magnetic metallic leads,
viz, source and drain. The diamond network is composed of two types of atoms
labeled by filled green and blue circles, where each diamond plaquette is
penetrated by an AB flux $\phi$.
Diamond Chain (DC)). The diamond array is connected symmetrically to two semi-
infinite one-dimensional ($1$D) non-magnetic metallic leads, commonly known as
source and drain which are characterized by the electrochemical potentials
$\mu_{1}$ and $\mu_{2}$ under the non-equilibrium condition when a bias
voltage is applied.
The full Hamiltonian for the complete system, i.e., source-DN-drain can be
written as,
$H=H_{D}+H_{L}+H_{R}+H_{LD}+H_{DR}$ (1)
where, $H_{D}$ represents the Hamiltonian for the diamond network. $H_{L(R)}$
corresponds to the Hamiltonian for the left (right) lead, i.e., source
(drain), and $H_{LD(DR)}$ is the Hamiltonian describing the chain-lead
coupling.
We model the diamond network by the nearest-neighbor tight-binding Hamiltonian
which in Wannier basis can be written as,
$\displaystyle H_{D}$ $\displaystyle=$
$\displaystyle\sum_{l,m}\mbox{\boldmath$c_{l,m}^{\dagger}\epsilon_{0}c_{l,m}$}+\sum_{l,m}\left(\mbox{\boldmath$c_{l,m}^{\dagger}t$}e^{i\alpha}\mbox{\boldmath$c_{l+1,m}$}+h.c.\right)$
(2)
$\displaystyle+\sum_{l,m}\left(\mbox{\boldmath$c_{l,m}^{\dagger}t$}e^{i\alpha}\mbox{\boldmath$c_{l,m+1}$}+h.c.\right)$
$\displaystyle+\sum_{l,m}\left(\mbox{\boldmath$c_{l,m}^{\dagger}(i\sigma_{y})~{}t_{so}$}~{}e^{i\alpha}\mbox{\boldmath$c_{l+1,m}$}+h.c.\right)$
$\displaystyle-\sum_{l,m}\left(\mbox{\boldmath$c_{l,m}^{\dagger}(i\sigma_{x})~{}t_{so}$}~{}e^{i\alpha}\mbox{\boldmath$c_{l,m+1}$}+h.c.\right)$
where,
$c_{l,m}^{\dagger}$=$\left(\begin{array}[]{cc}c_{l,m\uparrow}^{\dagger}&c_{l,m\downarrow}^{\dagger}\end{array}\right);$
$c_{l,m}$=$\left(\begin{array}[]{c}c_{l,m\uparrow}\\\
c_{l,m\downarrow}\end{array}\right);$
$\epsilon_{0}$=$\left(\begin{array}[]{cc}\epsilon_{0}&0\\\
0&\epsilon_{0}\end{array}\right);$ $t$=$t\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right);$ $t_{so}$=$\left(\begin{array}[]{cc}t_{so}&0\\\
0&t_{so}\end{array}\right)$.
Here $\epsilon_{0}$ is the site energy of each atomic site of the diamond
chain. For $A$ type of atoms $\epsilon_{0}=\epsilon_{A}$, while for $B$ type
of atoms we call $\epsilon_{0}$ as $\epsilon_{B}$ (see Fig. 1). The second and
third terms represent the electron hopping along $X$ and $Y$ directions,
respectively, where $t$ is the nearest-neighbor hopping strength and
$\alpha=\frac{2\pi\phi}{4\phi_{0}}$ is the phase factor due to the magnetic
flux $\phi$ threaded by each diamond plaquette. Here we use double indexing to
describe the location of lattice sites in the diamond network, as illustrated
in Fig. 2 for a single plaquette. The fourth and fifth terms are associated
with the spin dependent Rashba interaction, where $t_{so}$ is the isotropic
nearest-neighbor transfer integral that measures the strength of Rashba SO
coupling.
Similarly, the Hamiltonian $H_{L(R)}$ for the two leads can be written as,
$H_{L(R)}=\sum_{i}\mbox{\boldmath$c_{i}^{\dagger}\epsilon_{L(R)}c_{i}$}+\sum_{i}\left(\mbox{\boldmath$c_{i}^{\dagger}t_{L(R)}c_{i+1}$}+h.c.\right).$
(3)
Here also,
$\epsilon_{L(R)}$=$\left(\begin{array}[]{cc}\epsilon_{L(R)}&0\\\
0&\epsilon_{L(R)}\end{array}\right);$
$t_{L(R)}$=$\left(\begin{array}[]{cc}t_{L(R)}&0\\\
0&t_{L(R)}\end{array}\right)$
where, $\epsilon_{L(R)}$ is the site energy and $t_{L(R)}$ is the hopping
strength between the nearest-neighbor sites in the left (right) lead.
The diamond chain-to-lead coupling Hamiltonian is described by,
$H_{LD(DR)}=\left(\mbox{\boldmath$c_{0(NN)}^{\dagger}t_{LD(DR)}c_{11(N+1)}$}+h.c.\right)$
(4)
where, $t_{LD(DR)}$ being the chain-lead coupling strength.
In order to calculate spin dependent transmission probabilities through the
quantum network, we use single particle Green’s
Figure 2: (Color online). Index convention for representing the co-ordinates.
$l$ and $m$ denote the co-ordinates along the $X$ and $Y$ directions,
respectively.
function formalism. Within the regime of coherent transport and in absence of
Coulomb interaction this technique is well applied.
The single particle Green’s function representing the full system for an
electron with energy $E$ is defined as,
$G=(E-H+i\eta)^{-1}$ (5)
where $\eta\rightarrow 0^{+}$.
The matrix representation for the Hamiltonian can be expressed as
$\mbox{\boldmath$H$}=\left(\begin{array}[]{ccc}\mbox{\boldmath$H_{L}$}&\mbox{\boldmath$H_{LD}$}&0\\\
\mbox{\boldmath$H_{LD}^{\dagger}$}&\mbox{\boldmath$H_{D}$}&\mbox{\boldmath$H_{DR}$}\\\
0&\mbox{\boldmath$H_{DR}^{\dagger}$}&\mbox{\boldmath$H_{R}$}\\\
\end{array}\right)$ (6)
where, ${H_{L}}$, ${H_{D}}$ and ${H_{R}}$ are the Hamiltonian matrices for the
left lead, diamond network and right lead, respectively. ${H_{LD}}$ and
${H_{DR}}$ are the coupling matrices between diamond network and the leads.
Since there is no direct coupling between the leads themselves, the corner
elements of $H$ are null matrices. A similar definition holds true for the
Green’s function matrix $G$ as well.
$\mbox{\boldmath$G$}=\left(\begin{array}[]{ccc}\mbox{\boldmath$G_{L}$}&\mbox{\boldmath$G_{LD}$}&0\\\
\mbox{\boldmath$G_{DL}$}&\mbox{\boldmath$G_{D}$}&\mbox{\boldmath$G_{DR}$}\\\
0&\mbox{\boldmath$G_{RD}$}&\mbox{\boldmath$G_{R}$}\\\ \end{array}\right)$ (7)
The problem of finding $G$ in the full Hilbert space of $H$ can be mapped
exactly to a Green’s function $G_{D}^{eff}$ corresponding to an effective
Hamiltonian in the reduced Hilbert space of diamond network and we have
$\mbox{\boldmath${\mathcal{G}}$=$G_{D}^{eff}$}=\left(\mbox{\boldmath$E-H_{D}-\Sigma_{L}-\Sigma_{R}$}\right)^{-1}$
(8)
where, $\Sigma_{L}$ and $\Sigma_{R}$ represent the contact self-energies
introduced to incorporate the effects of semi-infinite leads coupled to the
system. The self-energies are expressed by the relations,
$\Sigma_{L}$ $\displaystyle=$ $H_{LD}^{{\dagger}}G_{L}H_{LD}$ $\Sigma_{R}$
$\displaystyle=$ $H_{DR}^{{\dagger}}G_{R}H_{DR}$. (9)
Thus, the form of self-energies are independent of the nano-structure itself
through which transmission is studied and they completely describe the
influence of the two leads attached to the system. Now, the transmission
probability $(T_{\sigma\sigma^{\prime}})$ of an electron with energy $E$ is
related to the Green’s function as,
$\displaystyle T_{\sigma\sigma^{\prime}}$ $\displaystyle=$
$\displaystyle\Gamma^{1}_{L(\sigma\sigma)}{\mathcal{G}}^{1N}_{r(\sigma\sigma^{\prime})}{\mathcal{G}}^{N1}_{a(\sigma^{\prime}\sigma)}\Gamma^{N}_{R(\sigma^{\prime}\sigma^{\prime})}$
(10) $\displaystyle=$
$\displaystyle\Gamma^{1}_{L(\sigma\sigma)}|{\mathcal{G}}^{1N}_{(\sigma\sigma^{\prime})}|^{2}\Gamma^{N}_{R(\sigma^{\prime}\sigma^{\prime})}$
where, $\Gamma^{1}_{L(\sigma\sigma)}$ = $\langle
11\sigma|{\bf\Gamma_{L}}|11\sigma\rangle$,
$\Gamma^{N}_{R(\sigma^{\prime}\sigma^{\prime})}$ = $\langle
NN\sigma^{\prime}|{\bf\Gamma_{R}}|NN\sigma^{\prime}\rangle$ and
${\mathcal{G}}^{1N}_{\sigma\sigma^{\prime}}=\langle
11\sigma|\mbox{\boldmath${\mathcal{G}}$}|NN\sigma^{\prime}\rangle$. Here,
${\mathcal{G}}_{r}$ and ${\mathcal{G}}_{a}$ are the retarded and advanced
single particle Green’s functions for an electron with energy $E$.
$\bf{\Gamma_{L}}$ and $\bf{{\Gamma_{R}}}$ are the coupling matrices,
representing the coupling of the quantum network to the left and right leads,
respectively, and they are defined by the relation,
$\mbox{\boldmath$\Gamma_{L(R)}$}=i\mbox{\boldmath$\left[\Sigma^{r}_{L(R)}-\Sigma^{a}_{L(R)}\right]$}$
(11)
Here, ${\Sigma^{r}_{L(R)}}$ and ${\Sigma^{a}_{L(R)}}$ are the retarded and
advanced self-energies, respectively, and they are conjugate to each other. It
is shown in literature by Datta et al. green1 that the self-energy can be
expressed as a linear combination of a real and an imaginary part in the form,
$\mbox{\boldmath${\Sigma^{r}_{L(R)}}$}=\mbox{\boldmath$\Lambda_{L(R)}$}-i\mbox{\boldmath$\Delta_{L(R)}$}$
(12)
The real part of self-energy describes the shift of the energy levels and the
imaginary part corresponds to the broadening of the levels. The finite
imaginary part appears due to incorporation of the semi-infinite leads having
continuous energy spectrum. Therefore, the coupling matrices can easily be
obtained from the self-energy expression and is expressed as,
$\mbox{\boldmath$\Gamma_{L(R)}$}=-2~{}{\mbox{Im}}(\mbox{\boldmath$\Sigma_{L(R)}$})$
(13)
Considering linear transport regime, conductance $(g_{\sigma})$ is obtained
using two-terminal Landauer conductance formula,
$g_{\sigma\sigma^{\prime}}=\frac{e^{2}}{h}T_{\sigma\sigma^{\prime}}$ (14)
Throughout our study we choose $c=e=h=1$ for simplicity.
## III Numerical results and discussion
In this section we study spin dependent transport through a diamond chain in
presence of Rashba spin orbit interaction and magnetic flux and investigate
the interplay between them. An array of diamonds is a bipartite structure
Figure 3: (Color online). Energy dispersion ($E$-$k$) curves for an infinite
diamond chain with $\epsilon_{A}=\epsilon_{B}=0$. The upper, middle and lower
spectra in the $1$st column correspond to $\phi=0$, $0.2$ and $0.4$,
respectively, when $t_{so}=0$. In the $2$nd column three different spectra
from the top represent the results for $t_{so}=0$, $2$ and $4$, respectively,
when $\phi$ is set to $0$. Finally, the three different figures in the last
column refer to the results for the identical values of $t_{so}$ considered in
the $2$nd column when $\phi$ is fixed at $0.4$.
with lattice sites having different co-ordination numbers. Electron
localization plays a significant role even in the absence of disorder in this
kind of geometry due to quantum interference effect. First we obtain
analytically the dispersion relation for an infinite diamond chain in the
presence of magnetic flux and Rashba interaction. Next, we simulate
numerically various features of spin transport using a finite size diamond
chain. Before analyzing the results first we specify the values of the
parameters those are used in the numerical simulations. We consider that the
two non-magnetic side-attached leads are made up of identical materials. The
on-site energies in the two leads ($\epsilon_{L(R)}$) are set to $0$. Hopping
strength between the sites in the leads is chosen as $t_{L(R)}=4$, whereas in
the diamond chain it is set as $t=3$. The Rashba strength ($t_{so}$) is chosen
to be uniform along $X$ and $Y$ directions and throughout the calculation its
magnitude is considered as comparable to $t$. Energy scale is fixed in unit of
$t$. Throughout the analysis we present all the results considering the chain-
to-electrode coupling strength as $t_{LD}=t_{DR}=2.5$.
### III.1 Energy dispersion relation in presence of Rashba SO interaction and
magnetic flux
The energy dispersion relation for an infinite diamond chain clearly depicts
several significant features of this kind of topology. In order to study the
$E$-$k$ relation theoretically, first we map the quasi one-dimensional diamond
network into a linear chain with modified site energy and hopping strength.
We start with the Schrodinger equation which can be cast in the form of a
difference equation. For an arbitrary site ($n,p$), where $n$ and $p$ denote
the indexing along $X$ and $Y$ directions, respectively, the difference
equation can be expressed as
${(E-\epsilon)\psi_{np}}$ $\displaystyle=$ $\displaystyle e^{\mp
i\alpha}\mbox{\boldmath$t_{x_{+}}\psi_{n+1,p}$}+e^{\mp
i\alpha}\mbox{\boldmath$t_{x_{-}}\psi_{n-1,p}$}+$ (15) $\displaystyle e^{\pm
i\alpha}\mbox{\boldmath$t_{y_{+}}\psi_{n,p+1}$}+e^{\pm
i\alpha}\mbox{\boldmath$t_{y_{-}}\psi_{n,p-1}$}$
where,
$t_{x_{+}}$=$\left(\begin{array}[]{cc}t&t_{so}\\\
-t_{so}&t\end{array}\right);$
$t_{x_{-}}$=$\left(\begin{array}[]{cc}t&-t_{so}\\\
t_{so}&t\end{array}\right);$
$t_{y_{+}}$=$\left(\begin{array}[]{cc}t&it_{so}\\\
it_{so}&t\end{array}\right);$
$t_{y_{-}}$=$\left(\begin{array}[]{cc}t&-it_{so}\\\
-it_{so}&t\end{array}\right);$
$E$=$\left(\begin{array}[]{cc}E&0\\\ 0&E\end{array}\right)$;
$\epsilon$=$\left(\begin{array}[]{cc}\epsilon_{0}&0\\\
0&\epsilon_{0}\end{array}\right)$; and
$\psi_{np}$=$\left(\begin{array}[]{c}\psi_{np,\uparrow}\\\
\psi_{np,\downarrow}\end{array}\right)$
$\phi$ being the AB flux enclosed by each diamond plaquette, $\psi_{np\sigma}$
being the wave function amplitude at the np-th site with spin $\sigma$.
$\phi_{0}=ch/e$, the elementary flux-quantum.
We begin the decimation technique by writing down the difference equations at
the sites containing A and B type atoms (see Fig. 2). The equations are given
below
$(E-\epsilon_{B})\psi_{12}$ $\displaystyle=$ $\displaystyle
e^{i\alpha}\mbox{\boldmath$t_{x_{+}}\psi_{22}$}+e^{-i\alpha}\mbox{\boldmath$t_{y_{-}}\psi_{11}$}$
$(E-\epsilon_{B})\psi_{21}$ $\displaystyle=$ $\displaystyle
e^{i\alpha}\mbox{\boldmath$t_{x_{-}}\psi_{11}$}+e^{-i\alpha}\mbox{\boldmath$t_{y_{+}}\psi_{22}$}$
$(E-\epsilon_{B})\psi_{23}$ $\displaystyle=$ $\displaystyle
e^{i\alpha}\mbox{\boldmath$t_{x_{+}}\psi_{33}$}+e^{-i\alpha}\mbox{\boldmath$t_{y_{-}}\psi_{22}$}$
$(E-\epsilon_{B})\psi_{32}$ $\displaystyle=$ $\displaystyle
e^{i\alpha}\mbox{\boldmath$t_{x_{-}}\psi_{22}$}+e^{-i\alpha}\mbox{\boldmath$t_{y_{+}}\psi_{33}$}$
(16)
and,
$(E-\epsilon_{A})\psi_{22}$ $\displaystyle=$ $\displaystyle
e^{-i\alpha}\mbox{\boldmath$t_{x_{+}}\psi_{32}$}+e^{-i\alpha}\mbox{\boldmath$t_{x_{-}}\psi_{12}$}$
(17)
$\displaystyle+e^{i\alpha}\mbox{\boldmath$t_{y_{+}}\psi_{23}$}+e^{i\alpha}\mbox{\boldmath$t_{y_{-}}\psi_{33}$}$
Substituting $\psi_{32}$, $\psi_{12}$, $\psi_{23}$ and $\psi_{33}$ from Eq.
(16) in Eq. (17) we get,
$\mbox{\boldmath$(E-\epsilon^{\prime})\psi_{22}$}=\mbox{\boldmath$t_{b}\psi_{11}+t_{f}\psi_{33}$}$
(18)
This represents the difference equation for an infinite linear chain with
modified site energy $\epsilon^{\prime}$ and the forward and backward hopping
strengths ${\bf t_{f}}$ and ${\bf t_{b}}$, respectively. These quantities are
expressed as follows.
${\epsilon^{\prime}}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\epsilon_{A}$}+\mbox{\boldmath$t_{x_{+}}.(E-\epsilon_{B})^{-1}.t_{x_{-}}$}$
$\displaystyle+\mbox{\boldmath$t_{x_{-}}.(E-\epsilon_{B})^{-1}.t_{x_{+}}$}+\mbox{\boldmath$t_{y_{+}}.(E-\epsilon_{B})^{-1}.t_{y_{-}}$}$
$\displaystyle+\mbox{\boldmath$t_{y_{-}}.(E-\epsilon_{B})^{-1}.t_{y_{+}}$}$
$t_{b}$ $\displaystyle=$ $\displaystyle
e^{-2i\alpha}\mbox{\boldmath$t_{x_{-}}.(E-\epsilon_{B})^{-1}.t_{y_{-}}$}$
$\displaystyle+e^{2i\alpha}\mbox{\boldmath$t_{y_{-}}.(E-\epsilon_{B})^{-1}.t_{x_{-}}$}$
$t_{f}$ $\displaystyle=$ $\displaystyle
e^{-2i\alpha}\mbox{\boldmath$t_{x_{+}}.(E-\epsilon_{B})^{-1}.t_{y_{+}}$}$
$\displaystyle+e^{2i\alpha}\mbox{\boldmath$t_{y_{+}}.(E-\epsilon_{B})^{-1}.t_{x_{+}}$}$
As the translational invariance is preserved in this decimated infinite linear
chain, the solution will be of Bloch form and can be written as,
$\mbox{\boldmath$\psi_{n}$}=\sum_{k}e^{ikna}\left(\begin{array}[]{c}\psi_{k,\uparrow}\\\
\psi_{k,\downarrow}\end{array}\right)$ (20)
$\psi_{n}$ being a short form of $\psi_{nn}$.
Using this form of $\psi_{n}$, the difference equation for an arbitrary site
$n$ can be expressed as,
$\displaystyle\sum_{k}\mbox{\boldmath$(E-\epsilon^{\prime})$}\left(\begin{array}[]{c}\psi_{k,\uparrow}\\\
\psi_{k,\downarrow}\end{array}\right)e^{ikna}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$t_{f}$}\sum_{k}\left(\begin{array}[]{c}\psi_{k,\uparrow}\\\
\psi_{k,\downarrow}\end{array}\right)e^{ik(n+1)a}$ (25) $\displaystyle+$
$\displaystyle\mbox{\boldmath$t_{b}$}\sum_{k}\left(\begin{array}[]{c}\psi_{k,\uparrow}\\\
\psi_{k,\downarrow}\end{array}\right)e^{ik(n-1)a}$ (28)
For the non-trivial solution of Eq. (LABEL:d7) we have the relation,
${\bf Det[M]}=0$ (30)
where,
$M$=$(\mbox{\boldmath$E$}-\mbox{\boldmath$\epsilon^{\prime}$}-\mbox{\boldmath$t_{f}$}e^{ika}-\mbox{\boldmath$t_{b}$}e^{-ika})$
.
Expanding Eq. (30) we obtain a $4$-th degree polynomial in $E$ and solving it
we get the $E$-$k$ dispersion relation in terms of the parameters $\phi$ and
$t_{so}$. The solutions correspond to energy eigenstates which are linear
combinations of up $(|k\uparrow\rangle)$ and down $(|k\downarrow\rangle)$
states.
Following the above analytical treatment, in Fig. 3 we show the $E$ versus $k$
dispersion curves for some typical parameter values of $\phi$ and $t_{so}$.
The first column corresponds to the results for some specific values of AB
flux $\phi$ in the absence of Rashba SO coupling, i.e., $t_{so}=0$. It is
observed that for zero magnetic flux the spectrum is degenerate and gapless,
whereas a small non-zero flux $(\phi=0.2)$ opens a gap symmetrically around
$E=0$ preserving the degeneracy. Here we choose $\epsilon_{A}=\epsilon_{B}=0$
and the gap appears symmetrically as long as $\phi$ is introduced, but the
point is that for unequal values of $\epsilon_{A}$ and $\epsilon_{B}$ gap
always appears even in the absence of $\phi$ (which is not shown in the
figure). The width of the gap increases symmetrically with the rise in $\phi$.
In the second column of Fig. 3 we present the energy dispersion curves for the
three different values of Rashba SO coupling strength keeping $\phi=0$, where
the upper, middle and lower spectra correspond to $t_{so}=0$, $2$ and $4$,
respectively. The upper spectrum is gapless and degenerate as described
earlier. For non-zero values of $t_{so}$, the energy spectra get splitted
vertically and all the degeneracies are removed except at the points $k=n\pi$,
where $n=0$, $\pm 1$, $\pm 2$, $\dots$. The gap becomes widened with the
increase in Rashba strength as clearly noticed from the middle and lower
spectra. The above features seem to be more interesting when a non-zero
magnetic flux is applied. In this case, each sub-band gets separated
vertically as illustrated in the third column of Fig. 3. With a
Figure 4: (Color online). Variations of (a) $g_{\uparrow\uparrow}$ and (b)
$g_{\uparrow\downarrow}$ with AB flux $\phi$ for a diamond chain considering
$5$ plaquettes at the typical energy $E=5$. The green and red curves
correspond to $t_{so}=0$ and $2$, respectively. Here we set
$\epsilon_{A}=\epsilon_{B}=0$.
sufficiently high magnetic flux ($\phi=0.4$) and Rashba strength ($t_{so}=4$)
two additional gaps occur in the dispersion spectrum along with the previous
one, and these gaps can be controlled externally by tuning the AB flux or the
Rashba strength. We will show that these results are quite significant so far
as the designing of nanoscale spintronic devices are concerned.
### III.2 Variation of conductances with AB flux
In Fig. 4 we plot conductance-flux characteristics for a diamond network both
in the presence and absence of Rashba SO interaction. The results are computed
at a typical energy $E=5$ considering five diamond plaquettes, where the green
and red curves correspond to $t_{so}=0$ and $2$, respectively. It is observed
that in the absence of Rashba coupling spin flip conductance
$g_{\uparrow\downarrow}$ drops exactly to zero for the entire range of $\phi$
(green curve in Fig. 4(b), coincides with the $\phi$ axis), while spin
conserved conductance $g_{\uparrow\uparrow}$ persists and it provides
$\phi_{0}$ flux-quantum periodicity as a function of $\phi$. Interestingly we
see that $g_{\uparrow\uparrow}$ completely vanishes at $\phi=\phi_{0}/2$ (see
green curve of Fig. 4(a)) due to the complete destructive interference among
the electronic waves passing through different arms of the plaquettes. On the
other hand, in the presence of Rashba SO interaction both
$g_{\uparrow\uparrow}$ and $g_{\uparrow\downarrow}$ have values for wide
ranges of $\phi$ and a significant change in their amplitudes takes place
compared to the case where $t_{so}=0$. In the presence of the SO interaction,
the oscillatory character of the conductances is still preserved providing
traditional $\phi_{0}$ flux-quantum periodicity. The important feature is that
even for non-zero value of $t_{so}$, spin flip conductance disappears at
$\phi=\phi_{0}/2$. Since $g_{\downarrow\downarrow}$ and
$g_{\downarrow\uparrow}$ exhibit exactly identical behavior to those mentioned
for $g_{\uparrow\uparrow}$ and $g_{\uparrow\downarrow}$, respectively, we do
not display these results explicitly.
The above numerical results can be justified from the following mathematical
analysis.
To illustrate the behaviors of AB oscillation both in the presence and absence
of Rashba SO interaction, we consider an ideal $1$D square loop threaded by an
AB flux $\phi$, as shown schematically in Fig. 5. Two semi-infinite one-
dimensional leads are connected at the vertices P and Q of the square loop.
Figure 5: (Color online). A single diamond plaquette threaded by an AB flux
$\phi$. $\psi_{i}$ and $\psi_{o}$ denote the incoming and outgoing waves,
respectively.
Rashba SO interaction is considered to be present only in the loop and not in
the leads. If $\psi_{i}$ and $\psi_{o}$ describe the incoming and outgoing
wave functions at the respective vertices P and Q, then $\psi_{o}$ can be
obtained considering only $1$st order tunneling processes as xie ,
$\mbox{\boldmath$\psi_{0}$}=\frac{1}{2}\left(e^{-i\frac{\gamma}{2}}\mbox{\boldmath$R_{x}$}(\theta)\mbox{\boldmath$R_{y}$}(\theta)+e^{i\frac{\gamma}{2}}\mbox{\boldmath$R_{y}$}(\theta)\mbox{\boldmath$R_{x}$}(\theta)\right)\mbox{\boldmath$\psi_{i}$}$
(31)
where, $\gamma=\frac{2\pi\phi}{\phi_{0}}$. $R_{\hat{r}}$$(\theta)$ is the
rotation operator defined by the relation,
$\mbox{\boldmath$R_{\hat{r}}$}(\theta)=\mbox{\boldmath$I$}\cos\frac{\theta}{2}-i\hat{r}.\vec{\sigma}\sin\frac{\theta}{2}$
(32)
where, $\theta=\frac{2m^{*}\alpha_{R}L}{\hbar^{2}}$ is the spin precession
angle. $\alpha_{R}$ is the strength of Rashba SO interaction and $L$
represents the length of each side of the square loop. The wave functions
$\psi_{i}$ and $\psi_{o}$ used in Eq. (31) are defined as follows.
$\psi_{o}$ $=\left(\begin{array}[]{c}\psi_{o,\uparrow}\\\
\psi_{o,\downarrow}\end{array}\right)$ and $\psi_{i}$
$=\left(\begin{array}[]{c}\psi_{i,\uparrow}\\\
\psi_{i,\downarrow}\end{array}\right)$.
Following Eq. (32) the matrices $R_{x}$$(\theta)$ and $R_{y}$$(\theta)$ can be
written as given below.
$R_{x}$$(\theta)=\left(\begin{array}[]{cc}\cos\frac{\theta}{2}&-i\sin\frac{\theta}{2}\\\
-i\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{array}\right)$.
and
$R_{y}$$(\theta)=\left(\begin{array}[]{cc}\cos\frac{\theta}{2}&-\sin\frac{\theta}{2}\\\
\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{array}\right)$.
With these matrix forms we can express the wave functions
$|\psi_{o,\uparrow}\rangle$ and $|\psi_{o,\downarrow}\rangle$ as linear
combinations of $|\psi_{i,\uparrow}\rangle$ and $|\psi_{i,\downarrow}\rangle$
by expanding Eq. (31) as,
$\displaystyle|\psi_{o,\uparrow}\rangle$ $\displaystyle=$ $\displaystyle
c_{\uparrow\uparrow}|\psi_{i,\uparrow}\rangle+c_{\downarrow\uparrow}|\psi_{i,\downarrow}\rangle$
$\displaystyle|\psi_{o,\downarrow}\rangle$ $\displaystyle=$ $\displaystyle
c_{\uparrow\downarrow}|\psi_{i,\uparrow}\rangle+c_{\downarrow\downarrow}|\psi_{i,\downarrow}\rangle$
(33)
where, the co-efficients $c_{\sigma\sigma^{\prime}}$ are functions of $\theta$
and $\phi$.
Now the probability of getting an up spin electron at the point Q, for the
incidence of an electron with up spin at the point P, i.e., the spin conserved
transmission probability $T_{\uparrow\uparrow}$ is proportional to
$|\langle\psi_{i,\uparrow}|\psi_{o,\uparrow}\rangle|^{2}$, viz,
$|c_{\uparrow\uparrow}|^{2}$. Similarly, the probability of getting a down
spin electron with up spin incidence, i.e., the spin flip transmission
probability $T_{\uparrow\downarrow}$ is proportional to
$|\langle\psi_{i,\uparrow}|\psi_{o,\downarrow}\rangle|^{2}$, viz,
$|c_{\uparrow\downarrow}|^{2}$. After a few mathematical steps the quantities
$|c_{\uparrow\uparrow}|^{2}$ and $|c_{\uparrow\downarrow}|^{2}$ are expressed
as,
$\displaystyle|c_{\uparrow\uparrow}|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{8}e^{-i\frac{2\pi\phi}{\phi_{0}}}\left[1+i\cos\theta+e^{i\frac{2\pi\phi}{\phi_{0}}}(i+\cos\theta)\right]$
(34)
$\displaystyle\times\left[\cos\theta-i+e^{i\frac{2\pi\phi}{\phi_{0}}}(1-i\cos\theta)\right]$
and
$|c_{\uparrow\downarrow}|^{2}=\frac{1}{8}e^{-i\frac{2\pi\phi}{\phi_{0}}}\left(1+e^{i\frac{2\pi\phi}{\phi_{0}}}\right)^{2}\sin^{2}\theta.$
(35)
In the absence of Rashba SO interaction $\theta=0$ and the above two equations
can be simplified as follows.
$|c_{\uparrow\uparrow}|^{2}=\frac{1}{2}\left[1+\cos\left(\frac{2\pi\phi}{\phi_{0}}\right)\right]$
(36)
and
$|c_{\uparrow\downarrow}|^{2}=0$ (37)
With the last four mathematical expressions (Eqs. (34)-(37)) we can clearly
justify the essential features those are presented in Fig. 4. In the absence
of Rashba SO interaction, spin flip conductance vanishes for the entire range
of $\phi$ (coincident green curve of Fig. 4(b) with $\phi$ axis) in accordance
with Eq. (37). On the other hand, a oscillatory character of up spin
conductance with $\phi_{0}$ periodicity in the absence of $t_{so}$ (green
curve of Fig. 4(a)) follows from Eq. (36). The vanishing behavior of
$g_{\uparrow\uparrow}$ at the typical flux $\phi=\phi_{0}/2$ is also justified
from Eq. (36). In the presence of SO interaction, both pure spin transmission
and spin flip transmission get modified satisfying Eqs. (34) and (35),
respectively. For finite value of $t_{so}$, spin flip conductance always
vanishes at $\phi=\phi_{0}/2$ obeying Eq. (35).
### III.3 Conductance-energy characteristics
Now we focus our attention on the conductance-energy characteristics of a
finite sized diamond network for some specific values of AB flux $\phi$ and
Rashba SO interaction strength $t_{so}$.
In Fig. 6 we plot up spin conductances ($g_{\uparrow\uparrow}$) as a function
of injecting electron energy ($E$) for a diamond network
Figure 6: (Color online). Conductance-energy ($g_{\uparrow\uparrow}$-$E$)
characteristics in the absence of Rashba SO strength for a diamond network
considering $15$ diamond plaquettes with $\epsilon_{A}=\epsilon_{B}=0$. (a),
(b) and (c) correspond to $\phi=0$, $0.2$ and $0.4$, respectively.
considering $15$ identical plaquettes in the absence of Rashba interaction.
The top, middle and bottom spectra correspond to AB flux $\phi=0$, $0.2$ and
$0.4$, respectively. When $\phi=0$, the spectrum is gapless (Fig. 6(a)). The
presence of $\phi$ opens a gap and the width of the gap increases with the
rise in $\phi$ as evident from
Figure 7: (Color online). $g_{\uparrow\uparrow}$ and $g_{\uparrow\downarrow}$
as a function of energy $E$ for a diamond chain with $15$ plaquettes
considering $\epsilon_{A}=\epsilon_{B}=0$ in the absence of AB flux $\phi$.
The $1$st, $2$nd and $3$rd rows represent the results when $t_{so}=0$, $2$ and
$4$, respectively.
Figure 8: (Color online). Spin conserved ($g_{\uparrow\uparrow}$,
$g_{\downarrow\downarrow}$) and spin flip conductances
($g_{\uparrow\downarrow}$, $g_{\downarrow\uparrow}$) as a function of energy
$E$ for a diamond chain with $15$ plaquettes when $\phi$ and $t_{so}$ are
fixed to $0.4$ and $4$, respectively. The parameters $\epsilon_{A}$ and
$\epsilon_{B}$ are set to $0$.
Figs. 6(b) and (c). This gap is symmetric around the energy $E=0$ with the
choice $\epsilon_{A}=\epsilon_{B}=0$. On the other hand, if the site energies
$\epsilon_{A}$ and $\epsilon_{B}$ are unequal then a gap in the conductance
spectrum appears (not shown in the figure) even in the absence of magnetic
flux both for an infinite as well as for a finite sized diamond array. This
gap will be symmetric across $E=0$ provided $\epsilon_{A}$ and $\epsilon_{B}$
are identical, and the width of the gap increases symmetrically about the
center of the gap with the enhancement in $\phi$. In this particular case we
do not consider any Rashba interaction, and therefore, no spin flip
transmission takes place.
In Fig. 7 we show the conductance-energy characteristics of a diamond network
with $15$ plaquettes for different values of $t_{so}$ in the absence of
magnetic flux $\phi$. The $1$st, $2$nd and $3$rd rows correspond to the
results when $t_{so}=0$, $2$ and $4$, respectively. The spin conserved
conductances ($g_{\uparrow\uparrow}$) are plotted in the first column, while
in the second column spin flip conductances ($g_{\uparrow\downarrow}$) are
given. In absence of $t_{so}$, gapless spectrum is observed for spin conserved
conductance, while spin flip conductance vanishes for the entire energy range.
For all other cases, a gap appears in the spectrum and its width can be
regulated by tuning the Rashba coupling strength.
The most interesting features in the conductance-energy characteristics are
observed when we consider the effects of both the AB flux $\phi$ and Rashba SO
coupling $t_{so}$. The results are shown in Fig. 8 for a diamond chain with
$15$ identical diamond plaquettes for $\phi=0.4$ and $t_{so}=4$. For
sufficiently high AB flux and Rashba strength, two additional energy gaps
occur at the flanks on both sides of the conductance spectrum in addition to
the central gap. The energy gaps are positioned identically in all these
spectra.
It is important to note that when anyone of $\phi$ and $t_{so}$ is zero and
other is non-zero, $g_{\uparrow\uparrow}$ becomes exactly identical to
$g_{\downarrow\downarrow}$, and so is $g_{\uparrow\downarrow}$ and
$g_{\downarrow\uparrow}$. On the other hand, when both are non-zero, spin
conserved conductances differ in magnitude, but the spin flip conductances
remain identical. All these conductance-energy characteristics shown in Figs.
6-8 are compatible with the $E$-$k$ diagrams presented in Fig. 3. The gaps of
the conductance spectra of finite sized diamond chain compare well with those
of the dispersion curves obtained earlier for an infinite sized diamond chain.
### III.4 DOS-energy characteristics
To gain insight into the nature of energy eigenstates of such a quantum
network we address the behavior of average density of states. It is expressed
as,
$\rho_{av}(E)=-\frac{1}{\pi N}{\mbox{Im}}[{\mbox{Tr}}[{\bf G}]]$ (38)
where, $N$ being the total number of atomic sites in the diamond chain.
As illustrative examples, in Fig. 9 we present the variations of average
density of states as a function of energy $E$ for a diamond chain
Figure 9: (Color online). Average density of states as a function of energy
$E$ for a diamond chain considering $15$ plaquettes with different values of
$\phi$ and $t_{so}$ when $\epsilon_{A}=\epsilon_{B}=0$. (a) $\phi=0$,
$t_{so}=0$; (b) $\phi=0.2$, $t_{so}=0$; (c) $\phi=0$, $t_{so}=2$ and (d)
$\phi=0.4$, $t_{so}=4$.
consisting of $15$ plaquettes for different values of AB flux $\phi$ and
Rashba SO coupling strength $t_{so}$. In (a), $\rho$-$E$ spectrum is given
when both $\phi$ and $t_{so}$ are fixed at zero. The spectrum does not exhibit
any gap as expected when $\epsilon_{A}=\epsilon_{B}=0$. A sharp peak is
observed at the band center, i.e., at $E=0$ due to localized states. These
localized states are highly degenerate and in general pinned at the energy
$E=\epsilon_{B}$. The existence of the localized state is a characteristic
feature of diamond network as mentioned in an earlier work sil . In (b) and
(c), energy gaps appear symmetrically around the central peak at $E=0$ by the
AB flux $\phi$ and Rashba coupling strength $t_{so}$, respectively. By tuning
the AB flux $\phi$ or Rashba coupling strength $t_{so}$, the width of the gap
can be controlled. Finally, in (d) we display average DOS when both $\phi$ and
$t_{so}$ are finite. In such situation two extra gaps appear together with the
central ones. The localized states are still situated at the same place as
earlier.
### III.5 Effect of Rashba spin-orbit interaction on localization
In such a quantum network, AB flux can induce complete localization. At
$\phi=\phi_{0}/2$, conductance drops exactly to zero in the absence of Rashba
SO interaction. This is due to the complete
Figure 10: (Color online). Average density of states as a function of energy
for a diamond network considering $15$ plaquettes with
$\epsilon_{A}=\epsilon_{B}=0$ when AB flux $\phi$ is set at $\phi_{0}/2$. (a)
$t_{so}=0$ and (b) $t_{so}=2$.
destructive interference between the electronic waves passing through
different arms of the network. At $\phi=\phi_{0}/2$, two more sharp peaks
appear in the $\rho$-$E$ characteristics due to localized states (Fig. 10(a))
in addition to the previous one pinned at $E=0$. The positions of these
localized states can be evaluated exactly and they are expressed
mathematically as,
$E=\frac{1}{2}\left[(\epsilon_{A}+\epsilon_{B})\pm\sqrt{(\epsilon_{A}+\epsilon_{B})^{2}-4(\epsilon_{A}\epsilon_{B}-4t^{2})}\right].$
(39)
In the presence of Rashba spin orbit interaction, the interference is not
completely destructive anymore at $\phi=\phi_{0}/2$. The two additional peaks
at the opposite sides of the central one disappear for non-zero Rashba
strength as clearly seen from Fig. 10(b). Rashba spin-orbit coupling affects
the spin dynamics significantly resulting in a non-zero conductance at
$\phi=\phi_{0}/2$.
### III.6 Rashba induced semi-conducting behavior
Here we address how Rashba SO interaction can induce semi-conducting behavior
in such a quantum network in the absence of $\phi$. A similar type of semi-
conducting nature controlled by AB flux has been established in such a system,
where SO interaction was not considered sil . To establish our idea, in
Figure 11: (Color online). Average density of states as a function of energy
for a diamond chain considering $15$ plaquettes in the absence of AB flux
$\phi$ where $\epsilon_{A}$ and $\epsilon_{B}$ are fixed at $0$ and $2$,
respectively. (a) $t_{so}=0$ and (b) $t_{so}=2$.
Fig. 11 we plot the average density of states as a function of energy
considering $15$ plaquettes where $\epsilon_{A}$ and $\epsilon_{B}$ are set at
$0$ and $2$, respectively. When $\epsilon_{A}$ and $\epsilon_{B}$ are not
same, the diamond network possesses an intrinsic gap in the energy spectrum
even in the absence of $\phi$ and $t_{so}$, as evident from Fig. 11(a). The
sharp peak in the DOS is situated at the edge of the gap, $E=\epsilon_{B}=2$
and it actually corresponds to the localized states. By controlling the
external gate potential, i.e., tuning the Rashba strength to a non-zero value,
the width of the gap can be increased arbitrarily as shown in Fig. 11(b),
keeping the position of the localized states invariant. Now, if the Fermi
level $E_{F}$ is fixed at $E=2$ where we have localized states (see Fig.
11(b)), then for small Rashba coupling strength the gap between the localized
level and the bottom of the right sub-band can be made small enough for the
electrons to bridge. Therefore, the system behaves as a $n$-type
semiconductor. Similarly, if $\epsilon_{B}$ is fixed at $-2$ and the Fermi
level is set at the top of the left sub-band, then the system can be
implemented equivalently as a $p$-type semiconductor. In this case holes are
created in the left sub-band. It is important to mention that when the site
energies ($\epsilon_{A}$ and $\epsilon_{B}$) are fixed at the same value, then
also the system can be used as a semi-conductor depending on the electron
concentration. The detailed analysis is available in Ref. sil .
### III.7 Spin filtering action
With proper tuning of the external parameters like, magnetic flux $\phi$ and
Rashba strength $t_{so}$, a diamond network can achieve a high degree of spin
polarization as discussed earlier in a theoretical work by Aharony et al.
aharony . Here, we discuss this feature from a different point of view.
When there is no external magnetic field or magnetic flux, time reversal
symmetry is not broken, and the Hamiltonian of the system remains
Figure 12: (Color online). Variation of conductances as a function of energy
for a diamond chain with $3$ plaquettes considering $\phi=0.3$ and $t_{so}=4$
where $\epsilon_{A}=\epsilon_{B}=0$. In (a) $g_{\uparrow\uparrow}$ and
$g_{\downarrow\downarrow}$ and in (b) $g_{\uparrow\downarrow}$ and
$g_{\downarrow\uparrow}$ are superposed to each other.
invariant under time reversal operation. Mathematically it is expressed as
$[H_{SO},T]=0$, where $T$ is the time reversal operator. The Rashba
Hamiltonian ($H_{SO}$) is usually written as,
$\displaystyle H_{SO}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{R}}{\hbar}\left(\vec{\sigma}\times\vec{p}\right)_{Z}$
(40) $\displaystyle=$ $\displaystyle
i\alpha_{R}\left(\sigma_{y}\frac{\partial}{\partial
x}-\sigma_{x}\frac{\partial}{\partial y}\right)$
and $T=i\sigma_{y}\hat{C}$, $\hat{C}$ being the complex conjugation operator.
The second quantized form of Eq. (40) is given by the fourth and fifth terms
of Eq. (2). As electrons are spin-$\frac{1}{2}$ particles, so following
Kramer’s theorem, each eigenstate is at least two-fold degenerate and spin is
no longer a good quantum number in the presence of spin-orbit interaction. In
many cases the degeneracy implied by Kramer’s theorem is merely the degeneracy
between states of spin up and spin down, or something equally obvious. The
theorem is non-trivial for a system with spin-orbit coupling in an
unsymmetrical electric field, so that neither nor angular momentum is
conserved. Kramer’s theorem implies that no such field can split the
degenerate pairs of energy levels ballen .
However, the degeneracy can be removed by applying external magnetic flux or
magnetic field as in this case time reversal symmetry is not conserved anymore
and therefore spin polarization can be achieved. The degree of polarization of
the transmitted electrons is conventionally defined as,
$P(E)=\left|\frac{(g_{\uparrow\uparrow}+g_{\downarrow\uparrow})-(g_{\downarrow\downarrow}+g_{\uparrow\downarrow})}{(g_{\uparrow\uparrow}+g_{\downarrow\uparrow})+(g_{\downarrow\downarrow}+g_{\uparrow\downarrow})}\right|.$
(41)
For our illustrative purpose, in Fig. 12 we plot the variations of
conductances as a function of energy for a diamond network with three
plaquettes considering $\phi=0.3$ and $t_{so}=4$. A significant change is
observed in the magnitudes of spin conserved conductances
($g_{\uparrow\uparrow}$ and $g_{\downarrow\downarrow}$) (Fig. 12(a)), while
the spin flip conductances ($g_{\uparrow\downarrow}$ and
$g_{\downarrow\uparrow}$) are identical as shown in Fig. 12(b). Therefore,
applying a non-zero flux spin polarization is clearly obtained. Following Eq.
(41) we calculate the degree of polarization for an arbitrary energy $E=-5$
(say), and it is about $44\,\%$.
## IV Closing remarks
To conclude, in the present work we have explored spin dependent transport
through an array of diamonds where Rashba SO interaction is present and each
diamond plaquette is threaded by an AB flux $\phi$. The diamond chain is
directly coupled to two semi-infinite $1$D non-magnetic metallic leads,
namely, source and drain. We have adopted a discrete lattice model within the
tight-binding framework to describe the system and present calculations based
on Green’s function formalism. We have obtained analytical expression for the
$E$-$k$ dispersion relation for an infinite diamond network with Rashba SO
interaction, and, show explicitly the interplay of spin-orbit interaction and
magnetic flux on its band structure. This analysis also gives insight about
the presence of spin dependent localized and extended eigenstates which
crucially controls the spin dependent transport through such device. This
analytical study, in fact, provides us a very good understanding about the
transport behavior of spins across a finite sized array of diamonds. It has
been clearly established that how delocalizing effect sets in due to Rashba SO
interaction when the AB flux $\phi$ is $\phi_{0}/2$. Quite interestingly we
show that depending on the specific choices of SO interaction strength and AB
flux, the quantum network can be utilized as a spin filter.
In the present work we have ignored the effects of temperature, electron-
electron correlation, electron-phonon interaction, disorder, etc. Here, we set
the temperature at $0$K, but the basic features will not change significantly
even at low temperature as long as thermal energy ($k_{B}T$) is less than the
average level spacing of the diamond chain. In this model it is also assumed
that the two side-attached non-magnetic leads have negligible resistance. Our
presented results may be useful in designing spin based nano electronic
devices.
## References
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* (16) J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys. Rev. Lett. 78, 1335 (1997).
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* (23) D. Bercioux, M. Governale, V. Cataudella, and V. M. Ramaglia, Phys. Rev. B 72, 075305 (2005).
* (24) A. Aharony, O. E. -Wohlman, Y. Tokura, and S. Katsumoto, Phys. Rev. B 78, 125328 (2008).
* (25) S. Sil, S. K. Maiti, and A. Chakrabarti, Phys. Rev. B 79, 193309 (2009).
* (26) Z. Gulácsi, A. Kampf, and D Vollhardt, Phys. Rev. Lett. 99, 026404 (2007).
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* (28) P. Földi, O. Kálmán, M. G. Benedict, and F. M. Peeters, Nano Lett. 8, 2556 (2008).
* (29) P. Földi, O. Kálmán, and F. M. Peeters, Phys. Rev. B 80, 125324 (2009).
* (30) B. Molnár, P. Vasilopoulos, and F. M. Peeters, Phys. Rev. B 72, 075330 (2005).
* (31) S. Datta, Quantum Transport: Atom to Transistor, Cambridge University Press, Cambridge (2005).
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|
arxiv-papers
| 2010-07-06T16:32:05 |
2024-09-04T02:49:11.445064
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Moumita Dey, Santanu K. Maiti and S. N. Karmakar",
"submitter": "Santanu Maiti K.",
"url": "https://arxiv.org/abs/1007.0943"
}
|
1007.1110
|
arxiv-papers
| 2010-07-07T11:56:59 |
2024-09-04T02:49:11.456974
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mehdi Nadjafikhah and Parastoo Kabinejad",
"submitter": "Mehdi Nadjafikhah",
"url": "https://arxiv.org/abs/1007.1110"
}
|
|
1007.1111
|
###### Abstract
Classification of $n-$th $(n\geq 2)$ order linear ODEs is considered. The
equation reduced to Laguerre Forsyth form by a point transformation then, the
other calculations would have done on this form. This method is due to V.A.
Yumaguzhin.
Keywords: Linear ODE, symmetry, Lie algebra, projective transformations.
© 200x Published by Islamic AZAD University-Karaj Branch.
_Mathematical Sciences_ _Vol. 1, No. 1,2 (2007) 01-12_
Classification of $n-$th order linear ODEs up to projective transformations
Mehdi Nadjafikhaha,111Corresponding Author. E-mail Address:
m$\\_$nadjafikhah@iust.ac.ir , Seyed Reza Hejazib
aFact. of Math., Dept. of Pure Math., Iran University of Science and
Technology, Narmak, Tehran, I.R. Iran.
bSame address.
## Introduction
The local classification of linear ODEs up to projective transformations is
obtained in this article. For $n\leq 2$, it is well known that any $n-$th
order linear ODE can be transformed locally to the form $y^{(n)}=0$ by a point
transformation. For $n\geq 0$, this statement is incorrect: there is finite
number of different equivalence classes of linear ODEs.
First this problem was posed by classics of the 15 century E. Laguerre, G.H.
Halphen and others. They obtain results concerning classification of third and
fourth orders linear ODE. Here, this problem is solved for $n\geq 0$ in a
neighborhood of regular germs.
Consider a general $n-$th order ODE which is solved by the higher order
derivative
$\displaystyle y^{(n)}=\sum_{i=1}^{n}a_{n-i}(x)y^{(n-i)},$ (1)
where $y(x)$ is a smooth function of $x$.
Lie shows that the point symmetry group of a second ordinary linear
differential equation has dimension at most eight, conversely the equation
admits an eight-dimensional symmetry group if and only if it can be mapped, by
a point transformation, to the linear equation $y^{\prime\prime}=0$. Thus, the
main result is any linear second ordinary differential equation can mapped to
the equation $y^{\prime\prime}=0$. So, the condition of second ordinary linear
differential equation is specified.
A same result shows that for $n\geq 3$, any linear ODE admits at most an
$(n+4)-$dimensional symmetry group of point transformation, therefore, the
symmetry group is $(n+4)-$dimensional if and only if the equation is
equivalent to the linear equation $y^{(n)}=0$. In continuation we will work on
the general form of linear ODE in the form of (1) once $n\geq 3$.
## 1 Laguerre-Forsyth form
The classification of linear differential equations is a special case of the
general problem of classifying differential operators, which has a variety of
important applications. Consider an $n-$th order ordinary differential
operator corresponding to (1)
$\displaystyle{\cal
D}=a_{n}(x)D_{x}^{n}+a_{n-1}(x)D_{x}^{n-1}+\cdots+a_{1}(x)D_{x}+a_{0}.$ (2)
The aim is finding out when two operators, or two linear ODE, of type (2), can
be mapped to each other by a suitable change of variables. To preserve
linearity, we restrict to those of the form
$\displaystyle\bar{x}=\varphi(x),\qquad\bar{y}=\psi(x)y,$ (3)
the chain rule action shows that $D_{\bar{x}}=(\varphi(x))^{-1}D_{x}$, and
with a rescaling of the dependent variable by
$\displaystyle\psi(x)=e^{\varphi(x)}$ we obtain the gauge factor. So, two
differential operator $\overline{\cal D}$ and $\cal D$ is called gauge
equivalent if they satisfy
$\displaystyle{\overline{\cal D}}=\psi\cdot{\cal
D}\cdot\frac{1}{\psi},\qquad\bar{x}=\varphi(x).$ (4)
A straightforward calculation shows that the change of variables (3) is given
by
$\displaystyle\bar{x}=\varphi(x)=\int\frac{dx}{\sqrt[n]{|a_{n}(x)|}},\;\;\;\;\psi(x)=|a_{n}(x)|^{\frac{1-n}{2n}}\exp\Big{\\{}\int^{x}\frac{a_{n-1}(y)}{na_{n}(y)}dy\Big{\\}},$
thus (1) is gauge equivalent to an operator of the form
$\displaystyle{\cal D}=\pm D_{x}^{n}+a_{n-2}(x)D_{x}^{n-2}+\cdots+a_{0}(x).$
(5)
If $\rho(x)$ be a nonvanishing smooth function, two differential operator
$\overline{\cal D}$ and $\cal D$ is called projective equivalence if they
satisfy
$\displaystyle{\overline{\cal D}}=\rho\cdot\psi\cdot{\cal
D}\cdot\frac{1}{\psi},\qquad\bar{x}=\varphi(x).$ (6)
A nonsingular $n-$th order linear operator of type (5) is projectively
equivalent to one in Laguerre-Forsyth form
$\displaystyle{\cal D}=D_{x}^{n}+a_{n-3}(x)D_{x}^{n-3}+\cdots+a_{0}(x),$ (7)
with change of variable (6) in the form of
$\displaystyle\bar{x}=\varphi(x),\qquad\bar{y}=\varphi_{x}^{\frac{n-1}{2}}y,\qquad\rho=\varphi_{x}^{-n},$
where $\varphi(x)$ is a solution of the Schwarzian equation
$\displaystyle\frac{n(n^{2}-1)}{12}\frac{\varphi_{x}\varphi_{xxx}-\frac{3}{2}\varphi_{xx}^{2}}{\varphi_{x}^{2}}=a_{n-2}(x).$
## 2 Classification of linear ODEs of Laguerre-Forsyth form
A useful theorem help us to reduce the classification of ODEs up to a special
transformation.
###### Theorem 2.1
Let $\Delta_{1}$ and $\Delta_{2}$ be ODEs of the form (7). If there is a point
transformation that takes $\Delta_{1}$ to $\Delta_{2}$, that is
$\displaystyle
f(x)=\frac{ax+b}{cx+d},\quad\hat{f}(x,y)=|f^{\prime}|^{\frac{n-1}{2}}\cdot
y,\quad a,b,c,d\in\bf{R}.$ (8)
A transformation $(f,\hat{f})$ of the form (8) is generated by a projective
transformation $f$ on $\bf R$. The isomorphisms $f\rightarrow(f,\hat{f})$
makes a group of point transformations in the form of (8). Consider these
projective transformations in a group $G$ and denoted by all projective
transformations of $\bf R$,i.e.,
$\displaystyle G=\Big{\\{}f(x)=\frac{ax+b}{cx+d}\Big{|}a,b,c,d\in{\bf
R}\;\mbox{and}\;ad\neq bc\Big{\\}}.$
It is easy to check that $G$ has two connected component $G_{1}=\\{f\in
G|f^{\prime}>0\\}$ and $G_{2}=\\{f\in G|f^{\prime}<0\\}$, thus, $G=G_{1}\cup
G_{2}$.
### 2.1 Bundles of Laguerre-Forsyth form
Consider $x$ as a coordinate on $\bf R$ and $a_{n-3},a_{n-2},...,a_{0}$
coordinates on ${\bf R}^{n-2}$. Then, we can construct a fiber bundle
corresponding to (7) in the form of
$\displaystyle p:{\bf R}\times{\bf R}^{n-2}\rightarrow{\bf R}.$ (9)
Any ODE of type (7) identifies with
$\Delta=\\{p_{n}=a_{n-3}(x)p_{n-3}+\cdots+a_{0}(x)p_{0}\\}$ is a section of
(9) denoted by $S_{\Delta}:x\rightarrow(x,a_{n-3}(x),...,a_{0}(x))$, where the
identification $\Delta\rightarrow S_{\Delta}$ is a bijection.
Let
$\Delta_{2}=\\{\tilde{p}_{n}=\tilde{a}_{n-3}(\tilde{x})\tilde{p}_{n-3}+\cdots+\tilde{a}_{0}(\tilde{x})\tilde{p}_{0}\\}$
be an ODE of the form (7). Subjecting $\Delta_{2}$ to an transformation (8),
the, we obtain linear ODE
$\Delta_{1}=\\{p_{n}=a_{n-3}(x)p_{n-3}+\cdots+a_{0}(x)p_{0}\\}$. The
coefficients $\Delta_{2}$ are expressed in terms of coefficients of
$\Delta_{1}$ and projective transformation $f^{-1}$ by the equation
$\displaystyle\tilde{a}_{n-i}=F_{n-i}\Bigg{(}a_{n-3},...,a_{n-i};\frac{df^{-1}}{d\tilde{x}},...,\frac{d^{i+1}f^{-1}}{d\tilde{x}^{i+1}}\Bigg{)},\quad
i=3,4,...,n.$ (10)
The equation (10) is a lifting of a projective transformation $f$ to
diffeomorphism $\bar{f}:{\bf R}\times{\bf R}^{n-2}\rightarrow{\bf R}\times{\bf
R}^{n-2}$ such that $p\circ\bar{f}=f\circ p$.
For any $f\in G$, a transformation of sections of $p$ defined by the formula
$\displaystyle S\rightarrow f(S)=\bar{f}\circ S\circ f^{-1},$
then, equation (10) can be represented as $S_{\Delta_{2}}=f(S_{\Delta_{1}})$.
###### Lemma 2.2
Consider two equation of the form (7). Then a transformation $(f,\hat{f})$ of
the form (8) maps $\Delta_{1}$ to $\Delta_{2}$ if and only if
$S_{\Delta_{2}}=f(S_{\Delta_{1}})$.
The main result of the lemma (2.2) is the classification of ODEs of the form
(7) up to transformation (8) reduces to classification of germs of sections of
$p$ up to projective transformation on $\bf R$.
### 2.2 Classification of regular germs
Let $S$ be a section of $p$ and $a$ be a point in domain of $S$. Denoted by
$\\{S\\}_{a}$ the germ of $p$ at $a$. Let $\\{S\\}_{a_{1}}$ and
$\\{S\\}_{a_{2}}$ be germs of sections $S_{1}$ and $S_{2}$ respectively. We
say that $\\{S\\}_{a_{1}}$ and $\\{S\\}_{a_{2}}$ are $G_{+}$-equivalent if
there exist $f\in G_{+}$ such that
$\\{f(S_{1})\\}_{f(a_{1})}=\\{S\\}_{a_{2}}$. A germ $\\{S\\}_{a}$ is regular
of class i if there exist a neighborhood $\cal O$ of $a$ and subbundle $E_{i}$
such that Im$S|_{\cal O}\subset E_{i}$. If $\\{S\\}_{a}$ is a regular germ of
class $i\geq 0$, then in a neighborhood of $a$ we have
$S(x)=(x,0,...,0,a_{i}(x),...,a_{0}(x))$. In the rest of the paper we will
often denoted $\\{S\\}_{a}$ by $\\{a_{i},...,a_{0}\\}_{a}$. If $\\{S\\}_{a}$
is a regular germ, then $a$ is a regular point of $S$.
###### Definition 2.3
Let $S$ be a section of $p$ and v be a vector field of the Lie algebra of
group $G$, if $\theta_{t}$ be the flow of v, we say v is a projective symmetry
of S if one of the following statements satisfied:
* 1)
$\theta_{t}(S)=\overline{\theta_{t}}\circ S\circ\theta_{t}^{-1}=S$,
* 2)
$\displaystyle\frac{d}{dt}\theta_{t}(S)\Big{|}_{t=0}=0.$
Denote by ${\cal P}(S)$ the Lie algebra of all projective symmetries of $S$.
Let $\Upsilon$ be the set of all regular germs at $0\in\bf R$ of sections of
$p$. Define
$\displaystyle\Upsilon_{i}=\Big{\\{}\\{S\\}_{a}|\mbox{dim}{{\cal
P}(S)}=i\Big{\\}},\quad i=0,1,3,$
and denote $\Upsilon=\Upsilon_{0}\cup\Upsilon_{1}\cup\Upsilon_{3}$. If $G_{0}$
be the isotropic subgroup of $G$ in 0, then, $\Upsilon_{i}$’s are
$G_{0}$-invariant.
Define $\Upsilon_{r,i}\subset\Upsilon_{r}$ be the subset of all regular germs
of class $i$. It follows from the invariance of subbundle $E_{i}$’s under
$G_{0}$, $\Upsilon_{r,i}$ is $G_{0}-$invariant. Consequently we have
$\displaystyle\Upsilon_{r}=\bigcup_{i=0}^{n-3}\Upsilon_{r,i},$
where this union is separated invariant subsets.
Let ${\bf R}_{+}$ and ${\bf R}_{-}$ be the set off positive and negative real
numbers respectively. If $\ell_{r,i}:\Upsilon_{r,i}\rightarrow({\bf
R}\backslash\\{0\\})\times{\bf R}$ be a map by the formula
$\\{a_{i},...,a_{0}\\}\mapsto(a_{i}(0),a_{i}^{\prime}(0))$ and
$\displaystyle G_{0+}\times\Upsilon_{r,i}\rightarrow\Upsilon_{r,i}$ (11)
$\displaystyle(f,\\{S\\}_{0})\mapsto\\{f(S)\\}_{0},$
be the action of $G_{0+}$ on $\Upsilon_{r,i}$ then,
###### Lemma 2.4
The map $\ell_{r,i}|_{\Theta}$ is a bijection from the orbit $\Theta$ of the
action (11) either to $({\bf R}_{+})\times{\bf R}$ or to $({\bf
R}_{-})\times{\bf R}$.
Let $\Omega_{r,i}^{+}=\ell_{r,i}^{-1}((1,0))$ and
$\Omega_{r,i}^{-}=\ell_{r,i}^{-1}((-1,0))$. Denote by $\Gamma_{r,i}$ the
subset of $\Omega_{r,i}^{+}\cup\Omega_{r,i}^{-}$ defined in the following way:
* 1)
$\Gamma_{r,0}=\Omega_{r,0}^{+}\cup\Omega_{r,0}^{-}$ for $i=0$,
* 2)
if $i>0$, then, $\Omega_{r,i}$ consists of all germs $\\{a_{i},...,a_{0}\\}$
from $\Omega_{r,i}^{+}\cup\Omega_{r,i}^{-}$ satisfying one of the following
conditions:
* i)
$a_{i-j}=0$ for all odd numbers $j$ with $1\leq j\leq i$,
* ii)
there exist an odd number $r$ with $1\leq r\leq i$ such that $a_{i-r}(0)>0$
and if $r>1$, then $a_{i-j}(0)=0$ for all odd numbers $j$ with $1\leq j<r$.
### 2.3 Classification of regular germs from the family
$\displaystyle\bf\Omega_{r,i}$
Let $\mu\in G_{-}$ defined by $\mu(x)=-x$ for all $x\in\bf R$, then, due to
lemma (2.4) and attentive to $\mu(\Omega_{r,i}^{-})=\Omega_{r,i}^{+}$ we have:
###### Theorem 2.5
* 1)
The set $\Omega_{r,i}^{+}\cup\Omega_{r,i}^{-}$ is a family of all germs from
$\Upsilon_{r,i}$ nonequivalent with respect to $G_{0+}$.
* 2)
If $n-i$ is odd, then $\Omega_{r,i}^{+}$ is a family of all germs from
$\Upsilon_{r,i}$ nonequivalent with respect to $G_{0}$.
* 3)
If $n-i$ is even, $\Gamma_{r,i}$ is a family of all germs from
$\Upsilon_{r,i}$ nonequivalent with respect to $G_{0}$.
an important corollary concludes this section as follows:
###### Corollary 2.6
Classification of regular germs of sections of (7) is:
* 1)
The family of germs of the form
$\displaystyle\\{\pm 1+a(x)x^{2},a_{i-1}(x),...,a_{0}\\}_{0}$
is a family of all regular germs of class $i$ nonequivalent with respect to
$G_{0+}.$
* 2)
If $n-i$ is odd, then the family of germs of the form
$\displaystyle\\{1+a(x)x^{2},a_{i-1}(x),...,a_{0}\\}_{0}$
is a family of all regular germs of class $i$ nonequivalent with respect to
$G_{0}.$
* 3)
If $n-i$ is even, then the family of germs of the form
$\displaystyle\\{\pm 1+a(x)x^{2},a_{i-1}(x),...,a_{0}\\}_{0},$
satisfying one of the following conditions:
* a)
$a_{i-j}(0)=0$ for all odd numbers $j$ with $1\leq j\leq i$,
* b)
there exist an odd number $r$ with $1\leq r\leq i$ such that $a_{i-r}(0)>0$
and if $r>1$, then $a_{i-j}(0)=0$ for all odd number $j$ with $1\leq j\leq r$,
is the family of germs of class $i$ nonequivalent with respect to $G_{0}.$
## 3 Conclusion
This article was a qualification of classification of linear ODEs due to V.A.
Yumaguzhin. First we transform the general form of ODEs to Laguerre-Forsyth
form, then by a suitable change of variable up to projective transformation we
reduce this classification to classification of the sections of bundles, next
by construction germs and specially regular germs of this sections near
identity, the classification reduced to classifying of regular germs by
providing some invariant subsets of the bundles.
## References
* [1] Lee, John,M., Introduction to Smooth Manifolds, Springer Verlage, New York, 2002.
* [2] Olver P.J., Equivalence, Invariant and Symmetry, Cambridge University Press, Cambridge 1995.
* [3] Olver P.J., Applications of Lie Groups to Differential equations, Seconed Edition, GTM, Vol. 107, Springer Verlage, New York, 1993.
* [4] Olver P.J., Differential Invariant and Differential Invariant Equations, University of Min-nesuta, 1994.
* [5] Olver P.J., Differential Invariants: Algebraic and Geometric Structure in Differential Equa-tions, P.H.M. Kersten and I.S. Krasil’shchik, eds., Proceeding, University of Twente, 1993, to appear.
* [6] Ovsiannikov, L.V., Group Analysis of Differential Equations, Academic press, New York, 1982.
* [7] Yumaguzhin V.A., Contact classification of linear differential equations. I., Program System Institute, M. Botik, Preslavl-Zalessky, 152020, Russia.
* [8] Yumaguzhin V.A., Contact classification of linear differential equations. II., Program System Institute, M. Botik, Preslavl-Zalessky, 152020, Russia.
* [9] Yumaguzhin V.A., Point transformation and classification of 3rd-order linear ODEs, Russian journal of Mathematical Physics, 4 (1996) No. 3, 403-410.
* [10] Yumaguzhin V.A., Classification of 3rd-order linear ODEs up to equivalence, Journal of Differential Geometry and its applications, 6 (1996) No. 4, 343-350.
|
arxiv-papers
| 2010-07-07T11:57:07 |
2024-09-04T02:49:11.460430
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mehdi Nadjafikhah and Seyed Reza Hejazi",
"submitter": "Mehdi Nadjafikhah",
"url": "https://arxiv.org/abs/1007.1111"
}
|
1007.1113
|
# Group analysis for
generalized reaction-diffusion convection equation
M. Nadjafikhah m_nadjafikhah@iust.ac.ir S. Dodangeh
s_dodangeh@mathdep.iust.ac.ir School of Mathematics, Iran University of
Science and Technology, Narmak, Tehran 1684613114, Iran.
###### Abstract
In this paper we discuss about group classification for non-linear generalized
reaction-diffusion convection equation:
$u_{t}=(f(x,u)u_{x})_{x}+h(x,u)u_{x}+k(x,u)$, by using Lie-classical symmetry
method. For this, first we find its symmetry group and then we find
differential invariants for resulted group by using infinitesimal criterion
method and at the end reduce modeling equations by using resulted invariants.
We present application of this group classification in group classification
and obtaining related similarity solution of KPP equation, too.
###### keywords:
symmetry, group classification, differential invariants, Lie-classical
method,infinitesimal criterion method, RDC equation, KPP equation, similarity
solutions.
††thanks: Corresponding author: Tel. +9821-73913426. Fax +9821-77240472.
, ,
## 1 Introduction
This paper devoted to group classification of Generalized Reaction-Diffusion
Convection (G-RDC) equation, by using Lie-classical method.
$\displaystyle\Delta\;:\;u_{t}=(f(x,u)u_{x})_{x}+h(x,u)u_{x}+k(x,u),$ (1.1)
Where $u(x,t)$ is unknown function and $f(x,u),h(x,u),k(x,u)$ are arbitrary
functions. The equation (1.1) generalizes a number of the well known second-
order evolution equations, describing various process in physics, chemistry
and biology. Symmetry group method plays an important role in the analysis of
differential equations. The history of group classification methods goes back
to Sophus Lie [9]. (See [2, 4, 7, 6]). His work devoted to finding symmetry
groups, differential invariants and linearized or reduced equation for given
model. There are several approach to group classification of differential
equation, we apply infinitesimal method (See [2, 4, 10]) for this. There are
another useful articles and accounts about group classification for similar
equations of (1.1) via other methods, (See [11, 12, 13]). In this paper we
generalize RDC equation to G-RDC equation and applay Lie-classical symmetry
method via applied approach. we hope this work be useful to applied and
theoretical readers.
## 2 Group classification for modeling equation
Let following one-parameter group
$\displaystyle\overline{x}=x+\varepsilon\xi(x,t,u)+O(\varepsilon^{2}),\qquad\overline{t}=t+\varepsilon\eta(x,t,u)+O(\varepsilon^{2}),\qquad\overline{u}=u+\varepsilon\varphi(x,t,u)+O(\varepsilon^{2}),$
(2.2)
be symmetry group of modeling equation $\Delta$. We can obtain $\xi$, $\eta$
and $\varphi$, by using infinitesimal method.
Consider the vector field
$X:=\xi\partial_{x}+\eta\partial_{t}+\varphi\partial_{u}$ in total space
$M=(x,t,u)$ with $p=2$ and $q=1$. If this vector field be an infinitesimal
generator of $\Delta$’s symmetry group, then
$\displaystyle X^{(2)}\Delta=0,\qquad\textrm{whenever}\qquad\Delta=0.$ (2.3)
Where $X^{(2)}$ is second prolong of $X$ and has following form:
$\displaystyle
X^{(2)}=X+\varphi^{x}\partial_{u_{x}}+\varphi^{t}\partial_{u_{t}}+\varphi^{xx}\partial_{xx}+\varphi^{xt}\partial_{u_{xt}}+\varphi^{tt}\partial_{u_{tt}},$
(2.4)
Where $\varphi^{x},\varphi^{t},\varphi^{xx},\varphi^{xt}$ and $\varphi^{tt}$
are respectively:
$\displaystyle\varphi^{x}=D_{x}Q+\xi u_{xx}+\eta u_{xt},\hskip
45.52458pt\varphi^{t}=D_{t}Q+\xi u_{xt}+\eta u_{tt},$
$\displaystyle\varphi^{xx}=D_{xx}Q+\xi u_{xxx}+\eta
u_{xxt},\qquad\varphi^{xt}=D_{xt}Q+\xi u_{xxt}+\eta
u_{xtt},\qquad\varphi^{tt}=D_{tt}Q+\xi u_{xtt}+\eta u_{ttt}.$
Where $D_{x}$, $D_{t}$ are total derivative with respect to specified
variables, $D_{xx}=D_{x}D_{x}$, $D_{xt}=D_{x}D_{t}$ and $D_{tt}=D_{t}D_{t}$,
and $Q=\varphi-\xi u_{x}-\eta u_{t}$ is the corresponding characteristic of
$X$ (See [2, 4, 7, 6]). By using criterion (2.4), we find:
$\displaystyle\varphi^{t}=(f_{xx}u_{x}+f_{xu}u_{x}^{2}+f_{x}u_{xx}+k_{x}+h_{x}u_{x})\xi+(f_{xu}u_{x}+f_{uu}u_{x}^{2}+f_{u}u_{xx}+h_{u}u_{x}+k_{u})\varphi+$
$\displaystyle\hskip
25.6073pt+(f_{x}+2f_{u}u_{x}+h)\varphi^{x}+f\varphi^{xx}.$ (2.5)
By substituting $\varphi^{x},\varphi^{t},\varphi^{xx},\varphi^{xt}$ and
$\varphi^{tt}$ in (2.5), we have following results:
coefficient monomial $1$
$\varphi_{t}-f_{x}\varphi_{x}-f\varphi_{xx}-h\varphi_{x}-k_{x}\xi-
k_{u}\varphi$ $u_{x}$
$\xi_{t}+f_{xx}\xi+f_{xu}\varphi+f_{x}(\varphi_{u}-\xi_{x})+2f_{u}\varphi_{x}+f(2\varphi_{xu}-\xi_{xx})+h(\varphi_{u}-\xi_{x})+h_{x}\xi+h_{u}\varphi$
$u_{t}$ $\varphi_{u}-\eta_{t}+f_{x}\eta_{x}+f\eta_{xx}+h\eta_{x}$ $u_{x}u_{t}$
$\xi_{u}-f_{x}\eta_{u}-2f_{u}\eta_{x}-f\eta_{xu}-h\eta_{u}$ $u_{t}^{2}$
$\eta_{u}$ $u_{x}^{2}$
$-f_{x}\xi_{u}+f_{xu}\xi+f_{uu}\varphi+2f_{u}(\varphi_{u}-\xi_{x})-h\xi_{u}+f(\varphi_{uu}-2\xi_{xu})$
$u_{x}^{3}$ $2f_{u}\xi_{u}+f\xi_{uu}$ $u_{x}^{2}u_{t}$
$2f_{u}\eta_{u}-f\eta_{uu}$ $u_{xx}$
$f_{x}\xi+f_{u}\varphi+f(\varphi_{u}-2\xi_{x})$ $u_{x}u_{xt}$ $2f\eta_{x}$
$u_{x}u_{xx}$ $3f\xi_{u}$ $u_{t}u_{xx}$ $f\eta_{u}$ $u_{xx}u_{x}^{2}$
$2f\eta_{u}$ (Table 1)
By simplifying above equations we obtain:
$\displaystyle\eta=\eta(t),\qquad\xi=\xi(x,t),\qquad\varphi_{u}-\eta_{t}=0,\qquad
f_{x}\xi+f_{u}\varphi+f(\varphi_{u}-2\xi_{x})=0,$
$\displaystyle\varphi_{t}-f_{x}\varphi_{x}-f\varphi_{xx}-h\varphi_{x}-k_{x}\xi-
k_{u}\varphi=0,\qquad f_{xu}\xi+f_{uu}\varphi+2f_{u}(\varphi_{u}-\xi_{x})=0,$
(2.6)
$\displaystyle\xi_{t}+f_{xx}\xi+f_{xu}\varphi+(f_{x}+h)(\varphi_{u}-\xi_{x})+2f_{u}\varphi_{x}-f\xi_{xx}+h_{x}\xi+h_{u}\varphi=0.$
## 3 Group classification in special cases
In this section we consider four special case of modeling equation and obtain
differential invariants for them by using (2) and infinitesimal criterion
method.
#### A:
$f(x,u)=xu^{-1}$, $h(x,u)=-2/u$, $k(x,u)=au+b$, where $a,b$ are constant real
numbers. In this case we have: $\eta=\frac{1}{a}e^{at}.e^{ac_{1}}+c_{2}$,
$\xi=c_{3}\sqrt{x}$, and $\varphi=u.e^{at}.e^{ac_{1}}$. As a result we find 3
independent vector fields: $X_{1}=e^{at}\partial_{t}+ae^{at}\partial_{u}$,
$X_{2}=\partial_{t}$, $X_{3}=\sqrt{x}\partial_{x}$.
#### B:
$f(x,u)=ax^{4}u$, $h(x,u)={\frac{bx}{u}}$, $k(x,u)=xu$, where $a\neq 0,b$ are
real numbers. In this case we have: $\eta=-c_{1}t+c_{2}$, $\xi=c_{1}x$,
$\varphi=-c_{1}u$. As a result we find 2 independent vector fields:
$X_{1}=-t\partial_{t}+x\partial_{x}-u\partial_{u}$, $X_{2}=\partial_{t}$.
#### C:
$f(x,u)=ax\exp{(-u/b)}$, $h(x,u)=xu$, $k(x,u)=c-bu$, where $a\neq,b\neq 0,c$
are real constant numbers. In this case we have: $\eta=c_{1}$,
$\xi=-\frac{c_{2}}{c}x\exp{(bt)}$, $\varphi=c_{2}\exp{(bt)}$. As a result we
find 2 independent vector fields:
$X_{1}=-\frac{1}{b}x\exp{(bt)}\partial_{x}+\exp{(bt)}\partial_{u},$
$X_{2}=\partial_{t}$.
#### D:
$f(x,u)=ax^{2}u$, $h(x,u)=xu$, $k(x,u)=u$, where $a$ is real nonzero constant.
In this case we have: $\eta=c_{1}$, $\xi=c_{2}x$, $\varphi=0$. As a result we
find 2 independent vector fields: $X_{1}=\partial_{t}$, $X_{2}=x\partial_{x}$.
Similar to above, the reader can use above procedure for finding her or him
interested modeling equation where has form (1.1), with interested $f,h$ and
$k$.
## 4 Resulted differential invariants
In this section we obtain differential invariants for above resulted symmetry
groups in several major and complicated cases. For example we compute
differential equation for B, $X_{1}$ and C, $X_{1}$.
#### B,
$X_{1}$: In this case we have following determination equation:
$\frac{dx}{x}=\frac{dt}{-t}=\frac{du}{-u}$, and by solving this equation we
find: $xt=c_{1}$, $xu=c_{2}$, $u/t=c_{3}$; and we choose $r=xt$ and $w=xu$ as
independent invariants. (we note $u/x=w/r$ and as a result obtain from $r,w$.)
#### C,
$X_{1}$: In this case we have following determination equation:
$\frac{bdx}{x\exp{(bt)}}=\frac{dt}{0}=\frac{du}{\exp{(bt)}}$, and by solving
this equation we find : $t=c_{1}$, $c_{2}=u+b\ln{x}$; and we choose $r=t$ and
$w=u+b\ln{x}$ as independent invariants.
case interested vector field differential invariants A $X_{1}$ $r=x$, $w=u-at$
$X_{2}$ $r=x$, $w=u$ $X_{3}$ $r=t$, $w=u$ B $X_{1}$ $r=xt$, $w=xu$ $X_{2}$
$r=x$, $w=u$ C $X_{1}$ $r=t$, $w=u+b\ln{x}$ $X_{2}$ $r=x$, $w=u$ D $X_{1}$
$r=x$, $w=u$ $X_{2}$ $r=t$, $w=u$ (Table 2)
In the above table, $F$ is an arbitrary function.
In the sequel, we obtain reduced equation respect to specified group symmetry
with infinitesimal generator $X$, (solution of this reduced equation called
$X$-invariants solution of original equation) by using resulted differential
invariants in the above table in two case.
For example, consider $u_{t}=(ax^{4}u)_{x}+bx/uu_{x}+xu$ (case B). By
considering $w=w(r)$, we find: $u_{t}=w_{r}$, $u_{x}=(xtw_{r}-w)/x^{2}$ and
$u_{xx}=(x^{2}(tw_{r}+xt^{2}w_{rr}-tw_{r})-2x(xtw_{r}-w))/x^{4}$. By
substituting this values in the given equation, we find following
$X_{1}$-reduced equation:
$\displaystyle
w_{r}=b+(1-4a)w+(6a+ar^{3})w^{2}+(4ar-2aw+awr-2arw)w_{r}+awr(r-a)w_{rr}$
As and second example, consider
$u_{t}=\big{(}{\frac{ax}{\exp{(u/b)}}}\big{)}_{x}+xuu_{x}+c-bu$, By
considering $w=w(r)$, we find: $u_{t}=w_{r}$, $u_{x}=-1/x$ and
$u_{xx}=1/x^{2}$. By substituting this values in the given equation, we find
following $X_{1}$-reduced equation:
$\displaystyle w_{r}=c+ab-2a,$
## 5 Some Applications
The Kolomogorov-Petrovskii-Piskonov (KPP) equation, (See [1, 8])
$\displaystyle E(u)\equiv bu_{t}-u_{xx}+\gamma uu_{x}+f(u),$ (5.7)
with ($b$,$\gamma$) real numbers, is encountered in reaction-diffusion systems
and prey-predator models. The optional convection term $uu_{x}$ [1, 4]) is
quite important in physical applications to prey-predator models.
### 5.1 Classical symmetries and Differential invaiants
If we let $b\neq 0$, then we have following equation:
$\displaystyle u_{t}=\frac{1}{b}(u_{xx}-\gamma uu_{x}-f(u)),$ (5.8)
By substituting this value in (2), we have following results.
#### Case I:
$b={\frac{\alpha\gamma}{\exp(\beta\alpha)}},f(u)={\frac{(1/2)\gamma\kappa\alpha
u}{\exp(\alpha\beta)}}+s$; Where $\alpha,\beta,\kappa$ and s are arbitrary
constants.
In this case we have:
$\displaystyle\xi=\frac{\exp(\alpha
t)\exp(\alpha\beta)}{\alpha}+c_{2},\qquad\eta=c_{1},\qquad\varphi=\kappa\exp(\alpha
t),$ (5.9)
For symmetry algebra we find:
$\displaystyle X_{1}=\partial_{t}\qquad X_{2}=\partial_{x},$ (5.10)
#### Case II:
$b\neq{\frac{\alpha\gamma}{\exp(\beta\alpha)}},f(u)\neq{\frac{(1/2)\gamma\kappa\alpha
u}{\exp(\alpha\beta)}}+s$; Where $\alpha,\beta,\kappa$ and s are arbitrary
constants.
In this case we have:
$\displaystyle\xi=c_{1}\qquad\eta=c_{2},\qquad\varphi=0,$ (5.11)
For symmetry algebra we find:
$\displaystyle X_{1}=\partial_{t}$ $\displaystyle\qquad X_{2}=\partial_{x},$
(5.12)
As a result we have following theorem:
###### Theorem 1.
Some exact solutions for modeling equation (5.7) invariant under a translation
group respect to $x$ and some solutions of this equation invariant under
translation respect to $t$.
### 5.2 Similarity solutions
In this subsection we find similarity solution of equation (5.8) by using
above resulted symmetry algebra.
#### similarity solution respect to $X=\partial_{t}$.
In this case we have following equation as $X$-reduced equation:
$\displaystyle w_{rr}-\gamma ww_{r}-f(w)=0,$ (5.13)
If we solve equation (5.13) with MAPLE, then we find: $w(x)=c$. Where
$\displaystyle\frac{d}{dc}{F(c)}F(c)-\gamma(cF(c))-f(c)=0\hskip
14.22636pt\mbox{or}\hskip 14.22636pt\frac{d}{dr}w(r)=F(c),\hskip
14.22636pt\mbox{or}\hskip 14.22636ptr=\int\frac{1}{F(c)}dc+C$ (5.14)
Where $F$ is arbitrary function with specified arguments and $c,C$ are
arbitrary constants.
#### similarity solution respect to $Y=\partial_{x}$.
In this case we have following equation as $Y$-reduced equation:
$\displaystyle w_{r}+\frac{1}{b}f(w)=0,$ (5.15)
If we solve equation (5.15) with MAPLE, then we find following solution:
$\displaystyle x-\int^{w(x)}\frac{b}{f(c_{1})}dc_{1}+c_{2}=0,$ (5.16)
Where $c_{1}$ and $c_{2}$ are arbitrary constants.
## Conclusion
In this paper first we find system of equations to finding symmetry group and
symmetry algebra for (G-RDC) equation, then obtain these symmetry groups in
several special cases and at the end we establish symmetry classification for
KPP equation by using group classification of (G-RDC) equation and we find its
similarity solution respect to resulted symmetry algebra.
## References
* [1] Kolmogorov, A.N. and Petrovskii I.G. and Piskunov, N. SThe study of a diffusion equation, related to the increase of the quantity of matter, and its application to one biological problem, Bulletin de l Universit e d Etat de Moscou, s erie internationale, section A Math. M ec. 1 1937, 1 26.
* [2] Olver, P.J. Applications of Lie Groups to Differential Equations, New York, Springer, 1986.
* [3] Newell, A.C. and Whitehead, J.A. Finite bandwidth, finite amplitude convection, J. Fluid Mech. 38, 1969, 279-303.
* [4] Olver, P.J. Equivalence, Invariants and Symmetry, Cambridge University Press, 1995.
* [5] Satsuma, J. Exact solutions of Burgers equation with reaction terms, Topics in soliton theory and exact solvable nonlinear equations,, World Scientific, Singapore, 1987, 255-262.
* [6] Ovsiannikov, L.V. Group Analysis of differential equations, Academic press, 1982.
* [7] Stephani, H. Differential Equations, Cambridge University Press, 1989.
* [8] Conte, R. and Musette, M. The Painlev Handbook, Springer Science and Business Media B.V, 2008,
* [9] Lie, S Arch. Math. 6 (1881) 328.
* [10] Olver, P.J. and Rosenau, P. Group-Invariant solutions of Differentil equations, SIAM J. APPL. MATH. Vol. 47, No. 2, April 1987.
* [11] Popovycha, R.O. and Sophocleousc, C. and Vaneevaa, O.O. Exact solutions of a remarkable fin equation, Applied Mathematics Letters 21 (2008) 209 214.
* [12] Cherniha, R. and Serov, M. and Rassokha, I Lie symmetries and form-preserving transformations of reaction diffusion convection equations, Journal of Mathematical Analysis and Applications, Volume 342, Issue 2, Pages 1363-1379.
* [13] Cherniha, R and Pliukhin, O New conditional symmetries and exact solutions of nonlinear reaction diffusion convection equations, J. Phys. A: Math. Theor. 40, 10049 10070, 2007.
|
arxiv-papers
| 2010-07-07T12:13:18 |
2024-09-04T02:49:11.464898
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mehdi Nadjafikhah and Saeed Dodangeh",
"submitter": "Mehdi Nadjafikhah",
"url": "https://arxiv.org/abs/1007.1113"
}
|
1007.1212
|
# Group Analysis via Weak Symmetries For
Benjamin-Bona-Mahony Equation
M. Nadjafikhah m_nadjafikhah@iust.ac.ir F. Ahangari fa_ahangari@iust.ac.ir
S. Dodangeh s_dodangeh@mathdep.iust.ac.ir Corresponding author: School of
Mathematics, Iran University of Science and Technology, Narmak, Tehran
1684613114, Iran.
###### Abstract
In this paper, weak symmetries of the Benjamin-Bona-Mahony (BBM) equation have
been investigated. Indeed, this method has been performed by applying the non-
classical symmetries of the BBM equation and the infinitesimal generators of
the classical symmetry algebra of the KdV equation as the starting
constraints. Similarity reduced equations as well as some exact solutions of
the BBM equation are obtained via this method.
###### keywords:
Weak symmetry, Non-classical symmetry, Similarity reduced Equation, Benjamin-
Bona-Mahony Equation.
## 1 Introduction
The Benjamin-Bona-Mahony equation
$\displaystyle{\rm BBM}\;:\;\;u_{t}+u_{x}+uu_{x}-u_{xxt}=0,$ (1.1)
used to model an approximation for surface water waves in a uniform channel
[1]. If we note the KdV type equation
$\displaystyle u_{t}+u_{x}+uu_{x}+u_{xxx}=0,$ (1.2)
then we find out the likeness between these equations. Indeed, this similarity
is not stochastic. Both of them used to model the waves appear in liquids,
compressible fluids, cold plasma and enharmonic crystals which are of surface,
hydro-magnetics, acoustic-gravity and acoustic types, respectively. The
interesting point is that the main difference between equations (1.1) and
(1.2) occurs in the case of short waves (Find more information in [1, 3]).
The physical applications and mathematical properties of the BBM equation
(1.1) have been motivated many investigations such as obtaining the exact
solutions via finite difference discrete process, global attractor and etc.
In this paper, we find the similarity reduced ODEs as well as resulted
similarity solutions of this equation via weak symmetry implementation.
Indeed, the organization of the present paper is as follows: Some historical
information on the weak symmetry method are given in section 2. In section 3,
we follow [10] in order to describe the theory of weak symmetries. Section 4
is devoted to performing this new class of symmetry methods using the
invariant surface condition of the BBM equation (which is indeed the non-
classical symmetry method) and infinitesimal generators of the classical
symmetry algebra of the KdV type equation as the starting points in the weak
symmetry method implementation. Finally, we have compared our results with
those related papers using the classical symmetry method in order to clarify
the advantages and disadvantages of the both strong and weak symmetry methods.
## 2 Background
Symmetry methods for differential equations, was originally developed by S.
Lie [7]. These methods without any doubt are very useful and algorithmic for
analyzing and solving linear and non-linear differential equations.
Classification of differential equations as well as linearization of them are
some other important applications of the symmetry transformation approach.
First G.W. Bluman and J.D. Cole introduced the notion of the non-classical
symmetry group of differential equations specially for the heat equation in
1969 (Find more information in [2]). For the non-classical method, we seek the
invariance of both the original equation and its invariant surface condition,
exactly this constraint (i.e invariance surface condition) causes the non-
classical solutions which are more general than the classical ones. There are
various implementations for performing the non-classical symmetry method, for
example, using the compatibility condition has been suggested by G. Cai and X.
Ling [5].
First the weak symmetries have been introduced by P.J. Olver, and P. Rosenau
in 1986 as a generalization of the non-classical symmetries with motivation of
finding every solutions of the given system. In principle, not only the
invariant solutions corresponding to arbitrary transformation groups can be
found by the reduction method, but also every possible solution of the system
can be found by using some transformation groups. In other words, there are no
conditions that need to be placed on the transformation group in order to
apply the basic reduction procedure (Find more information in [10]). In the
next section, we have an attempt to explain the notation and implementation of
the weak symmetry method by considering the BBM equation as an example in
order to prepare an appropriate setting.
## 3 On the weak symmetry method
Symmetry groups of a system of partial differential equations can be defined
in two types (see [10]).
#### Definition
Let $\Delta$ be a system of partial differential equations. A strong symmetry
group of $\Delta$ is a group of transformations $G$ on the space of
independent and dependent variables which has the following two properties:
* a)
The elements of $G$ transform solutions of the system to other solutions of
the system.
* b)
The $G-$invariant solutions of the system are found from a reduced system of
differential equations involving a fewer number of independent variables than
the original system $\Delta$.
#### Definition
A weak symmetry group of the system $\Delta$ is a group of transformations
which satisfies the reduction property (b), but no longer transforms solutions
to solutions.
Indeed, there are several transformation groups which don’t transform
solutions of given equations again to solutions, but their differential
invariants enable us to reduce them. In continuation we would illustrate the
procedure of performing this method. For this purpose, first consider an
arbitrary one-parameter transformation group, then substitute its related
differential invariants and their derivatives into the original equation,
finally, you will encounter with three different possible cases which in
continuation have been illustrated for the (BBM) equation using an appropriate
one-parameter transformation group.
### 3.1 Reduced equation has no parametric variables
Consider the one-parameter group
$(x,t,u)\mapsto(x+\lambda,t+\lambda,u),$
So, we have the characteristic equation $dx=dt=du/0$. By substituting the
resulted differential invariants i.e. $r=x-t$ and $w=u$, into equation (1.1),
we have $w_{rrr}+ww_{r}=0$. As we see, this equation has no parameter
variable.
### 3.2 Reduced equation isn’t incompatible and has parametric variables
Consider the one-parameter group
$(x,t,u)\mapsto(\lambda x,t,\lambda u).$
So, the characteristic equation is $dx/x=dt/0=du/u$. By substituting the
resulted differential invariants $r=t$ and $w=u/x$, in equation (1.1), we have
$x(w^{2}+w_{r})+w=0$, where $w=0$ is it’s solution and this equation has $x$
as the parametric variable.
### 3.3 Reduced equation is incompatible and has parametric variables
Consider the one-parameter group
$(x,t,u)\mapsto(x+2\lambda t+\lambda^{2},t+\lambda,u+8\lambda
t+4\lambda^{2}).$
By substituting the resulted differential invariants i.e. $r=x-t^{2}$ and
$w=u-4t^{2}$, in equation (1.1), we have
$ww_{r}+w_{r}+(8-2w_{rrr}-2w_{r})t+4w_{r}t^{2}$, where this equation has $t$
as the parametric variable and it is incompatible. Indeed, from the
coefficient of $t^{2}$ we have $w_{r}=0$ and from the coefficient of $t$ we
have $w_{rrr}+w_{r}=4$, this means that these equations are incompatible.
## 4 Implementation of the weak symmetry method for the BBM equation
Since, the weak symmetry method is based on conjecture, so here, the several
ideas of performing this method as well as some of its aspects are presented.
### 4.1 Non-classical symmetries of the BBM equation
There are several implementations to find the non-classical symmetries. Here,
we follow the procedure presented by G. Cai et al. which they obtained the
non-classical symmetries of the Burgers-Fisher equation based on the
compatibility conditions [4].
Consider the following one-parameter group:
$\displaystyle{}\tilde{x}$ $\displaystyle=$ $\displaystyle
x+\varepsilon\xi(x,t,u)+O(\varepsilon^{2}),$ $\displaystyle\tilde{t}$
$\displaystyle=$ $\displaystyle t+\varepsilon\eta(x,t,u)+O(\varepsilon^{2}),$
(4.3) $\displaystyle\tilde{u}$ $\displaystyle=$ $\displaystyle
u+\varepsilon\varphi(x,t,u)+O(\varepsilon^{2}),$
Assume that the equation $\Delta_{1}(x,u^{(n)}):=\mbox{eq}(\ref{eq:1})$ is
invariant under the transformation group (4.1) with the following invariant
surface condition:
$\displaystyle{}\Delta_{2}(x,u^{(n)}):=\eta u_{t}+\xi u_{x}-\varphi=0$ (4.4)
This means that ${X^{(4)}\Delta_{1}}|_{\Delta_{1}=0,\Delta_{2}=0}=0$, where
$X=\xi(x,t,u)\partial_{x}+\eta(x,t,u)\partial_{t}+\varphi(x,t,u)\partial_{u},$
is the infinitesimal generator of (4.1), and
$X^{(4)}=X+\varphi^{x}\partial_{u_{x}}+...+\varphi^{tttt}\partial_{u_{tttt}},$
is the fourth prolongation of $X$, with the coefficients defined as
$\varphi^{J}=D_{J}Q+\xi u_{Jx}+\eta u_{Jt}$, where $Q=\varphi-\xi u_{x}-\eta
u_{t}$ is the Lie characteristic and
$D_{J}=\sum_{i=0}u_{Ji}\,\partial_{u_{J}}$ is the total derivative w.r.t. $J$
(Find more information in [8, 9])
Without loss of generality in condition (4.4), two cases $\eta=0$ and $\eta=1$
must be considered.
Case I $\eta=1$: In this case we have $u_{t}=\varphi-\xi u_{x}$. Substituting
this expression in (1.1) we have $D_{t}(\varphi-\xi
u_{x})=D_{t}(u_{xxt}-u_{x}-uu_{x})$, where $D_{t}$ is total derivative w.r.t.
$t$. By substituting $\xi u_{xx}$ in both sides of above, we find
$\displaystyle\varphi^{t}$ $\displaystyle=$ $\displaystyle
u_{xxtt}-u_{xt}-u_{t}u_{x}-uu_{xt}+\xi u_{xx}$ $\displaystyle=$ $\displaystyle
D_{xxt}(u_{t})-(u+1)D_{x}(u_{t})+(\xi u_{x}-\varphi)u_{x}+\xi u_{xx},$
$\displaystyle=$ $\displaystyle D_{xxt}(\varphi-\xi
u_{x})-(u+1)D_{x}(\varphi-\xi u_{x})+(\xi u_{x}-\varphi)u_{x}+\xi u_{xx},$
$\displaystyle=$ $\displaystyle\varphi^{xxt}-\xi
u_{xxxt}-(u+1)\varphi^{x}+(u+1)\xi u_{xx}+(\xi u_{x}-\varphi)u_{x}+\xi
u_{xx},$
By virtue of $D_{x}(u_{t})=D_{x}(u_{xxt}-u_{x}-uu_{x})$, we have
$u_{xt}=u_{xxxt}-u_{xx}-uu_{xx}-u_{x}^{2}$. Finally, we find the following
governing equation:
$\displaystyle\varphi^{t}=\varphi^{xxt}-(u+1)\varphi^{x}-\varphi u_{x},$ (4.6)
where $\varphi^{t}=D_{t}(\varphi-\xi u_{x})+\xi u_{xt}$,
$\varphi^{x}=D_{x}(\varphi-\xi u_{x})+\xi u_{xx}$, and
$\varphi^{xxt}=D_{xxt}(\varphi-\xi u_{x})+\xi u_{xxxt}.$
By substituting the coefficient functions
$\varphi^{t},\varphi^{x},\varphi^{xxt}$ into invariance condition (4.6), we
are left with a polynomial equation involving the various derivatives of
$u(x,t)$ whose coefficients are certain derivatives of $\xi$ and $\varphi$.
Since, $\xi$ and $\varphi$ depend only on $x$, $t$, $u$ we can equate the
individual coefficients to zero, leading to these complete set of determining
equations: $\xi_{x}=\xi_{t}=\xi_{u}=0$, $\varphi=0$. So, we have $\xi=c_{1}$,
$\varphi=0$. So, we find the infinitesimal generators of the non-classical
symmetries using the above results as follows, when $c_{1}=1$, we have
$\sigma_{1}=u_{x}+u_{t}$, and for $c_{1}\neq 0$ the symmetries are
$\sigma_{2}=u_{x}$, $\sigma_{3}=u_{t}$. As a result we can state the following
proposition:
#### Proposition
The non-classical symmetries of the BBM equation in the case of $\eta=1$,
spanned by
$\displaystyle\sigma_{1}=u_{x}+u_{t},\qquad\sigma_{2}=u_{x},\qquad\sigma_{3}=u_{t}.$
(4.7)
As a result of above proposition we have the following group-invariant
solutions:
* 1)
For $\mathbf{\sigma_{1}}=u_{x}+u_{t}$, substituting it into $\sigma_{1}(u)$ we
find $u=F(x-t)$, where $F$ must satisfy in:
$FF^{\prime}-F^{\prime\prime\prime}=0$
* 2)
For $\mathbf{\sigma_{2}}=u_{x}$, substituting it into $\sigma_{2}(u)=0$ we
find $u=F(t)$ for an arbitrary $F$, so from equation (1.1) we obtain: $u=0$.
* 3)
For $\mathbf{\sigma_{3}}=u_{t}$, substituting it into $\sigma_{3}(u)=0$ we
find $u=F(x)$, where from equation (1.1) $F$ satisfies this equation:
$F^{\prime}+FF^{\prime}+F^{\prime\prime\prime}=0$.
Case II $\eta=0$: In this case, without lose of generality we can let $\xi=1$,
so we have: $u_{x}=\varphi$. Using this we can deduce
$A(x,t,u)=\varphi_{xt}-\varphi-u\varphi$. Subsisting this in the determining
equation $A\varphi_{u}+\varphi_{t}-A_{u}\varphi-A_{x}=0$, we obtain:
$\displaystyle\varphi_{xt}\varphi_{u}-2\varphi\varphi_{u}-u\varphi\varphi_{u}+\varphi_{t}=\varphi_{xtu}\varphi+u\varphi_{u}\varphi+2\varphi^{2}+\varphi_{xxt}+\varphi_{x}+u\varphi_{x}.$
(4.8)
By assuming $\varphi=\varphi(x,t)$ above equation changes into
$\varphi_{t}-2\varphi^{2}-\varphi_{xxt}-\varphi_{x}-u\varphi_{x}=0.$
So we have: $\varphi=1/(c-2x)$. As a result, we deduce that
$u(x,t)=x/(c-2t)+F(t)$ (where $F$ is an arbitrary function) is a solution of
(1.1).
### 4.2 Using the classical symmetries of KdV type equation (1.2)
Since the appearance forms of equation (1.1) and (1.2) are similar, we want to
try our chance in order to obtain new similarity reduced ODEs for BBM equation
through infinitesimal generators of the classical symmetries (CS) of KdV type
equation as the starting constraint. For the classical symmetries of the KdV
type equation using Lie classical symmetry we have the next theorem (Since,
the proof is computational, to keep scope we don’t present it here. Find more
information in [8, 9]).
#### Theorem
If we consider
$X=\xi(x,t,u)\partial_{x}+\eta(x,t,u)\partial_{t}+\eta(x,t,u)\partial_{u}$ as
the infinitesimal generator of the classical symmetry group of the KdV type
equation (1.2), then we have
$\displaystyle\eta=c_{1}t+c_{2},\qquad\xi=\frac{1}{3}c_{1}(x+2t)+c_{3}t+c_{4},\qquad\varphi=-\frac{2}{3}c_{1}u+c_{3},$
(4.9)
where $c_{1}$, $c_{2}$, $c_{3}$ and $c_{4}$ are arbitrary constants.
Hence the next corollary could be stated:
#### Corollary
The classical symmetries of equation (1.2) i.e. KdV type equation, spanned by:
$\displaystyle
X_{1}=(x+2t)\partial_{x}+3t\partial_{t}-2u\partial_{u},\;\;X_{2}=\partial_{t},\;\;X_{3}=t\partial_{x}+\partial_{u},\;\;X_{4}=\partial_{x}.$
(4.10)
So, we can consider any linear combinations of given vector fields in the
above corollary as the starting constraint of the weak symmetry method. In
continuation, we will illustrate the weak symmetry method using some linear
combination of $X_{1}$, $X_{2}$, $X_{3}$ and $X_{4}$ as the starting point.
#### Example
Consider the one-parameter transformation group with the infinitesimal
generator $X_{2}+X_{3}=t\partial_{x}+\partial_{t}+\partial_{u}$. The
characteristic equation is $dx/t=dt=du$. So, we find the differential
invariants as $r=t^{2}-2x$, $w=u/t$. By substituting these new variables in
the original equation (1.1) we deduce
$(2w_{r}-2ww_{r}-w_{rrr})t^{2}=2w_{r}t+4w_{rr}+w$, where $t$ can be considered
as the differential parameter. Note that solving the above ODE doesn’t give
new solution.
#### Example
Consider the one-parameter transformation group with the infinitesimal
generator $X_{3}=t\partial_{x}+\partial_{u}$. The characteristic equation is
$dx/t=dt/0=du$. So, we can obtain the differential invariants as $r=t$,
$w=u-x/t$. By substituting these new variables in the original equation (1.1)
we find: $rw_{r}+w-1=0,$, solving this reduced equation we obtain $w=r/(r+c)$.
So we can find $u=(tx+x^{2}+cx)/(t(x+c))$ as the solution of equation (1.1).
### 4.3 Some other suggestions
Some other ideas may be useful to reach other solutions of the BBM equation.
For example, non-classical potential symmetry method or using classical and
non-classical symmetries of other equations which have the similar forms as
the BBM equation. Meanwhile, Physical knowledge of the model framework can be
so effective in order to reach favorite solutions via weak symmetries. For
example if you know your desired solution may be invariant under some scale of
specific variables then the weak symmetry method can be started with an
appropriate scaling transformation. Since the main goal of this paper was
introducing weak symmetry method for BBM equation, we lay away performing of
above approaches.
## 5 More discussions
Now, we want to compare our results with other related papers. Paper [6] is
concentrated on the classical symmetries and optimal Lie system of the BBM
equation. Comparing with [6], we deduce that in this paper by applying the
weak symmetry method we have obtained more similarity solutions and other
useful suggestions are presented in order to reach more other solutions.
Taking into account the sections 2 and 3 of [6], the next theorem can be
resulted (Find more information in ([8], Chapter 3).
#### Theorem
If $u=f(x,t)$ is solution of the BBM equation (1.1), so are the functions
$\displaystyle u=f(x-\varepsilon,t),\quad
u=f(x-\alpha\varepsilon,t-\varepsilon),\quad
u=e^{(u+1)\varepsilon}f(x-\alpha\varepsilon,e^{-\varepsilon t}t),$
where $\varepsilon\ll 1$ and $\alpha$ are arbitrary constants.
Indeed, above theorem characterizes the invariant solutions of the BBM
equation, for instance if $u=c$ is a solution of equation (1.1), then from
this theorem we obtain $u=ce^{\varepsilon(u+1)}$ as a solution of the BBM
equation. For another example, if we consider the solution
$u=(tx+x^{2}+cx)/(t(x+c))$ of equation (1.1), from this theorem we deduce that
$u=\displaystyle{\frac{(t-\varepsilon)(x-\alpha\varepsilon)+(x-\alpha\varepsilon)^{2}+c(x-\alpha\varepsilon)}{(t-\varepsilon)(x-\alpha\varepsilon+c)}},$
(where $\varepsilon\ll 1$ and $\alpha$, $c$ are arbitrary constants), is again
a solution of BBM equation. By using such approach, we are enable to obtain
more new solutions for the BBM equation.
## Conclusions
In this paper, we have presented a comprehensive explanation of the weak
symmetry method as the generalization of the classical Lie symmetry method.
Indeed, we have performed the weak symmetry method for the BBM equation which
has been fulfilled by applying the non-classical symmetries of the BBM
equation and using the classical symmetries of the KdV type equation as the
starting constraints. Also, the similarity reduced equations as well as some
exact solutions of the BBM equation are obtained via this method. Finally, we
have compared our results with papers using the classical symmetry method.
Other suggestions for finding new exact solutions are also presented.
## References
* [1] Benjamin, T.B. and Bona, J.L. and Mahony, J.J., Model equations for long waves in nonlinear dispersive systems Phil. Trans. R. Soc. 1972, 272 47 78.
* [2] Bluman, G.W. and Cole, J.D., The general similarity solution of the heat equation, J. Math. Mech. 18 A969X 1025-1042.
* [3] Bona, J.L. and Bryant, P.J., A mathematical model for long waves generated by wave makers in nonlinear dispersive systems Proc. Cambridge Phil. Soc. 1973, 73 391 405.
* [4] Cai, G. and Wang, Y. and Zhang, F., Nonclassical symmetries and group invariant solutions of Burgers-Fisher equations. World Journal of Modelling and Simulation Vol. 3 (2007) No. 4, pp. 305-309.
* [5] Cai, G. and Ling, X., Nonclassical symmetries of a class of nonlinear partial differential equations and compatibility. World Journal of Modelling and Simulation Vol. 3. 2007. No. 1, pp. 51-57.
* [6] Karaca, M.A. and Hizel, E., Similarity reductions of Benjamin-Bona-Mahony equation, Applied Mathematical Sciences, Vol. 2, 2008, no. 10, 463 - 469
* [7] Lie, S., Uber die integration durch bestimmte integrale von einer klasse linear partieller differentialgleichung, Arch, for Math. 6 A881), pp. 328-368; also Gesammelte Abhandlungen, vol. 3, B.G. Teubner, Leipzig, 1922, pp. 492-523.
* [8] Olver, P.J., Applications of Lie groups to differential equations, New York, Springer, 1986.
* [9] Olver, P.J., Equivalence, invariants and symmetry, Cambridge University Press, 1995.
* [10] Olver, P.J. and Rosenau, P., Group-invariant solutions of differential equations Siam J., Applied Mathematics, Vol 47, No 2, April 1987\.
|
arxiv-papers
| 2010-07-07T18:12:31 |
2024-09-04T02:49:11.471525
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mehdi Nadjafikhah and Fatemeh Ahangari and Saeed Dodangeh",
"submitter": "Mehdi Nadjafikhah",
"url": "https://arxiv.org/abs/1007.1212"
}
|
1007.1253
|
# Efficient Sketches for the Set Query Problem††thanks: This research has been
supported in part by the David and Lucille Packard Fellowship, MADALGO (Center
for Massive Data Algorithmics, funded by the Danish National Research
Association), NSF grant CCF-0728645, a Cisco Fellowship, and the NSF Graduate
Research Fellowship Program.
Eric Price MIT CSAIL
###### Abstract
We develop an algorithm for estimating the values of a vector
$x\in\mathbb{R}^{n}$ over a support $S$ of size $k$ from a randomized sparse
binary linear sketch $Ax$ of size $O(k)$. Given $Ax$ and $S$, we can recover
$x^{\prime}$ with
$\left\|x^{\prime}-x_{S}\right\|_{2}\leq\epsilon\left\|x-x_{S}\right\|_{2}$
with probability at least $1-k^{-\Omega(1)}$. The recovery takes $O(k)$ time.
While interesting in its own right, this primitive also has a number of
applications. For example, we can:
1. 1.
Improve the linear $k$-sparse recovery of heavy hitters in Zipfian
distributions with $O(k\log n)$ space from a $1+\epsilon$ approximation to a
$1+o(1)$ approximation, giving the first such approximation in $O(k\log n)$
space when $k\leq O(n^{1-\epsilon})$.
2. 2.
Recover block-sparse vectors with $O(k)$ space and a $1+\epsilon$
approximation. Previous algorithms required either $\omega(k)$ space or
$\omega(1)$ approximation.
## 1 Introduction
In recent years, a new “linear” approach for obtaining a succinct approximate
representation of $n$-dimensional vectors (or signals) has been discovered.
For any signal $x$, the representation is equal to $Ax$, where $A$ is an
$m\times n$ matrix, or possibly a random variable chosen from some
distribution over such matrices. The vector $Ax$ is often referred to as the
measurement vector or linear sketch of $x$. Although $m$ is typically much
smaller than $n$, the sketch $Ax$ often contains plenty of useful information
about the signal $x$.
A particularly useful and well-studied problem is that of stable sparse
recovery. The problem is typically defined as follows: for some norm
parameters $p$ and $q$ and an approximation factor $C>0$, given $Ax$, recover
a vector $x^{\prime}$ such that
(1) $\displaystyle\left\|x^{\prime}-x\right\|_{p}$ $\displaystyle\leq
C\cdot\mathrm{Err}_{q}(x,k),$ $\displaystyle\mbox{\ where\
}\mathrm{Err}_{q}(x,k)$ $\displaystyle=\min_{k\mbox{-sparse
}\hat{x}}\left\|\hat{x}-x\right\|_{q}$
where we say that $\hat{x}$ is $k$-sparse if it has at most $k$ non-zero
coordinates. Sparse recovery has applications to numerous areas such as data
stream computing [Mut03, Ind07] and compressed sensing [CRT06, Don06], notably
for constructing imaging systems that acquire images directly in compressed
form (e.g., [DDT+08, Rom09]). The problem has been a subject of extensive
study over the last several years, with the goal of designing schemes that
enjoy good “compression rate” (i.e., low values of $m$) as well as good
algorithmic properties (i.e., low encoding and recovery times). It is known
that there exist distributions of matrices $A$ and associated recovery
algorithms that for any $x$ with high probability produce approximations
$x^{\prime}$ satisfying Equation (1) with $\ell_{p}=\ell_{q}=\ell_{2}$,
constant approximation factor $C=1+\epsilon$, and sketch length
$m=O(k\log(n/k))$;111In particular, a random Gaussian matrix [CD04] or a
random sparse binary matrix ([GLPS09], building on [CCF02, CM04]) has this
property with overwhelming probability. See [GI10] for an overview. it is also
known that this sketch length is asymptotically optimal [DIPW10, FPRU10].
Similar results for other combinations of $\ell_{p}$/$\ell_{q}$ norms are
known as well.
Because it is impossible to improve on the sketch size in the general sparse
recovery problem, recently there has been a large body of work on more
restricted problems that are amenable to more efficient solutions. This
includes _model-based compressive sensing_ [BCDH10], which imposes additional
constraints (or _models_) on $x$ beyond near-sparsity. Examples of models
include _block sparsity_ , where the large coefficients tend to cluster
together in blocks [BCDH10, EKB09]; _tree sparsity_ , where the large
coefficients form a rooted, connected tree structure [BCDH10, LD05]; and being
_Zipfian_ , where we require that the histogram of coefficient size follow a
_Zipfian_ (or _power law_) distribution.
A sparse recovery algorithm needs to perform two tasks: locating the large
coefficients of $x$ and estimating their value. Existing algorithms perform
both tasks at the same time. In contrast, we propose decoupling these tasks.
In models of interest, including Zipfian signals and block-sparse signals,
existing techniques can locate the large coefficients more efficiently or
accurately than they can estimate them. Prior to this work, however,
estimating the large coefficients after finding them had no better solution
than the general sparse recovery problem. We fill this gap by giving an
optimal method for estimating the values of the large coefficients after
locating them. We refer to this task as the _Set Query Problem_ 222The term
“set query” is in contrast to “point query,” used in e.g. [CM04] for
estimation of a single coordinate..
Main result. (Set Query Algorithm.) We give a randomized distribution over
$O(k)\times n$ binary matrices $A$ such that, for any vector
$x\in\mathbb{R}^{n}$ and set $S\subseteq\\{1,\dotsc,n\\}$ with
$\left|S\right|=k$, we can recover an $x^{\prime}$ from $Ax+\nu$ and $S$ with
$\left\|x^{\prime}-x_{S}\right\|_{2}\leq\epsilon(\left\|x-x_{S}\right\|_{2}+\left\|\nu\right\|_{2})$
where $x_{S}\in\mathbb{R}^{n}$ equals $x$ over $S$ and zero elsewhere. The
matrix $A$ has $O(1)$ non-zero entries per column, recovery succeeds with
probability $1-k^{-\Omega(1)}$, and recovery takes $O(k)$ time. This can be
achieved for arbitrarily small $\epsilon>0$, using $O(k/\epsilon^{2})$ rows.
We achieve a similar result in the $\ell_{1}$ norm.
The set query problem is useful in scenarios when, given a sketch of $x$, we
have some alternative methods for discovering a “good” support of an
approximation to $x$. This is the case, e.g., in block-sparse recovery, where
(as we show in this paper) it is possible to identify “heavy” blocks using
other methods. It is also a natural problem in itself. In particular, it
generalizes the well-studied _point query problem_ [CM04], which considers the
case that $S$ is a singleton. We note that, although the set query problem for
sets of size $k$ can be reduced to $k$ instances of the point query problem,
this reduction is less space-efficient than the algorithm we propose, as
elaborated below.
Techniques. Our method is related to existing sparse recovery algorithms,
including Count-Sketch [CCF02] and Count-Min [CM04]. In fact, our sketch
matrix $A$ is almost identical to the one used in Count-Sketch—each column of
$A$ has $d$ random locations out of $O(kd)$ each independently set to $\pm 1$,
and the columns are independently generated. We can view such a matrix as
“hashing” each coordinate to $d$ “buckets” out of $O(kd)$. The difference is
that the previous algorithms require $O(k\log k)$ measurements to achieve our
error bound (and $d=O(\log k)$), while we only need $O(k)$ measurements and
$d=O(1)$.
We overcome two obstacles to bring $d$ down to $O(1)$ and still achieve the
error bound with high probability333In this paper, “high probability” means
probability at least $1-1/k^{c}$ for some constant $c>0$.. First, in order to
estimate the coordinates $x_{i}$, we need a more elaborate method than, say,
taking the median of the buckets that $i$ was hashed into. This is because,
with constant probability, all such buckets might contain some other elements
from $S$ (be “heavy”) and therefore using any of them as an estimator for
$y_{i}$ would result in too much error. Since, for super-constant values of
$|S|$, it is highly likely that such an event will occur for at least one
$i\in S$, it follows that this type of estimation results in large error.
We solve this issue by using our knowledge of $S$. We know when a bucket is
“corrupted” (that is, contains more than one element of $S$), so we only
estimate coordinates that lie in a large number of uncorrupted buckets. Once
we estimate a coordinate, we subtract our estimation of its value from the
buckets it is contained in. This potentially decreases the number of corrupted
buckets, allowing us to estimate more coordinates. We show that, with high
probability, this procedure can continue until it estimates every coordinate
in $S$.
The other issue with the previous algorithms is that their analysis of their
probability of success does not depend on $k$. This means that, even if the
“head” did not interfere, their chance of success would be a constant (like
$1-2^{-\Omega(d)}$) rather than high probability in $k$ (meaning
$1-k^{-\Omega(d)}$). We show that the errors in our estimates of coordinates
have low covariance, which allows us to apply Chebyshev’s inequality to get
that the total error is concentrated around the mean with high probability.
Related work. A similar recovery algorithm (with $d=2$) has been analyzed and
applied in a streaming context in [EG07]. However, in that paper the authors
only consider the case where the vector $y$ is $k$-sparse. In that case, the
termination property alone suffices, since there is no error to bound.
Furthermore, because $d=2$ they only achieve a constant probability of
success. In this paper we consider general vectors $y$ so we need to make sure
the error remains bounded, and we achieve a high probability of success.
The recovery procedure also has similarities to recovering LDPCs using belief
propagation, especially over the binary erasure channel. The similarities are
strongest for exact recovery of $k$-sparse $y$; our method for bounding the
error from noise is quite different.
Applications. Our efficient solution to the set query problem can be combined
with existing techniques to achieve sparse recovery under several models.
We say that a vector $x$ follows a _Zipfian_ or _power law_ distribution with
parameter $\alpha$ if
$\left|x_{r(i)}\right|=\Theta(\left|x_{r(1)}\right|i^{-\alpha})$ where $r(i)$
is the location of the $i$th largest coefficient in $x$. When $\alpha>1/2$,
$x$ is well approximated in the $\ell_{2}$ norm by its sparse approximation.
Because a wide variety of real world signals follow power law distributions
([Mit04, BKM+00]), this notion (related to “compressibility”444A signal is
“compressible” when
$\left|x_{r(i)}\right|=O(\left|x_{r(1)}\right|i^{-\alpha})$ rather than
$\Theta(\left|x_{r(1)}\right|i^{-\alpha})$ [CT06]. This allows it to decay
very quickly then stop decaying for a while; we require that the decay be
continuous.) is often considered to be much of the reason why sparse recovery
is interesting [CT06, Cev08]. Prior to this work, sparse recovery of power law
distributions has only been solved via general sparse recovery methods:
$(1+\epsilon)\mathrm{Err}_{2}(x,k)$ error in $O(k\log(n/k))$ measurements.
However, locating the large coefficients in a power law distribution has long
been easier than in a general distribution. Using $O(k\log n)$ measurements,
the Count-Sketch algorithm [CCF02] can produce a candidate set
$S\subseteq\\{1,\dotsc,b\\}$ with $\left|S\right|=O(k)$ that includes all of
the top $k$ positions in a power law distribution with high probability (if
$\alpha>1/2$). We can then apply our set query algorithm to recover an
approximation $x^{\prime}$ to $x_{S}$. Because we already are using $O(k\log
n)$ measurements on Count-Sketch, we use $O(k\log n)$ rather than $O(k)$
measurements in the set query algorithm to get an $\epsilon/\sqrt{\log n}$
rather than $\epsilon$ approximation. This lets us recover a $k$-sparse
$x^{\prime}$ with $O(k\log n)$ measurements with
$\left\|x^{\prime}-x\right\|_{2}\leq\left(1+\frac{\epsilon}{\sqrt{\log
n}}\right)\mathrm{Err}_{2}(x,k).$
This is especially interesting in the common regime where $k<n^{1-c}$ for some
constant $c>0$. Then no previous algorithms achieve better than a
$(1+\epsilon)$ approximation with $O(k\log n)$ measurements, and the lower
bound in [DIPW10] shows that any $O(1)$ approximation requires $\Omega(k\log
n)$ measurements555The lower bound only applies to geometric distributions,
not Zipfian ones. However, our algorithm applies to more general _sub-Zipfian_
distributions (defined in Section 4.1), which includes both.. This means at
$\Theta(k\log n)$ measurements, the best approximation changes from
$\omega(1)$ to $1+o(1)$.
Another application is that of finding block-sparse approximations. In this
application, the coordinate set $\\{1\ldots n\\}$ is partitioned into $n/b$
blocks, each of length $b$. We define a $(k,b)$-block-sparse vector to be a
vector where all non-zero elements are contained in at most $k/b$ blocks. An
example of block-sparse data is time series data from $n/b$ locations over $b$
time steps, where only $k/b$ locations are “active”. We can define
$\mathrm{Err}_{2}(x,k,b)=\min_{(k,b)-\mbox{\scriptsize block-sparse
}\hat{x}}\left\|x-\hat{x}\right\|_{2}.$
The block-sparse recovery problem can now be formulated analogously to
Equation 1. Since the formulation imposes restrictions on the sparsity
patterns, it is natural to expect that one can perform sparse recovery from
fewer than $O(k\log(n/k))$ measurements needed in the general case. Because of
that reason and the prevalence of approximately block-sparse signals, the
problem of stable recovery of variants of block-sparse approximations has been
recently a subject of extensive research (e.g., see [EB09, SPH09, BCDH10,
CIHB09]). The state of the art algorithm has been given in [BCDH10], who gave
a probabilistic construction of a single $m\times n$ matrix $A$, with
$m=O(k+\frac{k}{b}\log n$), and an $n\log^{O(1)}n$-time algorithm for
performing the block-sparse recovery in the $\ell_{1}$ norm (as well as other
variants). If the blocks have size $\Omega(\log n)$, the algorithm uses only
$O(k)$ measurements, which is a substantial improvement over the general
bound. However, the approximation factor $C$ guaranteed by that algorithm was
super-constant.
In this paper, we provide a distribution over matrices $A$, with
$m=O(k+\frac{k}{b}\log n)$, which enables solving this problem with a constant
approximation factor and in the $\ell_{2}$ norm, with high probability. As
with Zipfian distributions, first one algorithm tells us where to find the
heavy hitters and then the set query algorithm estimates their values. In this
case, we modify the algorithm of [ABI08] to find block heavy hitters, which
enables us to find the support of the $\frac{k}{b}$ “most significant blocks”
using $O(\frac{k}{b}\log n)$ measurements. The essence is to perform
dimensionality reduction of each block from $b$ to $O(\log n)$ dimensions,
then estimate the result with a linear hash table. For each block, most of the
projections are estimated pretty well, so the median is a good estimator of
the block’s norm. Once the support is identified, we can recover the
coefficients using the set query algorithm.
## 2 Preliminaries
### 2.1 Notation
For $n\in\mathbb{Z}^{+}$, we denote $\\{1,\dotsc,n\\}$ by $[n]$. Suppose
$x\in\mathbb{R}^{n}$. Then for $i\in[n]$, $x_{i}\in\mathbb{R}$ denotes the
value of the $i$-th coordinate in $x$. As an exception,
$e_{i}\in\mathbb{R}^{n}$ denotes the elementary unit vector with a one at
position $i$. For $S\subseteq[n]$, $x_{S}$ denotes the vector $x^{\prime}\in
R^{n}$ given by $x^{\prime}_{i}=x_{i}$ if $i\in S$, and $x^{\prime}_{i}=0$
otherwise. We use $\operatorname{supp}(x)$ to denote the support of $x$. We
use upper case letters to denote sets, matrices, and random distributions. We
use lower case letters for scalars and vectors.
### 2.2 Negative Association
This paper would like to make a claim of the form “We have $k$ observations
each of whose error has small expectation and variance. Therefore the average
error is small with high probability in $k$.” If the errors were independent
this would be immediate from Chebyshev’s inequality, but our errors depend on
each other. Fortunately, our errors have some tendency to behave even better
than if they were independent: the more noise that appears in one coordinate,
the less remains to land in other coordinates. We use _negative dependence_ to
refer to this general class of behavior. The specific forms of negative
dependence we use are _negative association_ and _approximate negative
correlation_ ; see Appendix A for details on these notions.
## 3 Set-Query Algorithm
###### Theorem 3.1.
There is a randomized sparse binary sketch matrix $A$ and recovery algorithm
$\mathscr{A}$, such that for any $x\in\mathbb{R}^{n}$, $S\subseteq[n]$ with
$\left|S\right|=k$, $x^{\prime}=\mathscr{A}(Ax+\nu,S)\in\mathbb{R}^{n}$ has
$\operatorname{supp}(x^{\prime})\subseteq S$ and
$\left\|x^{\prime}-x_{S}\right\|_{2}\leq\epsilon(\left\|x-x_{S}\right\|_{2}+\left\|\nu\right\|_{2})$
with probability at least $1-1/k^{c}$. $A$ has $O(\frac{c}{\epsilon^{2}}k)$
rows and $O(c)$ non-zero entries per column, and $\mathscr{A}$ runs in $O(ck)$
time.
One can achieve
$\left\|x^{\prime}-x_{S}\right\|_{1}\leq\epsilon(\left\|x-x_{S}\right\|_{1}+\left\|\nu\right\|_{1})$
under the same conditions, but with only $O(\frac{c}{\epsilon}k)$ rows.
We will first show Theorem 3.1 for a constant $c=1/3$ rather than for general
$c$. Parallel repetition gives the theorem for general $c$, as described in
Section 3.7. We will also only show it with entries of $A$ being in
$\\{0,1,-1\\}$. By splitting each row in two, one for the positive and one for
the negative entries, we get a binary matrix with the same properties. The
paper focuses on the more difficult $\ell_{2}$ result; see Appendix B for
details on the $\ell_{1}$ result.
### 3.1 Intuition
We call $x_{S}$ the “head” and $x-x_{S}$ the “tail.” The head probably
contains the heavy hitters, with much more mass than the tail of the
distribution. We would like to estimate $x_{S}$ with zero error from the head
and small error from the tail with high probability.
Our algorithm is related to the standard Count-Sketch [CCF02] and Count-Min
[CM04] algorithms. In order to point out the differences, let us examine how
they perform on this task. These algorithms show that hashing into a single
$w=O(k)$ sized hash table is good in the sense that each point $x_{i}$ has:
1. 1.
Zero error from the head with constant probability (namely $1-\frac{k}{w}$).
2. 2.
A small amount of error from the tail in expectation (and hence with constant
probability).
They then iterate this procedure $d$ times and take the median, so that each
estimate has small error with probability $1-2^{-\Omega(d)}$. With $d=O(\log
k)$, we get that all $k$ estimates in $S$ are good with $O(k\log k)$
measurements with high probability in $k$. With fewer measurements, however,
some $x_{i}$ will probably have error from the head. If the head is much
larger than the tail (such as when the tail is zero), this is a major problem.
Furthermore, with $O(k)$ measurements the error from the tail would be small
only in expectation, not with high probability.
We make three observations that allow us to use only $O(k)$ measurements to
estimate $x_{S}$ with error relative to the tail with high probability in $k$.
1. 1.
The total error from the tail over a support of size $k$ is concentrated more
strongly than the error at a single point: the error probability drops as
$k^{-\Omega(d)}$ rather than $2^{-\Omega(d)}$.
2. 2.
The error from the head can be avoided if one knows where the head is, by
modifying the recovery algorithm.
3. 3.
The error from the tail remains concentrated after modifying the recovery
algorithm.
For simplicity this paper does not directly show (1), only (2) and (3). The
modification to the algorithm to achieve (2) is quite natural, and described
in detail and illustrated in Section 3.2. Rather than estimate every
coordinate in $S$ immediately, we only estimate those coordinates which mostly
do not overlap with other coordinates in $S$. In particular, we only estimate
$x_{i}$ as the median of at least $d-2$ positions that are not in the image of
$S\setminus\\{i\\}$. Once we learn $x_{i}$, we can subtract $Ax_{i}e_{i}$ from
the observed $Ax$ and repeat on $A(x-x_{i}e_{i})$ and $S\setminus\\{i\\}$.
Because we only look at positions that are in the image of only one remaining
element of $S$, this avoids any error from the head. We show in Section 3.3
that this algorithm never gets stuck; we can always find some position that
mostly doesn’t overlap with the image of the rest of the remaining support.
We then show that the error from the tail has low expectation, and that it is
strongly concentrated. We think of the tail as noise located in each “cell”
(coordinate in the image space). We decompose the error of our result into two
parts: the “point error” and the “propagation”. The point error is error
introduced in our estimate of some $x_{i}$ based on noise in the cells that we
estimate $x_{i}$ from, and equals the median of the noise in those cells. The
“propagation” is the error that comes from point error in estimating other
coordinates in the same connected component; these errors propagate through
the component as we subtract off incorrect estimates of each $x_{i}$.
Section 3.4 shows how to decompose the total error in terms of point errors
and the component sizes. The two following sections bound the expectation and
variance of these two quantities and show that they obey some notions of
negative dependence. We combine these errors in Section 3.7 to get Theorem 3.1
with a specific $c$ (namely $c=1/3$). We then use parallel repetition to
achieve Theorem 3.1 for arbitrary $c$.
### 3.2 Algorithm
We describe the sketch matrix $A$ and recovery procedure in Algorithm 1.
Unlike Count-Sketch [CCF02] or Count-Min [CM04], our $A$ is not split into $d$
hash tables of size $O(k)$. Instead, it has a single
$w=O(d^{2}k/\epsilon^{2})$ sized hash table where each coordinate is hashed
into $d$ unique positions. We can think of $A$ as a random $d$-uniform
hypergraph, where the non-zero entries in each column correspond to the
terminals of a hyperedge. We say that $A$ is drawn from $\mathbb{G}^{d}(w,n)$
with random signs associated with each (hyperedge, terminal) pair. We do this
so we will be able to apply existing theorems on random hypergraphs.
Figure 1 shows an example $Ax$ for a given $x$, and Figure 2 demonstrates
running the recovery procedure on this instance.
Figure 1: An instance of the set query problem. There are $n$ vertices on the
left, corresponding to $x$, and the table on the right represents $Ax$. Each
vertex $i$ on the left maps to $d$ cells on the right, randomly increasing or
decreasing the value in each cell by $x_{i}$. We represent addition by black
lines, and subtraction by red lines. We are told the locations of the heavy
hitters, which we represent by blue circles; the rest is represented by yellow
circles.
(a) (b)
(c) (d)
Figure 2: Example run of the algorithm. Part (a) shows the state as considered
by the algorithm: $Ax$ and the graph structure corresponding to the given
support. In part (b), the algorithm chooses a hyperedge with at least $d-2$
isolated vertices and estimates the value as the median of those isolated
vertices multiplied by the sign of the corresponding edge. In part (c), the
image of the first vertex has been removed from $Ax$ and we repeat on the
smaller graph. We continue until the entire support has been estimated, as in
part (d). Algorithm 1 Recovering a signal given its support.
Definition of sketch matrix $A$. For a constant $d$, let $A$ be a $w\times
n=O(\frac{d^{2}}{\epsilon^{2}}k)\times n$ matrix where each column is chosen
independently uniformly at random over all exactly $d$-sparse columns with
entries in $\\{-1,0,1\\}$. We can think of $A$ as the incidence matrix of a
random $d$-uniform hypergraph with random signs.
Recovery procedure.
1:procedure SetQuery($A,S,b$)$\triangleright$ Recover approximation
$x^{\prime}$ to $x_{S}$ from $b=Ax+\nu$
2: $T\leftarrow S$
3: while $\left|T\right|>0$ do
4: Define $P(q)=\\{j\mid A_{qj}\neq 0,j\in T\\}$ as the set of hyperedges in
$T$ that contain $q$.
5: Define $L_{j}=\\{q\mid A_{qj}\neq 0,\left|P(q)\right|=1\\}$ as the set of
isolated vertices in hyperedge $j$.
6: Choose a random $j\in T$ such that $\left|L_{j}\right|\geq d-1$. If this is
not possible, find a random $j\in T$ such that $\left|L_{j}\right|\geq d-2$.
If neither is possible, abort.
7: $x^{\prime}_{j}\leftarrow\operatorname*{median}_{q\in L_{j}}A_{qj}b_{q}$
8: $b\leftarrow b-x^{\prime}_{j}Ae_{j}$
9: $T\leftarrow T\setminus\\{j\\}$
10: end while
11: return $x^{\prime}$
12:end procedure
###### Lemma 3.1.
Algorithm 1 runs in time $O(dk)$.
###### Proof.
$A$ has $d$ entries per column. For each of the at most $dk$ rows $q$ in the
image of $S$, we can store the preimages $P(q)$. We also keep track of the
sets of possible next hyperedges, $J_{i}=\\{j\mid\left|L_{j}\right|\geq
d-i\\}$ for $i\in\\{1,2\\}$. We can compute these in an initial pass in
$O(dk)$. Then in each iteration, we remove an element $j\in J_{1}$ or $J_{2}$
and update $x^{\prime}_{j}$, $b$, and $T$ in $O(d)$ time. We then look at the
two or fewer non-isolated vertices $q$ in hyperedge $j$, and remove $j$ from
the associated $P(q)$. If this makes $\left|P(q)\right|=1$, we check whether
to insert the element in $P(q)$ into the $J_{i}$. Hence the inner loop takes
$O(d)$ time, for $O(dk)$ total. ∎
### 3.3 Exact Recovery
The random hypergraph $\mathbb{G}^{d}(w,k)$ of $k$ random $d$-uniform
hyperedges on $w$ vertices is well studied in [KŁ02]. We use their results to
show that the algorithm successfully terminates with high probability, and
that most hyperedges are chosen with at least $d-1$ isolated vertices:
###### Lemma 3.2.
With probability at least $1-O(1/k)$, Algorithm 1 terminates without aborting.
Furthermore, in each component at most one hyperedge is chosen with only $d-2$
isolated vertices.
We will show this by building up a couple lemmas. We define a connected
hypergraph $H$ with $r$ vertices on $s$ hyperedges to be a _hypertree_ if
$r=s(d-1)+1$ and to be _unicyclic_ if $r=s(d-1)$. Then Theorem 4 of [KŁ02]
shows that, if the graph is sufficiently sparse, $\mathbb{G}^{d}(w,k)$ is
probably composed entirely of hypertrees and unicyclic components. The precise
statement is as follows666Their statement of the theorem is slightly
different. This is the last equation in their proof of the theorem.:
###### Lemma 3.3 (Theorem 4 of [KŁ02]).
Let $m=w/d(d-1)-k$. Then with probability $1-O(d^{5}w^{2}/m^{3})$,
$\mathbb{G}^{d}(w,k)$ is composed entirely of hypertrees and unicyclic
components.
We use a simple consequence:
###### Corollary 3.1.
If $d=O(1)$ and $w\geq 2d(d-1)k$, then with probability $1-O(1/k)$,
$\mathbb{G}^{d}(w,k)$ is composed entirely of hypertrees and unicyclic
We now prove some basic facts about hypertrees and unicyclic components:
###### Lemma 3.4.
Every hypertree has a hyperedge incident on at least $d-1$ isolated vertices.
Every unicyclic component either has a hyperedge incident on $d-1$ isolated
vertices or has a hyperedge incident on $d-2$ isolated vertices, the removal
of which turns the unicyclic component into a hypertree.
###### Proof.
Let $H$ be a connected component of $s$ hyperedges and $r$ vertices.
If $H$ is a hypertree, $r=(d-1)s+1$. Because $H$ has only $ds$ total
(hyperedge, incident vertex) pairs, at most $2(s-1)$ of these pairs can
involve vertices that appear in two or more hyperedges. Thus at least one of
the $s$ edges is incident on at most one vertex that is not isolated, so some
edge has $d-1$ isolated vertices.
If $H$ is unicyclic, $r=(d-1)s$ and so at most $2s$ of the (hyperedge,
incident vertex) pairs involve non-isolated vertices. Therefore on average,
each edge has $d-2$ isolated vertices. If no edge is incident on at least
$d-1$ isolated vertices, every edge must be incident on exactly $d-2$ isolated
vertices. In that case, each edge is incident on exactly two non-isolated
vertices and each non-isolated vertex is in exactly two edges. Hence we can
perform an Eulerian tour of all the edges, so removing any edge does not
disconnect the graph. After removing the edge, the graph has $s^{\prime}=s-1$
edges and $r^{\prime}=r-d+2$ vertices; therefore
$r^{\prime}=(d-1)s^{\prime}+1$ so the graph is a hypertree. ∎
Corollary 3.1 and Lemma 3.4 combine to show Lemma 3.2.
### 3.4 Total error in terms of point error and component size
Define $C_{i,j}$ to be the event that hyperedges $i$ and $j$ are in the same
component, and $D_{i}=\sum_{j}C_{i,j}$ to be the number of hyperedges in the
same component as $i$. Define $L_{i}$ to be the cells that are used to
estimate $i$; so $L_{i}=\\{q\mid A_{qj}\neq 0,\left|P(q)\right|=1\\}$ at the
round of the algorithm when $i$ is estimated. Define
$Y_{i}=\operatorname*{median}_{q\in L_{i}}A_{qi}(b-Ax_{S})_{q}$ to be the
“point error” for hyperedge $i$, and $x^{\prime}$ to be the output of the
algorithm. Then the deviation of the output at any coordinate $i$ is at most
twice the sum of the point errors in the component containing $i$:
###### Lemma 3.5.
$\left|(x^{\prime}-x_{S})_{i}\right|\leq 2\sum_{j\in
S}\left|Y_{j}\right|C_{i,j}.$
###### Proof.
Let $T_{i}=(x^{\prime}-x_{S})_{i}$, and define $R_{i}=\\{j\mid j\neq i,\exists
q\in L_{i}\mbox{ s.t. }A_{qj}\neq 0\\}$ to be the set of hyperedges that
overlap with the cells used to estimate $i$. Then from the description of the
algorithm, it follows that
$\displaystyle T_{i}$ $\displaystyle=\operatorname*{median}_{q\in
L_{i}}A_{qi}((b-Ax_{S})_{q}-\sum_{j}A_{qj}T_{j})$
$\displaystyle\left|T_{i}\right|$
$\displaystyle\leq\left|Y_{i}\right|+\sum_{j\in R_{i}}\left|T_{j}\right|.$
We can think of the $R_{i}$ as a directed acyclic graph (DAG), where there is
an edge from $j$ to $i$ if $j\in R_{i}$. Then if $p(i,j)$ is the number of
paths from $i$ to $j$,
$\left|T_{i}\right|\leq\sum_{j}p(j,i)\left|Y_{i}\right|.$
Let $r(i)=\left|\\{j\mid i\in R_{j}\\}\right|$ be the outdegree of the DAG.
Because the $L_{i}$ are disjoint, $r(i)\leq d-\left|L_{i}\right|$. From Lemma
3.2, $r(i)\leq 1$ for all but one hyperedge in the component, and $r(i)\leq 2$
for that one. Hence $p(i,j)\leq 2$ for any $i$ and $j$, giving the result. ∎
We use the following corollary:
###### Corollary 3.2.
$\left\|x^{\prime}-x_{S}\right\|_{2}^{2}\leq 4\sum_{i\in S}D_{i}^{2}Y_{i}^{2}$
###### Proof.
$\displaystyle\left\|x^{\prime}-x_{S}\right\|_{2}^{2}$
$\displaystyle=\sum_{i\in S}(x^{\prime}-x_{S})_{i}^{2}\leq 4\sum_{i\in
S}(\sum_{\begin{subarray}{c}j\in S\\\
C_{i,j}=1\end{subarray}}\left|Y_{j}\right|)^{2}$ $\displaystyle\leq
4\sum_{i\in S}D_{i}\sum_{\begin{subarray}{c}j\in S\\\
C_{i,j}=1\end{subarray}}\left|Y_{j}\right|^{2}=4\sum_{i\in
S}D_{i}^{2}Y_{i}^{2}$
where the second inequality is the power means inequality. ∎
The $D_{j}$ and $Y_{j}$ are independent from each other, since one depends
only on $A$ over $S$ and one only on $A$ over $[n]\setminus S$. Therefore we
can analyze them separately; the next two sections show bounds and negative
dependence results for $Y_{j}$ and $D_{j}$, respectively.
### 3.5 Bound on point error
Recall from Section 3.4 that based entirely on the set $S$ and the columns of
$A$ corresponding to $S$, we can identify the positions $L_{i}$ used to
estimate $x_{i}$. We then defined the “point error”
$Y_{i}=\operatorname*{median}_{q\in
L_{i}}A_{qi}(b-Ax_{S})_{q}=\operatorname*{median}_{q\in
L_{i}}A_{qi}(A(x-x_{S})+\nu)_{q}$
and showed how to relate the total error to the point error. Here we would
like to show that the $Y_{i}$ have bounded moments and are negatively
dependent. Unfortunately, it turns out that the $Y_{i}$ are not negatively
associated so it is unclear how to show negative dependence directly. Instead,
we will define some other variables $Z_{i}$ that are always larger than the
corresponding $Y_{i}$. We will then show that the $Z_{i}$ have bounded moments
and negative association.
We use the term “NA” throughout the proof to denote negative association. For
the definition of negative association and relevant properties, see Appendix
A.
###### Lemma 3.6.
Suppose $d\geq 7$ and define
$\mu=O(\frac{\epsilon^{2}}{k}(\left\|x-x_{S}\right\|_{2}^{2}+\left\|\nu\right\|_{2}^{2}))$.
There exist random variables $Z_{i}$ such that the variables $Y_{i}^{2}$ are
stochastically dominated by $Z_{i}$, the $Z_{i}$ are negatively associated,
$\operatorname{E}[Z_{i}]=\mu$, and $\operatorname{E}[Z_{i}^{2}]=O(\mu^{2})$.
###### Proof.
The choice of the $L_{i}$ depends only on the values of $A$ over $S$; hence
conditioned on knowing $L_{i}$ we still have $A(x-x_{S})$ distributed randomly
over the space. Furthermore the distribution of $A$ and the reconstruction
algorithm are invariant under permutation, so we can pretend that $\nu$ is
permuted randomly before being added to $Ax$. Define $B_{i,q}$ to be the event
that $q\in\operatorname{supp}(Ae_{i})$, and define $D_{i,q}\in\\{-1,1\\}$
independently at random. Then define the random variable
$V_{q}=(b-Ax_{S})_{q}=\nu_{q}+\sum_{i\in[n]\setminus S}x_{i}B_{i,q}D_{i,q}.$
Because we want to show concentration of measure, we would like to show
negative association (NA) of the $Y_{i}=\operatorname*{median}_{q\in
L_{i}}A_{qi}V_{q}$. We know $\nu$ is a permutation distribution, so it is NA
[JP83]. The $B_{i,q}$ for each $i$ as a function of $q$ are chosen from a
Fermi-Dirac model, so they are NA [DR96]. The $B_{i,q}$ for different $i$ are
independent, so all the $B_{i,q}$ variables are NA. Unfortunately, the
$D_{i,q}$ can be negative, which means the $V_{q}$ are not necessarily NA.
Instead we will find some NA variables that dominate the $V_{q}$. We do this
by considering $V_{q}$ as a distribution over $D$.
Let $W_{q}=\operatorname{E}_{D}[V_{q}^{2}]=\nu_{q}^{2}+\sum_{i\in[n]\setminus
S}x_{i}^{2}B_{i,q}$. As increasing functions of NA variables, the $W_{q}$ are
NA. By Markov’s inequality $\Pr_{D}[V_{q}^{2}\geq cW_{q}]\leq\frac{1}{c}$, so
after choosing the $B_{i,q}$ and as a distribution over $D$, $V_{q}^{2}$ is
dominated by the random variable $U_{q}=W_{q}F_{q}$ where $F_{q}$ is,
independently for each $q$, given by the p.d.f. $f(c)=1/c^{2}$ for $c\geq 1$
and $f(c)=0$ otherwise. Because the distribution of $V_{q}$ over $D$ is
independent for each $q$, the $U_{q}$ jointly dominate the $V_{q}^{2}$.
The $U_{q}$ are the componentwise product of the $W_{q}$ with independent
positive random variables, so they too are NA. Then define
$Z_{i}=\operatorname*{median}_{q\in L_{i}}U_{q}.$
As an increasing function of disjoint subsets of NA variables, the $Z_{i}$ are
NA. We also have that
$\displaystyle Y_{i}^{2}$ $\displaystyle=(\operatorname*{median}_{q\in
L_{i}}A_{qi}V_{q})^{2}\leq(\operatorname*{median}_{q\in
L_{i}}\left|V_{q}\right|)^{2}$ $\displaystyle=\operatorname*{median}_{q\in
L_{i}}V_{q}^{2}\leq\operatorname*{median}_{q\in L_{i}}U_{q}=Z_{i}$
so the $Z_{i}$ stochastically dominate $Y_{i}^{2}$. We now will bound
$\operatorname{E}[Z_{i}^{2}]$. Define
$\displaystyle\mu$
$\displaystyle=E[W_{q}]=\operatorname{E}[\nu_{q}^{2}]+\sum_{i\in[n]\setminus
S}x_{i}^{2}E[B_{i,q}]$
$\displaystyle=\frac{d}{w}\left\|x-x_{S}\right\|_{2}^{2}+\frac{1}{w}\left\|\nu\right\|_{2}^{2}$
$\displaystyle\leq\frac{\epsilon^{2}}{k}(\left\|x-x_{S}\right\|_{2}^{2}+\left\|\nu\right\|_{2}^{2}).$
Then we have
$\displaystyle\Pr[W_{q}\geq c\mu]$ $\displaystyle\leq\frac{1}{c}$
$\displaystyle\Pr[U_{q}\geq c\mu]$
$\displaystyle=\int_{0}^{\infty}f(x)\Pr[W_{q}\geq c\mu/x]dx$
$\displaystyle\leq\int_{1}^{c}\frac{1}{x^{2}}\frac{x}{c}dx+\int_{c}^{\infty}\frac{1}{x^{2}}dx=\frac{1+\ln
c}{c}$
Because the $U_{q}$ are NA, they satisfy marginal probability bounds [DR96]:
$\Pr[U_{q}\geq t_{q},q\in[w]]\leq\prod_{i\in[n]}\Pr[U_{q}\geq t_{q}]$
for any $t_{q}$. Therefore
$\displaystyle\Pr[Z_{i}\geq c\mu]$
$\displaystyle\leq\sum_{\begin{subarray}{c}T\subset L_{i}\\\
\left|T\right|=\left|L_{i}\right|/2\end{subarray}}\prod_{q\in T}Pr[U_{q}\geq
c\mu]$ $\displaystyle\leq 2^{\left|L_{i}\right|}\left(\frac{1+\ln
c}{c}\right)^{\left|L_{i}\right|/2}$ (2) $\displaystyle\Pr[Z_{i}\geq c\mu]$
$\displaystyle\leq\left(4\frac{1+\ln c}{c}\right)^{d/2-1}$
If $d\geq 7$, this makes $\operatorname{E}[Z_{i}]=O(\mu)$ and
$\operatorname{E}[Z_{i}^{2}]=O(\mu^{2})$. ∎
### 3.6 Bound on component size
###### Lemma 3.7.
Let $D_{i}$ be the number of hyperedges in the same component as hyperedge
$i$. Then for any $i\neq j$,
$\mbox{Cov}(D_{i}^{2},D_{j}^{2})=\operatorname{E}[D_{i}^{2}D_{j}^{2}]-\operatorname{E}[D_{i}^{2}]^{2}\leq
O(\frac{\log^{6}k}{\sqrt{k}}).$
Furthermore, $\operatorname{E}[D_{i}^{2}]=O(1)$ and
$\operatorname{E}[D_{i}^{4}]=O(1)$.
###### Proof.
The intuition is that if one component gets larger, other components tend to
get smaller. Also the graph is very sparse, so component size is geometrically
distributed. There is a small probability that $i$ and $j$ are connected, in
which case $D_{i}$ and $D_{j}$ are positively correlated, but otherwise
$D_{i}$ and $D_{j}$ should be negatively correlated. However analyzing this
directly is rather difficult, because as one component gets larger, the
remaining components have a lower average size but higher variance. Our
analysis instead takes a detour through the hypergraph where each hyperedge is
picked independently with a probability that gives the same expected number of
hyperedges. This distribution is easier to analyze, and only differs in a
relatively small $\tilde{O}(\sqrt{k})$ hyperedges from our actual
distribution. This allows us to move between the regimes with only a loss of
$\tilde{O}(\frac{1}{\sqrt{k}})$, giving our result.
Suppose instead of choosing our hypergraph from $\mathbb{G}^{d}(w,k)$ we chose
it from $\mathbb{G}^{d}(w,\frac{k}{\binom{w}{d}})$; that is, each hyperedge
appeared independently with the appropriate probability to get $k$ hyperedges
in expectation. This model is somewhat simpler, and yields a very similar
hypergraph $\overline{G}$. One can then modify $\overline{G}$ by adding or
removing an appropriate number of random hyperedges $I$ to get exactly $k$
hyperedges, forming a uniform $G\in\mathbb{G}^{d}(w,k)$. By the Chernoff
bound, $\left|I\right|\leq O(\sqrt{k}\log k)$ with probability
$1-\frac{1}{k^{\Omega(1)}}$.
Let $\overline{D}_{i}$ be the size of the component containing $i$ in
$\overline{G}$, and $H_{i}=D_{i}^{2}-\overline{D}_{i}^{2}$. Let $E$ denote the
event that any of the $D_{i}$ or $\overline{D}_{i}$ is more than $C\log k$, or
that more than $C\sqrt{k}\log k$ hyperedges lie in $I$, for some constant $C$.
Then $E$ happens with probability less than $\frac{1}{k^{5}}$ for some $C$, so
it has negligible influence on $\operatorname{E}[D_{i}^{2}D_{j}^{2}]$. Hence
the rest of this proof will assume $E$ does not happen.
Therefore $H_{i}=0$ if none of the $O(\sqrt{k}\log k)$ random hyperedges in
$I$ touch the $O(\log k)$ hyperedges in the components containing $i$ in
$\overline{G}$, so $H_{i}=0$ with probability at least
$1-O(\frac{\log^{2}k}{\sqrt{k}})$. Even if $H_{i}\neq 0$, we still have
$\left|H_{i}\right|\leq(D_{i}^{2}+D_{j}^{2})\leq O(\log^{2}k)$.
Also, we show that the $\overline{D}_{i}^{2}$ are negatively correlated, when
conditioned on being in separate components. Let $\overline{D}(n,p)$ denote
the distribution of the component size of a random hyperedge on
$\mathbb{G}^{d}(n,p)$, where $p$ is the probability an hyperedge appears. Then
$\overline{D}(n,p)$ dominates $\overline{D}(n^{\prime},p)$ whenever
$n>n^{\prime}$ — the latter hypergraph is contained within the former. If
$\overline{C}_{i,j}$ is the event that $i$ and $j$ are connected in
$\overline{G}$, this means
$\operatorname{E}[\overline{D}_{i}^{2}\mid\overline{D}_{j}=t,\overline{C}_{i,j}=0]$
is a decreasing function in $t$, so we have negative correlation:
$\displaystyle\operatorname{E}[\overline{D}_{i}^{2}\overline{D}_{j}^{2}\mid\overline{C}_{i,j}=0]$
$\displaystyle\leq\operatorname{E}[\overline{D}_{i}^{2}\mid\overline{C}_{i,j}=0]\operatorname{E}[\overline{D}_{j}^{2}\mid\overline{C}_{i,j}=0]$
$\displaystyle\leq\operatorname{E}[\overline{D}_{i}^{2}]\operatorname{E}[\overline{D}_{j}^{2}].$
Furthermore for $i\neq j$,
$\Pr[\overline{C}_{i,j}=1]=\operatorname{E}[\overline{C}_{i,j}]=\frac{1}{k-1}\sum_{l\neq
i}\operatorname{E}[\overline{C}_{i,l}]=\frac{\operatorname{E}[\overline{D}_{i}]-1}{k-1}=O(1/k)$.
Hence
$\displaystyle\operatorname{E}[\overline{D}_{i}^{2}\overline{D}_{j}^{2}]=$
$\displaystyle\operatorname{E}[\overline{D}_{i}^{2}\overline{D}_{j}^{2}\mid\overline{C}_{i,j}=0]\Pr[\overline{C}_{i,j}=0]+$
$\displaystyle\operatorname{E}[\overline{D}_{i}^{2}\overline{D}_{j}^{2}\mid\overline{C}_{i,j}=1]\Pr[\overline{C}_{i,j}=1]$
$\displaystyle\leq$
$\displaystyle\operatorname{E}[\overline{D}_{i}^{2}]\operatorname{E}[\overline{D}_{j}^{2}]+O(\frac{\log^{4}k}{k}).$
Therefore
$\displaystyle\operatorname{E}[D_{i}^{2}D_{j}^{2}]$ $\displaystyle=$
$\displaystyle\operatorname{E}[(\overline{D}_{i}^{2}+H_{i})(\overline{D}_{j}^{2}+H_{j})]$
$\displaystyle=$
$\displaystyle\operatorname{E}[\overline{D}_{i}^{2}\overline{D}_{j}^{2}]+2\operatorname{E}[H_{i}\overline{D}_{j}^{2}]+\operatorname{E}[H_{i}H_{j}]$
$\displaystyle\leq$
$\displaystyle\operatorname{E}[\overline{D}_{i}^{2}]\operatorname{E}[\overline{D}_{j}^{2}]+O(2\frac{\log^{2}k}{\sqrt{k}}\log^{4}k+\frac{\log^{2}k}{\sqrt{k}}\log^{2}k)$
$\displaystyle=$
$\displaystyle\operatorname{E}[D_{i}^{2}-H_{i}]^{2}+O(\frac{\log^{6}k}{\sqrt{k}})$
$\displaystyle=$
$\displaystyle\operatorname{E}[D_{i}^{2}]^{2}-2\operatorname{E}[H_{i}]\operatorname{E}[D_{i}^{2}]+\operatorname{E}[H_{i}]^{2}+O(\frac{\log^{6}k}{\sqrt{k}})$
$\displaystyle\leq$
$\displaystyle\operatorname{E}[D_{i}^{2}]^{2}+O(\frac{\log^{6}k}{\sqrt{k}})$
Now to bound $\operatorname{E}[D_{i}^{4}]$ in expectation. Because our
hypergraph is exceedingly sparse, the size of a component can be bounded by a
branching process that dies out with constant probability at each step. Using
this method, Equations 71 and 72 of [COMS07] state that $\Pr[\overline{D}\geq
k]\leq e^{-\Omega(k)}$. Hence $\operatorname{E}[\overline{D}_{i}^{2}]=O(1)$
and $\operatorname{E}[\overline{D}_{i}^{4}]=O(1)$. Because $H_{i}$ is $0$ with
high probability and $O(\log^{2}k)$ otherwise, this immediately gives
$\operatorname{E}[D_{i}^{2}]=O(1)$ and $\operatorname{E}[D_{i}^{4}]=O(1)$. ∎
### 3.7 Wrapping it up
Recall from Corollary 3.2 that our total error
$\left\|x^{\prime}-x_{S}\right\|_{2}^{2}\leq 4\sum_{i}Y_{i}^{2}D_{i}^{2}\leq
4\sum_{i}Z_{i}D_{i}^{2}.$
The previous sections show that $Z_{i}$ and $D_{i}^{2}$ each have small
expectation and covariance. This allows us to apply Chebyshev’s inequality to
concentrate $4\sum_{i}Z_{i}D_{i}^{2}$ about its expectation, bounding
$\left\|x^{\prime}-x_{S}\right\|_{2}$ with high probability:
###### Lemma 3.8.
We can recover $x^{\prime}$ from $Ax+\nu$ and $S$ with
$\left\|x^{\prime}-x_{S}\right\|_{2}\leq\epsilon(\left\|x-x_{S}\right\|_{2}+\left\|\nu\right\|_{2})$
with probability at least $1-\frac{1}{c^{2}k^{1/3}}$ in $O(k)$ recovery time.
Our $A$ has $O(\frac{c}{\epsilon^{2}}k)$ rows and sparsity $O(1)$ per column.
###### Proof.
Our total error is
$\left\|x^{\prime}-x_{S}\right\|_{2}^{2}\leq 4\sum_{i}Y_{i}^{2}D_{i}^{2}\leq
4\sum_{i}Z_{i}D_{i}^{2}.$
Then by Lemma 3.6 and Lemma 3.7,
$\displaystyle\operatorname{E}[4\sum_{i}Z_{i}D_{i}^{2}]=4\sum_{i}\operatorname{E}[Z_{i}]\operatorname{E}[D_{i}^{2}]=k\mu$
where
$\mu=O(\frac{\epsilon^{2}}{k}(\left\|x-x_{S}\right\|_{2}^{2}+\left\|\nu\right\|_{2}^{2}))$.
Furthermore,
$\displaystyle\operatorname{E}[(\sum_{i}Z_{i}D_{i}^{2})^{2}]$
$\displaystyle=\sum_{i}\operatorname{E}[Z_{i}^{2}D_{i}^{4}]+\sum_{i\neq
j}\operatorname{E}[Z_{i}Z_{j}D_{i}^{2}D_{j}^{2}]$
$\displaystyle=\sum_{i}\operatorname{E}[Z_{i}^{2}]\operatorname{E}[D_{i}^{4}]+\sum_{i\neq
j}\operatorname{E}[Z_{i}Z_{j}]\operatorname{E}[D_{i}^{2}D_{j}^{2}]$
$\displaystyle\leq\sum_{i}O(\mu^{2})+\sum_{i\neq
j}\operatorname{E}[Z_{i}]\operatorname{E}[Z_{j}](\operatorname{E}[D_{i}^{2}]^{2}+O(\frac{\log^{6}k}{\sqrt{k}}))$
$\displaystyle=O(\mu^{2}k\sqrt{k}\log^{6}k)+k(k-1)\operatorname{E}[Z_{i}D_{i}^{2}]^{2}$
$\displaystyle\mbox{Var}(\sum_{i}Z_{i}D_{i}^{2})$
$\displaystyle=\operatorname{E}[(\sum_{i}Z_{i}D_{i}^{2})^{2}]-k^{2}\operatorname{E}[Z_{i}D_{i}^{2}]^{2}$
$\displaystyle\leq O(\mu^{2}k\sqrt{k}\log^{6}k)$
By Chebyshev’s inequality, this means
$\displaystyle\Pr[4\sum_{i}Z_{i}D_{i}^{2}\geq(1+c)\mu k]\leq
O(\frac{\log^{6}k}{c^{2}\sqrt{k}})$
$\displaystyle\Pr[\left\|x^{\prime}-x_{S}\right\|_{2}^{2}\geq(1+c)C\epsilon^{2}(\left\|x-x_{S}\right\|_{2}^{2}+\left\|\nu\right\|_{2}^{2})]\leq
O(\frac{1}{c^{2}k^{1/3}})$
for some constant $C$. Rescaling $\epsilon$ down by $\sqrt{C(1+c)}$, we can
get
$\left\|x^{\prime}-x_{S}\right\|_{2}\leq\epsilon(\left\|x-x_{S}\right\|_{2}+\left\|\nu\right\|_{2})$
with probability at least $1-\frac{1}{c^{2}k^{1/3}}$: ∎
Now we shall go from $k^{-1/3}$ probability of error to $k^{-c}$ error for
arbitrary $c$, with $O(c)$ multiplicative cost in time and space. We simply
perform Lemma 3.8 $O(c)$ times in parallel, and output the pointwise median of
the results. By a standard parallel repetition argument, this gives our main
result:
###### Theorem 3.1.
We can recover $x^{\prime}$ from $Ax+\nu$ and $S$ with
$\left\|x^{\prime}-x_{S}\right\|_{2}\leq\epsilon(\left\|x-x_{S}\right\|_{2}+\left\|\nu\right\|_{2})$
with probability at least $1-\frac{1}{k^{c}}$ in $O(ck)$ recovery time. Our
$A$ has $O(\frac{c}{\epsilon^{2}}k)$ rows and sparsity $O(c)$ per column.
###### Proof.
Lemma 3.8 gives an algorithm that achieves $O(k^{-1/3})$ probability of error.
We will show here how to achieve $k^{-c}$ probability of error with a linear
cost in $c$, via a standard parallel repetition argument.
Suppose our algorithm gives an $x^{\prime}$ such that
$\left\|x^{\prime}-x_{S}\right\|_{2}\leq\mu$ with probability at least $1-p$,
and that we run this algorithm $m$ times independently in parallel to get
output vectors $x^{1},\dotsc,x^{m}$. We output $y$ given by
$y_{i}=\operatorname*{median}_{j\in[m]}(x^{j})_{i}$, and claim that with high
probability $\left\|y-x_{S}\right\|_{2}\leq\mu\sqrt{3}$.
Let $J=\\{j\in[m]\mid\left\|x^{j}-x_{S}\right\|_{2}\leq\mu\\}$. Each $j\in[m]$
lies in $J$ with probability at least $1-p$, so the chance that
$\left|J\right|\leq 3m/4$ is less than $\binom{m}{m/4}p^{m/4}\leq(4ep)^{m/4}$.
Suppose that $\left|J\right|\geq 3m/4$. Then for all $i\in S$, $\left|\\{j\in
J\mid(x^{j})_{i}\leq
y_{i}\\}\right|\geq\left|J\right|-\frac{m}{2}\geq\left|J\right|/3$ and
similarly $\left|\\{j\in J\mid(x^{j})_{i}\geq
y_{i}\\}\right|\geq\left|J\right|/3$. Hence for all $i\in S$,
$\left|y_{i}-x_{i}\right|$ is smaller than at least $\left|J\right|/3$ of the
$\left|(x^{j})_{i}-x_{i}\right|$ for $j\in J$. Hence
$\displaystyle\left|J\right|\mu^{2}$ $\displaystyle\geq\sum_{i\in S}\sum_{j\in
J}((x^{j})_{i}-x_{i})^{2}\geq\sum_{i\in
S}\frac{\left|J\right|}{3}(y_{i}-x_{i})^{2}$
$\displaystyle=\frac{\left|J\right|}{3}\left\|y-x\right\|_{2}^{2}$
or
$\left\|y-x\right\|_{2}\leq\sqrt{3}\mu$
with probability at least $1-(4ep)^{m/4}$.
Using Lemma 3.8 to get $p=\frac{1}{16k^{1/3}}$ and
$\mu=\epsilon(\left\|x-x_{S}\right\|_{2}+\left\|\nu\right\|_{2})$, with
$m=12c$ repetitions we get Theorem 3.1. ∎
## 4 Applications
We give two applications where the set query algorithm is a useful primitive.
### 4.1 Heavy Hitters of sub-Zipfian distributions
For a vector $x$, let $r_{i}$ be the index of the $i$th largest element, so
$\left|x_{r_{i}}\right|$ is non-increasing in $i$. We say that $x$ is _Zipfian
with parameter $\alpha$_ if
$\left|x_{r_{i}}\right|=\Theta(\left|x_{r_{1}}\right|i^{-\alpha})$. We say
that $x$ is _sub-Zipfian with parameters ( $k$, $\alpha$)_ if there exists a
non-increasing function $f$ with
$\left|x_{r_{i}}\right|=\Theta(f(i)i^{-\alpha})$ for all $i\geq k$. A Zipfian
with parameter $\alpha$ is a sub-Zipfian with parameter $(k,\alpha)$ for all
$k$, using $f(i)=\left|x_{r_{1}}\right|$.
The Zipfian heavy hitters problem is, given a linear sketch $Ax$ of a Zipfian
$x$ and a parameter $k$, to find a $k$-sparse $x^{\prime}$ with minimal
$\left\|x-x^{\prime}\right\|_{2}$ (up to some approximation factor). We
require that $x^{\prime}$ be $k$-sparse (and no more) because we want to find
the heavy hitters themselves, not to find them as a proxy for approximating
$x$.
Zipfian distributions are common in real-world data sets, and finding heavy
hitters is one of the most important problems in data streams. Therefore this
is a very natural problem to try to improve; indeed, the original paper on
Count-Sketch discussed it [CCF02]. They show a result complementary to our
work, namely that one can find the support efficiently:
###### Lemma 4.1 (Section 4.1 of [CCF02]).
If $x$ is sub-Zipfian with parameter $(k,\alpha)$ and $\alpha>1/2$, one can
recover a candidate support set $S$ with $\left|S\right|=O(k)$ from $Ax$ such
that $\\{r_{1},\dotsc,r_{k}\\}\subseteq S$. $A$ has $O(k\log n)$ rows and
recovery succeeds with high probability in $n$.
###### Proof sketch.
Let $S_{k}=\\{r_{1},\dotsc,r_{k}\\}$. With $O(\frac{1}{\epsilon^{2}}k\log n)$
measurements, Count-Sketch identifies each $x_{i}$ to within
$\frac{\epsilon}{k}\left\|x-x_{S_{k}}\right\|_{2}$ with high probability. If
$\alpha>1/2$, this is less than $\left|x_{r_{k}}\right|/3$ for appropriate
$\epsilon$. But $\left|x_{r_{9k}}\right|\leq\left|x_{r_{k}}\right|/3$. Hence
only the largest $9k$ elements of $x$ could be estimated as larger than
anything in $x_{S_{k}}$, so the locations of the largest $9k$ estimated values
must contain $S_{k}$. ∎
It is observed in [CCF02] that a two-pass algorithm could identify the heavy
hitters exactly. However, with a single pass, no better method has been known
for Zipfian distributions than for arbitrary distributions; in fact, the lower
bound [DIPW10] on linear sparse recovery uses a geometric (and hence sub-
Zipfian) distribution.
As discussed in [CCF02], using Count-Sketch777Another analysis ([CM05]) uses
Count-Min to achieve a better polynomial dependence on $\epsilon$, but at the
cost of using the $\ell_{1}$ norm. Our result is an improvement over this as
well. with $O(\frac{k}{\epsilon^{2}}\log n)$ rows gets a $k$-sparse
$x^{\prime}$ with
$\left\|x^{\prime}-x\right\|_{2}\leq(1+\epsilon)\mathrm{Err}_{2}(x,k)=\Theta(\frac{\left|x_{r_{1}}\right|}{\sqrt{2\alpha-1}}k^{1/2-\alpha}).$
where, as in Section 1,
$\mathrm{Err}_{2}(x,k)=\min_{k\text{-sparse
}\hat{x}}\left\|\hat{x}-x\right\|_{2}.$
The set query algorithm lets us improve from a $1+\epsilon$ approximation to a
$1+o(1)$ approximation. This is not useful for approximating $x$, since
increasing $k$ is much more effective than decreasing $\epsilon$. Instead, it
is useful for finding $k$ elements that are quite close to being the actual
$k$ heavy hitters of $x$.
Naïve application of the set query algorithm to the output set of Lemma 4.1
would only get a close $O(k)$-sparse vector, not a $k$-sparse vector. To get a
$k$-sparse vector, we must show a lemma that generalizes one used in the proof
of sparse recovery of Count-Sketch (first in [CM06], but our description is
more similar to [GI10]).
###### Lemma 4.2.
Let $x,x^{\prime}\in\mathbb{R}^{n}$. Let $S$ and $S^{\prime}$ be the locations
of the largest $k$ elements (in magnitude) of $x$ and $x^{\prime}$,
respectively. Then if
$\left\|(x^{\prime}-x)_{S\cup
S^{\prime}}\right\|_{2}\leq\epsilon\mathrm{Err}_{2}(x,k),$
for $\epsilon\leq 1$, we have
$\left\|x^{\prime}_{S^{\prime}}-x\right\|_{2}\leq(1+3\epsilon)\mathrm{Err}_{2}(x,k).$
Previous proofs have shown the following weaker form:
###### Corollary 4.1.
If we change the condition (*) to
$\left\|x^{\prime}-x\right\|_{\infty}\leq\frac{\epsilon}{\sqrt{2k}}\mathrm{Err}_{2}(x,k)$,
the same result holds.
The corollary is immediate from Lemma 4.2 and $\left\|(x^{\prime}-x)_{S\cup
S^{\prime}}\right\|_{2}\leq\sqrt{\left|S\cup
S^{\prime}\right|}\left\|(x^{\prime}-x)_{S\cup S^{\prime}}\right\|_{\infty}$.
###### Proof of Lemma 4.2.
We have
(3) $\displaystyle\left\|x^{\prime}_{S^{\prime}}-x\right\|_{2}^{2}$
$\displaystyle=\left\|(x^{\prime}-x)_{S^{\prime}}\right\|_{2}^{2}+\left\|x_{S\setminus
S^{\prime}}\right\|_{2}^{2}+\left\|x_{[n]\setminus(S\cup
S^{\prime})}\right\|_{2}^{2}$
The tricky bit is to bound the middle term $\left\|x_{S\setminus
S^{\prime}}\right\|_{2}^{2}$. We will show that it is not much larger than
$\left\|x_{S^{\prime}\setminus S}\right\|_{2}^{2}$.
Let $d=\left|S\setminus S^{\prime}\right|$, and let $a$ be the $d$-dimensional
vector corresponding to the absolute values of the coefficients of $x$ over
$S\setminus S^{\prime}$. That is, if $S\setminus
S^{\prime}=\\{j_{1},\dots,j_{d}\\}$, then $a_{i}=\left|x_{j_{i}}\right|$ for
$i\in[d]$. Let $a^{\prime}$ be analogous for $x^{\prime}$ over $S\setminus
S^{\prime}$, and let $b$ and $b^{\prime}$ be analogous for $x$ and
$x^{\prime}$ over $S^{\prime}\setminus S$, respectively.
Let $E=\mathrm{Err}_{2}(x,k)=\left\|x-x_{S}\right\|_{2}$. We have
$\displaystyle\left\|x_{S\setminus
S^{\prime}}\right\|_{2}^{2}-\left\|x_{S^{\prime}\setminus S}\right\|_{2}^{2}$
$\displaystyle=\left\|a\right\|_{2}^{2}-\left\|b\right\|_{2}^{2}$
$\displaystyle=(a-b)\cdot(a+b)$
$\displaystyle\leq\left\|a-b\right\|_{2}\left\|a+b\right\|_{2}$
$\displaystyle\leq\left\|a-b\right\|_{2}(2\left\|b\right\|_{2}+\left\|a-b\right\|_{2})$
$\displaystyle\leq\left\|a-b\right\|_{2}(2E+\left\|a-b\right\|_{2})$
So we should bound $\left\|a-b\right\|_{2}$. We know that
$\left|\left|p\right|-\left|q\right|\right|\leq\left|p-q\right|$ for all $p$
and $q$, so
$\displaystyle\left\|a-a^{\prime}\right\|_{2}^{2}+\left\|b-b^{\prime}\right\|_{2}^{2}$
$\displaystyle\leq\left\|(x-x^{\prime})_{S\setminus
S^{\prime}}\right\|_{2}^{2}+\left\|(x-x^{\prime})_{S^{\prime}\setminus
S}\right\|_{2}^{2}$ $\displaystyle\leq\left\|(x-x^{\prime})_{S\cup
S^{\prime}}\right\|_{2}^{2}\leq\epsilon^{2}E^{2}.$
We also know that $a-b$ and $b^{\prime}-a^{\prime}$ both contain all
nonnegative coefficients. Hence
$\displaystyle\left\|a-b\right\|_{2}^{2}$
$\displaystyle\leq\left\|a-b+b^{\prime}-a^{\prime}\right\|_{2}^{2}$
$\displaystyle\leq\left(\left\|a-a^{\prime}\right\|_{2}+\left\|b^{\prime}-b\right\|_{2}\right)^{2}$
$\displaystyle\leq
2\left\|a-a^{\prime}\right\|_{2}^{2}+2\left\|b-b^{\prime}\right\|_{2}^{2}$
$\displaystyle\leq 2\epsilon^{2}E^{2}$ $\displaystyle\left\|a-b\right\|_{2}$
$\displaystyle\leq\sqrt{2}\epsilon E.$
Therefore
$\displaystyle\left\|x_{S\setminus
S^{\prime}}\right\|_{2}^{2}-\left\|x_{S^{\prime}\setminus S}\right\|_{2}^{2}$
$\displaystyle\leq\sqrt{2}\epsilon E(2E+\sqrt{2}\epsilon E)$
$\displaystyle\leq(2\sqrt{2}+2)\epsilon E^{2}$ $\displaystyle\leq 5\epsilon
E^{2}.$
Plugging into Equation 3, and using
$\left\|(x^{\prime}-x)_{S^{\prime}}\right\|_{2}^{2}\leq\epsilon^{2}E^{2}$,
$\displaystyle\left\|x^{\prime}_{S^{\prime}}-x\right\|_{2}^{2}$
$\displaystyle\leq\epsilon^{2}E^{2}+5\epsilon
E^{2}+\left\|x_{S^{\prime}\setminus
S}\right\|_{2}^{2}+\left\|x_{[n]\setminus(S\cup S^{\prime})}\right\|_{2}^{2}$
$\displaystyle\leq 6\epsilon E^{2}+\left\|x_{[n]\setminus S}\right\|_{2}^{2}$
$\displaystyle=(1+6\epsilon)E^{2}$
$\displaystyle\left\|x^{\prime}_{S^{\prime}}-x\right\|_{2}$
$\displaystyle\leq(1+3\epsilon)E.$
∎
With this lemma in hand, on Zipfian distributions we can get a $k$-sparse
$x^{\prime}$ with a $1+o(1)$ approximation factor.
###### Theorem 4.1.
Suppose $x$ comes from a sub-Zipfian distribution with parameter $\alpha>1/2$.
Then we can recover a $k$-sparse $x^{\prime}$ from $Ax$ with
$\left\|x^{\prime}-x\right\|_{2}\leq\frac{\epsilon}{\sqrt{\log
n}}\mathrm{Err}_{2}(x,k).$
with $O(\frac{c}{\epsilon^{2}}k\log n)$ rows and $O(n\log n)$ recovery time,
with probability at least $1-\frac{1}{k^{c}}$.
###### Proof.
By Lemma 4.1 we can identify a set $S$ of size $O(k)$ that contains all the
heavy hitters. We then run the set query algorithm of Theorem 3.1 with
$\frac{\epsilon}{3\sqrt{\log n}}$ substituted for $\epsilon$. This gives an
$\hat{x}$ with
$\displaystyle\left\|\hat{x}-x_{S}\right\|_{2}$
$\displaystyle\leq\frac{\epsilon}{3\sqrt{\log n}}\mathrm{Err}_{2}(x,k).$
Let $x^{\prime}$ contain the largest $k$ coefficients of $\hat{x}$. By Lemma
4.2 we have
$\displaystyle\left\|x^{\prime}-x\right\|_{2}\leq(1+\frac{\epsilon}{\sqrt{\log
n}})\mathrm{Err}_{2}(x,k).$
∎
### 4.2 Block-sparse vectors
In this section we consider the problem of finding block-sparse
approximations. In this case, the coordinate set $\\{1\ldots n\\}$ is
partitioned into $n/b$ blocks, each of length $b$. We define a $(k,b)$-block-
sparse vector to be a vector where all non-zero elements are contained in at
most $k/b$ blocks. That is, we partition $\\{1,\dotsc,n\\}$ into
$T_{i}=\\{(i-1)b+1,\dotsc,ib\\}$. A vector $x$ is $(k,b)$-block-sparse if
there exist $S_{1},\dotsc,S_{k/b}\in\\{T_{1},\dotsc,T_{n/b}\\}$ with
$\operatorname{supp}(x)\subseteq\bigcup S_{i}$. Define
$\mathrm{Err}_{2}(x,k,b)=\min_{(k,b)-\mbox{\scriptsize block-sparse
}\hat{x}}\left\|x-\hat{x}\right\|_{2}.$
Finding the support of block-sparse vectors is closely related to finding
block heavy hitters, which is studied for the $\ell_{1}$ norm in [ABI08]. The
idea is to perform dimensionality reduction of each block into $\log n$
dimensions, then perform sparse recovery on the resulting $\frac{k\log
n}{b}$-sparse vector. The differences from previous work are minor, so we
relegate the details to Appendix C.
###### Lemma 4.3.
For any $b$ and $k$, there exists a family of matrices $A$ with
$O(\frac{k}{\epsilon^{5}b}\log n)$ rows and column sparsity
$O(\frac{1}{\epsilon^{2}}\log n)$ such that we can recover a support $S$ from
$Ax$ in $O(\frac{n}{\epsilon^{2}b}\log n)$ time with
$\left\|x-x_{S}\right\|_{2}\leq(1+\epsilon)\mathrm{Err}_{2}(x,k,b)$
with probability at least $1-n^{-\Omega(1)}$.
Once we know a good support $S$, we can run Algorithm 1 to estimate $x_{S}$:
###### Theorem 4.2.
For any $b$ and $k$, there exists a family of binary matrices $A$ with
$O(\frac{1}{\epsilon^{2}}k+\frac{k}{\epsilon^{5}b}\log n)$ rows such that we
can recover a $(k,b)$-block-sparse $x^{\prime}$ in
$O(k+\frac{n}{\epsilon^{2}b}\log n)$ time with
$\left\|x^{\prime}-x\right\|_{2}\leq(1+\epsilon)\mathrm{Err}_{2}(x,k,b)$
with probability at least $1-\frac{1}{k^{\Omega(1)}}$.
###### Proof.
Let $S$ be the result of Lemma 4.3 with approximation $\epsilon/3$, so
$\left\|x-x_{S}\right\|_{2}\leq(1+\frac{\epsilon}{3})\mathrm{Err}_{2}(x,k,b).$
Then the set query algorithm on $x$ and $S$ uses $O(k/\epsilon^{2})$ rows to
return an $x^{\prime}$ with
$\left\|x^{\prime}-x_{S}\right\|_{2}\leq\frac{\epsilon}{3}\left\|x-x_{S}\right\|_{2}.$
Therefore
$\displaystyle\left\|x^{\prime}-x\right\|_{2}$
$\displaystyle\leq\left\|x^{\prime}-x_{S}\right\|_{2}+\left\|x-x_{S}\right\|_{2}$
$\displaystyle\leq(1+\frac{\epsilon}{3})\left\|x-x_{S}\right\|_{2}$
$\displaystyle\leq(1+\frac{\epsilon}{3})^{2}\mathrm{Err}_{2}(x,k,b)$
$\displaystyle\leq(1+\epsilon)\mathrm{Err}_{2}(x,k,b)$
as desired. ∎
If the block size $b$ is at least $\log n$ and $\epsilon$ is constant, this
gives an optimal bound of $O(k)$ rows.
## 5 Conclusion and Future Work
We show efficient recovery of vectors conforming to Zipfian or block sparse
models, but leave open extending this to other models. Our framework
decomposes the task into first locating the heavy hitters and then estimating
them, and our set query algorithm is an efficient general solution for
estimating the heavy hitters once found. The remaining task is to efficiently
locate heavy hitters in other models.
Our analysis assumes that the columns of $A$ are fully independent. It would
be valuable to reduce the independence needed, and hence the space required to
store $A$.
We show $k$-sparse recovery of Zipfian distributions with $1+o(1)$
approximation in $O(k\log n)$ space. Can the $o(1)$ be made smaller, or a
lower bound shown, for this problem?
## Acknowledgments
I would like to thank my advisor Piotr Indyk for much helpful advice, Anna
Gilbert for some preliminary discussions, and Joseph O’Rourke for pointing me
to [KŁ02].
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## Appendix A Negative Dependence
Negative dependence is a fairly common property in balls-and-bins types of
problems, and can often cleanly be analyzed using the framework of _negative
association_ ([DR96, DPR96, JP83]).
###### Definition 1 (Negative Association).
Let $(X_{1},\dotsc,X_{n})$ be a vector of random variables. Then
$(X_{1},\dotsc,X_{n})$ are _negatively associated_ if for every two disjoint
index sets, $I,J\subseteq[n]$,
$\displaystyle\operatorname{E}[f(X_{i},i\in I)g(X_{j},j\in J)]$
$\displaystyle\leq$ $\displaystyle\operatorname{E}[f(X_{i},i\in
I)]E[g(X_{j},j\in J)]$
for all functions $f\colon\mathbb{R}^{\left|I\right|}\to\mathbb{R}$ and
$g\colon\mathbb{R}^{\left|J\right|}\to\mathbb{R}$ that are both non-decreasing
or both non-increasing.
If random variables are negatively associated then one can apply most standard
concentration of measure arguments, such as Chebyshev’s inequality and the
Chernoff bound. This means it is a fairly strong property, which makes it hard
to prove directly. What makes it so useful is that it remains true under two
composition rules:
###### Lemma A.1 ([DR96], Proposition 7).
1. 1.
If $(X_{1},\dotsc,X_{n})$ and $(Y_{1},\dotsc,Y_{m})$ are each negatively
associated and mutually independent, then
$(X_{1},\dotsc,X_{n},Y_{1},\dotsc,Y_{m})$ is negatively associated.
2. 2.
Suppose $(X_{1},\dotsc,X_{n})$ is negatively associated. Let
$I_{1},\dotsc,I_{k}\subseteq[n]$ be disjoint index sets, for some positive
integer $k$. For $j\in[k]$, let
$h_{j}\colon\mathbb{R}^{\left|I_{j}\right|}\to\mathbb{R}$ be functions that
are all non-decreasing or all non-increasing, and define
$Y_{j}=h_{j}(X_{i},i\in I_{j})$. Then $(Y_{1},\dotsc,Y_{k})$ is also
negatively associated.
Lemma A.1 allows us to relatively easily show that one component of our error
(the point error) is negatively associated without performing any computation.
Unfortunately, the other component of our error (the component size) is not
easily built up by repeated applications of Lemma A.1888This paper considers
the component size of each hyperedge, which clearly is not negatively
associated: if one hyperedge is in a component of size $k$ than so is every
other hyperedge. But one can consider variants that just consider the
distribution of component sizes, which seems plausibly negatively associated.
However, this is hard to prove.. Therefore we show something much weaker for
this error, namely _approximate negative correlation_ :
$\operatorname{E}[X_{i}X_{j}]-\operatorname{E}[X_{i}]E[X_{j}]\leq\frac{1}{k^{\Omega(1)}}\operatorname{E}[X_{i}]\operatorname{E}[X_{j}]$
for all $i\neq j$. This is still strong enough to use Chebyshev’s inequality.
## Appendix B Set Query in the $\ell_{1}$ norm
This section works through all the changes to prove the set query algorithm
works in the $\ell_{1}$ norm with $w=O(\frac{1}{\epsilon}k)$ measurements.
We use Lemma 3.5 to get an $\ell_{1}$ analog of Corollary 3.2:
(4) $\displaystyle\left\|x^{\prime}-x_{S}\right\|_{1}$
$\displaystyle=\sum_{i\in S}\left|(x^{\prime}-x_{S})_{i}\right|$
$\displaystyle\leq\sum_{i\in S}2\sum_{j\in
S}C_{i,j}\left|Y_{j}\right|=2\sum_{i\in S}D_{i}\left|Y_{i}\right|.$
Then we bound the expectation, variance, and covariance of $D_{i}$ and
$\left|Y_{i}\right|$. The bound on $D_{i}$ works the same as in Section 3.6:
$\operatorname{E}[D_{i}]=O(1)$, $\operatorname{E}[D_{i}^{2}]=O(1)$,
$\operatorname{E}[D_{i}D_{j}]-\operatorname{E}[D_{i}]^{2}\leq
O(\log^{4}k/\sqrt{k})$.
The bound on $\left|Y_{i}\right|$ is slightly different. We define
$U_{q}^{\prime}=\left|\nu_{q}\right|+\sum_{i\in[n]\setminus
S}\left|x_{i}\right|B_{i,q}$
and observe that $U_{q}^{\prime}\geq\left|V_{q}\right|$, and $U_{q}^{\prime}$
is NA. Hence
$Z_{i}^{\prime}=\operatorname*{median}_{q\in L_{i}}U_{q}^{\prime}$
is NA, and $\left|Y_{i}\right|\leq Z_{i}^{\prime}$. Define
$\displaystyle\mu$
$\displaystyle=\operatorname{E}[U_{q}^{\prime}]=\frac{d}{w}\left\|x-x_{S}\right\|_{1}+\frac{1}{w}\left\|\nu\right\|_{1}$
$\displaystyle\leq\frac{\epsilon}{k}(\left\|x-x_{S}\right\|_{1}+\left\|\nu\right\|_{1})$
then
$\Pr[Z_{i}^{\prime}\geq c\mu]\leq
2^{\left|L_{i}\right|}(\frac{1}{c})^{\left|L_{i}\right|/2}\leq\left(\frac{4}{c}\right)^{d-2}$
so $\operatorname{E}[Z_{i}^{\prime}]=O(\mu)$ and
$\operatorname{E}[Z_{i}^{\prime 2}]=O(\mu^{2})$.
Now we will show the analog of Section 3.7. We know
$\left\|x^{\prime}-x_{S}\right\|_{2}\leq 2\sum_{i}D_{i}Z_{i}^{\prime}$
and
$\operatorname{E}[2\sum_{i}D_{i}Z_{i}^{\prime}]=2\sum_{i}\operatorname{E}[D_{i}]\operatorname{E}[Z_{i}^{\prime}]=k\mu^{\prime}$
for some
$\mu^{\prime}=O(\frac{\epsilon}{k}(\left\|x-x_{S}\right\|_{1}+\left\|\nu\right\|_{1}))$.
Then
$\displaystyle\operatorname{E}[(\sum D_{i}Z_{i}^{\prime})^{2}]$
$\displaystyle=\sum_{i}\operatorname{E}[D_{i}^{2}]\operatorname{E}[Z_{i}^{\prime
2}]+\sum_{i\neq
j}\operatorname{E}[D_{i}D_{j}]\operatorname{E}[Z_{i}^{\prime}Z_{j}^{\prime}]$
$\displaystyle\leq\sum_{i}O(\mu^{\prime 2})+\sum_{i\neq
j}(\operatorname{E}[D_{i}]^{2}+O(\log^{4}k/\sqrt{k}))\operatorname{E}[Z_{i}^{\prime}]^{2}$
$\displaystyle=O(\mu^{\prime
2}k\sqrt{k}\log^{4}k)+k(k-1)\operatorname{E}[D_{i}Z_{i}^{\prime}]^{2}$
$\displaystyle\mbox{Var}(2\sum_{i}Z_{i}^{\prime}D_{i})$ $\displaystyle\leq
O(\mu^{\prime 2}k\sqrt{k}\log^{4}k).$
By Chebyshev’s inequality, we get
$\Pr[\left\|x^{\prime}-x_{S}\right\|_{1}\geq(1+\alpha)k\mu^{\prime}]\leq
O(\frac{\log^{4}k}{\alpha^{2}\sqrt{k}})$
and the main theorem (for constant $c=1/3$) follows. The parallel repetition
method of Section 3.7 works the same as in the $\ell_{2}$ case to support
arbitrary $c$.
## Appendix C Block Heavy Hitters
###### Lemma 4.3.
For any $b$ and $k$, there exists a family of matrices $A$ with
$O(\frac{k}{\epsilon^{5}b}\log n)$ rows and column sparsity
$O(\frac{1}{\epsilon^{2}}\log n)$ such that we can recover a support $S$ from
$Ax$ in $O(\frac{n}{\epsilon^{2}b}\log n)$ time with
$\left\|x-x_{S}\right\|_{2}\leq(1+\epsilon)\mathrm{Err}_{2}(x,k,b)$
with probability at least $1-n^{-\Omega(1)}$.
###### Proof.
This proof follows the method of [ABI08], but applies to the $\ell_{2}$ norm
and is in the (slightly stronger) sparse recovery framework rather than the
heavy hitters framework. The idea is to perform dimensionality reduction, then
use an argument similar to those for Count-Sketch (first in [CM06], but we
follow more closely the description in [GI10]).
Define $s=k/b$ and $t=n/b$, and decompose $[n]$ into equal sized blocks
$T_{1},\dotsc,T_{t}$. Let $x_{(T_{i})}\in\mathbb{R}^{b}$ denote the
restriction of $x_{T_{i}}$ to the coordinates $T_{i}$. Let $U\subseteq[t]$
have $\left|U\right|=s$ and contain the $s$ largest blocks in $x$, so
$\mathrm{Err}_{2}(x,k,b)=\left\|\sum_{i\notin U}x_{T_{i}}\right\|_{2}$.
Choose an i.i.d. standard Gaussian matrix $\rho\in\mathbb{R}^{m\times b}$ for
$m=O(\frac{1}{\epsilon^{2}}\log n)$. Define $y_{q,i}=(\rho x_{(T_{q})})_{i}$,
so as a distribution over $\rho$, $y_{q,i}$ is a Gaussian with variance
$\left\|x_{(T_{q})}\right\|_{2}^{2}$.
Let $h_{1},\dotsc,h_{m}\colon[t]\to[l]$ be pairwise independent hash functions
for some $l=O(\frac{1}{\epsilon^{3}}s)$, and
$g_{1},\dotsc,g_{m}\colon[t]\to\\{-1,1\\}$ also be pairwise independent. Then
we make $m$ hash tables $H^{(1)},\dotsc,H^{(m)}$ of size $l$ each, and say
that the value of the $j$th cell in the $i$th hash table $H^{(i)}$ is given by
$H^{(i)}_{j}=\sum_{q:h_{i}(q)=j}g_{i}(q)y_{q,i}$
Then the $H^{(i)}_{j}$ form a linear sketch of
$ml=O(\frac{k}{\epsilon^{5}b}\log n)$ cells. We use this sketch to estimate
the mass of each block, and output the blocks that we estimate to have the
highest mass. Our estimator for $\left\|x_{T_{i}}\right\|_{2}$ is
$z_{i}^{\prime}=\alpha\operatorname*{median}_{j\in[m]}\left|H^{(j)}_{h_{j}(i)}\right|$
for some constant scaling factor $\alpha\approx 1.48$. Since we only care
which blocks have the largest magnitude, we don’t actually need to use
$\alpha$.
We first claim that for each $i$ and $j$ with probability $1-O(\epsilon)$,
$(H^{(j)}_{h_{j}(i)}-y_{i,j})^{2}\leq
O(\frac{\epsilon^{2}}{s}(\mathrm{Err}_{2}(x,k,b))^{2})$. To prove it, note
that the probability any $q\in U$ with $q\neq i$ having $h_{j}(q)=h_{j}(i)$ is
at most $\frac{s}{l}\leq\epsilon^{3}$. If such a collision with a heavy hitter
does not happen, then
$\displaystyle\operatorname{E}[(H^{(j)}_{h_{j}(i)}-y_{i,j})^{2}]$
$\displaystyle=\operatorname{E}[\sum_{p\neq i,h_{j}(p)=h_{j}(i)}y_{p,j}^{2}]$
$\displaystyle\leq\sum_{p\notin U}\frac{1}{l}\operatorname{E}[y_{p,j}^{2}]$
$\displaystyle=\frac{1}{l}\sum_{p\notin U}\left\|x_{T_{p}}\right\|_{2}^{2}$
$\displaystyle=\frac{1}{l}(\mathrm{Err}_{2}(x,k,b))^{2}$
By Markov’s inequality and the union bound, we have
$\Pr[(H^{(j)}_{h_{j}(i)}-y_{i,j})^{2}\geq\frac{\epsilon^{2}}{s}(\mathrm{Err}_{2}(x,k,b))^{2}]\leq\epsilon+\epsilon^{3}=O(\epsilon)$
Let $B_{i,j}$ be the event that
$(H^{(j)}_{h_{j}(i)}-y_{i,j})^{2}>O(\frac{\epsilon^{2}}{s}(\mathrm{Err}_{2}(x,k,b))^{2})$,
so $\Pr[B_{i,j}]=O(\epsilon)$. This is independent for each $j$, so by the
Chernoff bound $\sum_{j=1}^{m}B_{i,j}\leq O(\epsilon m)$ with high probability
in $n$.
Now, $\left|y_{i,j}\right|$ is distributed according to the positive half of a
Gaussian, so there is some constant $\alpha\approx 1.48$ such that
$\alpha\left|y_{i,j}\right|$ is an unbiased estimator for
$\left\|x_{T_{i}}\right\|_{2}$. For any $C\geq 1$ and some
$\delta=O(C\epsilon)$, we expect less than $\frac{1-C\epsilon}{2}m$ of the
$\alpha\left|y_{i,j}\right|$ to be below
$(1-\delta)\left\|x_{T_{i}}\right\|_{2}$, less than $\frac{1-C\epsilon}{2}m$
to be above $(1+\delta)\left\|x_{T_{i}}\right\|_{2}$, and more than $C\epsilon
m$ to be in between. Because $m\geq\Omega(\frac{1}{\epsilon^{2}}\log n)$, the
Chernoff bound shows that with high probability the actual number of
$\alpha\left|y_{i,j}\right|$ in each interval is within
$\frac{\epsilon}{2}m=O(\frac{1}{\epsilon}\log n)$ of its expectation. Hence
$\left|\left\|x_{T_{i}}\right\|_{2}-\alpha\operatorname*{median}_{j\in[m]}\left|y_{i,j}\right|\right|\leq\delta\left\|x_{T_{i}}\right\|_{2}=O(C\epsilon)\left\|x_{T_{i}}\right\|_{2}.$
even if $\frac{(C-1)\epsilon}{2}m$ of the $y_{i,j}$ were adversarially
modified. We can think of the events $B_{i,j}$ as being such adversarial
modifications. We find that
$\displaystyle\left|\left\|x_{T_{i}}\right\|_{2}-z_{i}\right|$
$\displaystyle=\left|\left\|x_{T_{i}}\right\|_{2}-\alpha\operatorname*{median}_{j\in[m]}\left|H_{h_{j}(i)}^{(j)}\right|\right|$
$\displaystyle\leq
O(\epsilon)\left\|x_{T_{i}}\right\|_{2}+O(\frac{\epsilon}{\sqrt{s}}\mathrm{Err}_{2}(x,k,b)).$
$(\left\|x_{T_{i}}\right\|_{2}-z_{i})^{2}\leq
O(\epsilon^{2}\left\|x_{T_{i}}\right\|_{2}^{2}+\frac{\epsilon^{2}}{s}(\mathrm{Err}_{2}(x,k,b))^{2})$
Define $w_{i}=\left\|x_{T_{i}}\right\|_{2}$, $\mu=\mathrm{Err}_{2}(x,k,b)$,
and $\hat{U}\subseteq[t]$ to contain the $s$ largest coordinates in $z$. Since
$z$ is computed from the sketch, the recovery algorithm can compute $\hat{U}$.
The output of our algorithm will be the blocks corresponding to $\hat{U}$.
We know $\mu^{2}=\sum_{i\notin U}w_{i}^{2}=\left\|w_{[t]\setminus
U}\right\|_{2}^{2}$ and $\left|w_{i}-z_{i}\right|\leq O(\epsilon
w_{i}+\frac{\epsilon}{\sqrt{s}}\mu)$ for all $i$. We will show that
$\left\|w_{[t]\setminus\hat{U}}\right\|_{2}^{2}\leq(1+O(\epsilon))\mu^{2}.$
This is analogous to the proof of Count-Sketch, or to Corollary 4.1. Note that
$\displaystyle\left\|w_{[t]\setminus\hat{U}}\right\|_{2}^{2}$
$\displaystyle=\left\|w_{U\setminus\hat{U}}\right\|_{2}^{2}+\left\|w_{[t]\setminus(U\cup\hat{U})}\right\|_{2}^{2}$
For any $i\in U\setminus\hat{U}$ and $j\in\hat{U}\setminus U$, we have
$z_{j}>z_{i}$, so
$w_{i}-w_{j}\leq O(\frac{\epsilon}{\sqrt{s}}\mu+\epsilon w_{i})$
Let $a=\max_{i\in U\setminus\hat{U}}w_{i}$ and $b=\min_{j\in\hat{U}\setminus
U}w_{j}$. Then $a\leq b+O(\frac{\epsilon}{\sqrt{s}}\mu+\epsilon a)$, and
dividing by $(1-O(\epsilon))$ we get $a\leq
b(1+O(\epsilon))+O(\frac{\epsilon}{\sqrt{s}}\mu)$. Furthermore
$\left\|w_{\hat{U}\setminus U}\right\|_{2}^{2}\geq b^{2}\left|\hat{U}\setminus
U\right|$, so
$\displaystyle\left\|w_{U\setminus\hat{U}}\right\|_{2}^{2}\leq$
$\displaystyle\left(\left\|w_{\hat{U}\setminus
U}\right\|_{2}\frac{1+O(\epsilon)}{\sqrt{\left|\hat{U}\setminus
U\right|}}+O(\frac{\epsilon}{\sqrt{s}}\mu)\right)^{2}\left|\hat{U}\setminus
U\right|$ $\displaystyle\leq$ $\displaystyle\left(\left\|w_{\hat{U}\setminus
U}\right\|_{2}(1+O(\epsilon))+O(\epsilon\mu)\right)^{2}$ $\displaystyle=$
$\displaystyle\left\|w_{\hat{U}\setminus
U}\right\|_{2}^{2}(1+O(\epsilon))+(2+O(\epsilon))\left\|w_{\hat{U}\setminus
U}\right\|_{2}O(\epsilon\mu)$ $\displaystyle+O(\epsilon^{2}\mu^{2})$
$\displaystyle\leq$ $\displaystyle\left\|w_{\hat{U}\setminus
U}\right\|_{2}^{2}+O(\epsilon\mu^{2})$
because $\left\|w_{\hat{U}\setminus U}\right\|_{2}\leq\mu$. Thus
$\displaystyle\left\|w-w_{\hat{U}}\right\|_{2}=\left\|w_{[t]\setminus\hat{U}}\right\|_{2}^{2}$
$\displaystyle\leq O(\epsilon\mu^{2})+\left\|w_{\hat{U}\setminus
U}\right\|_{2}^{2}+\left\|w_{[t]\setminus(U\cup\hat{U})}\right\|_{2}^{2}$
$\displaystyle=O(\epsilon\mu^{2})+\mu^{2}=(1+O(\epsilon))\mu^{2}.$
This is exactly what we want. If $S=\bigcup_{i\in\hat{U}}T_{i}$ contains the
blocks corresponding to $\hat{U}$, then
$\left\|x-x_{S}\right\|_{2}=\left\|w-w_{\hat{U}}\right\|_{2}\leq(1+O(\epsilon))\mu=(1+O(\epsilon))\mathrm{Err}_{2}(x,k,b)$
Rescale $\epsilon$ to change $1+O(\epsilon)$ into $1+\epsilon$ and we’re done.
∎
|
arxiv-papers
| 2010-07-07T22:03:06 |
2024-09-04T02:49:11.480726
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Eric Price",
"submitter": "Eric Price",
"url": "https://arxiv.org/abs/1007.1253"
}
|
1007.1281
|
11institutetext: Department of Systems Science, School of Management, Beijing
Normal University - Beijing 100875, PRC
Networks and genealogical trees Structures and organization in complex systems
# Exact Solution for Optimal Navigation with Total Cost Restriction
Y. Li D. Zhou Y. Hu111E-mail: yanqing.hu.sc@gmail.com J. Zhang Z. Di
###### Abstract
Recently, Li et al. have concentrated on Kleinberg’s navigation model with a
certain total length constraint $\Lambda=cN$, where $N$ is the number of total
nodes and $c$ is a constant. Their simulation results for the 1- and
2-dimensional cases indicate that the optimal choice for adding extra long-
range connections between any two sites seems to be $\alpha=d+1$, where $d$ is
the dimension of the lattice and $\alpha$ is the power-law exponent. In this
paper, we prove analytically that for the 1-dimensional large networks, the
optimal power-law exponent is $\alpha=2$ Further, we study the impact of the
network size and provide exact solutions for time cost as a function of the
power-law exponent $\alpha$. We also show that our analytical results are in
excellent agreement with simulations.
###### pacs:
89.75.Hc
###### pacs:
89.75.Fb
## 1 Introduction
Since Milgram and his cooperators conducted the first small-world experiment
in the 1960s, much attention has been dedicated to the problem of navigation
in real social networks. The first navigation model was proposed by
Kleinberg[1]. He employed an $L\times L$ square lattice, where in addition to
the links between nearest neighbors each node $i$ was connected to a random
node $j$ with a probability $P_{ij}\propto r_{ij}^{-\alpha}$ ($r_{ij}$ denotes
the lattice distance between nodes $i$ and $j$). Kleinberg has proved that
when $\alpha=d$, where $d$ is the dimension of the lattice, the optimal time
cost of navigation with a decentralized algorithm is at most $O(\log^{2}L)$.
The optimal case indicates that individuals are able to find short paths
effectively with only local information, which can explain six degrees of
separation quite well. In recent years, further studies on Kleinberg’s
navigation model have been developed [2, 4, 6, 3, 7, 5, 8]. Roberson et al.
study the navigation problem in fractal small-world networks [2], where they
prove that $\alpha=d$ is also the optimal power-law exponent in the fractal
case. Cartozo et al. use dynamical equations to study the process of Kleinberg
navigation [4, 3]. They provide an exact solution for the asymptotic behavior
of such a greedy algorithm as a function of the dimension $d$ of the lattice
and the power-law exponent $\alpha$. Yang et al. construct a network with
limited cost $\Lambda=C$. The limited cost $\Lambda$ represents the total
length of the long-range connections which are added with power-law distance
distribution $P(r)=a{r^{-\alpha}}$ [5] in one-dimensional space. They find
that the network has the smallest average shortest path when $\alpha=2$. More
recently, Li et al. have considered Kleinberg’s navigation model with a total
cost constraint [6]. In their model, the total length of the long-range
connections is restricted to $\Lambda=c\cdot N$, where $N$ represents the
total number of nodes in the lattice based network and $c$ is a positive
constant. Their results show that the best transport condition(minimal number
of steps to reach the target) is obtained with a power-law exponent
$\alpha=d+1$ for constrained navigation in a $d$-dimensional lattice in the 1
and 2-dimensional cases. In this paper, we give a rigorous theoretical
analysis of the optimal condition $\alpha=2$ for navigation and provide the
exact time cost for various power-law exponents $\alpha$ on the 1-dimensional
cost constrained network.
## 2 Dynamical Equations for One-Dimensional Navigation with Cost Restriction
We consider the one-dimensional navigation problem on a cycle with $N=4n$
nodes. For the sake of simplicity, we assume that each node has only two
short-range connections to its two nearest neighbors, and the probability of
having a long-range connection satisfies to a certain power-law distribution.
Obviously, the largest possible length of a long-range connection is $2n$ in
this cycle. We number all nodes inclusively from $0$ to $4n-1$ and assume that
the navigation process starts from the node $0$ and ends at the node $n$ for
further simplification. The network is illustrated in FIG.1.
According to the discussion above, the probability of a long-range connection
between any given pair of nodes with a distance $r$ is
$p(r,\alpha)=\frac{{r^{-\alpha}}}{2{\sum\limits_{r=1}^{2n}{r^{-\alpha}}}},\alpha\geq
0,$ (1)
where $\alpha$ is the power exponent of the power-law distribution. The
expected length of the long-range connection from any node satisfies
$E(L_{\alpha})=2\sum\limits_{r=1}^{2n}{r\cdot p(r,\alpha)}$. To be consistent
with Li’s work[6], in this paper we set the total cost limit to
$\Lambda=c\cdot 4n$, where $c$ is a positive constant and $4n$ is the number
of nodes on the cycle. Subject to this limit, the expected number of long-
range connections on the whole cycle can be written as
$E(N_{\alpha}){\rm{=}}\frac{\Lambda}{{E(L_{\alpha})}}$.
Since all nodes are homogeneous, we know that the number of directed long-
range connections from each node should obey the Poisson distribution with a
parameter $\lambda=\frac{{E(N_{\alpha})}}{4n}$. Thus, the probability of a one
long-range connection from each node can be given by $\lambda e^{-\lambda}$.
When $\lambda$ is small enough, the probability of the existence two or more
than two long-range connections for a arbitrary node can be ignored. More
over, for a given node and distance $r$, there are only two nodes which
satisfy the condition that the distances between them and the given node be
$r$. So, if $\lambda$ is too large, we cannot construct a spatial network on
which the length of long-rang connections is power law distribution. According
to the above two reasons, in this paper we only consider the case where
navigation process is carried out by at most one long-range connection ($c$ is
small) for each node.
Figure 1: The navigation model in this paper. Node $0$ is the starting node,
and node $n$ is the target.
If we use $E(L_{s}^{\alpha})$ to denote the expected distance by a long-range
connection from a node $s$ toward the target node $n$, then we have
$E(L_{s}^{\alpha})=\sum\limits_{r=1}^{n-s}{r\cdot
p(r,\alpha)}+\sum\limits_{r=n-s+1}^{2n-2s-1}{(2n-2s-r)\cdot p(r,\alpha)}.$ (2)
We further denote the expected distance by an edge (long or short range) from
a node $s$ toward the target node $n$ as $E(J_{s})$. Then we have
$E(J_{s})=\lambda e^{-\lambda}\cdot E(L_{s}^{\alpha})+1-\lambda
e^{-\lambda}\cdot[\sum\limits_{r=1}^{n-s}{p(r,\alpha)}+\sum\limits_{r=n-s+1}^{2n-2s-1}{p(r,\alpha)}]$
(3)
For simplicity, we consider the continuous form of all equations provided
above. Then $\lambda$ can be written as
$\lambda=\begin{cases}c\frac{2}{{2n+1}},&\alpha=0\\\ c\frac{{\ln
2n}}{{2n}},&\alpha=1\\\ c\frac{{1-\frac{1}{{2n}}}}{{\ln 2n}},&\alpha=2\\\
c\frac{{\alpha-2}}{{\alpha-1}}\cdot\frac{{{{(2n)}^{1-\alpha}}-1}}{{{{(2n)}^{2-\alpha}}-1}},&else.\\\
\end{cases}$ (4)
When the network size is large enough, Eq.(4) can be simplified to
$\lambda=\begin{cases}0,&0\leq\alpha\leq 2\\\
c\frac{{\alpha-2}}{{\alpha-1}},&else.\\\ \end{cases}$ (5)
The method of dynamical equations is used to deduce the searching time with
limited total cost. Suppose that at time $t$, the corresponding position is
$s(t)$. Obviously, $s(0)=0$ holds. The dynamical equation can be written as
$\left\\{\begin{array}[]{l}\frac{{ds}}{{dt}}=E(J_{s}),\\\ s(0)=0.\\\
\end{array}\right.$ (6)
Before solving Eq.(6), we first study the optimal power-law exponent $\alpha$
by comparing $E(J_{s})$ (Eq.(3)) under different values of $\alpha$. We
rewrite the distance to the target $n-s$ as $\varepsilon n$, where
$0<\varepsilon\leq 1$ is a constant. For any given $0<\varepsilon\leq 1$, we
assume $\varepsilon n$ is large enough, such that long-range connections will
be used in the search process. Finally, Eq.(3) can be simplified to the
following forms,
$E(J_{s})\sim\begin{cases}\frac{1}{4}\varepsilon^{2}c+1,&\alpha=0\\\
\frac{{\ln 2}}{2}\varepsilon c+1,&\alpha=1\\\ \frac{1}{2}c+1,&\alpha=2\\\
\frac{e^{-c\frac{{\alpha-2}}{{\alpha-1}}}}{{2(\alpha-1)}}c+1,&\alpha>2\\\
\frac{{2^{\alpha-1}-1}}{{2(\alpha-1)}}\varepsilon^{2-\alpha}c+1,&else\\\
\end{cases}$ (7)
It is not difficult to show the right side of the Eq.(7) monotonically
increases with $\alpha$ for $0\leq\alpha\leq 2$ and decreases with $\alpha$
for $\alpha>2$. We can also prove that $E(J_{s})$ is continuous at the point
$\alpha=2$. Overall, $E(J_{s})$ is continuous and reaches its maximal value at
$\alpha=2$ for any given $\varepsilon$. It has been revealed that the optimal
condition for navigation with limited cost is a tradeoff between the length
and the number of long-range connections added to the cycle. Prior to solving
the dynamical equations theoretically, we have already shown that the optimal
power-law exponent is $\alpha=2$ with some proper simplifications.
Figure 2: Effects of the network size $n$ on the numerical results. The
network size is $10^{\beta}$, and $\beta$ is chosen from 3 to 10 from bottom
to top. It can be seen that the optimal choice of $\alpha$ gets closer to
$\alpha=2$ as $n$ increases.
## 3 Results
Figure 3: Comparison between numerical results and simulation results for
$c=0.5$ and $c=1$ respectively. The curves represent the numerical results
while the squares denote the simulation results with $n=10000$. It is shown
that our numerical solutions are consistent with the simulation results and
both of them have the optimal value at $\alpha=2$ approximately.
As discussed above, the optimal power-law exponent for navigation in a
1-dimensional large network is $\alpha=2$. FIG.2 represents the size effect on
$E(J_{s})$. It can be found that $\alpha=2$ is the optimal power-law exponent
when the network size goes to infinity.
To obtain exact time cost of navigation with cost restriction, Eq.(6) should
be solved. In the following, we will first give its numerical results and then
derive its exact solutions for various values of $\alpha$. The Ronge-Kutta
method has been introduced to solve the dynamical equation numerically and the
results are presented in FIG.3. It shows that the optimal power-law exponent
is $\alpha=2$. To check up our method, we also perform search experiments on
the one-dimensional cycle. The comparison between our analytical results and
the simulation results with $c=0.5$ and $c=1$ are given by FIG.3. As can be
seen, they agree quite well and both of them obtain the optimal navigation at
$\alpha=2$.
It is able to get the exact solutions of navigation with limited cost for
various values of $\alpha$. For instance, the dynamical equation in the case
$\alpha=0$ is,
$\frac{{ds}}{{dt}}=\frac{{c(n-s-1)^{2}}}{{4n^{2}}}+1.$ (8)
Based on the initial condition $s(0)=0$, we can get the exact solution of
Eq.(8).
Here, we use $T$ to denote the time cost for getting the destination node $n$.
We should have $T=\frac{2n}{{\sqrt{c}}}\arctan\frac{{\sqrt{c}}}{2}$ for large
enough $n$. Thus, the average required time to navigate from the source node
to the target satisfies
$\frac{T}{n}=\frac{2}{{\sqrt{c}}}\arctan\frac{{\sqrt{c}}}{2}.$ (9)
Analogously, the exact solution for exponent $\alpha$ when $n$ approaches
infinity are acquired as
$\frac{T}{n}=\begin{cases}\frac{2}{{\sqrt{c}}}\arctan\frac{{\sqrt{c}}}{2},&\alpha=0\\\
\frac{2}{{c\ln 2}}\ln\frac{{c\ln 2+2}}{2},&\alpha=1\\\
\frac{{2(\alpha-1)}}{{2(\alpha-1)+ce^{-c\frac{{\alpha-2}}{{\alpha-1}}}}},&\alpha\geq
2.\\\ \end{cases}$ (10)
The above results suggest that the relationship between the exact time cost
and the distance is linear for most values of $\alpha$. Meanwhile, we have
studied the size effect on navigation with cost constraint. Based on Eq.(4),
we know that $\lambda$ approaches its limit much more slower when $\alpha$
gets closer to 2 as $n$ increases. The time cost of navigation with different
network sizes are provided in FIG.4. It can be verified that it will approach
its limit as $n$ goes to infinity, which is given by Eq.(10).
In summary, we constructed a the dynamical equation for the 1-dimensional
navigation with limited cost. Based on the equation, we proved that for large
networks and comparatively small cost the optimal power-law exponent is
$\alpha=2$. Our analytical results confirm the previous simulations[6] well.
Figure 4: Size effect on navigation. The exact solutions are obtained on the
1-dimensional cycle with parameter $c=1$. The network size $n$ is
$10^{\beta}$, and parameter $\beta$ values of the upper three solid curves are
chosen from 3 to 5 from top to bottom. It is shown that $\alpha=2$ is always
the optimal choice for various values of $n$. The bottom line, obtained when
$n$ goes infinity, represents the exact solutions of the navigation process
with limited cost. Notice that we only provide the exact solutions at
$\alpha=0$ and $\alpha=1$ when $\alpha<2$, thus we only connect the two exact
points with dashed lines.
Acknowledgement.We wish to thank Prof. Shlomo Havlin for some useful
discussions and two anonymous referees for their helpful suggestions. This
work is partially supported by the Fundamental Research Funds for the Central
Universities and NSFC under Grant No. 70771011 and 60974084. Y. Hu is
supported by Scientific Research Foundation and Excellent Ph.D Project of
Beijing Normal University.
## References
* [1] Kleinberg J. Nature4062000845.
* [2] Roberson M. R. Ben-Avraham D. Phys. Rev. E74200617101.
* [3] Caretta Cartozo C. De Los Rios P. Phys. Rev. Lett.1022009238703.
* [4] Carmi S., Carter S., Sun J. Ben-Avraham D. Phys. Rev. Lett.1022009238702.
* [5] Yang H., Nie Y., Zeng A., Fan Y., Hu Y. Di Z. EPL89201058002.
* [6] Li G., Reis S. D. S., Moreira A. A., Havlin S., Stanley H. E. Andrade, Jr. J. S. Phys. Rev. Lett.1042010018701.
* [7] Barri re L.,Fraigniaud P., Kranakis E., Krizanc D. Springer,New York2001.
* [8] Hu Y., Wang Y., Li D., Havlin S., Di Z. arXiv:1002.18022010.
|
arxiv-papers
| 2010-07-08T02:35:31 |
2024-09-04T02:49:11.492264
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yong Li, Dong Zhou, Yanqing Hu, Jiang Zhang, Zengru Di",
"submitter": "Li Yong",
"url": "https://arxiv.org/abs/1007.1281"
}
|
1007.1323
|
# A note on the invariance in the nonabelian tensor product
Francesco G. Russo Laboratorio di Dinamica Strutturale e Geotecnica (StreGa)
Universitá del Molise, via Duca degli Abruzzi, 86039, Termoli (CB).
francescog.russo@yahoo.com
###### Abstract.
In the nonabelian tensor product $G\otimes H$ of two groups $G$ and $H$ many
properties pass from $G$ and $H$ to $G\otimes H$. There is a wide literature
for different properties involved in this passage. We look at weak conditions
for which such a passage may happen.
###### Key words and phrases:
Nonabelian tensor product; classes of groups; universal property
Mathematics Subject Classification 2010: Primary 20J99; Secondary 20F18
## 1\. Terminology and statement of the result
Let $G$ and $H$ be two groups acting upon each other in a $compatible$ $way$:
(1.1) $~{}^{{}^{g}h}g^{\prime}=~{}^{g}(^{h}(^{{}^{g^{-1}}}h^{\prime})),\ \ \ \
\ ~{}^{{}^{h}g}h^{\prime}=~{}^{h}(^{g}(^{{}^{h^{-1}}}h^{\prime})),$
for $g,g^{\prime}\in G$ and $h,h^{\prime}\in H$, and acting upon themselves by
conjugation. The $nonabelian$ $tensor$ $product$ $G\otimes H$ of $G$ and $H$
is the group generated by the symbols $g\otimes h$ with defining relations
(1.2) $gg^{\prime}\otimes h=(~{}^{g}g^{\prime}\otimes~{}^{g}h)(g\otimes h),\ \
\ \ g\otimes hh^{\prime}=(g\otimes h)(~{}^{h}g\otimes~{}^{h}h^{\prime}).$
When $G=H$ and all actions are by conjugations, $G\otimes G$ is called
$nonabelian$ $tensor$ $square$ of $G$. These notions were introduced in [3, 4]
and some significant contributions can be found in [1, 2, 5, 6, 8, 9, 10, 12,
13].
From the defining relations in $G\otimes H$,
(1.3) $\kappa:g\otimes h\in G\otimes H\mapsto\kappa(g\otimes
h)=[g,h]\in[G,H]=\langle g^{-1}h^{-1}gh\ |\ g\in G,h\in H\rangle$
is an epimorphism of groups. Still from [3, 4], if $G$ and $H$ act trivially
upon each other, then $G\otimes H$ is isomorphic to the usual tensor product
$G^{ab}\otimes_{\mathbb{Z}}H^{ab}$. If they act compatibly upon each other,
then their actions induce an action of the free product $G*H$ on $G\otimes H$
given by ${}^{x}(g\otimes h)=^{x}g\otimes^{x}h$, where $x\in G*H$.
The $exterior$ $product$ $G\wedge H$ is the group obtained with the additional
relation $g\otimes h=1_{\otimes}$ on $G\otimes H$, that is,
(1.4) $G\wedge H=(G\otimes H)/D,$
where $D=\langle g\otimes g:g\in G\cap H\rangle$. Now it is easy to check that
(1.5) $\kappa^{\prime}:g\wedge h\in G\wedge H\mapsto\kappa^{\prime}(g\wedge
h)=[g,h]\in[G,H]$
is a well–defined epimorphism of groups. For convenience of the reader, we
recall that there is a famous commutative diagram with exact rows and central
extensions as columns in [3, (1)]: It correlates the second homology group
$H_{2}(G)$ of $G$ with the third homology group $H_{3}(G)$ of $G$, the
Whitehead’s quadratic functor $\Gamma$, the Whitehead’s function $\psi$ and
$\ker\kappa=J_{2}(G)$ (see also [3, 4, 14]).
Now we get to the purpose of the present paper. Given a class of groups
$\mathfrak{X}$, many authors answered the question:
(1.6) $\mathrm{If}\ \ G,H\in\mathfrak{X},\ \ \mathrm{then}\ \ G\otimes
H\in\mathfrak{X}$
In case $\mathfrak{X}=\mathfrak{F}$ is the class of all finite groups, see
[5]. In case $\mathfrak{X}=\mathfrak{N}$ is the class of all nilpotent groups,
see [2, 13]. In case $\mathfrak{X}=\mathfrak{S}$ is the class of all soluble
groups, see [10, 13]. In case $\mathfrak{X}=\mathfrak{P}$ is the class of all
polycyclic groups, see [8]. In case $\mathfrak{X}=\mathbf{L}\mathfrak{F}$ is
the class of all locally finite groups, see [9]. In case
$\mathfrak{X}=\check{\mathfrak{C}}$ (resp., $\mathfrak{X}=\mathfrak{S}_{2}$)
is the class of all Chernikov (resp., soluble minimax) groups, see [11]. Some
topological properties are also closed with respect to forming the nonabelian
tensor product, as observed in [3, 4].
We recall some notations from [7].
* –
$\mathfrak{X}=\mathbf{S}\mathfrak{X}$ means that $\mathfrak{X}$ is closed with
respect to forming subgroups.
* –
$\mathfrak{X}=\mathbf{H}\mathfrak{X}$ means that $\mathfrak{X}$ is closed with
respect to forming homomorphic images.
* –
$\mathfrak{X}=\mathbf{P}\mathfrak{X}$ means that $\mathfrak{X}$ is closed with
respect to forming extensions, i.e.: if $N\in\mathfrak{X}$ is a normal
subgroup of $G$ and $G/N\in\mathfrak{X}$, then $G\in\mathfrak{X}$.
* –
$\mathfrak{X}=\mathbf{H_{2}}\mathfrak{X}$ means that $\mathfrak{X}$ is closed
with respect to forming the second homology group, i.e.: if
$G\in\mathfrak{X}$, then $H_{2}(G)\in\mathfrak{X}$.
* –
$\mathfrak{X}=\mathbf{H_{3}}\mathfrak{X}$ means that $\mathfrak{X}$ is closed
with respect to forming the third homology group, i.e.: if $G\in\mathfrak{X}$,
then $H_{3}(G)\in\mathfrak{X}$.
* –
$\mathfrak{X}=\mathbf{T}\mathfrak{X}$ means that $\mathfrak{X}$ is closed with
respect to forming (usual) abelian tensor products , i.e.: if
$A,B\in\mathfrak{X}$ are abelian, then
$A\otimes_{\mathbb{Z}}B\in\mathfrak{X}$.
Our main contribution is below.
Main Theorem. Let $G$ and $H$ be two groups, acting compatibly upon each other
and
$\mathfrak{X}=\mathbf{S}\mathfrak{X}=\mathbf{H}\mathfrak{X}=\mathbf{P}\mathfrak{X}=\mathbf{H_{2}}\mathfrak{X}=\mathbf{H_{3}}\mathfrak{X}=\mathbf{T}\mathfrak{X}$.
If $G,H,\Gamma((G\cap H)^{ab})\in\mathfrak{X}$, then $G\otimes
H\in\mathfrak{X}$.
In [2, 5, 8, 9, 10, 11, 13], the quoted results follow from Main Theorem, when
we choose $\mathfrak{X}$ among
$\mathfrak{F},\mathfrak{N},\mathfrak{S},\mathfrak{P},\mathbf{L}\mathfrak{F},\mathfrak{\check{C}},\mathfrak{S}_{2}$.
## 2\. Proof and some consequences
We illustrate that it is possible to adapt an argument in [8, Section 2].
###### Proof of Main Theorem.
Let $P=G*H/IJ$ be the Pfeiffer product of $G$ and $H$, where $I$ and $J$ are
the normal closures in $G*H$ of $\langle{{}^{h}}ghg^{-1}h^{-1}:g\in G,h\in
H\rangle$ and $\langle{{}^{g}}hgh^{-1}g^{-1}:g\in G,h\in H\rangle$,
respectively. See [8, 14]. Note that $P$ is a homomorphic image of $G\ltimes
H$, hence $P\in\mathfrak{X}$. Here we have used
$\mathfrak{X}=\mathbf{H}\mathfrak{X}$. Let $\mu:G\rightarrow P$ and
$\nu:H\rightarrow P$ be inclusions. Denote $\overline{G}=\mu(G)$ and
$\overline{H}=\nu(H)$. Then $\overline{G}$ and $\overline{H}$ are normal
subgroups of $P$ and $P=\overline{G}\ \overline{H}$. Of course, $\ker\mu\leq
Z(G)$ and $\ker\nu\leq Z(H)$. An argument as in [3, Proposition 9] shows that
the following sequence is exact:
(2.1) $(G\otimes\ker\nu)\times(\ker\mu\otimes H)\stackrel{{\scriptstyle
i}}{{\longrightarrow}}G\otimes
H\longrightarrow\overline{G}\otimes\overline{H}\longrightarrow 1,$
where $i$ is the inclusion $(g\otimes h^{\prime},g^{\prime}\otimes
h)\mapsto(g\otimes h^{\prime})(g^{\prime}\otimes h)$. It is easy to see that
$\textrm{Im}~{}i\leq Z(G\otimes H)$. Since ${}^{h}g=^{\nu(g)}g$ and
${}^{g}h=^{\mu(g)}h$, $\ker\mu$ and $\ker\nu$ act trivially on $H$ and $G$,
respectively.
Therefore,
(2.2) $G\otimes\ker\nu\simeq
G^{ab}\otimes_{\mathbb{Z}}\ker\nu^{ab}=G^{ab}\otimes_{\mathbb{Z}}\ker\nu$
and
(2.3) $\ker\mu\otimes
H\simeq\ker\mu^{ab}\otimes_{\mathbb{Z}}H^{ab}=\ker\mu\otimes_{\mathbb{Z}}H^{ab}.$
In particular, $G\otimes\ker\nu\in\mathfrak{X}$. Here we have used
$\mathfrak{X}=\mathbf{T}\mathfrak{X}$. Analogously, $\ker\mu\otimes
H\in\mathfrak{X}$. It follows that $\textrm{Im}~{}i\in\mathfrak{X}$ because it
is a homomorphic image of $(G\otimes\ker\nu)\times(\ker\mu\otimes
H)\in\mathfrak{X}$. Still we have used $\mathfrak{X}=\mathbf{H}\mathfrak{X}$.
Since $\overline{G}\otimes\overline{H}\simeq(G\otimes H)/\textrm{Im}~{}i$, it
is enough to prove that $\overline{G}\otimes\overline{H}\in\mathfrak{X}$. Here
we have used $\mathfrak{X}=\mathbf{P}\mathfrak{X}$. We may work with
$\overline{G}$ instead of $G$ and with $\overline{H}$ instead of $H$ in order
to get our result. Then there is no loss of generality in assuming that $G$
and $H$ are normal subgroups of $P$, $P=GH$, and all actions are induced by
conjugation in $P$. Note that $(G\wedge H)/\ker\kappa^{\prime}$ is isomorphic
to $[G,H]\leq G\cap H\leq G\in\mathfrak{X}$ and so $(G\wedge
H)/\ker\kappa^{\prime}\in\mathfrak{X}$. Here we have used
$\mathfrak{X}=\mathbf{H}\mathfrak{X}=\mathbf{S}\mathfrak{X}$. If we prove
$\ker\kappa^{\prime}\in\mathfrak{X}$, then $G\wedge H\in\mathfrak{X}$ by
$\mathfrak{X}=\mathbf{P}\mathfrak{X}$. If we prove also $D\in\mathfrak{X}$,
then $G\otimes H\in\mathfrak{X}$, still by
$\mathfrak{X}=\mathbf{P}\mathfrak{X}$ and we are done.
By [4, Theorem 4.5], we have an exact sequence:
(2.4) $\longrightarrow H_{3}(P/G)\oplus
H_{3}(P/H)\longrightarrow\ker\kappa^{\prime}\longrightarrow
H_{2}(P)\longrightarrow.$
Since $P,P/G,P/H\in\mathfrak{X}$, we have
$H_{2}(P),H_{3}(P/G),H_{3}(P/H)\in\mathfrak{X}$. Here we have used
$\mathfrak{X}=\mathbf{H}\mathfrak{X}=\mathbf{H_{2}}\mathfrak{X}=\mathbf{H_{3}}\mathfrak{X}$.
On the other hand, $\ker\kappa^{\prime}$ is an extension of $H_{3}(G/M)\oplus
H_{3}(G/N)\in\mathfrak{X}$ by $H_{2}(G)\in\mathfrak{X}$. Therefore,
$\ker\kappa^{\prime}\in\mathfrak{X}$, as claimed. Here we have used
$\mathfrak{X}=\mathbf{P}\mathfrak{X}$.
Having in mind the famous diagram [3, (1)], it is easy to check that there
exists a well–defined homomorphism of groups $\psi:\Gamma((G\cap
H)^{ab})\rightarrow(G\cap H)\otimes(G\cap H)$. See [3, p.181] or [4]. Then
$\textrm{Im}~{}\psi=D\in\mathfrak{X}$, as claimed. Here we have used
$\mathfrak{X}=\mathbf{H}\mathfrak{X}$ and $\Gamma((G\cap
H)^{ab})\in\mathfrak{X}$.
The result follows.
∎
Note that $\Gamma(G^{ab})$ plays a fundamental role in deciding if $G\otimes
G\in\mathfrak{X}$. This was already noted in [2, Section 3] for the class of
all free nilpotent groups of finite rank. Then it is clear that the following
corollary extends many results in [2, 5, 8, 9, 10, 11, 13] in case of the
nonabelian tensor square.
Corollary.
Assume $G=H$ in Main Theorem. If $G,\Gamma(G^{ab})\in\mathfrak{X}$, then
$G\otimes G\in\mathfrak{X}$.
We end with two observations on the invariance with respect to the nonabelian
tensor product.
Remark 1. Sometimes it is enough that $[G,H]\in\mathfrak{X}$ in order to
decide whether $G\otimes H\in\mathfrak{X}$. In case of
$\mathfrak{X}=\mathfrak{N}$, or $\mathfrak{X}=\mathfrak{S}$, this can be found
in [3, 10, 13].
The second deals with the universal property of the nonabelian tensor
products.
Remark 2. In a certain sense the universal property of the nonabelian tensor
products (see [4]) justifies Main Theorem, because it shows that we need at
least
$\mathfrak{X}=\mathbf{S}\mathfrak{X}=\mathbf{H}\mathfrak{X}=\mathbf{P}\mathfrak{X}$,
if we hope to answer (1.6) positively.
## References
* [1] M. Bacon, L.-C. Kappe and R. F. Morse, On the nonabelian tensor square of a 2-Engel group, Arch. Math. 69(1997), 353–364.
* [2] R. D. Blyth, P. Moravec and R. F. Morse, On the nonabelian tensor squares of free nilpotent groups of finite rank, In: Computational group theory and the theory of groups, Contemp. Math., 470, Amer. Math. Soc., Providence, RI, 2008, pp. 27–43.
* [3] R. Brown, D. L. Johnson and E. F. Robertson, Some computations of non-abelian tensor products of groups, J. Algebra 111 (1987), 177–202.
* [4] R. Brown and J. -L. Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987), 311–335.
* [5] G. Ellis, The nonabelian tensor product of finite groups is finite, J. Algebra 111 (1987), 203–205.
* [6] N. Inassaridze, Nonabelian tensor products and nonabelian homology of groups, J. Pure Appl. Algebra 112(1996), 191–205.
* [7] J. C. Lennox and D. J. S. Robinson, The Theory of Infinite Soluble Groups, Clarendon Press, Oxford, 2004.
* [8] P. Moravec, The nonabelian tensor product of polycyclic groups is polycyclic, J. Group Theory 10 (2007), 795–798.
* [9] P. Moravec, The exponents of nonabelian tensor products of groups, J. Pure Appl. Algebra 212 (2008), 1840–1848.
* [10] I. Nakaoka, Nonabelian tensor product of solvable groups, J. Group Theory 3 (2000), 157–167.
* [11] F.G. Russo, Nonabelian tensor product of soluble minimax groups, In: Computational Group Theory and Cohomology, Contemp. Math., pp. 179–184.
* [12] N. H. Sarmin, Infinite two generator groups of class two and their non-abelian tensor squares, Int. J. Math. Math. Sci. 32 (10) (2002), 615–625.
* [13] M. P. Visscher, On the nilpotency class and solvability length of nonabelian tensor product of groups, Arch. Math. (Basel) 73 (1999), 161–171.
* [14] J. H. C. Whitehead, A certain exact sequence. Ann. of Math. 52 (1950), 51–110.
|
arxiv-papers
| 2010-07-08T09:39:02 |
2024-09-04T02:49:11.498052
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Francesco G. Russo (Universita' degli Studi di Palermo, Palermo,\n Italy)",
"submitter": "Francesco G. Russo",
"url": "https://arxiv.org/abs/1007.1323"
}
|
1007.1415
|
# Sensitivity of Quantum Walks with Perturbation
Chen-Fu Chiang School of Electrical Engineering and Computer Science,
University of Central Florida, Orlando, FL 32816, USA. Email:
cchiang@eecs.ucf.edu
###### Abstract
Quantum computers are susceptible to noises from the outside world. We
investigate the effect of perturbation on the hitting time of a quantum walk
and the stationary distribution prepared by a quantum walk based algorithm.
The perturbation comes from quantizing a transition matrix Q with perturbation
E (errors). We bound the perturbed quantum walk hitting time from above by
applying Szegedy’s work and the perturbation bounds with Weyl’s perturbation
theorem on classical matrix. Based on an efficient quantum sample preparation
approach invented in speed-up via quantum sampling and the perturbation bounds
for stationary distribution for classical matrix, we find an upper bound for
the total variation distance between the prepared quantum sample and the true
quantum sample.
## 1 Introduction
Markov chains and random walks have been useful tools in classical
computation. One can use random walks to obtain the final stationary
distribution of a Markov chain to sample from. In such an application the time
the Markov chain takes to converge, i.e., convergence time, is of interest
because shorter convergence time means lower cost in generating a sample.
Sampling from stationary distributions of Markov chains combined with
simulated annealing is the core of many clever classical approximation
algorithms. For instance, approximating the volume of convex bodies [1],
approximating the permanent of a non-negative matrix [2], and the partition
function of statistical physics models such as the Ising model [3] and the
Potts model [4]. In addition, one can also use the random walks to search for
the marked state in the Markov chain, in which the hitting time is of
interest. It is because hitting time indicates the time it requires to find
the marked state.
The Markov Chain Monte Carlo (MCMC) method is a centerpiece of many efficient
classical algorithms. It allows us to approximately sample from a particular
distribution $\pi$ over a large state space $\Omega$. Perturbations of
classical Markov chains are widely studied with respect to hitting time and
stationary distribution. The variation of a stationary distributuion caused by
perturbation would deteriorate the accuracy of our sampling. With the above
facts, we are interested to know what effect perturbation has on currently
existing quantum walk based algorithms.
The note is organized as follows. In section 2.1 we present the deviation
effect of perturbation on spectral gap of Markov chain and we apply it to the
hitting time of a quantum walk in section 2.2. In section 2.3 we state the
result of total variation distance of classical stationary distributions. By
using an efficient algorithm [5] for preparing quantum samples, in section 2.4
we obtain the total variation distance between the prepared quantum sample and
the true quantum sample. Finally in section 3, we make our conclusion and
suggest future works.
## 2 The Perturbation
Given a stochastic symmetric matrix $P\in\mathbb{C}^{n\times n}$, we can
quantize the Markov chain [6] and we showed that the implementation of one
step of quantum walk [7] can be achieved efficiently. However, the above
settings always are under the assumption of perfect scenarios. In real life
there are many sources of errors that would perturb the process. Noises might
be propagated along with the input source or they might be introduced during
the process. Here we solely look at the noises that are introduced at the
beginning of the process.
The noises can be introduced due to the precision limitation and the noisy
environment. For instance, not all numbers have a perfect binary
representation and the approximated numbers would cause perturbation. Suppose
our input decoding mechanism can always take the input matrix and represent it
in a symmetric transition matrix $Q$ where $Q$ can be perfectly represented by
the limited precision and it is the matrix closes to the original matrix $P$
that the system can prepare.
Let $E$ be the noise that is introduced because of system’s precision
limitation and the environment, we can express the transistion matrix as
$Q=P+E.$ (1)
### 2.1 Classical Spectral Gap Perturbation
Classically many researches [8, 9, 10, 11, 12, 13] are focused on the spectral
gaps and stationary distributions of the matrice with perturbation. In a
recent work by Ipsen and Nadler [8] , they refined the perturbation bounds for
eigenvalues of Hermitians. Throughout the rest of the note, $\|\cdot\|$ always
denotes $l_{2}$ norm, unless otherwise specified. Based on their result, we
summarized the following:
###### Corollary 1.
Suppose $P$ and $Q$ $\in\mathbb{C}^{n\times n}$ are Hermitian symmetric
transition matrices with respective eigenvalues
$\lambda_{n}(P)\leq\ldots\leq\lambda_{1}(P)=1,\quad\quad\lambda_{n}(Q)\leq\ldots\leq\lambda_{1}(Q)=1,$
(2)
and $Q=P+E$, then
$\max_{1\leq i\leq n}|\lambda_{i}(P)-\lambda_{i}(Q)|\leq\|E\|.$ (3)
Furthermore, the spectral gap $\delta$ of $P$ and the spectral gap $\Delta$ of
$Q$ have the following relationship
$\delta-\|E\|\leq\Delta\leq\delta+\|E\|.$ (4)
###### Proof.
Eq. (3) is a direct result from the Weyl’s Perturbation Theorem. The Weyl’s
Perturbation Theorem bounds the worst-case absolute error between the $i$th
exact and the $i$th perturbed eigenvalues of Hermitian matrices in terms of
the $l_{2}$ norm [10, 11]. And since
$1-\lambda_{2}(P)=\delta,\quad\quad 1-\lambda_{2}(Q)=\Delta,$
by eq. (3) we have $|\delta-\Delta|\leq\|E\|$. Therefore, in general we can
bound the perturbed spectral gap $\Delta$ as
$\delta-\|E\|\leq\Delta\leq\delta+\|E\|.$
∎
Generally speaking, the global norm of $E$ might be very large when the
dimensions $n>>1$ [14]. However, in our case because $E$ is the difference
between two very close stochastic symmetric matrices, its global norm would
never become large.
### 2.2 Quantum Hitting Time Perturbation
Given two Hermitian stochastic matrices, $P$ and $Q$, we explore the
difference between walk operators, $W(P)$ and $W(Q)$, with respect to their
hitting time. Denote the set of marked elements as $|M|$. Based on the result
from Corollary 1, we have the following:
###### Corollary 2.
Given two symmetric reversible ergodic transition matrices $P$ and $Q$
$\in\mathbb{C}^{n\times n}$, where $Q=P+E$, let $W(P)$ and $W(Q)$ be quantum
walks based on $P$ and $Q$, respectively. Let $M$ be the set of marked
elements in the state space. Denote $HT(P)$ the hitting time of walk $W(P)$
and $HT(Q)$ the hitting time of walk $W(Q)$. Suppose $|M|=\epsilon N$. If the
second largest eigenvalues of $P$ and $Q$ are at most $1-\delta$ and
$1-\Delta$, respectively, then in general
$HT(P)=O\Big{(}\sqrt{\frac{1}{\delta\epsilon}}\Big{)},\quad\quad
HT(Q)=O\Big{(}\sqrt{\frac{1}{(\delta-\|E\|)\epsilon}}\Big{)}$ (5)
where $\delta-\|E\|\leq\Delta\leq\delta+\|E\|$.
###### Proof.
Suppose the Markov chain $P$, $Q$ and matrix $E$ are in the following block
structure
$P=\left(\begin{array}[]{cc}P_{1}&P_{2}\\\ P_{3}&P_{4}\end{array}\right),\quad
Q=\left(\begin{array}[]{cc}Q_{1}&Q_{2}\\\ Q_{3}&Q_{4}\end{array}\right),\quad
E=\left(\begin{array}[]{cc}E_{1}&E_{2}\\\ E_{3}&E_{4}\end{array}\right)$ (6)
where we order the elements such that the marked ones come last, i.e.,
$P_{4}$, $Q_{4}$ and $E_{4}\in\mathbb{C}_{|M|\times|M|}$. The corresponding
modified Markov chains [6] would be
$\tilde{Q}=\left(\begin{array}[]{cc}Q_{1}&0\\\
Q_{3}&I\end{array}\right)=\left(\begin{array}[]{cc}P_{1}+E_{1}&0\\\
P_{3}+E_{3}&I\end{array}\right).$ (7)
By [6], we have $HT(P)=O(\sqrt{\frac{1}{1-\|P_{1}\|}})$ and
$HT(Q)=O(\sqrt{\frac{1}{1-\|Q_{1}\|}})$. Since we know
$\|P_{1}\|\leq 1-\frac{\delta\epsilon}{2}\quad\mbox{and}\quad\|Q_{1}\|\leq
1-\frac{\Delta\epsilon}{2}$ (8)
by [6] and by Cauchy’s interlacing theorem we have $\|E\|\geq\|E_{1}\|$ [15,
Cor.III.1.5], we then obtain
$\|Q_{1}\|\leq\min\left\\{\|P_{1}\|+\|E\|,1-\frac{(\delta-\|E\|)\epsilon}{2}\right\\}$
(9)
as $\delta-\|E\|\leq\Delta\leq\delta+\|E\|$. Therefore, the hitting times for
$P$ and $Q$ are derived. ∎
### 2.3 Classical Sample Perturbation
In this section we adapt the results from the work [9] to bound the stationary
distriubtion $\pi(Q)$ of a perturbed matrix $Q$ with respect to the
perturbation $E$ and the true stationary distribution $\pi(P)$, i.e.,
$Q\cdot\pi(Q)=\pi(Q),\quad P\cdot\pi(P)=\pi(P).$ (10)
Let $\Omega$ be the state space and $\Omega^{\prime}=\Omega\cup\\{0\\}$. The
total variation distance between two probability distributions over $\Omega$
is defined as
$D(\pi(P),\pi(Q))=\frac{1}{2}\sum_{x\in\Omega}\|\pi(P)_{x}-\pi(Q)_{x}\|_{1}=\max_{S\subseteq\Omega^{\prime}}|\pi(P)_{S}-\pi(Q)_{S}|.$
(11)
Here $\pi(P)$ denotes the stationary distribution of matrix $P$, $\pi(P)_{x}$
is the $x$th element of $\pi(P)$ and $\pi(P)_{S}$ denotes the sum of
$\pi(P)_{x}$ where $x\in S$, i.e., $\sum_{x\in S}\pi(P)_{x}=\pi(P)_{S}$.
In [9] it is assumed that the transition matrix is row-wise stochastic. Our
matrix is column-wise stochastic (see eqn.(10)) but since it is symmetric, it
is also row-stochastic. By chooseing condition number $\kappa_{5}$ in [9], the
ergodicity coefficient, using the $l_{p}$ norm, is defined as
$\tau_{p}(P)=\sup_{\|v\|_{p}=1,\\\ v^{T}e=0}\|v^{T}P\|_{p}$ (12)
where $e$ is a column vector of all ones. Since $P$ is a stochastic matrix,
the ergodic coefficient satisfies $0\leq\tau_{1}(P)\leq 1$. In case of
$\tau_{1}(P)<1$, we have a perturbation bound in terms of the ergodic
coefficient of $P$:
${}D(\pi(P),\pi(Q))=\frac{1}{2}\|\pi(P)-\pi(Q)\|_{1}\leq\frac{1}{2(1-(\tau_{1}{P}))}\|E\|_{\infty}.$
(13)
### 2.4 Quantum Sample Perturbation
While there are several methods that make use of Szegedy’s quantum walk
operators to prepare quantum samples [5, 16, 17], we choose [5] as the main
approach to analyze as it leads to an overall speed-up in the general case.
The other approaches [16, 17] take advantage of the quantum Zeno effect but
the problem is that the quantum Zeno effect would result in an exponential
slow-down in the general case.
The work by Wocjan and Abeyesinghe [5] showed an approach to prepare the
coherent stationary distribution of a Markov Chain via modified quantum walk
and Grover’s $\frac{\pi}{3}$-amplitude amplication techniques. The theorem
listed below is the main theorem in Speed-up via Quantum Sampling. We refer
the interested readers to [5] for details on this algorithm for the
construction techniques and the computational complexity.
###### Theorem 1 (Speed-up via quantum sampling [5]).
Let $Q_{0},Q_{1},\ldots.,Q_{r}=Q$ be a sequence of classical Markov chains
with stationary distributions $\pi_{0},\pi_{1},\ldots,\pi_{r}$ and spectral
gap $\delta_{0},\ldots,\delta_{r}$. Assume that the stationary distributuions
of adjacent Markov chains are close to each other in the sense that
$|\langle\pi_{i}|\pi_{i+1}\rangle|^{2}\geq c$ where c is some constant, for
$i=0,\ldots,r-1.$. Then for any $\eta>0$, there is an efficient quantum
sampling algorithm, making it possible to sample according to a probability
distributuion $\tilde{\pi}_{r}$ that is close to $\pi_{r}$ with respect to the
total variation distance, i.e., $D(\tilde{\pi}_{r},\pi_{r})\leq\eta$.
Based on the theorem above, we can immediately conclude the following
corollary:
###### Corollary 3.
When the coherent quantum sample based on the perturbed Markov chain is
prepared by using techniques of [5] with precision $\eta$, the total variation
distance between the prepared quantum sample $\tilde{\pi}(Q)$ and the true
quantum sample $\pi(P)$ is less than
$\eta+\frac{1}{2(1-(\tau_{1}{P}))}\|E\|_{\infty}$.
###### Proof.
By Theorem 1 we can efficiently construct a quantum sample
$\tilde{\pi}{(Q)}_{r}$ that is $\eta$ close to $\pi(Q)$. Then by triangle
inequality we obtain
$D(\pi(P),\tilde{\pi}(Q))\leq
D(\pi(P),\pi(Q))+D(\pi(Q)+\tilde{\pi}(Q))\leq\frac{1}{2(1-(\tau_{1}{P}))}\|E\|_{\infty}+\eta.$
(14)
∎
## 3 Conclusion and Discussion
We apply the existing classical results from matrix perturbation to quantum
walk based algorithms. With the noise introduced at the input, as quantum
system is susceptible to the outside world and some other precision limitation
problem, we bounded the quantum hitting time with perturbation from the above
that the perturbed quantum walk hitting time is
$HT(Q)=O\Big{(}\sqrt{\frac{1}{(\delta-\|E\|)\epsilon}}\Big{)}.$
In the meanwhile, we also showed that how the quantum sample prepared by using
the approach in [5] would fluctuate from the true quantum sample when
perturbation is present. The analysis is based on the assumption that we have
a series of Markov chains $Q1,\ldots,Q_{r}=Q$. Hence, we have
$D(\pi(P),\tilde{\pi}(Q))\leq\frac{1}{2(1-(\tau_{1}{P}))}\|E\|_{\infty}+\eta.$
Intuitively from the analysis we can see that the total variation distance for
$D(\pi(\tilde{Q}),\pi(Q))$ is simply additive and $D(\pi(P),\pi(Q))$ cannot be
eliminated. However, if the matrix $P=Q_{i}$ is inside the sequence
$Q_{1},\ldots,Q_{r}$ where $1<i<r$, can we invent a procedure to detect to
avoid such overshoot? Future study is to find the relation between quantum
mixing time, the time it takes to get $\eta$-close to the true stationary
distribution, and the quantum hitting time as it was studied so classically.
Furthermore, another possible analysis approach would be assuming that we have
a series of Markov chains $P_{1},\ldots,P_{r}=P$ (without the noise). We can
adapt the analysis in [5] to study how the noise would affect (i) accuracy
when blindly preparing the quantum sample without acknowledging the existence
of noise or (ii) complexity when the noise is acknowledged and desired
accuracy must be achieved.
## 4 Acknowledgments
C. C. gratefully acknowledges the support of NSF grants CCF-0726771 and
CCF-0746600.
## References
* [1] L. Lovász and S. Vempala, Simulated Annealing in Convex Bodies and an $O^{*}(n^{4})$ Volume Algorithm, Journal of Computer and System Sciences, vol. 72, issue 2, pp. 392–417, 2006.
* [2] M. Jerrum, A. Sinclair, and E. Vigoda, A Polynomial-Time Approximation Algorithm for the Permanent of a Matrix Non-Negative Entries, Journal of the ACM, vol. 51, issue 4, pp. 671–697, 2004.
* [3] M. Jerrum and A. Sinclair, Polynomial-Time Approximation Algorithms for the Ising Model, SIAM Journal on Computing, vol. 22, pp. 1087–1116, 1993.
* [4] I. Bezáková, D. Štefankovič, V. Vazirani and E. Vigoda, Accelerating Simulated Annealing for the Permanent and Combinatorial Counting Problems, SIAM Journal on Computing, vol. 37, No. 5, pp. 1429–1454, 2008.
* [5] P. Wocjan and A. Abeyesinghe, Speed-up via Quantum Sampling, Phys. Rev. A, vol. 78, pp. 042336, 2008.
* [6] M. Szegedy, Quantum Speed-up of Markov Chain Based Algorithms, Proc. of 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 32–41, 2004.
* [7] C.-F. Chiang, D. Nagaj, P. Wocjan, Efficient Circuits for the Quantum Walks, QIC vol. 10 no. 5&6 pp. 0420–0434, 2010.
* [8] I. Ipsen and B. Nadler, Refined Perturbation Bounds for Eigenvalues of Hermitian and Non-Hermitian Matrices, SIAM J. Matrix Anal. Appl., vol. 31, no. 1, pp. 40–53, 2009.
* [9] G. Cho and C. Meyer, Comparison of Perturbation Bounds for the Stationary Distribution of a Markov Chain, vol. 335, issue 1-3, pp.137 - 150, Linear Algebra and Its Applications, 2001.
* [10] G. Golub and C. Loan, Matrix Computations, 3rd ed., The Johns Hopkins University Press, 1996.
* [11] B. Parlett, The Symmetric Eigenvlaue Problems, SIAM, Philadelphia, 1998\.
* [12] F. Bauer and C. Fike, Norms and Exclusion Theorems, Numer. Math.,vol. 2, pp. 137 - 141, 1960.
* [13] S. Eisenstat and I. Ipsen, Three Absolute Perturbation Bounds for Matrix Eigenvalues Imply Relative Bounds, SIAM Journal on Matrix Analysis and Applications, vol. 20 , issue 1, pp. 149 - 158, 1999.
* [14] I. Johnstone, On the distribution of the largest eigenvalue in principal components analysis, vol. 29, no. 2, pp. 295 - 327, Annals of Statistics, 2001.
* [15] R. Bhatia, Matrix Analysis, Springer Verlag, New York, 1997.
* [16] R. Somma, S. Boixo, and H. Barnum, Quantum Simulated Annealing, arXiv:abs/0712.2008
* [17] R. Somma, S. Boixo, H. Barnum, E. Knill, Quantum Simulations of Classical Annealing Processes, Phys. Rev. Lett. vol. 101, pp. 130504, 2008.
|
arxiv-papers
| 2010-07-08T16:48:33 |
2024-09-04T02:49:11.504986
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chen-Fu Chiang",
"submitter": "Chen-Fu Chiang",
"url": "https://arxiv.org/abs/1007.1415"
}
|
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