# Refractive index in the JUNO liquid scintillator

H.S. Zhang  M. Beretta<EMAIL_ADDRESS>S. Cialdi  C.X. Yang  J.H.
Huang  F. Ferraro  G.F. Cao<EMAIL_ADDRESS>G. Reina  Z.Y. Deng  E. Suerra
S. Altilia  V. Antonelli  D. Basilico  A. Brigatti  B. Caccianiga  M.G.
Giammarchi  C. Landini  P. Lombardi  L. Miramonti  E. Percalli  G. Ranucci
A.C. Re  P. Saggese  M.D.C. Torri  S. Aiello  G. Andronico  A. Barresi  A.
Bergnoli  M. Borghesi  R. Brugnera  R. Bruno  A. Budano  A. Cammi  V. Cerrone
R. Caruso  D. Chiesa  C. Clementi  S. Dusini  A. Fabbri  G. Felici  A.
Garfagnini  N. Giudice  A. Gavrikov  M. Grassi  R.M. Guizzetti  N. Guardone
B. Jelmini  L. Lastrucci  I. Lippi  L. Loi  C. Lombardo  F. Mantovani  S.M.
Mari  A. Martini  M. Montuschi  M. Nastasi  D. Orestano  F. Ortica  A. Paoloni
F. Petrucci  E. Previtali  M. Redchuck  B. Ricci  A. Romani  G. Sava  A.
Serafini  C. Sirignano  M. Sisti  L. Stanco  E. Stanescu Farilla  V. Strati
A. Triossi  C. Tuve  C. Venettacci  G. Verde  L. Votano  Institute of High
Energy Physics, Chinese Academy of Sciences, Beijing 100049, China  University
of Chinese Academy of Sciences, Beijing 100049, China  University of Science
and Technology of China, Hefei 230026, China  INFN, Sezione di Milano e
Università degli Studi di Milano, Dipartimento di Fisica, Italy  INFN, Sezione
di Catania e Università di Catania, Dipartimento di Fisica e Astronomia, Italy
INFN, sezione di Padova e Università di Padova, Dipartimento di Fisica e
Astronomia, Italy  INFN, sezione di Roma Tre e Università degli Studi Roma
Tre, Dipartimento di Fisica e Matematica, Italy  INFN, Sezione di Milano
Bicocca e Dipartimento di Fisica Università di Milano Bicocca, Italy  INFN,
Sezione di Perugia e Università degli Studi di Perugia, Dipartimento di
Chimica, Biologia e Biotecnologie, Italy  Laboratori Nazionali dell’INFN di
Frascati, Italy  INFN, Sezione di Ferrara, Italy  Università degli Studi di
Ferrara, Dipartimento di Fisica e Scienze della Terra, Italy  INFN, Sezione di
Milano Bicocca e Dipartimento di Energetica, Politecnico di Milano, Italy

###### Abstract

In the field of rare event physics, it is common to have huge masses of
organic liquid scintillator as detection medium. In particular, they are
widely used to study neutrino properties or astrophysical neutrinos. Thanks to
its safety properties (such as low toxicity and high flash point) and easy
scalability, linear alkyl benzene is the most common solvent used to produce
liquid scintillators for large mass experiments. The knowledge of the
refractive index is a pivotal point to understand the detector response, as
this quantity (and its wavelength dependence) affects the Cherenkov radiation
and photon propagation in the medium. In this paper, we report the measurement
of the refractive index of the JUNO liquid scintillator between 260-1064 nm
performed with two different methods (an ellipsometer and a refractometer),
with a sub percent level precision. In addition, we used an interferometer to
measure the group velocity in the JUNO liquid scintillator and verify the
expected value derived from the refractive index measurements.

###### keywords:

Refractive index , linear alkyl benzene , Cherenkov light , scintillator ,
neutrino physics , JUNO experiment

††journal: Nuclear Instruments and Methods in Physics Research A

##  1 Introduction

The linear alkyl benzene, also known as LAB, is one of the most common
solvents to produce modern organic liquid scintillators. For example, the Daya
Bay experiment used 20x8 tons of LAB doped with Gadolinium to measure the
$\theta_{13}$ neutrino oscillation parameter with high accuracy [1]. Thanks to
its favorable safety properties (low toxicity and high flash point), large
scalability (a common solvent in the industry), and the possibility of being
purified successfully, LAB will be used in several future neutrino
experiments, like JUNO [2] or SNO+ [3] .

In particular, the JUNO experiment is a huge neutrino detector, under
construction in China, which will use 20 ktons of liquid scintillator, based
on LAB, contained in a 35 m diameter acrylic sphere. The primary scientific
goal of JUNO is the determination of the neutrino mass ordering which,
according to detailed sensitivity studies, is expected to require six years of
data taking.

To reach this challenging task, JUNO will need an unprecedented energy
resolution for a liquid scintillator based detector aiming to obtain $3\%$ at
1 MeV and a position resolution of $7\,\mathrm{cm}$ at 1 MeV [2]. In order to
reach the required performances on energy and position reconstruction, it is
mandatory to know with high accuracy all the optical properties of the
scintillator. In particular, this paper is focused on the refractive index of
the JUNO scintillator. In this paper, the results on the measurement of the
refractive index, together with the two techniques that were used to obtain
these results in different wavelength intervals, are presented.

The refractive index determines the velocity of photons in the medium. This is
a crucial input for the reconstruction of the event interaction vertex, which
is based on the arrival times of each photon produced in a given event to each
phototube. The position reconstruction is fundamental in the JUNO analysis for
several reasons: first of all, it allows the selection of an innermost region
of the detector (Fiducial Volume) where the external background is negligible
due to the self-shielding of the scintillator. Secondly, it is crucial to
select a couple of events spatially correlated due to the interaction of the
reactor anti-neutrinos in the detector111 Reactor anti-neutrinos mostly
interact via the Inverse Beta Decay (IBD) reaction, in which a positron and a
neutron are produced via the anti-neutrino capture by a proton. The two
emitted particles are time and spatial correlated hence looking at this
correlation it is possible to reduce the background contribution.. In addition
the refractive index is connected to other properties of the liquid
scintillator like the Rayleigh scattering length [4][5], which can spoils the
energy resolution if not well controlled. For these reasons, knowing the
refractive index over a large range of wavelengths and with high accuracy is
very important. Recently, the RENO collaboration measured the refractive index
of six wavelengths of LAB and PC [6]. And H. Wan Chan Tseung et al. used the
ellipsometer to measure the refractive index of LAB-OPP in the range of 210
nm-1000 nm with high accuracy [7]. In this study, the 260 nm-1064 nm
refractive index of LAB is measured by ellipsometer and refractometer.

The refractive index also has an impact on the capability to reconstruct the
energy of events in JUNO. The reconstruction of the energy of particles
interacting in JUNO is possible thanks to the collection of photons generated
in its active volume. Together with scintillation photons, which are the
majority, also Cherenkov photons are produced following the well-known Frank-
Tamm formula:

$\frac{d^{2}N}{dxd\lambda}=\frac{2\pi
e^{2}}{\lambda^{2}}\left(1-\frac{1}{\beta^{2}n^{2}(\lambda)}\right)$ (1)

hence, an accurate knowledge of the refractive index will help to constrain
the number of Cherenkov photons and, therefore, to precisely determine the
energy response of the detector.

Furthermore, the accurate knowledge of the Cherenkov contribution to the total
detected light is important when the Correlated and Integrated Directionality
(CID) method is applied. This method was originally developed by the Borexino
collaboration [8], to increase the sensitivity to solar neutrino [9].

##  2 Refractive index measurements

To measure the refractive index we used two different techniques covering
different parts of the spectrum. The region from 260 nm to 500 nm was covered
using an ellipsometer, located at the Institute of High Energy Physics (IHEP)
in Beijing, allowing to determine the refractive index in a spectral region in
which the LAB absorbs the incident light but still some light is reflected.
From 400 nm to 1064 nm, we used a refractometer, built in the Physics
department of the University of Milan, which allows to cross-check the
measurements in the region between 400 nm and 500 nm and extends the
measurements up to 1064 nm.

The JUNO liquid scintillator is composed of LAB + 2.5 g/L PPO + 3 mg/L bis-
MSB. Since the concentration of the fluor and wavelength shifter are very low,
we assume that the refractive index of the liquid scintillator is dominated by
the LAB one and we performed our measurements on the pure LAB. We checked our
assumption for $\lambda=405\,\mathrm{nm}$ and verified its validity.

###  2.1 Ellipsometric measurements

Spectroscopic ellipsometry utilizes the polarization of reflected light on a
sample to probe the dielectric properties of the material under investigation.
In ellipsometric measurements, the primary focus is on quantifying the
polarization of reflected light, thus mitigating concerns regarding the
absorption of the light by the medium. This facilitated precise measurement of
the refractive index in the vacuum ultraviolet region, in contrast to
refractometric measurements, which are constrained by the need for the
incident beam to pass through the sample, resulting in limitations due to
absorbance.

Figure 1 shows a schematic diagram and a photograph of the experimental setup
for the polarization measurements. The light emitted from a xenon lamp is
directed through a monochromator to select the wavelength of the incident
radiation. Subsequently, the light is collimated and directed through a beam
splitter, with a PMT (monitor PMT) positioned to correct the potential
instabilities of the xenon lamp. The incident light is polarized by a
polarizer and directed onto the sample through an aperture. The polarization
of the reflected light is examined via an analyzer (a second polarizer) and a
PMT (signal PMT). The polarizer and analyzer are both constructed with 16
calcium fluoride (CaF2) windows, organized in parallel stacks of 8 pieces, and
positioned symmetrically to prevent any displacement of the beam’s center as
the polarizer rotates. As the light passes through each CaF2 window at the
Brewster angle, the s-mode polarized light is diminished by reflections. This
enables the acquisition of diverse polarizations through the rotation of the
polarizer and analyzer. The apertures are used to reduce the stray light in
the system. The current output of each PMT is measured by a picoamperometer
and is proportional to the amout of light impinging on the photocathode.

Figure 1: Top: A schematic diagram of the experimental setup for the
ellipsometric measurements. Bottom: A photograph of the setup in the dark
room.

In a typical scenario, the current $I_{signal}$ of the signal PMT at a
specific rotated angle $P$ of the polarizer can be expressed as [10, 11]:

$\centering
I_{signal}\propto[1+\frac{1}{\eta}\alpha^{\prime}\cos{(2A)}+\frac{1}{\eta}\beta^{\prime}\sin{(2A)}]\@add@centering$
(2)

where $A$ represents the rotated angle of the analyzer, as shown in Figure 1,
$\eta=\frac{1+\xi^{2}}{1-\xi^{2}}$ where $\xi$ is the ratio of the s-mode
polarized light passing through the polarizer and analyzer. To account for
potential variations of $\eta$ at different rotated angles $P$ of the
polarizer, we replace $\eta$ with $\eta^{\prime}$ in Equation 2 by the
following empirical formula:

$\centering\eta^{\prime}=(1+\frac{a}{b+tan^{2}(P-P_{s})})\eta\@add@centering$
(3)

where $a$ and $b$ are two parameters that can be determined using the well-
established calibration method. Thus Equation 2 can be expressed as:

$\centering
I_{signal}\propto(1+\hat{\alpha}cos2A+\hat{\beta}sin2A)\@add@centering$ (4)
$\centering\hat{\alpha}=\frac{1}{\eta^{\prime}}\alpha^{\prime}\@add@centering$
(5)
$\centering\hat{\beta}=\frac{1}{\eta^{\prime}}\beta^{\prime}\@add@centering$
(6)

The $\alpha^{\prime}$ and $\beta^{\prime}$ in Equation 2 are the two Fourier
coefficients and can be determined via:

$\centering\alpha^{\prime}=\frac{1}{\eta}(\alpha cos(2A_{s})-\beta
sin(2A_{s}))\@add@centering$ (7)
$\centering\beta^{\prime}=\frac{1}{\eta}(\alpha sin(2A_{s})+\beta
cos(2A_{s}))\@add@centering$ (8)

where the coefficients $\alpha$ and $\beta$ can be parameterized with the
formula:

$\centering\alpha=\frac{tan^{2}(\psi)-tan^{2}(P-P_{s})}{tan^{2}(P-P_{s})+tan^{2}(\psi)}\@add@centering$
(9)

$\centering\beta=\frac{2\,tan\psi\,cos\Delta\,tan(P-P_{s})}{tan^{2}(P-P_{s})+tan^{2}(\psi)}\@add@centering$
(10)

$A_{s}$ and $P_{s}$ represent the angles between the incident light plane and
the p-polarization direction of the analyzer and polarizer, respectively,
which need to be calibrated along with $\eta$. $\Psi$ and $\Delta$ are
ellipsometric parameters [12], commonly utilized in commercial ellipsometers.
The refractive index $n$ of the sample can be determined via:

$\centering
n=n_{air}\sin^{2}(\theta_{i})\sqrt{1+\tan^{2}\theta_{i}\big{(}\frac{1-\rho}{1+\rho}\big{)}}\@add@centering$
(11)

where $\theta_{i}$ represents the angle of incidence and $\rho$ is the ratio
between the two polarization components of the reflected light, given by
$\rho=\frac{R_{p}}{R_{s}}$.

The regression calibration method is employed to obtain the aforementioned
parameters of $\eta$, $A_{s}$ and $P_{s}$ in this experiment [13] and the
calibration is performed using 400 nm light for the measured samples of LAB
and PC (pseudocumene), respectively. The current of the signal PMT is measured
as a function of the angle $A$ for a given angle $P$, with $P$ being scanned
from 0 to $2\pi$ with an interval of $5^{\circ}$. The Fourier coefficients of
$\hat{\alpha}$ and $\hat{\beta}$ are obtained by fitting the current curve
using Equation 4, and an example at $P$ = $155^{\circ}$ is shown in Figure 2.
The obtained $\hat{\alpha}$ and $\hat{\beta}$ are presented in Figure 3 as a
function of $P$ for pseudocumene (PC, an alternative solvent) and LAB.
Subsequently $\hat{\alpha}(P)$, $\hat{\beta}(P)$ and
$R(P)=1-\hat{\alpha}^{2}-\hat{\beta}^{2}$ curves are simultaneously fitted to
extract the calibration parameters, which are summarized in Table 1. The
fitting results demonstrate a good agreement with the experimental data.

Figure 2: Signal PMT current at the polarizer angle of $P=155^{\circ}$ as a
function of rotated angle $A$ of the analyzer.

Figure 3: Top: The Fourier coefficients of $\hat{\alpha}$, $\hat{\beta}$ and $R$ as a function of polarizer angle $P$ for PC. Bottom: The Fourier coefficients as a function of $P$ for LAB. The red dots are the obtained Fourier coefficients and the black lines represent the fitting results. Parameters | PC | LAB
---|---|---
$P_{s}$ | 0.95096 $\pm$ 0.00010 | 0.95055 $\pm$ 0.00011
$A_{s}$ | 0.11809 $\pm$ 0.00007 | 0.11805 $\pm$ 0.00007
$\eta$ | 1.06492 $\pm$ 0.00006 | 1.06496 $\pm$ 0.00006
$a$ | 0.01293 $\pm$ 0.00004 | 0.01368 $\pm$ 0.00004
$b$ | 0.01727 $\pm$ 0.00008 | 0.02166 $\pm$ 0.00011
Table 1: Summary of PC and LAB calibration parameters obtained from fitting.

With the obtained calibration parameters, the refractive index of PC and LAB
was measured within a range from 260 nm to 500 nm. Fixing the angle of the
polarizer $P$ at $54^{\circ}$, the analyzer is rotated from 0 to $2\pi$ with
an interval of $10^{\circ}$, and the current of the signal PMT is mapped to
the analyzer rotation angle. Subsequently, the $\rho$ ratio can be determined
by fitting the curve of $I_{signal}$ versus $A$ at each wavelength, with the
angle of incidence measured to be $67.06^{\circ}$. The obtained refractive
indices of LAB and PC are presented in Table 2 and illustrated in Figure 4. A
good agreement with the results reported in [14] is evident for PC.

Figure 4: The refractive indices of PC as a function of wavelength. The measured values are shown as red points and the black curve is from Borexino’s measurements [14]. Wavelength[nm] | Refractive index(LAB) | Refractive index(PC)
---|---|---
260 | 1.578 $\pm$ 0.005 | 1.653 $\pm$ 0.005
270 | 1.569 $\pm$ 0.005 | 1.637 $\pm$ 0.005
280 | 1.556 $\pm$ 0.005 | 1.637 $\pm$ 0.005
290 | 1.546 $\pm$ 0.005 | 1.612 $\pm$ 0.005
300 | 1.541 $\pm$ 0.005 | 1.595 $\pm$ 0.005
310 | 1.533 $\pm$ 0.005 | 1.584 $\pm$ 0.005
320 | 1.528 $\pm$ 0.005 | 1.566 $\pm$ 0.005
330 | 1.523 $\pm$ 0.005 | 1.562 $\pm$ 0.005
340 | 1.519 $\pm$ 0.005 | 1.550 $\pm$ 0.005
350 | 1.512 $\pm$ 0.005 | 1.552 $\pm$ 0.005
360 | 1.510 $\pm$ 0.005 | 1.542 $\pm$ 0.005
370 | 1.507 $\pm$ 0.005 | 1.539 $\pm$ 0.005
380 | 1.506 $\pm$ 0.005 | 1.538 $\pm$ 0.005
390 | 1.508 $\pm$ 0.005 | 1.542 $\pm$ 0.005
400 | 1.501 $\pm$ 0.005 | 1.532 $\pm$ 0.005
410 | 1.498 $\pm$ 0.005 | 1.529 $\pm$ 0.005
420 | 1.496 $\pm$ 0.005 | 1.525 $\pm$ 0.005
430 | 1.496 $\pm$ 0.005 | 1.521 $\pm$ 0.005
440 | 1.494 $\pm$ 0.005 | 1.521 $\pm$ 0.005
450 | 1.497 $\pm$ 0.005 | 1.519 $\pm$ 0.005
460 | 1.492 $\pm$ 0.005 | 1.517 $\pm$ 0.005
470 | 1.489 $\pm$ 0.005 | 1.513 $\pm$ 0.005
480 | 1.492 $\pm$ 0.005 | 1.504 $\pm$ 0.005
490 | 1.487 $\pm$ 0.005 | 1.505 $\pm$ 0.005
500 | 1.490 $\pm$ 0.005 | 1.515 $\pm$ 0.005
Table 2: Summary of LAB and PC refractive indices obtained with ellipsometric
measurements.

###  2.2 Refractometric measurements

To characterize the refractive index in the visible and Near Infrared spectra,
we decided to build a table-top experimental setup able to measure this
important property of the scintillator. We have employed a standard
refractometric technique aiming to investigate the wavelength range between
400 nm and 1100 nm.

Figure 5: Schematic view of the refractometer measurement.

Our refractometer consists of a collimated light source, a photosensor and a
support for the sample. In particular, we used several laser sources, listed
in Table 3, in the range in which the JUNO scintillator emits its fluorescence
light. The LS sample is contained in $1\times 5\times 5$ $\mathrm{m^{3}}$
quartz cuvette, a DCC1545M camera (CCD) as a photosensor, a DAQ software for
the acquisition, and other components for the correct propagation of light
from the source to the CCD, such as collimators, optical fibers, and
diffraction grating. A picture of the setup is shown in Figure 6.

From the Snell law, it is possible to retrieve the refractive index by looking
at the displacement of the position of a point-like source, when the beam
passes through different refractive indexes:

$d=l\frac{\sin\big{[}\theta_{i}-\arcsin\big{(}\frac{n_{a}}{n_{s}}\sin\theta_{i}\big{)}\big{]}}{\cos\big{[}\arcsin(\frac{n_{a}}{n_{s}}\sin\theta_{i})\big{]}}$
(12)

where $d$ is the observable displacement, $l$ is the width of the cuvette,
$\theta_{i}$ is the incident angle of the laser beam, $n_{a}$ is the
refractive index of the air and $n_{s}$ is the refractive index of the
scintillator.

Table 3: Laser source list Name | Wavelengths [nm]
---|---
diode | 405, 670
Argon | 475, 514
Ar + Ti-sapphire | 746, 823
Nd-YAG | 1064
Figure 6: In the red box it is possible to see the refractometer experimental
setup. In the left top edge, the single-mode optical fiber (in orange) is
connected to a collimator. Then, the laser beam passes through a quartz
optical cell and reaches a CDD camera.

For each measurement, we checked that the laser beam arrived perpendicular to
the CCD camera, within less than one degree precision which implies an error
order of $10^{-4}$ on the displacement measurement, negligible for our
purpose. Hence, two possible factors limiting the precision of the
measurements are the uncertainty on the incident angle on the cuvette
($\theta_{i}$) and the uncertainty on the displacement. The former can be
calibrated using a material, such as water, with a well-known refractive
index. Therefore, the main source of error is the uncertainty of the
displacement.

In order to avoid contamination between water and liquid scintillator in the
optical path of the laser beam in the cuvette, we measured the displacement
filling the cuvette first with the scintillator and then with distilled water.
Since the density of the LAB is lower than the one of the water, even if small
LAB residuals remain in the cell, it does not spoil the measurements since
they will float on top of water. If water was used first instead, some drops
of water could stay wrapped in the liquid scintillator causing some unwanted
reflections. We have tested and validated our procedure by measuring some
”well-known” refractive indexes, like Ethanol, finding it in agreement with
the value in literature within a 0.1$\%$ accuracy. The results obtained on the
LAB with this technique are shown in Table 4 and depicted in Figure 7 with red
dots.

Table 4: Experimental results of LAB refractive indexes by the refractometer measurements Wavelength [nm] | Refractive index | Error
---|---|---
405 | 1.505 | 0.007
476 | 1.486 | 0.007
514 | 1.490 | 0.007
633 | 1.473 | 0.003
670 | 1.478 | 0.007
746 | 1.473 | 0.007
823 | 1.469 | 0.007
1064 | 1.468 | 0.007

###  2.3 Results and analysis

Using these two techniques, it was possible to measure the refractive index in
the range between $260\,\mathrm{nm}$ and $1064\,\mathrm{nm}$ reaching a
precision ranging from $\sim 1\%$ to $\sim 0.2\%$. Using the Sellmeier law, we
can verify the trend of the data.

$n^{2}(\lambda)=1-\frac{B}{1-C/\lambda^{2}}$ (13)

The global fit of the two datasets is shown in Figure 7. As it is possible to
see, the data of the two datasets (ellipsometer in blue, refractometer in red)
are compatible in the region from $400\,\mathrm{nm}$ and $500\,\mathrm{nm}$
and all the experimental points are compatible with the fit. The results on
the Sellmeier fit are shown in Table 5. Combining the two methods, we increase
the accuracy and precision of this crucial parameter of the liquid
scintillator.

Figure 7: Combined analysis of the two datasets. As it is possible to see both the fits are in the error band of the global fit. Table 5: Summary of the fit results applying the Sellmeier law on the two datasets. Dataset | B | $\sigma_{B}$ | C [nm2] |  $\sigma_{C}$ [nm2]
---|---|---|---|---
Combined | 1.147 | 0.003 | 14.1$\times 10^{3}$ | 0.5 $\times 10^{3}$

##  3 Group velocity of the JUNO liquid scintillator

The refractive index impacts event reconstruction in JUNO mostly for what
concerns position reconstruction. We measured directly the group velocity in
the liquid scintillator to check our measurement of the refractive index.
Indeed, for a dispersive medium, it is possible to write the phase velocity as
the ratio between the speed of light and the refractive index. Hence the group
velocity is described by:

$v_{g}(\lambda)=\frac{c}{n(\lambda)}\bigg{(}1-\frac{\lambda}{n(\lambda)}\frac{dn(\lambda)}{d\lambda}\bigg{)}^{-1}$
(14)

To perform this measurement, we used a standard Michelson interferometer,
which is a device allowing us to study the interference of two beams. In
particular, a laser beam passes through a semi-reflective mirror (a beam
splitter), which splits it along two arms. One of these arms ends with a plane
mirror reflecting back the light. The other arm has a 5 cm cuvette in which
the beam passes reaching a mirror in order to be reflected back. Then the two
beams are recombined and seen by a fast photosensor which acquires the data.

Figure 8: Schematic view of group velocity measurement performed using a
Michelson interferometer.

Inserting a sample in the cell along the optical path, it is possible to see a
shift in the position of the interference fringes. Measuring this shift via a
micrometric screw, which moves the mirror, it is possible to retrieve the
group velocity. A scheme of the setup is shown in Figure 8.

Figure 9: In blue it is possible to see the phase velocity curve determined by
the refractive index measurements. The group velocity measurements are
depicted in green. The green line instead represents the group velocity curve
calculated from the refractive index measurements using the formula 14.

We measured the group velocity using two laser sources reaching three
different wavelengths: 340, 516, and 1036 nm. The results are shown in Figure
9 overlaying on the predicted group velocity curve obtained by using the
formula in Equation 14 and the Sellmeier fit on the refractive index
measurements. The data, shown in Table 6, follow the predicted law in the full
spectrum at $2\,\sigma$ level.

Table 6: Results of the group velocity measurements in a fraction of the light speed. Wavelength [nm] | $v_{g}/c$
---|---
340 | 0.6178$\pm$ 0.0069
516 | 0.6392$\pm$0.0066
1036 | 0.6746$\pm$0.0071

##  4 Conclusion

JUNO will be a very large detector based on 20,000 tons of organic liquid
scintillator. To reach its physics goals, the experiment needs an accurate
Monte Carlo code with the best possible description of the liquid
scintillator. Different groups of the collaboration started programs to
characterize the liquid scintillator accurately. In this work, we summarize
the experimental results on the refractive index obtained by two different
groups of collaboration in order to extend the results in the largest optical
range possible. These results on the refractive index will help to better
understand the Cherenkov contribution in the liquid scintillator and to
improve the position reconstruction in the JUNO detector. Furthermore, the
group velocity measurements support the measurements of the refractive index
and allows the best measurement of the light speed in the JUNO liquid
scintillator.

## Acknowledgments

The IHEP group gratefully acknowledges support from the National Natural
Science Foundation of China (NSFC) under grant No. 12375196. The Milan group
would like to thank Professor Cialdi of the University of Milan for the time
and the knowledge shared with us for both the refractometer and the
interferometer. Many thanks also to Milan University for the financial support
needed to build the experimental setup.

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# Stability of Contraction-Driven Cell Motion

C. A. Safsten†, V. Rybalko‡, L. Berlyand†

(† Department of Mathematics, Pennsylvania State University, University Park,
PA 16802
‡ B. Verkin Institute for Low Temperature Physics and Engineering of NASU, 47
Nauky ave, Khariv 61103, Ukraine.
E-mail<EMAIL_ADDRESS>
###### Abstract

We consider motility of keratocyte cells driven by myosin contraction and
introduce a 2D free boundary model for such motion. This model generalizes a
1D model from [12] by combining a 2D Keller-Segel model and a Hele-Shaw type
boundary condition with the Young-Laplace law resulting in a boundary
curvature term which provides a regularizing effect. We show that this model
has a family of traveling solutions with constant shape and velocity which
bifurcates from a family of radially symmetric stationary states. Our goal is
to establish observable steady motion of the cell with constant velocity.
Mathematically, this amounts to establishing stability of the traveling
solutions. Our key result is an explicit asymptotic formula for the stability-
determining eigenvalue of the linearized problem. This formula greatly
simplifies the task of numerically computing the sign of this eigenvalue and
reveals the physical mechanisms of stability. The derivation of this formula
is based on a special ansatz for the corresponding eigenvector which exhibits
an interesting singular behavior such that it asymptotically (in the small-
velocity limit) becomes parallel to another eigenvector. This reflects the
non-self-adjoint nature of the linearized problem, a signature of living
systems. Finally, our results describe the onset of motion via a transition
from unstable radial stationary solutions to stable asymmetric traveling
solutions.

##  1 Introduction

Sustained motion on a substrate has been observed in experiments on living
cells. Keratocytes in particular frequently exhibit motion. They are found
naturally moving on flat surfaces, e.g., the human cornea, making them ideal
subjects for experiment. Moreover, their flat shape lends itself toward two
dimensional modeling. Keratocytes are often observed in a stationary state
with a circular shape, or traveling with constant velocity and maintaining a
constant, asymmetric shape. This motion is explained by three mechanisms:
adhesion, protrusion, and contraction, whose effects are summarized as
follows. The moving cell contains actin and myosin proteins. Actin
polymerizes, forming a _cytoskeleton_ which provides structure for the cell.
Actin polymerizing near the edge of the cell causes protrusions of the cell
membrane. These protrusions then adhere to the substrate, stabilizing the cell
in its new shape. Myosin causes the actin polymers to contract. If the myosin
is concentrated on one side of the cell, the cell contracts on that side,
driving intracellular fluid to the other side of the cell and expanding the
cell on that side. The flow of intracellular fluid also carries the myosin to
the other side of the cell, continuing the process and resulting in net
motion. The study of cytoskeloton gel has led to the recent development of the
so-called “active gel physics” [10].

We propose a two dimensional free boundary PDE model for cell motility which
describes the evolution of the cell shape and the distribution of myosin
within the cell. As in [12], our model explains cell motility as being driven
primarily by contraction (as opposed to adhesion or protrusion). This model
exhibits bifurcation of a family of traveling solutions (modeling cells moving
with constant velocity and shape) from a family of stationary solutions
(modeling non-moving cells).

Of particular interest is the question of stability of traveling solutions to
this model. In order to be observed in experiment, steady cell motion must be
robust, not disrupted by small perturbations present in any experimental
setup. We show that traveling solutions to our model also have this property
by showing that they are mathematically stable. Stability is also important
for numerical computations. Since any computational model of a moving cell is
necessarily an approximation of a true cell, stability of traveling solutions
is necessary for numerical simulations of cell motion to converge.

A 1D contraction-driven free-boundary model is proposed in [12, 13], see more
recent work [11]. Our model generalizes this one to two dimensions, and also
answers the question of stability of traveling solutions. Our analysis shows
that asymmetry in the myosin distribution results in the net motion of the
cell. The essence of this phenomenon lies in the “motor effect” studied in [9]
in the context of a 1D model for myosin moving along filaments.

A 2D free-boundary model for cell motion driven by polymerization of actin (as
opposed to myosin contraction) is proposed in [5]. Like our model, this model
also possesses a branch of traveling solutions bifurcating from a family of
stationary solutions. Analysis shows that the bifurcation in this model is
subcritical, meaning that traveling solutions near the bifurcation point are
unstable (see also a 2D model in [3]).

Other 2D free boundary models of cell motility have been examined numerically,
e.g., [2, 5, 6]. For example, in [2], the authors propose a model for
keratocyte motility taking into account actin polymerization in addition to
myosin-driven contraction. Numerical analysis of this free boundary model
shows close agreement with both experimental results and theoretical results
in our model. Additionally, a 2D moving cell model where the boundary has
fixed shape was introduced and studied analytically and numerically in [7].
This model possesses several stationary solutions whose stability is proved
provided the total myosin mass is sufficiently small.

Phase-field models of cell motion provide an alternative to free boundary
models. Computational results of these models, shown in, e.g., [14], also
agree qualitatively with results from our free boundary model.

##  2 The Model

We consider a 2D model for a cell occupying a region $\Omega(t)$ with free
boundary. We model the flow of the acto-myosin network as a gel obeying
Darcy’s law $-\nabla p=\zeta u$, where $-p$ is scalar stress ($p$ being
pressure), $u$ is flow velocity, and $\zeta>0$ is the constant adhesion
coefficient. We take for the constitutive equation for scalar stress
$-p=\mu\,\text{div}u+km-p_{h}$ where $\mu>0$ is the constant bulk viscosity of
the gel, making $\mu\,\text{div}u$ the hydrodynamic stress; $m$ is the density
of myosin with constant contractility coefficient; and $p_{h}$ is the constant
hydrodynamic pressure at equilibrium. We will assume that $\mu$ and $k$ are
scaled such that $\mu=k=1$.

Following the Young-Laplace law, we assume that on the boundary
$p+p_{\ast}=\gamma\kappa$ where $\kappa$ is the curvature of the boundary
$\partial\Omega(t)$, $\gamma$ is the constant surface tension coefficient, and
$p_{e}$ is the effective elastic restoring force induced by membrane cortex
tension. With the idea of generalizing Hooke’s law for 1D springs, $p_{e}$
models the elastic restoring force nonlocally as proportional to the
difference between the area $|\Omega|$ and a reference area $|\Omega_{h}|$:

$p_{e}=-k_{e}\frac{|\Omega|-|\Omega_{h}|}{|\Omega_{h}|},$ (1)

where $k_{e}$ is constant inverse compressibility coefficient. The purpose of
$p_{e}$ is to serve as a regularization term, preventing $\Omega$ from
becoming arbitrarily large or collapsing to a point (c.f. vertex models, e.g.
[1]).

The evolution of the density of myosin is described by the advection-diffusion
equation $\partial_{t}m=\Delta m-\text{div}(um)$ with the no-flux boundary
condition $\partial_{\nu}m=0$ where $\nu$ is the outward normal vector to the
moving boundary $\partial\Omega(t)$. Finally, the motion of the cell boundary
is described a kinematic boundary condition $V_{\nu}=-\partial_{\nu}p/\zeta$,
where $V_{\nu}$ is the velocity of the boundary in the normal direction. This
boundary condition ensures that the normal velocity of the boundary matches
the normal velocity $\nu\cdot u$ of the intracellular fluid.

Combining the above equations and introducing the convenient potential
$\phi=-(p+p_{h})/\zeta$, we arrive at the free-boundary PDE model

$\displaystyle 0$ $\displaystyle=\Delta\phi+m-\zeta\phi$ $\displaystyle\text{
in }\Omega(t)$ (2) $\displaystyle\partial_{t}m$ $\displaystyle=\Delta
m-\text{div}(m\nabla\phi)$ $\displaystyle\text{ in }\Omega(t)$ (3)
$\displaystyle\zeta\phi$ $\displaystyle=p_{\ast}(|\Omega(t)|)-\gamma\kappa$
$\displaystyle\text{ on }\partial\Omega(t)$ (4) $\displaystyle\partial_{\nu}m$
$\displaystyle=0$ $\displaystyle\text{ on }\partial\Omega(t)$ (5)
$\displaystyle V_{\nu}$ $\displaystyle=\partial_{\nu}\phi$
$\displaystyle\text{ on }\partial\Omega(t),$ (6)

where $p_{\ast}=p_{h}+p_{e}$.

The system (2)-(6) has a family of solutions corresponding to stationary
cells. These solutions have constant stress $\phi_{0}$ and myosin density
$m_{0}$, and a circular shape:

$\displaystyle\phi_{0}$ $\displaystyle=\frac{p_{\ast}(\pi
R^{2})-\gamma/R}{\zeta}$ (7) $\displaystyle m_{0}$ $\displaystyle=p_{\ast}(\pi
R^{2})-\gamma/R$ (8) $\displaystyle\Omega_{0}(t)$
$\displaystyle=\\{(x,y)\in\mathbb{R}^{2}:\sqrt{x^{2}+y^{2}}<R\\}.$ (9)

The family of stationary solutions is parameterized by the radius $R$ of the
cell. The total myosin mass of the stationary solution with radius $R$ is

$M(R)=\pi R^{2}m_{0}=p_{\ast}(\pi R^{2})\pi R^{2}-\pi\gamma R.$ (10)

Provided

$p_{\ast}^{\prime}(\pi R^{2})<-(2m_{0}+\gamma/R)/(2\pi R^{2}),$ (11)

the function $M(R)$ is strictly decreasing, so it has an inverse $R(M)$.
Therefore, either $M$ or $R$ may be used to parameterize the family of
stationary solutions.

##  3 Traveling Solutions and Bifurcation

A traveling solution $(\phi,m,\Omega)$ to (2)-(6) with velocity
$\mathbf{V}\in\mathbb{R}^{2}$ has the ansatz
$\phi=\phi(\mathbf{x}-\mathbf{V}t)$, $m=m(\mathbf{x}-\mathbf{V}t)$ where
$\mathbf{x}\in\mathbb{R}^{2}$, and $\Omega=\Omega_{0}+\mathbf{V}t$.
Substituting this ansatz into (2)-(6), we find that (2), (4), and (5) are
unchanged while (3) and (6) become

$\displaystyle\partial_{t}m$ $\displaystyle=\mathbf{V}\cdot\nabla m+\Delta
m-\text{div}(m\nabla\phi),\text{ and}$ (12) $\displaystyle 0$
$\displaystyle=\partial_{\nu}(\phi-\mathbf{V}\cdot\mathbf{x})$ (13)

respectively. Solutions to (12) have the form $m=\Lambda
e^{\phi-\mathbf{V}\cdot\mathbf{x}}$ for some $\Lambda$ depending on
$\mathbf{V}$. Substituting this solution for $m$ reduces (2)-(6) to the system

$\displaystyle 0$ $\displaystyle=\Delta\phi+\Lambda
e^{\phi-\mathbf{V}\cdot\mathbf{x}}-\zeta\phi$ $\displaystyle\text{ in }\Omega$
(14) $\displaystyle\zeta\phi$ $\displaystyle=p_{\ast}(|\Omega|)-\gamma\kappa$
$\displaystyle\text{ on }\partial\Omega$ (15) $\displaystyle 0$
$\displaystyle=\partial_{\nu}(\phi-\mathbf{V}\cdot\mathbf{x})$
$\displaystyle\text{ on }\partial\Omega$ (16)

for unknowns $\phi$, $\Omega$, and $\Lambda$, each of which depend on
$\mathbf{V}$. When $\mathbf{V}=0$, we recover the stationary solution (7)-(9),
where $\Omega$ is a disk of radius $R$. Therefore, for $\mathbf{V}\neq 0$, we
take the cell boundary $\partial\Omega$ as the polar graph of the curve
$R+\rho(\theta,\mathbf{V})$ where $\rho(\theta,0)=0$. Choose coordinates such
that $\theta=0$ corresponds to the direction of $\mathbf{V}$ so $\phi$,
$\rho$, and $\Lambda$ depend only on $V=|\mathbf{V}|$. To approximate
solutions to (14)-(16) we take asymptotic expansions of the unknowns as
follows:

$\displaystyle\begin{split}\phi(r,\theta&,V)=\phi_{0}(r,\theta)+\phi_{1}(r,\theta)V+\phi_{2}(r,\theta)V^{2}\\\
&+\phi_{3}(r,\theta)V^{3}+O(V^{4})\end{split}$ (17)
$\displaystyle\rho(\theta,V)$
$\displaystyle=\rho_{1}(\theta)V+\rho_{2}(\theta)V^{2}+\rho_{3}(\theta)V^{3}+O(V^{4})$
(18) $\displaystyle\Lambda(V)$
$\displaystyle=\Lambda_{0}+\Lambda_{1}V+\Lambda_{2}V^{2}+\Lambda_{3}V^{3}+O(V^{4}).$
(19)

Substituting the expansions (17)-(19) into (14)-(16), we obtain coefficients
$\phi_{n}$, $\rho_{n}$ and $\Lambda_{n}$ via an iterative procedure. As part
of this procedure, we take into account specific features of the free
boundary, e.g we transform the free boundary condition (15) for $\phi(R+\rho)$
to a fixed boundary condition for $\phi(R)$ and its derivatives by expanding
$\phi$ about $r=R$. Solving for $\phi_{n}$, $\rho_{n}$, and $\Lambda_{n}$, we
find that $\phi_{0}$ is given by (7), $\phi_{1}$ is the product of an
explicitly known function of $r$ with $\cos\theta$, and $\rho_{1}$ is zero.
For $n\geq 2$, $\phi_{n}$ is the sum of Fourier modes up to $\cos(n\theta)$.
We solve for the Fourier coefficients numerically.

The coefficient $\phi_{1}$ is instrumental for determining at what parameter
values the branch of traveling solutions bifurcates from the family of
stationary state solutions. All of the coefficients $\phi_{n}$ are determined
by solving PDEs in a disk of radius $R$ with two boundary conditions derived
from (15) and (16). For $n\geq 2$, appropriate choices of $\rho_{n}$ and
$\Lambda_{n}$ allow $\phi_{n}$ to satisfy both boundary conditions, but
$\rho_{1}$ and $\Lambda_{1}$ are decided by other considerations111The system
(2)-(6) does not have unique solutions; if $\phi(\mathbf{x},t)$,
$m(\mathbf{x},t)$ and $\Omega(t)$ solve (2)-(6), then so does the translation
$\phi(\mathbf{x}-\mathbf{y},t)$, $m(\mathbf{x}-\mathbf{y},t)$ and
$\Omega(t)+\mathbf{y}$. Therefore, we take $\rho_{1}=0$ to select the
traveling solution centered at the origin. Furthermore, we expect $\Lambda(V)$
to be an even function of $V$: $\Lambda(V)=\Lambda(-V)$, so $\Lambda_{1}=0$.
Therefore the parameters on which $\phi_{1}$ depends must be chosen so that
$\phi_{1}$ meets both boundary conditions. Those parameters are $R$ and the
physical parameters $\zeta$, $\gamma$, $k_{e}$ and $p_{h}$. Specifically, $R$
and the physical parameters must meet the _bifurcation condition_

$F(R,\zeta,\gamma,k_{e},p_{h}):=\frac{\zeta
I_{1}(Rs)}{s^{3}I_{1}^{\prime}(Rs)}-\frac{m_{0}R}{s^{2}}=0,$ (20)

where $s=\sqrt{\zeta-p_{\ast}(\pi R^{2})+\gamma/R}$, and $I_{1}$ is the
modified Bessel function of the first kind with order 1, and $I_{1}^{\prime}$
is its derivative. For each choice of the physical parameters, (20) determines
the radius $R_{0}$ which is the radius of the stationary solution from which
the branch of traveling solutions bifurcates. To visualize this bifurcation,
we can plot the branches of traveling solutions and stationary solutions
together in a bifurcation diagram. The stationary solutions are parameterized
by their total total myosin mass $M$ while the traveling solutions are
parameterized by their speed $V$, so the bifurcation diagram will have $M$ and
$V$ on its axes. The bifurcation occurs when $V=0$ and $M=M_{0}$, the critical
myosin mass obtained by plugging $R_{0}$ into (10). The total myosin mass of
traveling solutions is calculated as

$\displaystyle M(V)$
$\displaystyle=\int_{0}^{2\pi}\int_{0}^{R+\rho(\theta,V)}\Lambda(V)e^{\phi(r,\theta,V)-Vr\cos\theta}r\,dr\,d\theta$
(21) $\displaystyle=m_{0}\pi R^{2}+M_{2}V^{2}+O(V^{4}),$ (22)

where

$\begin{split}M_{2}=2\pi\zeta\int_{0}^{2\pi}&\frac{m_{0}R^{2}}{\gamma-2k_{e}R}\phi_{2}(R,\theta)\,d\theta\\\
&+\int_{0}^{2\pi}\int_{0}^{R}\phi_{2}(r,\theta)r\,dr\,d\theta,\end{split}$
(23)

and $m_{0}$ comes from (8). An example bifurcation diagram plotting $M(V)$
when $M_{2}>0$ is shown in Figure 1. If $M_{2}<0$, the branch of steadily
moving cells opens to the left instead of the right. The sign of $M_{2}$
depends on the parameters $R$, $m_{0}$, $\zeta$, $\gamma$, and $k_{e}$. Figure
2 shows a typical example of how $M_{2}$ depends on the effective bulk
elasticity $k_{e}$. In particular, we note that for any combination of
parameter values $R$, $m_{0}$, $\zeta$, and $\gamma$, there are three special
values of $k_{e}$. The first and smallest of these three is $k_{0}$ at which
$M_{2}=0$. Assuming the higher order terms in (22) are bounded in a
neighborhood of $k_{0}$, there exists a small velocity $V_{0}\neq 0$ such that
$M^{\prime}(V_{0})=0$ when $k_{e}$ is close to $k_{0}$, giving rise to a
_bending point_ $(M(V_{0}),V_{0})$ on the branch of traveling solutions, at
which $M(V)$ changes from increasing to decreasing (or vice-versa). The second
special value of $k_{e}$ is $k_{\ast}$ at which $M_{2}$ is singular,
indicating that the bifurcation is not smooth (not twice differentiable) at
that parameter value. Finally, the third and largest special value of $k_{e}$
is $k_{c}$ which is the smallest value of $k_{e}$ such that the condition (11)
holds, which is a requirement for the stability results below.

Figure 1: Forward bifurcation diagram ($C>0$) showing the total total myosin
mass $M$ and speed $V$ of steadily moving cells. Total myosin mass increases
to the right. Blue corresponds to stability, orange to instability. Figure 2:
The dependence of the $V^{2}$ coefficient $C$ of the total total myosin mass
$M$ of steadily moving cells with velocity $V$ on the bulk elasticity $k_{e}$.

In addition to expansion coefficients for scalar stress $\phi_{n}$, we can
solve for the expansion coefficients $\rho_{n}$ for the bounding curve
$R+\rho$ of the region $\Omega$ modeling a cell traveling with constant
velocity. We find that $\rho_{1}=0$, while we find
$\rho_{2}(\theta)=a+b\cos(2\theta)$ and $\rho_{3}(\theta)=c\cos(3\theta)$ for
$a$, $b$, and $c$ depending on physical parameters and the numerically
calculated $\phi_{2}$ and $\phi_{3}$. Figure 3 shows the cell shape for
various velocities, along with the myosin density $m$ inside the cell. The
cell shape and myosin density agree qualitatively with simulations from a
similar free-boundary model in [8] (see Fig. 1) and a phase-field model in
[14] (See Fig. 3).

Figure 3: The cell shape and myosin density as speed increases from $V=0$ to
$V=0.3$. Motion is to the right. Blue colors indicate higher myosin density.

##  4 Stability of Traveling Solutions

In order to have observable steady motion with constant velocity, we need to
establish stability of the corresponding traveling solutions. However, the
notion of stability of traveling solutions is different from the usual notion
of Lyapunov stability. Namely, the best stability we can hope for is
_stability up to shifts and change in velocity_ , which can be understood as
follows. Let $u_{0}(\mathbf{V})$ and $\tilde{u}_{0}(\mathbf{V})$ be traveling
solutions with the same velocity such that $u_{0}(\mathbf{V})$ is centered at
the origin and $\tilde{u}_{0}(\mathbf{V})$ is centered at $(\varepsilon,0)$
for some small $\varepsilon$, i.e., $\tilde{u}_{0}(\mathbf{V})$ is a _shift_
of $u_{0}(V)$ in the $x$-direction. Since $u_{0}(\mathbf{V})$ and
$\tilde{u}_{0}(\mathbf{V})$ are close, a small perturbation of
$u_{0}(\mathbf{V})$ may become close to $\tilde{u}_{0}(\mathbf{V})$ after a
long time. Therefore, _stability up to shifts_ means that a perturbation of a
traveling solution with velocity $\mathbf{V}$ stabilizes to another traveling
solution with velocity $\mathbf{V}$, but shifted in the $x$ or $y$ direction.
Now consider a traveling solution $u_{0}(\mathbf{V}^{\prime})$ centered at the
origin with velocity $\mathbf{V}^{\prime}-\mathbf{V}=\delta$, $\|\delta\|\ll
1$ at time $t=0$. Then $u_{0}(\mathbf{V})$ and $u_{0}(\mathbf{V}^{\prime})$
have close shapes at $t=0$. After a long time $T\gg 1/\delta$, these solutions
are a distance $T\delta\gg 1$ apart. Therefore, a perturbation which is close
at $t=0$ to both solutions can only be close to one of the two at $t=T$. This
illustrates the concept of _stability up to change in velocity_. Another way
of understanding stability up to change in velocity is to observe that this
concept is equivalent to stability up to two scalar quantities: _rotation
angle_ and _speed_ (e.g., stability up to rotations is analogous to stability
up to shifts since both are coordinate changes). Note that speed is uniquely
determined by the total myosin mass $M$, and vice-versa. Therefore,
conservation of $M$ can be used to control the speed. We can summarize
stability up to shifts and change in velocity in the following way: a
perturbation of the traveling solution $u_{0}(\mathbf{V})$ eventually
stabilizes to another traveling solution $\tilde{u}_{0}(\mathbf{V}^{\prime})$
which is initially close to $u_{0}(\mathbf{V})$.

To establish stability up to shifts and change in velocity, we first write
(2)-(6) as an evolution equation for the myosin density $m$ and polar cell
boundary curve $R+\rho$:

$\frac{\partial}{\partial t}(m,\rho)=F(m,\rho).$ (24)

Here $F$ is a nonlinear operator derived from (3) and (6) (taking $\phi$ as an
auxiliary function determined by (2) and (4)). Next, we find the linear
operators $A_{S}(R)$ and $A_{T}(\mathbf{V})$ which are the linearizations of
$F$ about the stationary solution at $\mathbf{V}=0$ with radius $R$ and the
traveling solution with velocity $\mathbf{V}\neq 0$, respectively (to
linearize about a constant-in-time solution, $A_{T}(\mathbf{V})$ is found in
coordinates moving with velocity $\mathbf{V}$ so the traveling solution
appears stationary). At the bifurcation point, the family of traveling
solutions intersects with the family of stationary solutions, so the operators
are equal: $A_{S}(R_{0})=A_{T}(0)$. The signs of the real parts of the
eigenvalues of $A_{S}(R)$ and $A_{T}(\mathbf{V})$ determine the stability of
the stationary and traveling solutions. For $R\neq R_{0}$, all the eigenvalues
of $A_{S}(R)$ are negative except the zero eigenvalue (multiplicity $3$) and
an eigenvalue $E(R)$ whose sign depends on $R$, see (27). Similarly, for
$\mathbf{V}\neq 0$, all eigenvalues of $A_{T}(\mathbf{V})$ are negative except
the zero eigenvalue (multiplicity $4$), and an eigenvalue $\lambda(V)$ whose
sign depends on $V=|\mathbf{V}|$, and determines stability.

Away from the bifurcation point, each of the eigenvectors of $A_{S}(R)$ (or
$A_{T}(\mathbf{V})$) are a derivative of the stationary (or traveling)
solution in a parameter (e.g., $R$) with respect to which the class of
stationary (or traveling) solutions is invariant. That is, changing this
parameter in a stationary (or traveling) solution still results in a
stationary (or traveling) solution. For $R\neq R_{0}$ and $\mathbf{V}\neq 0$,
both $A_{S}(R)$ and $A_{T}(\mathbf{V})$ have two eigenvectors for the zero
eigenvalue corresponding to invariance with respect to translations in the $x$
and $y$ directions. Additionally, $A_{S}(R)$ has another eigenvector
corresponding to invariance with respect to a change in the radius $R$. For
$A_{T}(\mathbf{V})$, the other eigenvectors for the zero eigenvalue are
_generalized_ eigenvectors because their corresponding parameters change the
velocity $\mathbf{V}$ of a traveling solution–a traveling solution with
velocity $\mathbf{V}^{\prime}\neq\mathbf{V}$ is not constant-in-time in the
coordinates that move with velocity $\mathbf{V}$. These two generalized
eigenvectors correspond to changing the speed $V=|\mathbf{V}|$ of traveling
solutions and changing the direction of motion.

Since the eigenvectors of $A_{T}(\mathbf{V})$ corresponding to shifts both
have eigenvalue zero, any linear combination of these two is also an
eigenvector. Denote by $a(\mathbf{V})$ such a linear combination that has unit
length and corresponds to shifts in the direction of $\mathbf{V}$. Denote by
$b(\mathbf{V})$ the generalized eigenvector corresponding to invariance in
speed $V$ so that $A_{T}(\mathbf{V})b(\mathbf{V})=a(\mathbf{V})$. These two
eigenvectors are of particular interest because of their relationship with
$\lambda(V)$, the eigenvalue of $A_{T}(\mathbf{V})$ which is the deciding
factor in the stability of traveling solutions since it is the one eigenvalue
which may be positive. (By the rotational symmetry of $A_{T}(\mathbf{V})$, its
eigenvalues depend only on speed $V$, not the direction of $\mathbf{V}$.)
Specifically, we will see that the eigenvector $c(\mathbf{V})$ for
$\lambda(V)$ becomes parallel to $a(\mathbf{V})$ as $V\to 0$. This fact
emphasizes the non-self-adjoint nature of $A_{T}(V)$. Indeed, in the usual
self-adjoint case, eigenvectors are orthogonal to one another, so the fact
that $a(\mathbf{V})$ and $c(\mathbf{V})$ are asymptotically parallel is
surprising. Since $a(0)$ is parallel to $c(0)$, we conclude that their
eigenvalues are also equal: $\lambda(0)=0$. Next, we expect $\lambda(V)$ to
depend only on the cell’s speed, not its direction, so
$\lambda(V)=\lambda(-V)$ and thus $\lambda^{\prime}(0)=0$. Therefore, we have
the asymptotic expansion $\lambda(V)=\lambda_{2}V^{2}+O(V^{4})$. Thus, for
small $V$, the sign of $\lambda(V)$ is the same as the sign of $\lambda_{2}$,
and the question of stability hinges on this coefficient.

Both $a(\mathbf{V})$ and $b(\mathbf{V})$ are known explicitly in terms of
traveling solution approximated in the previous section, but $\lambda_{2}$ is
not, and neither is $c(\mathbf{V})$. Therefore, to find $\lambda_{2}$, we use
an ansatz for $c(\mathbf{V})$ derived from its relation to $a(\mathbf{V})$ and
$b(\mathbf{V})$:

$c(\mathbf{V})=a(\mathbf{V})+\lambda_{2}V^{2}b(\mathbf{V})+O(V^{3}).$ (25)

Plugging (25) into
$A_{T}(\mathbf{V})c(\mathbf{V})=\lambda(\mathbf{V})c(\mathbf{V})$ and
comparing terms of like power in $V$, we can solve for $\lambda_{2}$, though
doing so requires solving equations up to fifth order in $V$. We discover that
$\lambda_{2}$ bears an interesting and enlightening relationship with the
eigenvalue $E(R)$ of $A_{S}(R)$ and the total myosin mass $M(V)$ of traveling
solutions with velocity $V$:

$\lambda_{2}=-\frac{dE}{dR}\Big{|}_{R=R_{0}}\frac{dR}{dM}\Big{|}_{M=M(0)}\frac{d^{2}M}{dV^{2}}\Big{|}_{V=0}.$
(26)

Therefore, the question of stability comes down to the signs of the three
derivatives in (26). First $dR/dM$ can be calculated explicitly from (10) and
is always negative when (11) is met. Next, we observe $\partial^{2}M/\partial
V^{2}|_{V=0}=2M_{2}$, so its sign can be seen in Figure 2. In particular,
$\partial^{2}M/\partial V^{2}|_{V=0}>0$ when condition (11) is met. Finally,
$dE/dR$ is also explicitly known:

$\frac{dE}{dR}=-C\frac{\partial F}{\partial R},\;\;E(R_{0})=0$ (27)

where $C>0$ depends on $R$ and the physical constants, and $F$ is given
explicitly by (20). Observe that $E(R)$ is the largest nonzero eigenvalue for
the stationary solutions and it describes their _moveability_ in the following
sense. If $\text{Re}\,E(R)>0$, then stationary solutions “want to move.”
Otherwise, they do not. Figure 2 shows how $\partial F/\partial R|_{R=R_{0}}$
depends on $k_{e}$. We see that $\partial F/\partial R|_{R=R_{0}}=0$ precisely
when $k_{e}=k_{\ast}$, when $M_{2}$ has a singularity. When condition (11) is
met ($k_{e}>k_{c}>k_{\ast}$), $\partial F/\partial R|_{R=R_{0}}>0$, and thus,
$\partial E/\partial R|_{R=R_{0}}<0$. Of the three derivatives in (26), two of
them are negative, so $\lambda_{2}<0$.

Since $\lambda_{2}<0$, we conclude that all eigenvalues of $A_{T}(\mathbf{V})$
have negative real part except the zero eigenvalue. The eigenvectors for the
zero eigenvalue correspond to shifts in the $x$ and $y$ direction and changes
to a traveling solution’s velocity. Therefore, our analysis suggests that
traveling solutions are stable up to shifts and change in velocity, as
explained above. Although the velocity of traveling solutions to which our
perturbed solution asymptotically stabilizes is not known, its speed is
completely determined by the total myosin mass $M$, which is conserved in
time. In other words, the asymptotic velocity of a perturbed traveling
solution is determined up to a rotation.

Conclusions. We proposed a 2D free boundary model for cell motility which has
traveling solutions. We performed linear stability analysis of the traveling
solutions. Linearizing about the traveling solution of velocity $\mathbf{V}$
presented us with two challenges: the linearized problem is not self-adjoint
and it has zero eigenvalue of multiplicity four with both true and generalized
eigenvectors. The four eigenvectors corresponding to the zero eigenvalues
correspond to shifts in the $x$ and $y$ direction and to changes in speed and
rotation angle of traveling solutions. There is only one eigenvalue,
$\lambda(V)$, which may have positive real part. This led us to introducing a
notion of stability _up to shifts and rotation angle (translation and rotation
of coordinates)_ , since usual Lyapunov stability does not apply. Moreover,
the non-self-adjoint nature of the linearized problem manifests in the
eigenvector $c(\mathbf{V})$ for $\lambda(V)$ becoming asymptotically parallel
to one of the shift eigenvectors. This leads to our specific choice of ansatz
for $c(\mathbf{V})$, employing a shift eigenvector and its corresponding
generalized eigenvector as coefficients in an asymptotic expansion. This
ansatz yields an explicit asymptotic formula for $\lambda(V)$ in terms of
physical quantities which can be calculated numerically. We found numerically
that for most physical parameters, $\text{Re}\,\lambda(V)<0$ for small $V$.
Since one can show that all other nonzero eigenvalues have negative real part,
and the eigenvectors corresponding to zero correspond to shifts, rotations,
and changes in speed, we establish stability of traveling solutions up to
translations and rotations of the coordinate system.

Acknowledgemennts. The work of L. Berlyand and C. A. Safsten was partially
supported by NSF grant DMS-2005262. V. Rybalko is grateful to PSU Center for
Mathematics of Living and Mimetic Matter, and to PSU Center for
Interdisciplinary Mathematics for support of his two stays at Penn State. His
travel was also supported by NSF grant DMS-1405769. We thank our colleagues I.
Aronson, J. Casademunt, J.-F. Joanny, N. Meunier, A. Mogilner, J. Prost, R.
Alert, and L. Truskinovsky for many useful discussions on our results and
suggestions on the model. We also express our gratitude to the members of the
L. Berlyand’s PSU research team R. Creese and M. Potomkin for discussions.

## References

  * [1] S. Alt, P. Ganguly, G. Salbreux. _Vertex models: from cell mechanics to tissue morphogenesis_ , Philos Trans R Soc Lond B Biol Sci. 2017;372(1720):20150520. doi:10.1098/rstb.2015.0520
  * [2] E. Barnhart, K.-C. Lee, G. M. Allen, J. A. Theriot, A. Mogilner. _Balance between cell-substrate adhesion and myosin contraction determines the frequency of motility initiation in fish keratocytes_. Proc Natl Acad Sci USA (2015) 112(16):5045-50.
  * [3] L. Berlyand, J. Fuhrmann, V. Rybalko. _Bifurcation of traveling waves in a Keller-Segel type free boundary model of cell motility_ , Commun. Math. Sci. 16 (2018), No. 3, 735–762.
  * [4] L. Berlyand, V. Rybalko _Stability of steady states and bifurcation to traveling waves in a free boundary model of cell motility_ , arXiv:1905.03667 [math.AP] (May 2019).
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  * [6] A. C. Callan-Jones, J.-F. Joanny, J. Prost. _Viscous-Fingering-Like Instability of Cell Fragments_ , Phys. Rev. Lett., 100 (2008), 258106-4.
  * [7] C. Etchegaray , N. Meunier, R. Voituriez. _Analysis of a Nonlocal and Nonlinear Fokker-Planck Model for Cell Crawling Migration_. SIAM J. Appl. Math., 77(6), 2017, 2040-2065.
  * [8] K. Keren, Z. Pincus, G. M. Allen, E. L. Barnhart, G. Marriott, A. Mogilner, J. A. Theriot, _Mechanism of shape determination in motile cells_ , Nature, (2008) 453 (7194), 475–480,.
  * [9] B. Perthame, P. E. Souganidis. _Asymmetric potentials and motor effect: a homogenization approach_. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, (2009), 26-6 2055-2071
  * [10] J. Prost, F. Jülicher, J.-F. Joanny. _Active gel physics_ , Nature Physics 11 (2015)
  * [11] T. Putelat, P. Recho, L. Truskinovsky. _Mechanical stress as a regulator of cell motility_ , Phys. Rev. E, 97(1), 012410, 2018.
  * [12] P. Recho, T. Putelat, L. Truskinovsky, _Contraction-Driven Cell Motility_ , _Phys. Rev. Lett._ (2013) 111
  * [13] P. Recho, T. Putelat, and L. Truskinovsky. _Mechanics of motility initiation and motility arrest in crawling cells_. Journal of Mechanics Physics of Solids, (2015) 84:469505.
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# Information algebra system of soft sets

Guan Xuechong<EMAIL_ADDRESS>Li Yongming<EMAIL_ADDRESS>College of
Mathematical Science, Xuzhou Normal University, Xuzhou, 221116, China  College
of Computer Science, Shaanxi Normal University, Xi’an, 710062, China

###### Abstract

Information algebra is algebraic structure for local computation and
inference. Given an initial universe set and a parameter set, we show that a
soft set system over them is an information algebra. Moreover, in a soft set
system, the family of all soft sets with a finite parameter subset can form a
compact information algebra.

###### keywords:

soft set , complete lattice , information algebra , compact information
algebra.

††journal:

##  1 Introduction

The information algebra system introduced by Shenoy [1] was inspired by the
formulation of some basic axioms of local computation and inference under
uncertainty [2]. It gives a basic mathematical model for treating
uncertainties in information. Related studies [3, 4, 5] showed that the
framework of information algebras covers many instances from constraint
systems, Bayesian networks, Dempster-Shafer belief functions to relational
algebra, logic and etc. Considering about the feasibility of information
processing with computer, Kohlas [3, 4] presented a special information
algebra with approximation structure called compact information algebra
recently.

On the other hand, Molodtsov [6] initiated a novel concept, which is called
soft set, as a new mathematical tool for dealing with uncertainties [7]. In
fact, a soft set is a parameterized family of subsets of a given universe set.
The way of parameterization in problem solving makes soft set theory
convenient and simple for application. Now it has been applied in several
directions, such as operations research [8, 9], topology [10, 11, 12],
universal algebra [13, 14, 15, 16], especially decision-making [17, 18, 19,
20, 21].

It is thus evident that information algebra theory and soft set theory are
both theoretical research tools for dealing with non-deterministic phenomenon.
To study relationships between them is necessary. In this paper, we are
concerned about the problem that whether there exist the frameworks of
information algebras or even compact information algebras in soft sets. By
choosing some appropriate operators, we construct an information algebra
system of soft sets over an initial universe set and a parameter set. Then we
further prove that, in a soft set system, the family of soft sets with a
finite parameter subset can form a compact information algebra. These
conclusions obtained in this paper demonstrate that soft set systems are also
the instances of information algebras.

##  2 Preliminaries

In this section, first, we present some basic definitions about soft sets and
some notations in lattice theory.

Suppose that $(L,\leq)$ is a partially ordered set and $A\subseteq L$. We
write $\vee A$ and $\wedge A$ for the least upper bound and the greatest lower
bound of $A$ in $L$ respectively if they exist.

Let $L$ be a partially ordered set. If $a\vee b$ and $a\wedge b$ exist for all
$a,b\in L$, then we call $L$ a lattice. If $\vee A$ exists for every subset
$A\subseteq L$, we call $L$ a complete lattice. Clearly, a partially ordered
set $L$ is a complete lattice if, and only if, $L$ has the bottom element and
$\vee A$ exists for all nonempty subset $A\subseteq L$.

A set $A\subseteq L$ is said to be directed, if for all $a,b\in A$, there is a
$c\in A$ such that $a,b\leq c$. For $a,b\in L$, we call $a$ way-below $b$, in
symbols $a\ll b$, if and only if for all directed subsets $X\subseteq L$, if
$\vee X$exists and $b\leq\vee X$, then there exists an $x\in X$ such that
$a\leq x$.

Let $U$ be an initial universe set and $E$ be a set of parameters, which
usually are initial attributes, characteristics, or properties of objects in
the initial universe set. ${\cal P}(U)$ denotes the power set of $U$.

###### Definition 2.1

([6]) A pair $(F,A)$ is called a soft set over $U$, where $F$ is a mapping
given by $F:A\rightarrow{\cal P}(U)$.

Therefore a soft set is a tuple which associates with a set of parameters and
a mapping from the parameter set into the power set of an universe set. In
other words, a soft set over $U$ is a parameterized family of subsets of the
universe $U$. For $\varepsilon\in A$, $F(\varepsilon)$ may be considered as
the set of $\varepsilon$-approximate elements of the soft set $(F,A)$[6].

###### Definition 2.2

([7]) A soft set $(F,A)$ over $U$ is said to be a null soft set, if for all
$e\in A,F(e)=\emptyset$. We write it by $(\emptyset,A)$.

A soft set $(F,A)$ over $U$ is said to be an absolute soft set denoted by
$\tilde{A}$, if for all $e\in A,F(e)=U$.

###### Definition 2.3

([8]) The extended intersection of two soft sets $(F,A)$ and $(G,B)$ over a
common universe $U$ is the soft set $(H,C)$, where $C=A\cup B$, and $\forall
e\in C$,

$H(e)=\left\\{\begin{array}[]{ll}F(e),&{\mbox{if}}\ \ e\in A-B;\\\
G(e),&{\mbox{if}}\ \ e\in B-A;\\\ F(e)\cap G(e),&{\mbox{if}}\ \ e\in A\cap
B.\end{array}\right.$

We write $(F,A)\sqcap_{\varepsilon}(G,B)=(H,C)$.

In this paper we adopt the concept of information algebra given by Kohlas from
[3]. For a full introduction and these abundant examples of information
algebras, please refer to [3, 4, 5].

###### Definition 2.4

([3]) Let $(D,\leq)$ be a lattice. Suppose there are three operations defined
in the tuple $(\Phi,D)$:

1.Labeling $d$: $\Phi\rightarrow D;\phi\mapsto d(\phi)$, where $d(\phi)$ is
called the domain of $\phi$. For an $s\in D$, let $\Phi_{s}$ denote the set of
all valuations with domain $s$.

2.Combination $\otimes$:
$\Phi\times\Phi\rightarrow\Phi;(\phi,\psi)\mapsto\phi\otimes\psi$,

3.Marginalization $\downarrow$: $\Phi\times
D\rightarrow\Phi;(\phi,x)\mapsto\phi^{\downarrow x}$, for $x\leq d(\phi)$.

If the system $(\Phi,D)$ satisfies the following axioms, it is called an
information algebra:

1.Semigroup: $\Phi$ is associative and commutative under combination. For all
$s\in D$, there is a neutral element $e_{s}$ with $d(e_{s})=s$ such that for
all $\phi\in\Phi$ with $d(\phi)=s,e_{s}\otimes\phi=\phi$.

2\. Labeling: For $\phi,\psi\in\Phi$, $d(\phi\otimes\psi)=d(\phi)\vee
d(\psi)$.

3\. Marginalization: For $\phi\in\Phi,x\in D,x\leq d(\phi),d(\phi^{\downarrow
x})=x$.

4\. Transitivity: For $\phi\in\Phi$ and $x\leq y\leq d(\phi),(\phi^{\downarrow
y})^{\downarrow x}=\phi^{\downarrow x}$

5\. Combination: For $\phi,\psi\in\Phi$ with
$d(\phi)=x,d(\psi)=y,(\phi\otimes\psi)^{\downarrow
x}=\phi\otimes\psi^{\downarrow x\wedge y}$.

6\. Stability: For $x,y\in D,x\leq y$, $e_{y}^{\downarrow x}=e_{x}$.

7\. Idempotency: For $\phi\in\Phi$ and $x\in D,x\leq d(\phi)$,
$\phi\otimes\phi^{\downarrow x}=\phi$.

The items putting forward in the definition of information algebra can be seen
as the axiomatic presentations of some basic principles in local computation
and inference. Studies have shown this algebraic structure covers many
instances from belief functions, constraint systems, relational databases, and
possibility theory to relational algebra and logic([5]). For example, each
lattice $L$ is a simply information algebra on a domain set $L$ itself. The
operations are defined as follows:

1\. Labeling $d$: For $x\in L$, $d(x)=x$.

2\. Combination $\otimes$: $x\otimes y=x\vee y$.

3\. Projection $\downarrow$: If $x\leq y$, $x^{\downarrow y}=x\wedge y$.

For an information algebra $(\Phi,D)$, we introduce a order relation as
follows:

$\psi\leq\phi$, if $\psi\otimes\phi=\phi$.

This order relation induced by the operation combination is a partial order on
the set $\Phi$, if $(\Phi,D)$ is an information algebra.

##  3 Information algebra of soft sets

In this section, with these operations of soft sets defined above, we will
construct an information algebra of soft sets. Let $U$ be an initial universe
set and $E$ be a set of parameters. ${\cal S}_{U,E}$(or simply ${\cal S}$ when
this doesn’t lead to confusions) denotes the set of all soft sets $(F,A)$ over
$U$, where $A\subseteq E$, that is,

${\cal S}$=$\\{(F,A):(F,A)\mbox{\ is\ a\ soft\ set\ over\ $U$,\ where
$A\subseteq E$}\\}.$

Three operations are defined as follows:

1\. Labeling $d$: For a soft set $(F,A)$, we define $d((F,A))=A$.

2\. Projection $\downarrow$: If $B\subseteq A$, we define $(F,A)^{\downarrow
B}$ to be a soft set $(G,B)$ such that for all $b\in B$, $G(b)=F(b)$.

3\. Combination $\otimes$: For any two soft sets $(F,A),(G,B)\in{\cal S}$, we
define

$(F,A)\otimes(G,B)=(F,A)\sqcap_{\varepsilon}(G,B).$

We call a quintuple $({\cal S},{\cal
P}(E),d,\sqcap_{\varepsilon},\downarrow)$(abbreviated as $({\cal S},{\cal
P}(E))$ a soft set system over $U$ and $E$. Now we show this system is an
information algebra.

###### Theorem 3.1

The soft set system $({\cal S},{\cal P}(E))$ over $U$ and $E$ is an
information algebra.

Proof. Obviously, ${\cal P}(E)$ is a lattice composed by the domains of soft
sets in ${\cal S}$.

1\. Semigroup: Clearly ${\cal S}$ is commutative with respect to the operation
$\sqcap_{\varepsilon}$. For $A\subseteq E$, the absolute soft set $\tilde{A}$
is the neutral element such that $(F,A)\sqcap_{\varepsilon}\tilde{A}=(F,A)$
for all soft set $(F,A)$ with domain $A$.

Following we show the associative law holds in the set ${\cal S}$. Let
$(F,A),(G,B),(H,C)\in{\cal S}$. We write

$(F,A)\sqcap_{\varepsilon}(G,B)=(Q_{1},A\cup B)$,

$(G,B)\sqcap_{\varepsilon}(H,C)=(Q_{2},B\cup C)$,

$[(F,A)\sqcap_{\varepsilon}(G,B)]\sqcap_{\varepsilon}(H,C)=(Q_{3},A\cup B\cup
C)$,

$(F,A)\sqcap_{\varepsilon}[(G,B)\sqcap_{\varepsilon}(H,C)]=(Q_{4},A\cup B\cup
C)$.

We need to show that $Q_{3}=Q_{4}$. For any an $e\in A\cup B\cup C$, it can be
divided into seven conditions as follows: $e\in(A-B)-C,e\in(B-A)-C,e\in(A\cap
B)-C,e\in C-(A\cup B),e\in(A-B)\cap C,e\in(B-A)\cap C$ and $e\in A\cap B\cap
C$. Here we take the condition of $e\in(A\cap B)-C$ as an example to
illuminate the proof. Assume that $e\in(A\cap B)-C$. Since $(A\cap
B)-C=A\cap(B-C)$, we have $e\in A\cap(B-C)$. Then

$Q_{3}(e)=Q_{1}(e)=F(e)\cap G(e),$

and

$Q_{4}(e)=F(e)\cap Q_{2}(e)=F(e)\cap G(e).$

So $Q_{3}(e)=Q_{4}(e)$. The other conditions are also easy to show. Therefore,
the associative law holds.

2\. According to these related definitions, the proof of the axioms of
labeling, marginalization, transitivity and idempotency are directly.

3\. Stability: For $A\in{\cal P}(E)$, the neutral element with domain $A$ is
the absolute soft set $\tilde{A}$. Furthermore, if $B\subseteq A$, we have
$\tilde{A}^{\downarrow B}=\tilde{B}$. Thus the stability is true.

4\. Combination: For $(F,A),(G,B)\in{\cal S}$, we need to show

$((F,A)\sqcap_{\varepsilon}(G,B))^{\downarrow
S}=(F,A)\sqcap_{\varepsilon}(G,B)^{\downarrow{S\cap B}},$

if $A\subseteq S\subseteq A\cup B$.

In fact, let

$(F,A)\sqcap_{\varepsilon}(G,B)=(H,A\cup B),$
$(F,A)\sqcap_{\varepsilon}(G,B)^{\downarrow{S\cap B}}=(H^{{}^{\prime}},S).$

For all $e\in S$, we have

$H(e)=H^{{}^{\prime}}(e)=\left\\{\begin{array}[]{ll}F(e),&{\mbox{if}}\ \ e\in
S\cap(A-B);\\\ G(e),&{\mbox{if}}\ \ e\in S\cap(B-A);\\\ F(e)\cap
G(e),&{\mbox{if}}\ \ e\in A\cap B.\end{array}\right.$

Then $((F,A)\sqcap_{\varepsilon}(G,B))^{\downarrow
S}=(F,A)\sqcap_{\varepsilon}(G,B)^{\downarrow{S\cap B}}$.

Hence $({\cal S},{\cal P}(E),d,\sqcap_{\varepsilon},\downarrow)$ is an
information algebra.

##  4 Compact information algebra of soft sets

In general, only “finite” information can be treated in computers. Therefore,
a structure called compact information algebra has been proposed by Kohlas.
Its main character is that each information can be approximated by these
“finite” information with a same domain.

###### Definition 4.1

([4]) A system $(\Phi,\Phi_{f},D)$, where $(\Phi,D)$ is an information
algebra, the lattice $D$ has a top element,

$\Phi_{f}=\mathop{\bigcup}\limits_{x\in D}\Phi_{f,x}$

where the sets $\Phi_{f,x}\subseteq\Phi_{x}$ are closed under combination,
contain the neutral element $e_{x}\in\Phi_{f,x}$, and satisfy the following
axioms of convergence and density with respect to the ordering relation $\leq$
induced by the operation combination, is called a compact information algebra.

1\. Convergency: If $X\subseteq\Phi_{f,x}$ is directed, then the supremum
$\vee X$ over $\Phi$ exists and $\vee X\in\Phi_{x}$.

2\. Density: For all $\phi\in\Phi_{x}$,

$\phi=\bigvee\\{\psi\in\Phi_{f,x}:\psi\leq\phi\\}.$

3\. Compactness: If $X\subseteq\Phi_{f,x}$ is a directed set, and
$\phi\in\Phi_{f,x}$ such that $\phi\leq\vee X$ then there exists a $\psi\in X$
such that $\phi\leq\psi$.

###### Lemma 4.1

([4]) If $(\Phi,\Phi_{f},D)$ is a compact information algebra, then
$\phi\in\Phi_{f,x}$ if, and only if $\phi\ll\phi$ in set $\Phi_{x}$.

For convenience, we give an equivalent form for the order relation $\leq$
induced by the operation combination of soft sets.

###### Proposition 4.1

Let the order relation $\leq$ be induced by the operation combination in the
system $({\cal S},{\cal P}(E))$. For two soft sets $(F,A)$ and $(G,B)$ over a
common universe $U$, then $(F,A)\leq(G,B)$ if and only if,

(i) $A\subseteq B$, and

(ii) $\forall\varepsilon\in A$, $G(\varepsilon)\subseteq F(\varepsilon)$.

Proof. We write $(F,A)\sqcap_{\varepsilon}(G,B)=(H,A\cup B)$.

If $(F,A)\leq(G,B)$, then $(H,A\cup B)=(G,B)$. So $A\cup B=B$, that is,
$A\subseteq B$. For all $\varepsilon\in A$, by the definition of the operation
$\sqcap_{\varepsilon}$, we have $H(\varepsilon)=F(\varepsilon)\cap
G(\varepsilon)=G(\varepsilon)$. Then $G(\varepsilon)\subseteq F(\varepsilon)$
for all $\varepsilon\in A$.

The reverse is also obvious.

###### Proposition 4.2

Let $\\{(F_{i},A_{i}):i\in I\\}$ be soft sets over a same universe $U$. Then

$\mathop{\bigvee}\limits_{i\in
I}(F_{i},A_{i})=(H,\mathop{\bigcup}\limits_{i\in I}A_{i}),$

where $H:\mathop{\bigcup}\limits_{i\in I}A_{i}\rightarrow{\cal P}(U)$ is
defined as follows:

$\forall e\in\mathop{\bigcup}\limits_{i\in I}A_{i}$, let $J^{(e)}=\\{i\in
I:e\in A_{i}\\}$, $H(e)=\mathop{\bigcap}\limits_{i\in J^{(e)}}F_{i}(e)$.

Proof. Clearly $(H,\mathop{\bigcup}\limits_{i\in I}A_{i})$ is an upper bound
of $\\{(F_{i},A_{i}):i\in I\\}$. Suppose that $(G,B)$ is another upper bound
of $\\{(F_{i},A_{i}):i\in I\\}$. Thus $\mathop{\bigcup}\limits_{i\in
I}A_{i}\subseteq B$. $\forall e\in\mathop{\bigcup}\limits_{i\in I}A_{i},i\in
J^{(e)}$, we have $G(e)\subseteq F_{i}(e)$. Then

$G(e)\subseteq\mathop{\bigcap}\limits_{i\in J^{(e)}}F_{i}(e)=H(e).$

This proves that $(H,\mathop{\bigcup}\limits_{i\in I}A_{i})\leq(G,B)$. Thus
$\mathop{\bigvee}\limits_{i\in
I}(F_{i},A_{i})=(H,\mathop{\bigcup}\limits_{i\in I}A_{i})$.

###### Proposition 4.3

$({\cal S}_{A},\leq)$ is a complete lattice. The top element is
$(\emptyset,A)$, and the bottom element is $\tilde{A}$. Here ${\cal S}_{A}$ is
the set of all soft sets with domain $A$ in the system $({\cal S},{\cal
P}(E))$.

Proof. For all nonempty subset $\\{(F_{i},A):i\in I\\}\subseteq{\cal S}_{A}$,
by the conclusion of Proposition 4.2, we have

$\mathop{\bigvee}\limits_{i\in I}(F_{i},A)=(H,A)\in{\cal S}_{A},$

where $H:A\rightarrow{\cal P}(U)$ is defined as
$H(e)=\mathop{\bigcap}\limits_{i\in I}F_{i}(e)$ for all $e\in A$. Moreover,
$\tilde{A}$ is the bottom element in the set ${\cal S}_{A}$. Thus ${\cal
S}_{A}$ is a complete lattice.

###### Lemma 4.2

Let $(F,A)$ be a soft set over a universe $U$ and $A$ be a finite subset of
$E$. Then $(F,A)\ll(F,A)$ in ${\cal S}_{A}$ if, and only if $U-F(e)$ is a
finite subset of $U$ for all $e\in A$.

Proof. (1) “if”: Let $\\{(G_{i},A):i\in I\\}$ be a directed set and
$(F,A)\leq\bigvee\limits_{i\in I}(G_{i},A)$. We write $\bigvee\limits_{i\in
I}(G_{i},A)=(G,A)$. $\forall e\in A$, we have $G(e)=\bigcap\limits_{i\in
I}G_{i}(e)\subseteq F(e)$. Then

$U-F(e)\subseteq U-G(e)=\bigcup\limits_{i\in I}(U-G_{i}(e)).$

For any an $x\in U-F(e)$, there is an $i(x)\in I$ such that $x\in
U-G_{i(x)}(e)$. Now we get a finite set $\\{(G_{i(x)},A):x\in U-F(e)\\}$,
because $U-F(e)$ is finite. By the directness of $\\{(G_{i},A):i\in I\\}$,
there exists an $i^{(e)}\in I$ such that $(G_{i(x)},A)\leq(G_{i^{(e)}},A)$ for
all $x\in U-F(e)$. Thus $x\in U-G_{i(x)}(e)\subseteq U-G_{i^{(e)}}(e)$. We
obtain $U-F(e)\subseteq U-G_{i^{(e)}}(e)$, that is, $G_{i^{(e)}}(e)\subseteq
F(e)$.

Since $A$ is a finite set, it implies that $\\{(G_{i^{(e)}},A):e\in A\\}$ is
also finite. By the directness of $\\{(G_{i},A):i\in I\\}$ again, there exists
a $j\in I$ such that $(G_{i^{(e)}},A)\leq(G_{j},A)$ for all $e\in A$. We have
$G_{j}(e)\subseteq G_{i^{(e)}}(e)\subseteq F(e)$ for all $e\in A$. This
implies $(F,A)\leq(G_{j},A)$. Thus $(F,A)\ll(F,A)$ in ${\cal S}_{A}$.

(2) “only if”: For all $e\in A$, $U-F(e)$ can be represented as the supremum
of $\\{B_{i}:i\in I\\}$, where $\\{B_{i}:i\in I\\}$ is a directed family of
all the finite subsets of $U-F(e)$, i.e., $U-F(e)=\bigcup\limits_{i\in
I}B_{i}$. We define a family of soft sets $(H_{i},A)$ as follows:

$H_{i}(\varepsilon)=\left\\{\begin{array}[]{ll}U-B_{i},&{\mbox{if}}\ \
\varepsilon=e;\\\ F(\varepsilon),&{\mbox{otherwise}}.\end{array}\right.$

With respect to the order relation $\leq$, $\\{(H_{i},A):i\in I\\}$ is a
directed subsets of ${\cal S}_{A}$. Also we have $(F,A)=\bigvee\limits_{i\in
I}(H_{i},A)$ by Proposition 4.2. Since $(F,A)\ll(F,A)$ in ${\cal S}_{A}$,
there exists a $k\in I$ such that $(F,A)\leq(H_{k},A)$. Hence $U-F(e)\subseteq
U-H_{k}(e)=B_{k}$. Thus $U-F(e)$ is a finite subset of $U$. This proves what
we have stated.

Let ${\cal S}_{\cal F}\subseteq{\cal S}$ denote the set of all soft sets with
a finite subset of $E$, i.e.,

${\cal S}_{\cal F}=\\{(F,A):(F,A)\mbox{\ is\ a\ soft\ set\ over\
$U$},\mbox{where\ $A$\ is\ a\ finite\ subset\ of\ $E$ }\\}.$

The symbol ${\cal P}_{f}(E)$ denotes the set of all finite subsets of $E$.

Let ${\cal S}_{f,A}=\\{(F,A):\forall e\in A,U-F(e)\mbox{\ is\ a\ finite\
subset\ of\ }U\\}$. We denote

${\cal S}_{f}=\mathop{\bigcup}\limits_{A\in{\cal P}_{f}(E)}{\cal S}_{f,A}.$

###### Theorem 4.1

$({\cal S}_{\cal F},{\cal S}_{f},{\cal P}_{f}(E))$ is a compact information
algebra.

Proof. First, we have $({\cal S}_{\cal F},{\cal P}_{f}(E))$ is an information
algebra. It is similar as the proof of Theorem 3.1. By Proposition 4.3, we
know $({\cal S}_{A},\leq)$ is a complete lattice for all finite subset $A$ of
$E$. Hence the convergency in Definition 4.1 is also true.

By the definition of way-below relation $\ll$ and the conclusion of Lemma 4.2,
the compactness is also clear.

Now we need to show the following equation holds for all finite subset
$A\subseteq E$,

$(F,A)=\vee\\{(G,A)\in{\cal S}_{f,A}:(G,A)\leq(F,A)\\}.$

For all $e\in A$, $U-F(e)$ can be represented as the supremum of
$\\{B_{i}:i\in I^{(e)}\\}$, i.e., $U-F(e)=\bigcup\limits_{i\in I^{(e)}}B_{i}$,
where $\\{B_{i}:i\in I^{(e)}\\}$ is a directed family of all the finite
subsets of $U-F(e)$. We define a family of soft sets $(F_{i},A)$ as follows:

$F_{i}(\varepsilon)=\left\\{\begin{array}[]{ll}U-B_{i},&{\mbox{if}}\ \
\varepsilon=e;\\\ U,&{\mbox{otherwise}}.\end{array}\right.$

Let ${\cal B}=\\{(F_{i},A):i\in I^{(e)},e\in A\\}$. Clearly ${\cal
B}\subseteq{\cal S}_{f,A}$. Meanwhile, for all $(F_{i},A)\in{\cal B}$, we have
$(F_{i},A)\leq(F,A)$. In fact, for all $e\in A$, if $i\in I^{(e)}$, we have
$F(e)=U-\bigcup\limits_{i\in I^{(e)}}B_{i}\subseteq U-B_{i}=F_{i}(e)$.
Otherwise, $F_{i}(e)=U$. Thus $F(e)\subseteq F_{i}(e)$ is true. So
$(F_{i},A)\leq(F,A)$. We write $\bigvee\limits_{(F_{i},A)\in{\cal
B}}(F_{i},A)=(H,A)$. For all $d\in A$, we have

$\displaystyle\begin{array}[]{lll}H(d)&=&\bigcap\limits_{(F_{i},A)\in{\cal
B}}F_{i}(d)\\\ &=&(\bigcap\limits_{i\in
I^{(d)}}F_{i}(d))\cap(\bigcap\limits_{i\in I^{(e)},e\in A,e\not=d}F_{i}(d))\\\
&=&(\bigcap\limits_{i\in I^{(d)}}F_{i}(d))\cap U\\\ &=&\bigcap\limits_{i\in
I^{(d)}}(U-B_{i})\\\ &=&U-\bigcup\limits_{i\in I^{(d)}}B_{i}\\\
&=&F(d).\end{array}$

So we have $F=H$. Therefore,

$\displaystyle\begin{array}[]{lll}(F,A)&=&(H,A)\\\
&=&\bigvee\limits_{(F_{i},A)\in{\cal B}}(F_{i},A)\\\
&\leq&\vee\\{(G,A)\in{\cal S}_{f,A}:(G,A)\leq(F,A)\\}\\\
&\leq&(F,A).\end{array}$

Hence

$(F,A)=\vee\\{(G,A)\in{\cal S}_{f,A}:(G,A)\leq(F,A)\\}.$

According to the proof above, we obtain that $({\cal S}_{\cal F},{\cal
S}_{f},{\cal P}_{f}(E))$ is a compact information algebra.

##  5 Conclusion

In this paper, by defining the operations combination and projection of soft
sets, we obtained the structure of information algebras on the family of all
soft sets over an initial universe set and a parameter set. Therefore, a soft
set system can be subsumed under the specific instances of information algebra
systems. We also gave a model of compact information algebra in a soft set
system. We have shown the family of all soft sets with a finite parameter
subset can form a compact information algebra.

## Acknowledgments

This work is supported by National Science Foundation of China (Grant
No.60873119) and the Higher School Doctoral Subject Foundation of Ministry of
Education of China under Grant 200807180005.

## References

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# Property($K^{*}$) Implies $R(X)\leqslant 1+\frac{\displaystyle
1}{\displaystyle 1+r_{X^{*}}(1)}$

Tim Dalby<EMAIL_ADDRESS>
###### Abstract.

It is shown that if the dual of a Banach space satisfies Property($K^{*}$)
then $R(X)\leqslant 1+\frac{\displaystyle 1}{\displaystyle 1+r_{X^{*}}(1)}<2$
where $r_{X^{*}}(c)$ is Opial’s modulus for $X^{*}.$ Thus $X$ has the weak
fixed point property.

###### Key words and phrases:

weak fixed point property, property($K^{*}$), Opial’s condition, Opial’s
modulus

###### 2010 Mathematics Subject Classification:

46B10, 47H09, 47H10

##  1\. Introduction

The Banach space moduli of $R(X),r_{X}(c)$ were introduced within the context
of fixed point theory. This area of research looks at whether, in an infinite
dimensional real Banach space, every nonexpansive mappings on every weak
compact convex nonempty set has a fixed point. If this is so then the space is
said to have the weak fixed point property (w-FPP). It can be shown that this
propery is separably determined, so in this paper the Banach space is assumed
to be separable.

If the sets are not necessarily weak compact but closed then the property is
called the fixed point property (FPP).

Another aspect of this topic is that it involves weakly convergent sequences
and usually a translation is employed so that the sequence being studied is
weakly convergent to zero. Inherent in most results is the interplay between
weak null sequences and the norm. Much more background can be found in Goebel
and Kirk [7] and Kirk and Sims [8].

The effect of the dual space, $X^{*},$ on the outcomes in $X$ have also been
studied. See for example [3] and [4]. This paper continues along this line.

Definitions and notation will be presented in the next section.

In [3] Dalby showed that if $X$ is a separable Banach space where the dual,
$X^{*}$, has Property($K^{*}$) and the nonstrict *Opial condition then
$R(X)<2$. This last condition implies $X$ has the w-FPP. For the latter result
see [6].

In [1] Benavides showed that if $X$ is a reflexive Banach space and if there
exist $c_{0}\in(0,1)$ where $r_{X^{*}}(c_{0})>0$ then
$R(a,X)\leqslant\max\left\\{1+ac_{0},a+\frac{\displaystyle 1}{\displaystyle
1+r_{X^{*}}(c_{0})}\right\\}$ where $a\geqslant 0.$ This in turn implies
$R(a,X)<1+a\mbox{ if }r_{X^{*}}(1)>0.$ Here $r(c)$ is Opial’s modulus. Earlier
in that paper it was shown that $R(a,X)<1+a\mbox{ for some }a>0$ ensures that
$X$ has the w-FPP. Because of reflexivity, in this case $X$ has the FPP.

Recall from [5] that $R(a,X)<1+a$ for some $a>0$ is equivalent to $R(1,X)<2$
and that $R(X)<2$ implies $R(1,X)<2.$

So the result in [1] can be rewritten as: If $X$ is a reflexive Banach space
where $r_{X^{*}}(1)>0$ then $X$ has the FPP. But Dalby proved in [2] that
$r_{X}(1)>0$ is equivalent to $X$ having Property($K$). This proof can readily
be transferred to $X^{*}$. That is $r_{X^{*}}(1)>0$ is equivalent to $X^{*}$
having Property($K^{*}$). Hence the result in [1] can be again rewritten as:
If $X$ is a reflexive Banach space where $X^{*}$ has Property($K^{*}$) then
$X$ has the FPP.

So the results of Benavides and Dalby are similar.

In this paper the requirement that $X$ be reflexive has been changed to $X$
being separable and the condition that $X^{*}$ has the nonstrict *Opial
property has been removed.

##  2\. Definitions

######  Definition 2.1.

Sims, [11]

A Banach space $X$ has property($K$) if there exists $K\in[0,1)$ such that
whenever $x_{n}\rightharpoonup 0,\lim_{n\rightarrow\infty}\|x_{n}\|=1\mbox{
and }\liminf_{n\rightarrow\infty}\|x_{n}-x\|\leqslant 1\mbox{ then
}\|x\|\leqslant K.$

If the sequence is in $B_{X^{*}}$ and is weak* convergent to zero then the
property is called Property($K^{*}$).

######  Definition 2.2.

Opial [10]

A Banach space has Opial’s condition if

$x_{n}\rightharpoonup 0\ \mbox{and }x\not=0\mbox{ implies
}\limsup_{n}\|x_{n}\|<\limsup_{n}\|x_{n}-x\|.$

The condition remains the same if both the $\limsup$s are replaced by
$\liminf$s.

If the $<$ is replaced by $\leqslant$ then the condition is called nonstrict
Opial.

In $X^{*}$ the conditions are the same except that the sequence is weak* null.

Later a modulus was introduced to gauge the strength of Opial’s condition and
a stronger version of the condition was defined.

######  Definition 2.3.

Lin, Tan and Xu, [9]

Opial’s modulus is

$r_{X}(c)=\inf\\{\liminf_{n\rightarrow\infty}\|x_{n}-x\|-1:c\geqslant
0,\|x\|\geqslant c,x_{n}\rightharpoonup 0\mbox{ and
}\liminf_{n\rightarrow\infty}\|x_{n}\|\geqslant 1\\}.$

$X$ is said to have uniform Opial’s condition if $r_{X}(c)>0$ for all $c>0.$
See [9] for more details.

##  3\. Result

######  Proposition 3.1.

Let $X$ be a separable Banach space with $X^{*}$ having Property($K^{*}$).
Then $R(X)\leqslant 1+\frac{\displaystyle 1}{\displaystyle 1+r_{X^{*}}(1)}.$

###### Proof.

Let $X$ be a separable Banach space such that $X^{*}$ has Property($K^{*}$).
Then $r_{X^{*}}(1)>0.$ Also, let $x_{n}\in B_{X}\mbox{ for all
}n,x_{n}\rightharpoonup 0\mbox{ and }x\in B_{X}.$ Then for each $n\geqslant 1$
choose $x_{n}^{*}\in X^{*}$ such that $\|x_{n}^{*}\|=1\mbox{ and
}x_{n}^{*}(x_{n}+x)=\|x_{n}+x\|$. Using the property that $X$ is separable we
can assume, without loss of generality, that
$x_{n}^{*}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}x^{*}\mbox{ where
}\|x^{*}\|\leqslant 1$.

$\displaystyle\liminf_{n\rightarrow\infty}\|x_{n}+x\|$
$\displaystyle=\liminf_{n\rightarrow\infty}x_{n}^{*}(x_{n}+x)$
$\displaystyle=\liminf_{n\rightarrow\infty}x_{n}^{*}(x_{n})+x^{*}(x)$
$\displaystyle\leqslant\liminf_{n\rightarrow\infty}\|x_{n}^{*}\|\|x_{n}\|+\|x^{*}\|\|x\|$
$\displaystyle=\liminf_{n\rightarrow\infty}\|x_{n}\|+\|x^{*}\|\|x\|$
$\displaystyle\leqslant 1+\|x^{*}\|.\qquad\qquad{\dagger}$

The proof now splits into four cases.

Case 1 $\bf x^{*}=0$

From † we have $\liminf_{n\rightarrow\infty}\|x_{n}+x\|\leqslant 1.$ This
means that $R(X)\leqslant 1$ which in turn implies $R(X)=1\leqslant
1+\frac{\displaystyle 1}{\displaystyle 1+r_{X^{*}}(1)}.$

Case 2 $\bf
x^{*}\not=0,\|x^{*}\|\leqslant\liminf_{n\rightarrow\infty}\|x_{n}^{*}-x^{*}\|$

We have $\frac{\displaystyle
x_{n}^{*}}{\displaystyle\|x^{*}\|}-\frac{\displaystyle
x^{*}}{\displaystyle\|x^{*}\|}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}0\mbox{
and }\liminf_{n\rightarrow\infty}\left\|\frac{\displaystyle
x_{n}^{*}}{\displaystyle\|x^{*}\|}-\frac{\displaystyle
x^{*}}{\displaystyle\|x^{*}\|}\right\|\geqslant 1.$

Thus

$\displaystyle\liminf_{n\rightarrow\infty}\left\|\frac{x_{n}^{*}}{\|x^{*}\|}-\frac{x^{*}}{\|x^{*}\|}+\frac{x^{*}}{\|x^{*}\|}\right\|$
$\displaystyle\geqslant r_{X^{*}}(1)+1$ $\displaystyle
1=\lim_{n\rightarrow\infty}\|x_{n}^{*}\|$
$\displaystyle\geqslant\|x^{*}\|(r_{X}(1)+1)$ $\displaystyle\|x^{*}\|$
$\displaystyle\leqslant\frac{1}{\displaystyle r_{X^{*}}(1)+1}.$

From † we have $\liminf_{n\rightarrow\infty}\|x_{n}+x\|\leqslant
1+\frac{\displaystyle 1}{\displaystyle r_{X^{*}}(1)+1}.$

Therefore $R(X)\leqslant 1+\frac{\displaystyle 1}{\displaystyle
r_{X^{*}}(1)+1}.$

Case 3 $\bf
x^{*}\not=0,0<\liminf_{n\rightarrow\infty}\|x_{n}^{*}-x^{*}\|<\|x^{*}\|$

To make the presentation easier to read let
$a=\liminf_{n\rightarrow\infty}\|x_{n}^{*}-x^{*}\|$ then
$a<\|x^{*}\|\leqslant 1.$

Note that $\frac{\displaystyle x_{n}^{*}-x^{*}}{\displaystyle
a}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}0\mbox{ and
}\liminf_{n\rightarrow\infty}\left\|\frac{\displaystyle
x_{n}^{*}-x^{*}}{\displaystyle a}\right\|=1.$

Now $\liminf_{n\rightarrow\infty}\left\|\frac{\displaystyle
x_{n}^{*}}{\displaystyle a}-\frac{\displaystyle x^{*}}{\displaystyle
a}+\frac{\displaystyle x^{*}}{\displaystyle
a}\right\|=\liminf_{n\rightarrow\infty}\frac{\|\displaystyle
x_{n}^{*}\|}{\displaystyle a}=\frac{\displaystyle 1}{\displaystyle a}.$

Using $\frac{\displaystyle\|x^{*}\|}{\displaystyle
a}>1,\liminf_{n\rightarrow\infty}\left\|\frac{\displaystyle
x_{n}^{*}}{\displaystyle a}-\frac{\displaystyle x^{*}}{\displaystyle
a}+\frac{\displaystyle x^{*}}{\displaystyle a}\right\|\geqslant
r_{X^{*}}(\frac{\displaystyle\|x^{*}\|}{\displaystyle a})+1\geqslant
r_{X^{*}}(1)+1$ since $r_{X^{*}}(c)$ is nondecreasing.

Thus $1\geqslant a(r_{X^{*}}(1)+1)$ and $\frac{\displaystyle 1}{\displaystyle
r_{X^{*}}(1)+1}\geqslant a=\liminf_{n\rightarrow\infty}\|x_{n}^{*}-x^{*}\|.$

This leads to

$\displaystyle\liminf_{n\rightarrow\infty}\|x_{n}+x\|$
$\displaystyle=\liminf_{n\rightarrow\infty}x_{n}^{*}(x_{n}+x)$
$\displaystyle=\liminf_{n\rightarrow\infty}(x_{n}^{*}-x^{*})(x_{n}+x)+\liminf_{n\rightarrow\infty}x^{*}(x_{n}+x)$
$\displaystyle=\liminf_{n\rightarrow\infty}(x_{n}^{*}-x^{*})(x_{n})+x^{*}(x)$
$\displaystyle\leqslant\liminf_{n\rightarrow\infty}\|x_{n}^{*}-x^{*}\|\|x_{n}\|+\|x\|\qquad{\dagger}{\dagger}$
$\displaystyle\leqslant\frac{1}{r_{X^{*}}(1)+1}+1$ $\displaystyle\mbox{ So
}R(X)$ $\displaystyle\leqslant 1+\frac{1}{r_{X^{*}}(1)+1}.$

Case 4 $\bf x^{*}\not=0,\liminf_{n\rightarrow\infty}\|x_{n}^{*}-x^{*}\|=0$

From ††, $\liminf_{n\rightarrow\infty}\|x_{n}+x\|\leqslant\|x\|$ but weak
lower semicontinuity of the norm means
$\|x\|\leqslant\liminf_{n\rightarrow\infty}\|x_{n}+x\|.$

Therefore $\liminf_{n\rightarrow\infty}\|x_{n}+x\|=\|x\|\leqslant 1\leqslant
1+\frac{\displaystyle 1}{\displaystyle r_{X^{*}}(1)+1}$ leading to
$R(X)\leqslant 1+\frac{\displaystyle 1}{\displaystyle r_{X^{*}}(1)+1}.$ ∎

Note  Careful reading of the proof reveals that $R(X)\leqslant
1+\frac{\displaystyle 1}{\displaystyle 1+r_{X^{*}}(1)}$ is always  true.

So Opial’s modulus for $X^{*}$ plays a role in the behaviour of weak null
sequences in $X.$

Property($K^{*}$)’s role is ensure that $r_{X^{*}}(1)>0$ which in turn means
that $R(X)<2.$

Remark Using $R(1,X)\leqslant RW(1,X)\leqslant R(X)$ we have:

Property($K^{*}$) implies $RW(1,X)\leqslant 1+\frac{\displaystyle
1}{\displaystyle r_{X^{*}}(1)+1}<1$

and

Property( $K^{*}$) implies $R(1,X)\leqslant 1+\frac{\displaystyle
1}{\displaystyle r_{X^{*}}(1)+1}<1.$

In [5] it was shown that $RW(1,X)<2$ is equivalent to $RW(a,X)<1+a$ for some
$a>0.$ Similarly, $R(1,X)<2$ is equivalent to $R(a,X)<1+a$ for some $a>0.$

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