# CONSTRAINTS FROM TYPE IA SUPERNOVAE ON
$\Lambda$-CDM MODEL IN RANDERS-FINSLER SPACE

Zhe Chang1,2<EMAIL_ADDRESS>Ming-Hua Li1,2<EMAIL_ADDRESS>1Institute of
High Energy Physics, Chinese Academy of Sciences, 100049 Beijing, China  Xin
Li2,3<EMAIL_ADDRESS>2Theoretical Physics Center for Science Facilities,
Chinese Academy of Sciences
3Institute of Theoretical Physics, Chinese Academy of Sciences, 100190
Beijing, China

###### Abstract

Gravitational field equations in Randers-Finsler space of approximate Berwald
type are investigated. A modified Friedmann equation and a new luminosity
distance-redshift relation is proposed. A best-fit to the Type Ia supernovae
(SNe) observations yields that the $\Omega_{\Lambda}$ in the $\Lambda$-CDM
model is suppressed to almost zero. This fact indicates that the astronomical
observations on the Type Ia SNe can be described well without invoking any
form of dark energy. The best-fit age of the universe is given. It is in
agreement with the age of our galaxy.

###### pacs:

##  I I. Introdution

The phenomenon that our universe is expanding was first observed by E. Hubble
Hubble  in 1929, soon after the birth of General Relativity. Since 1990s,
rapid progress on the observations of the Type Ia SNe have been made riess
Perlmutter Bennett Kowalski . In order to explain the observations, dark
energy with the property of negative pressure must be involved in the
framework of standard cosmological model. Multiple kinds of models have been
proposed in the past decade Copeland . Some of them suggested an evolving
canonical scalar field with a potential peebles earlyqu Carroll:1998zi .
Others tried to modify the General Relativity to make up for the gap between
the theory and the observations Capo Carroll:2003wy odintsovl . One of the
most famous candidates of dark energy is the cosmological constant $\Lambda$.
However, it requires “fine tuning” in an early epoch of the universe
Steinhardt .

_Finsler geometry_ , which includes the Riemann geometry as its special case,
supplies a new systematic approach to the problems mentioned above. Gravity in
a Finsler space has been studied for a long time Takano Tavakol Bogoslovsky1 .
A theory of the gauge transformations in the context of Finsler space has been
discussed by G. Asanov Asanov1 Asanov2 Asanov3  and S. Ikeda Ikeda1 ; Ikeda2 ,
and its applications to General Relativity has been suggested by R. Beil Beil1
; Beil2 . The corresponding gravitational field equation was derived from the
Riemannian osculating metric in Asanov1 . Considering consistency with the
Bianchi identity in Finsler geometry and the general covariance principle of
Einstein, we gave a new gravitational field equation in a Berwald-Finsler
space Lixin . In the framework of Finsler geometry, a modified Newton s
gravity has been proposed, which agrees quite well with the asymptotically
flat rotation curves of spiral galaxies without invoking dark matter Finsler
DM .

In this paper, we test possible constraints from the Type Ia SNe on cosmology
in Finsler geometry. Based on preceding work Lixin ; Modified Friedmann model
, we present a new relation between the luminosity distance and redshift. A
best-fit to the Type Ia SNe data indicates that our universe is matter-
dominated in Finsler space. A reasonable age of our universe is proposed,
which is consistent with that obtained from radioactive dating of isotopes in
stars and from white dwarfs in our galaxy.

The paper is organized as follows. In section II, a brief introduction of
Finsler geometry is presented. In section III, we discuss the gravitational
field equation in a Randers-Finsler space of approximate Berwald type. A new
luminosity-redshift relation is set up in section IV. It is one of the keys to
understand the Type Ia SNe observations without invoking dark energy
hypothesis. The numerical study is carried out in section V. In section VI, we
give concluding remarks and the comparison bewtween our age prediction and
that of other models.

##  II II. Finsler Geometry

Let us first introduce basic concepts and notions in Finsler geometry. We use
$T_{x}M$ to denote the _tangent space_ at a point $x$ on a manifold $M$, i.e.
$x\in M$. $TM$ is the _tangent bundle_ of $M$. Each element of $TM$ is
described by $(x,y)$, where $x\in M$ and $y\in T_{x}M$. The _natural
projection_ $\pi:TM\rightarrow M$ is defined as $\pi(x,y)\equiv x$.

_Finsler geometry_ has its genesis in integrals of the form

$\displaystyle\int^{b}_{a}F\left(x^{1},\cdots,x^{n};y^{1},\cdots,y^{n}\right)dt\
,$ (1)

where $x^{i}$ stands for _position_ and $y^{i}\equiv dx^{i}/{dt}$ for
_velocity_. The integrand $F$ is called a _Finsler structure_.

A _Finsler structure_ of $M$ Book by Bao

$\displaystyle F:TM\rightarrow[0,\infty)$

is a function with the following properties:
(i) _Regularity_ : F is $C^{\infty}$ on the entire _slit tangent bundle_
$TM\backslash 0$.
(ii) _Positive homogeneity_ : $F(x,\lambda y)=\lambda F(x,y)$ for all
$\lambda>0$.
(iii) _Strong convexity_ : The $n\times n$ Hessian matrix

$\displaystyle g_{\mu\nu}\equiv\frac{\partial}{\partial
y^{\mu}}\frac{\partial}{\partial y^{\nu}}\left(\frac{1}{2}F^{2}\right)$ (2)

is positive-definite at every point of $TM\backslash 0$. And the lowering and
raising of indices in this paper are carried out by the _fundamental tensor_
$g_{\mu\nu}$ defined above and its inverse $g^{\mu\nu}$. The _Carten tensor_
$A_{\lambda\mu\nu}$, which is defined as

$\displaystyle A_{\lambda\mu\nu}\equiv\frac{F}{4}\frac{\partial}{\partial
y^{\lambda}}\frac{\partial}{\partial y^{\mu}}\frac{\partial}{\partial
y^{\nu}}(F^{2})\ ,$ (3)

is regarded as a measurement of deviation from the Riemannian manifold.

According to Chern’s work Chern2 , each Finsler manifold admits a unique
linear connection, called the _Chern connection_. It is torsion-free and
almost metric-compatible. The _connection coefficients_ take a form as

$\displaystyle\Gamma^{\alpha}_{~{}\mu\nu}=\gamma^{\alpha}_{~{}\mu\nu}-g^{\alpha\lambda}\left(A_{\lambda\mu\beta}\frac{N^{\beta}_{~{}\nu}}{F}-A_{\mu\nu\beta}\frac{N^{\beta}_{~{}\lambda}}{F}+A_{\nu\lambda\beta}\frac{N^{\beta}_{~{}\mu}}{F}\right),$
(4)

where $\gamma^{\alpha}_{~{}\mu\nu}$ is the _formal Christoffel symbols_ of the
second kind

$\displaystyle\gamma^{\alpha}_{~{}\mu\nu}\equiv\frac{1}{2}g^{\alpha
s}\left(\frac{\partial g_{s\mu}}{\partial x^{\nu}}-\frac{\partial
g_{\mu\nu}}{\partial x^{s}}+\frac{\partial g_{\nu s}}{\partial
x^{\mu}}\right)\ .$ (5)

$N^{\mu}_{~{}\nu}$ is defined as

$\displaystyle
N^{\mu}_{~{}\nu}\equiv\gamma^{\mu}_{~{}\nu\alpha}y^{\alpha}-\frac{A^{\mu}_{~{}\nu\lambda}}{F}\gamma^{\lambda}_{~{}\alpha\beta}y^{\alpha}y^{\beta}\
.$ (6)

The _curvature tensor_ of a Finsler space is given as Book by Bao

$\displaystyle R^{~{}\lambda}_{\kappa~{}\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{\delta\Gamma^{\lambda}_{~{}\kappa\nu}}{\delta
x^{\mu}}-\frac{\delta\Gamma^{\lambda}_{~{}\kappa\mu}}{\delta
x^{\nu}}+\Gamma^{\lambda}_{~{}\alpha\mu}\Gamma^{\alpha}_{~{}\kappa\nu}-\Gamma^{\lambda}_{~{}\alpha\nu}\Gamma^{\alpha}_{~{}\kappa\mu}\
,$ (7)

where

$\displaystyle\frac{\delta}{\delta x^{\mu}}=\frac{\partial}{\partial
x^{\mu}}-N^{\nu}_{~{}\mu}\frac{\partial}{\partial y^{\mu}}\ .$ (8)

The _Ricci scalar_ is defined as

$\displaystyle Ric\equiv g^{\mu\nu}R_{\mu\nu}\ ,$ (9)

where

$\displaystyle
R^{\mu}_{~{}\nu}\equiv\frac{y^{\lambda}}{F}R^{~{}\mu}_{\lambda~{}\nu\kappa}\frac{y^{\kappa}}{F}=\frac{y^{\lambda}}{F}\left(\frac{\delta}{\delta
x^{\nu}}\frac{N^{\mu}_{~{}\lambda}}{F}-\frac{\delta}{\delta
x^{\lambda}}\frac{N^{\mu}_{~{}\nu}}{F}\right)\ .$ (10)

The _Ricci tensor_ $Ric_{\mu\nu}$, first introduced by Akbar-Zadeh Akbar , is

$\displaystyle
Ric_{\mu\nu}\equiv\frac{\partial^{2}(\frac{1}{2}F^{2}Ric)}{\partial
y^{\mu}\partial y^{\nu}}=\left[\frac{1}{2}F^{2}Ric\right]_{y^{\mu}y^{\nu}}\ .$
(11)

A _Randers space_ is a specific type of Finsler space, whose Finsler structure
takes the form Randers

$\displaystyle F(x,y)\equiv\alpha(x,y)+\beta(x,y)\ ,$ (12)

where

$\displaystyle\alpha(x,y)$ $\displaystyle\equiv$
$\displaystyle\sqrt{\tilde{a}_{\mu\nu}(x)y^{\mu}y^{\nu}}\ ,$ (13)
$\displaystyle\beta(x,y)$ $\displaystyle\equiv$
$\displaystyle\tilde{b}_{\mu}(x)y^{\mu}\ .$ (14)

The $\tilde{a}_{\mu\nu}$ are the components of a Riemannian metric and the
$\tilde{b}_{\mu}$ are those of a 1-form. Lower case Greek indices run from
‘$0$’ to ‘$3$’. Specifically, the lowering and raising of indices for the
terms decorated with a tilde are carried out by $\tilde{a}_{\mu\nu}$ and its
inverse $\tilde{a}^{\mu\nu}$ instead of the fundamental tensor.

A Finsler structure $F$ is said to be of _Berwald type_ if the Chern
connection coefficients $\Gamma^{\alpha}_{~{}\mu\nu}$ in natural coordinates
have no $y$ dependence. A Randers space is said to be of Berwald type if and
only if Kikuchi

$\displaystyle\tilde{b}_{\mu|\nu}\equiv\frac{\partial\tilde{b}_{\mu}}{\partial
x^{\nu}}-\tilde{b}_{\kappa}\tilde{\gamma}^{\kappa}_{~{}\mu\nu}=0\ ,$ (15)

where $\tilde{\gamma}^{\kappa}_{~{}\mu\nu}$ is the Christoffel symbols of a
Riemannian metric $\tilde{\alpha}$. After some tedious calculations, one
obtains that

$\displaystyle\Gamma^{\kappa}_{~{}\mu\nu}=\tilde{\gamma}^{\kappa}_{~{}\mu\nu}\
.$ (16)

So the curvature tensor of a Randers space of Berwald type is given as

$\displaystyle R^{~{}\lambda}_{\kappa~{}\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{\partial\tilde{\gamma}^{\lambda}_{~{}\kappa\nu}}{\partial
x^{\mu}}-\frac{\partial\tilde{\gamma}^{\lambda}_{~{}\kappa\mu}}{\partial
x^{\nu}}+\tilde{\gamma}^{\lambda}_{~{}\alpha\mu}\tilde{\gamma}^{\alpha}_{~{}\kappa\nu}-\tilde{\gamma}^{\lambda}_{~{}\alpha\nu}\tilde{\gamma}^{\alpha}_{~{}\kappa\mu}\
,$ (17)

and the corresponding Ricci tensor $Ric_{\mu\nu}$ is

$\displaystyle
Ric_{\mu\nu}=\frac{1}{2}(R^{~{}\alpha}_{\mu~{}\alpha\nu}+R^{~{}\alpha}_{\nu~{}\alpha\mu})\
.$ (18)

##  III III. The Friedmann Model in the framework of Finsler Geometry

In order to investigate the Friedmann-Robertson-Walker (FRW) cosmology, we
take $\tilde{\alpha}_{\mu\nu}$ to be the form

$\displaystyle\tilde{a}_{\mu\nu}={\rm
diag}\left(1,-\frac{R^{2}(t)}{1-kr^{2}},-R^{2}(t)r^{2},-R^{2}(t)r^{2}\sin^{2}\theta\right)\
,$ (19)

where $k=0,+1,-1$ stands for a flat, closed or open universe respectively.
With the condition (15) in mind and assuming that the space of our universe is
almost homogeneous and isotropic, we take

$\displaystyle\tilde{b}_{\mu}=(\tilde{b}_{0},0,0,0)\ ,$ (20)

where $\tilde{b}_{0}$ is a small constant.

Using the identities (2), (17) and (18), one may directly calculates the Ricci
tensor in the Randers space of approximate Berwald type. Nonzero components
are listed below:

$\displaystyle Ric_{00}$ $\displaystyle=$
$\displaystyle-3\frac{\ddot{R}}{R}\tilde{a}_{00},$ $\displaystyle Ric_{ij}$
$\displaystyle=$
$\displaystyle-\left(\frac{\ddot{R}}{R}+2\frac{\dot{R}^{2}}{R^{2}}+\frac{2k}{R^{2}}\right)\tilde{a}_{ij}\
.$ (21)

The trace of the Ricci tensor $Ric_{\mu\nu}$ gives the _scalar curvature_
$S\equiv g^{\mu\nu}Ric_{\mu\nu}$ ,

$\displaystyle S$ $\displaystyle=$
$\displaystyle-6\frac{\alpha}{F}\left(\frac{\ddot{R}}{R}+\frac{\dot{R}^{2}}{R^{2}}+\frac{k}{R^{2}}\right)-3\frac{\ddot{R}}{R}\frac{\alpha^{2}}{F^{2}}\left(\frac{\beta}{F}\tilde{a}_{00}\frac{y^{0}}{\alpha}\frac{y^{0}}{\alpha}-2\tilde{a}_{00}\frac{y^{0}}{\alpha}\tilde{b}^{0}\right)$
(22)
$\displaystyle-\left(\frac{\ddot{R}}{R}+2\frac{\dot{R}^{2}}{R^{2}}+\frac{2k}{R^{2}}\right)\frac{\alpha^{2}}{F^{2}}\left(\frac{\beta}{F}\tilde{a}_{ij}\frac{y^{i}}{\alpha}\frac{y^{j}}{\alpha}\right).$

A new gravitational field equation in the Berwald-Finsler space is given as
Lixin

$\displaystyle\left[Ric_{\mu\nu}-\frac{1}{2}g_{\mu\nu}S\right]+\left\\{\frac{1}{2}B^{~{}\alpha}_{\alpha~{}\mu\nu}+B^{~{}\alpha}_{\mu~{}\nu\alpha}\right\\}=8\pi
GT_{\mu\nu}\ ,$ (23)

where

$\displaystyle
B_{\mu\nu\alpha\beta}=-A_{\mu\nu\lambda}R^{~{}\lambda}_{\theta~{}\alpha\beta}y^{\theta}/F\
.$ (24)

$T^{\mu}_{\nu}$ is the energy-momenta tensor as $T^{\mu}_{\nu}={\rm
diag}(\rho,-p,-p,-p)$, where $p\equiv p(x)$ and $\rho\equiv\rho(x)$ is
respectively the pressure and the energy density of the cosmic fluid.

In a Randers space of approximate Berwald type, the gravitational field
equation (23) reduces to

$\displaystyle\left[Ric_{\mu\nu}-\frac{1}{2}g_{\mu\nu}S\right]=8\pi
GT_{\mu\nu}\ ,$ (25)

because the terms $B^{~{}\alpha}_{\alpha~{}\mu\nu}$ and
$B^{~{}\alpha}_{\mu~{}\nu\alpha}$ are zero. $Ric_{\mu\nu}$ and $S$ are given
by the identities (21) and (22).

For the sake of simplicity, we introduce two parameters $A$ and $B$ as

$\displaystyle
A\equiv\frac{\alpha}{F}\left(\frac{\beta}{F}\tilde{a}_{00}\frac{y^{0}}{\alpha}\frac{y^{0}}{\alpha}-2\tilde{a}_{00}\frac{y^{0}}{\alpha}\tilde{b}^{0}\right)$
(26)

and

$\displaystyle
B\equiv\frac{\alpha}{F}\left(\frac{\beta}{F}\tilde{a}_{ij}\frac{y^{i}}{\alpha}\frac{y^{j}}{\alpha}\right)\
.$ (27)

The $0$-$0$ component of the field equation (25) gives the modified Friedmann
equation

$\displaystyle\frac{\alpha}{F}\left(\frac{\dot{R}^{2}}{R^{2}}+\frac{k}{R^{2}}\right)-\frac{1}{2}\frac{\alpha}{F}\frac{\ddot{R}}{R}A+\frac{1}{6}\frac{\alpha}{F}\left(\frac{\ddot{R}}{R}+2\frac{\dot{R}^{2}}{R^{2}}+2\frac{k}{R^{2}}\right)B=\frac{8\pi
G}{3}\rho$ (28)

and the $i$-$j$ component of (25) gives

$\displaystyle
3\frac{\alpha}{F}\left(2\frac{\ddot{R}}{R}+\frac{\dot{R}}{R^{2}}+\frac{k}{R^{2}}\right)+\frac{9}{2}\frac{\ddot{R}}{R}\frac{\alpha}{F}A+\frac{1}{2}\frac{\alpha}{F}\left(\frac{\ddot{R}}{R}+2\frac{\dot{R}}{R^{2}}+2\frac{k}{R^{2}}\right)B=8\pi
G(-3p)\ .$ (29)

Subtracting the above two equations, we have

$\displaystyle\frac{\alpha}{F}\frac{\ddot{R}}{R}(1+A)=-\frac{4\pi
G}{3}(\rho+3p)\ .$ (30)

The $0$-$0$ component of the field equation (28) can be rewritten into the
form

$\displaystyle\frac{\alpha}{F}\left(\frac{\dot{R}^{2}}{R^{2}}+\frac{k}{R^{2}}\right)\left(1+\frac{B}{3}\right)\left(1+A\right)=\frac{8\pi
G}{3}\rho\left(1+A\right)+\frac{4\pi
G}{3}\left(\rho+3p\right)\left(-\frac{A}{2}+\frac{B}{6}\right)\ .$ (31)

Assuming that $A$ and $B$ are both time-independent, implementation of time
derivative $\frac{d}{dt}$ on both sides of the equation (31) and using (30)
again leads us to

$\displaystyle-\frac{\dot{R}}{R}\left[\left(1+\frac{B}{3}\right)(\rho+3p)+\rho\left(2+\frac{3A}{2}+\frac{B}{6}\right)+p\left(-\frac{3A}{2}+\frac{B}{2}\right)\right]=\
$ $\displaystyle\dot{\rho}\left(1+\frac{3A}{4}+\frac{B}{12}\right)$ (32)
$\displaystyle+\dot{p}\left(-\frac{3A}{4}+\frac{B}{4}\right)\ .$

With the equations of state $p_{i}=w_{i}\rho_{i}$ of each individual component
$i$ (where the constant $w_{i}=0,-1,-1/3$ corresponds to _matter_ , _vacuum_
and ‘ _curvature_ ’ respectively), the equation (32) can be solved,

$\displaystyle\rho_{i}\propto
R^{-\displaystyle{3(1+w_{i})+\frac{3}{2}(1-w_{i})A+\frac{1}{2}(1+3w_{i})B\over
1+\frac{3}{4}(1-w_{i})A+\frac{1}{12}(1+3w_{i})B}}\ .$ (33)

##  IV IV. A New Luminosity-Redshift Relation

We adopt the conventional definitions

$\displaystyle H(a)\equiv\frac{\dot{R}}{R}\ ,\qquad\rho_{\rm
crit0}\equiv{{3H_{0}^{2}}\over{8\pi G}}\
,\qquad\Omega_{i0}\equiv{{\rho_{i0}}\over{\rho_{\rm crit0}}}=\left({{8\pi
G}\over 3H_{0}^{2}}\right)\rho_{i0}\ ,$ (34)

and

$\displaystyle\rho_{k}\equiv-{{3k}\over{8\pi GR_{0}^{2}a^{-2}}}\
,\qquad\rho_{\rm vac}=\rho_{\Lambda}\equiv{{\Lambda}\over{8\pi G}}\ .$ (35)

Combining the equations (31) and (33), one obtains

$\displaystyle
H(a)=H_{0}\left[\sum_{i(k)}\Omega_{i0}f_{i}(w_{i},A,B)a^{-n_{i}(w_{i},A,B)}\right]^{1/2}\
,$ (36)

where

$\displaystyle
f_{i}(w_{i},A,B)=\frac{1+\frac{3}{4}(1-w_{i})A+\frac{1}{12}(1+3w_{i})B}{(1+\frac{B}{3})(1+A)}\
,$ (37) $\displaystyle
n_{i}(w_{i},A,B)=-\frac{3(1+w_{i})+\frac{3}{2}(1-w_{i})A+\frac{1}{2}(1+3w_{i})B}{1+\frac{3}{4}(1-w_{i})A+\frac{1}{12}(1+3w_{i})B}\
.$ (38)

Here $\Omega_{\rm k}=1-\Omega_{\rm M}-\Omega_{\Lambda}$ . The notation
$\sum_{i(k)}$ denotes that the sum includes $\Omega_{k}$. As an approximation,
we do not take the radiation term into account due to its little influence on
predictions of the $\Lambda$-CDM model when using the Type Ia SNe data alone.

The _luminosity distance_ $d_{L}$ as a function of the redshift $z$ of a
supernova is Sean M. Carroll

$\displaystyle\begin{array}[]{rcl}d_{L}(z)=&\displaystyle{{(1+z)\over{H_{0}\sqrt{|\Omega_{k0}|}}}\,{\rm
sinn}\left[H_{0}\sqrt{|\Omega_{k0}|}\int^{1}_{1/(1+z)}{{da}\over{a^{2}H(a)}}\right]}\
,\end{array}$ (40)

where ‘$sinn$’ stands for ‘$sin$’ (if $k>0$), ‘$1$’ (if $k=0$) or ‘$sinh$’ (
if $k<0$ ) under certain circumstances. Substituting the equation (36) into
(40), we get a luminosity distance-redshift relation that looks like

$\displaystyle d_{L}(z)=d_{L}(z;\Omega_{\rm M},\Omega_{\Lambda},A,B)\ .$ (41)

##  V V. Numerical Study

The _distance modulus_ $\mu$ is related to the luminosity distance via

$\displaystyle\mu\equiv m-M=5\log_{10}[d_{L}({\rm Mpc})]+25\ ,$ (42)

where $m$ is the _apparent magnitude_ of the source and $M$ its _absolute
magnitude_. And it is $\mu$ and $z$ that the Supernova Project measured. A
total uncertainty of $\mu$ (denoted by ‘$\sigma_{\mu}$’) and the corresponding
redshift $z$ were presented in the reference Kowalski .

The $\chi^{2}$ statistic in our fit is

$\displaystyle\chi_{\rm SN}^{2}(\Omega_{\rm
M},\Omega_{\Lambda})\equiv\sum_{i=1}^{307}{[\mu_{\rm obs}(z_{i})-\mu_{\rm
th}(z_{i};\Omega_{\rm
M},\Omega_{\Lambda})]^{2}\over{{\sigma_{\mu}(z_{i})}^{2}}}\ ,$ (43)

where $\mu_{\rm th}(z_{i})$ is obtained by the equation(42), while $\mu_{\rm
obs}(z_{i})$ and $\sigma_{\mu}(z_{i})$ come from the observations. We employ
the Markov Chain Monte-Carlo (MCMC) techniques Antony Lewis  to explore the
parameter space. The likelihood function looks like ${\cal L}\propto
e^{-\chi_{\rm SN}^{2}(\Omega_{\rm M},\Omega_{\Lambda})/2}$. For simplicity, we
take $A=-3$ and $B=-1$, leaving $\Omega_{\Lambda}$(or $\Omega_{\rm M}$) to be
the _only free parameter_ in our model with the constraint $\Omega_{\rm
M}+\Omega_{\Lambda}=1$ of a $k=0$ flat universe.

The marginalized posterior and the mean likelihood distributions of the
density parameter $\Omega_{\rm M}$ are shown in Fig.[1]. The two contours in
Fig.[2] and Fig.[3] respectively line out the $68\%$ and $95\%$ confidence
regions of the marginalized distribution in the
$\Omega_{\Lambda}$-$\Omega_{\rm M}$ and Age-$\Omega_{\rm M}$ planes. Best-fit
values of $\Omega_{\rm M}=0.9997_{-0.0009}^{+0.0003}$ and
$\Omega_{\Lambda}=0.0003_{-0.0003}^{+0.0001}$ are obtained with $-ln{\cal
L}=182.1819$ for a total number of $307$ data points. The almost vanished
$\Omega_{\Lambda}$ indicates that, in our model, there is no need to invoke
the $\Omega_{\Lambda}$ term in the Einstein’s field equation to account for
the supernova observations. The best-fit age of the universe is
$18.298_{-0.013}^{+0.102}$ Gyr.

##  VI VI. Conclusions

In this paper, we have initiated an exploration on the possibility of a
modified Friedmann model in a Randers-Finsler space of approximate Berwald
type as an alternative to the dark energy hypothesis. Wondering whether the
space-time of our universe is a Randers-Finslerian manifold instead of a
Riemannian one, we have rewritten the Einstein’s field equation in such a
space and the new Friedmann equation was also given. A best-fit to the Type Ia
SNe data suppresses the effective density parameter $\Omega_{\Lambda}$ in the
$\Lambda$-CDM model to almost zero. This fact demonstrates that a Randers-
Finsler geometrical explanation of the ‘accelerated’ expanding universe
without invoking dark energy is possible.

Moreover, the best-fit age of the universe in our model is consistent with the
10 to 20 Gyr estimate obtained from radioactive dating of isotopes in stars
schramm truran  and the 6.5 to 10 Gyr minimum age given by the white dwarfs in
our Galactic disk oswalt96 bergeron . However, the change from the old Riemann
space-time to a new Randers-Finsler one may call for a redefinition of not
only the luminosity distance, but also probably other metric-related
quantities. This will be the subject of our future investigation.

###### Acknowledgements.

##  VII Acknowledgments

Our work was supported by the NSF of China under Grant No. 10575106 and No.
10875129.

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  * (39) D. Spergel et al., Astrophys. J. Suppl. 170, 377 (2007).

Figure 1: The posterior constraints of $\Omega_{\rm M}$ with $A=-3$ and $B=-1$
using all data. The solid line indicates the fully marginalized posterior of
$\Omega_{\rm M}$. The dotted line shows the mean likelihood of the samples.
The fact that the two matching well implies that the mean likelihood is well
constrained and our result is justifiable. We take the Hubble constant to be
$H_{0}=70.5$ $\rm km\cdot s^{-1}\cdot Mpc^{-1}$ instead of a base variable
parameter, because the Type Ia supernova data alone cannot put a well
constraint on it. The center value lies at $\Omega_{\rm M}=0.9997$. The best-
fit likelihood ${\cal L}$ for a total number of $307$ data points is $-ln{\cal
L}({\cal L}\propto e^{-\chi_{\rm SN}^{2}(\Omega_{\rm
M},\Omega_{\Lambda})/2})=182.1819$. Corresponding to a high best-fit
$\Omega_{\rm M}$, a low best-fit $\Omega_{\Lambda}=0.0003$ should be
anticipated, which indicates a matter-dominated universe.

Figure 2: Best-fit $68\%$ and $95\%$ confidence regions from the marginalized
posterior distributions in the $\Omega_{\Lambda}$-$\Omega_{\rm M}$ plane for
the Finsler cosmological model with $A=-3$ and $B=-1$. The inner contour
denotes the $68\%$ confidence limit and the outer one denotes the $95\%$ one.
The cross ‘$+$’ at the upper left corner denotes the best-fit values of
$(\Omega_{\Lambda},\Omega_{\rm M})=(0.0003,0.9997)$ in the modified Friedmann
model. Compared to the $(\Omega_{\Lambda},\Omega_{\rm M})=(0.28,0.72)$ in the
reference spergel , our result indicates that in a Randers-Finsler universe,
no dark energy but only matter components exist.

Figure 3: Best-fit $68\%$ and $95\%$ confidence regions from the marginalized
posterior distributions in the Age-$\Omega_{\rm M}$ plane for the Finsler
cosmological model with $A=-3$ and $B=-1$. The inner contour denotes the
$68\%$ confidence limit and the outer one denotes the $95\%$ one. The cross
‘$+$’ at the upper right corner in the above figure denotes the best-fit value
of the age of the universe in our modified Friedmann model is $18.298$ Gyr.
This prediction is consistent with the $10$ to $20$ Gyr estimate obtained from
radioactive dating of isotopes in stars schramm truran  and the $6.5$ to $10$
Gyr minimum age given by the white dwarfs in our Galactic disk oswalt96
bergeron . Thus, the Type Ia SNe data could be well explained by a Randers-
Finsler space-time without invoking any form of dark energy.


# Effect of nonlocal transformations on the linearizability and exact
solvability of the nonlinear generalized modified Emden type equations

Omar Mustafa<EMAIL_ADDRESS>Department of Physics, Eastern
Mediterranean University, G. Magusa, north Cyprus, Mersin 10 - Turkey.

###### Abstract

Abstract: The nonlinear generalized modified Emden type equations (GMEE) are
known to be linearizable into simple harmonic oscillator (HO) or damped
harmonic oscillators (DHO) via some nonlocal transformations. Hereby, we show
that the structure of the nonlocal transformation and the linearizability into
HO or DHO determine the nature/structure of the dynamical forces involved
(hence, determine the structure of the dynamical equation). Yet, a reverse
engineering strategy is used so that the exact solutions of the emerging GMEE
are nonlocally transformed to find the exact solutions of the HO and DHO
dynamical equations. Consequently, whilst the exact solution for the HO
remains a textbook one, the exact solution for the DHO (never reported
elsewhere, to the best of our knowledge) turns out to be manifestly the most
explicit and general solution that offers consistency and comprehensive
coverage for the associated under-damping, critical-damping, and over-damping
cases (i.e., no complex settings for the coordinates and/or the velocities are
eminent/feasible). Moreover, for all emerging dynamical system, we report
illustrative figures for each solution as well as the corresponding phase-
space trajectories as they evolve in time.

PACS numbers: 05.45.-a, 03.50.Kk, 03.65.-w

Keywords: Damped harmonic oscillator, Generalized modified Emden equations,
Nonlocal transformation, Linearizability and exact solvability, Euler-Lagrange
equations invariance.

##  I Introduction

Nonlinear oscillators are commonly used to describe a variety of physical
systems ranging from elementary pedagogical implementations to more advanced
physical phenomena like atmospheric, plasma, nonlinear optics, electronics,
biophysics, etc (c.f., e.g., Chandrasekar-PRE 2005  and related references
cited therein). They represent more realistic models than the simple harmonic
oscillator (HO). The so called generalized modified Emden equation (GMEE)

$\ddot{q}+\left(3k\,q+a_{{}_{1}}\right)\,\dot{q}+a_{{}_{2}}q+a_{{}_{1}}kq^{2}+k^{2}q^{3}=0$
(1)

is one of such nonlinear oscillators, where
$a_{{}_{1}},a_{{}_{2}},k\in\mathbb{R}$ Chandrasekar 2006 ; Chandrasekar 2012 .
It collapses into the modified Emden equation MEE for $a_{{}_{1}}=0$, into
damped harmonic oscillator (DHO) for $k=0$, and into simple harmonic
oscillator (HO) for $a_{{}_{1}}=k=0$. Its integrability properties are
discussed in details (e.g., Chandrasekar-PRE 2005 ; Chandrasekar 2006 ;
Chandrasekar 2012 ; Leach 1985 ; Leach 1993 ; Chandrasekar1 2006 ;
Chandrasekar 2007 ) and lie far beyond the scope of the current proposal. Such
a nonlinear equation finds its feasible applications in, for example, the
equilibrium configuration of spherical gas clouds Leach 1985 , the spherically
symmetric relativistically gravitating mass Mc Vittie 1980 , the Yang-Mills
boson gauge theory Yang-Mills 1954 , etc. The linearizability of such a
second-order ordinary differential equation (ODE) into either linear HO or
linear DHO is made possible through the interesting nonlocal transformation

$U\left(t\right)=q\left(t\right)exp\left({\int
f\left(q(t)\right)dt}\right)\Longleftrightarrow\frac{\dot{U}\left(t\right)}{U(t)}=\frac{\dot{q}(t)}{q(t)}+f\left(q(t)\right)$
(2)

by Chandrasekar et al. Chandrasekar 2006 . Obviously, not only the structure
of $f\left(q(t)\right)\in\mathbb{R}$ in (2) but also the linearization process
into either linear HO or DHO would determine the nature/structure of the
dynamical force involved in the problem at hand. Chandrasekar et al.
Chandrasekar 2006  have considered a variety of
$f\left(q\left(t\right)\right)$ structures and worked out the corresponding
integrability properties and solutions.

In the current methodical proposal, nevertheless, we shall use a linear form
for $f\left(q\left(t\right)\right)$ and consider

$f\left(q\left(t\right)\right)=\alpha q_{{}_{\pm}}\left(t\right)\pm\zeta,$ (3)

to imply that the transformation (2) now reads

$U_{{}_{\pm}}\left(t\right)=q_{{}_{\pm}}\left(t\right)\exp\left(\int\left(\alpha
q_{{}_{\pm}}\left(t\right)\pm\zeta\right)dt\right)\;\Longleftrightarrow\frac{\dot{U}_{{}_{\pm}}\left(t\right)}{U_{{}_{\pm}}\left(t\right)}=\frac{\dot{q}_{{}_{\pm}}\left(t\right)}{q_{{}_{\pm}}\left(t\right)}+\alpha
q_{{}_{\pm}}\left(t\right)\pm\zeta\text{ },$ (4)

where $\alpha,\zeta\in\mathbb{R}$ are positive constants to be determined in
the linearization process. The constant term $\pm\zeta$ would allow us to
indulge and study the effect of both dissipative and anti-dissipative Bender
2016  (boosters/anti-damping) external forces as viable and realistic
characterizations for dynamical systems. In the case of dissipative forces the
phase-space trajectories shrink, whereas in the case of anti-dissipative
forces the phase-space trajectories expand, as the dynamical systems evolve in
time (e.g., Chandrasekar 2007 ; Mustafa PS 2021 ). One should, hereby, be
aware that we take $U_{{}_{+}}(t),q_{{}_{+}}(t)$ for $+\zeta$ and
$U_{{}_{-}}(t),q_{{}_{-}}(t)$ for $-\zeta$.

Throughout we consider the motion of a classical particles with mass
$m_{\circ}=1$ moving under the influence of a conservative quartic anharmonic
potential force field

$V\left(q\right)=\frac{1}{2}\omega^{2}q^{2}+\frac{1}{4}\alpha^{2}q^{4}$ (5)

(often called the Duffing oscillator Amore 2005 ) and subjected to a non-
conservative dissipative (i.e, $\mathcal{R}_{{}_{+}}(q,\dot{q})$) and/or anti-
dissipative (i.e., $\mathcal{R}_{{}_{-}}(q,\dot{q})$) Rayleigh force fields

$\mathcal{R}_{\pm}\left(q,\dot{q}\right)=\frac{1}{2}\left[3\alpha
q\pm\gamma_{{}_{1}}\left(\zeta\right)\right]\dot{q}^{2}+\left[\gamma_{{}_{0}}\left(\zeta\right)q\pm\gamma_{{}_{1}}\left(\zeta\right)\alpha
q^{2}\right]\dot{q}.$ (6)

Then the standard Lagrangian describing this particle is given by

$L\left(q,\dot{q},t\right)=\frac{1}{2}\dot{q}^{2}-\left[\frac{1}{2}\omega^{2}q^{2}+\frac{1}{4}\alpha^{2}q^{4}\right],$
(7)

and the corresponding Euler-Lagrange dynamical equation reads

$\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}}\right)-\frac{\partial
L}{\partial q}+\frac{\partial\mathcal{R}}{\partial\dot{q}}=0,$ (8)

to cast the related dynamical equations

$\ddot{q}+\left[3\alpha\,q\pm\gamma_{{}_{1}}(\zeta)\right]\,\dot{q}+[\omega^{2}+\gamma_{{}_{0}}(\zeta)]q\pm\alpha\gamma_{{}_{1}}(\zeta)q^{2}+\alpha^{2}q^{3}=0,$
(9)

to be linearized. In the process, we shall use reverse engineering to come out
with exact solutions for the HO and for the DHO using the most general
solutions for the nonlinear GMEEs of (9). In so doing, we would be imposing no
initial conditions on the general solutions of (9) so that such solutions
would be the most explicit and general solutions to be adopted. Consequently,
one would then use the appropriate initial conditions that suit the problem at
hand.

The organization of the current methodical proposal is in order. In section 2,
we consider the linearization of some GMEE-type (9) dynamical equations into
HO (with $\gamma_{{}_{0}}(\zeta)=\zeta^{2}$ and
$\gamma_{{}_{1}}(\zeta)=2\zeta$) and DHO type ones (with
$\gamma_{{}_{0}}(\zeta)=3\zeta^{2}$ and $\gamma_{{}_{1}}(\zeta)=4\zeta$) and
report their exact solutions. Where, the details of our GMEE solutions suggest
that they are explicit, general and valid solutions not only for the GMEE-type
equations (9) but also for the HO and DHO type ones. No surprises for the
general solution for the HO are observed. However, for the DHO we obtain the
most explicit and general solution that offers consistency and comprehensive
coverage for the associated under-damping, critical-damping, and over-damping
cases. That is, no complex settings for $q_{{}_{\pm}}(t)$ and/or
$\dot{q}_{{}_{\pm}}(t)$ are eminent/feasible. In section 3, moreover, we
consider the linearization of an MEE-type (i.e., see (21) below, which is a
special case of (9) with $\gamma_{{}_{0}}(\zeta)=-\zeta^{2}$ and
$\gamma_{{}_{1}}(\zeta)=0$) dynamical equations into DHO-type ones and report
their exact solutions. Therein, the reported solution turns out to be
explicit, general and valid for the MEE-type (i.e., $q_{{}_{-}}(t)$) and for
the DHO-type (i.e., $U_{{}_{-}}(t)$) equations. Moreover, for each dynamical
system, we report illustrative figures for each $q_{{}_{\pm}}(t)$ and
$\\{q_{{}_{\pm}}(t),p_{{}_{\pm}}(t)\\}$ classical state (i.e., phase-space
trajectory) as they evolve in time. To the best of our knowledge, the reported
solutions, in both section 2 and 3, have never been reported elsewhere.
Section 4 is devoted for our concluding remarks.

##  II Linearization of GMEE-type into HO and DHO

###  II.1 Linearization into HO

Consider the linear harmonic oscillator dynamical equation

$\ddot{U}\left(t\right)+\omega^{2}U\left(t\right)=0.$ (10)

A nonlocal transformation in the form of (4) would transform (10) into

$\ddot{q}_{{}_{\pm}}\left(t\right)+\left(3\alpha q_{{}_{\pm}}\left(t\right)\pm
2\zeta\right)\;\dot{q}_{{}_{\pm}}\left(t\right)+\text{
}\Omega^{2}\,q_{{}_{\pm}}\left(t\right)\pm 2\zeta\alpha
q_{{}_{\pm}}\left(t\right)^{2}+\alpha^{2}q_{{}_{\pm}}\left(t\right)^{3}=0\text{
};\text{ }\Omega^{2}=\zeta^{2}+\omega^{2}.$ (11)

This dynamical equation is the GMEE (9) with
$\gamma_{{}_{0}}(\zeta)=-\zeta^{2}$ and $\gamma_{{}_{1}}(\zeta)=0$). It
describes a classical particle, with mass $m_{\circ}=1$, moving under the
influence of a quartic anharmonic oscillator potential force field

$V\left(q_{{}_{\pm}}\right)=\frac{1}{2}\omega^{2}q_{{}_{\pm}}^{2}+\frac{1}{4}\alpha^{2}q_{{}_{\pm}}^{4},$
(12)

and subjected to the Rayleigh dissipative and/or anti-dissipative force field

$\mathcal{R}_{{}_{\pm}}\left(q_{{}_{\pm}},\dot{q}_{{}_{\pm}}\right)=\frac{1}{2}\left(3\alpha
q_{{}_{\pm}}\pm 2\zeta\right)\;\dot{q}_{{}_{\pm}}^{2}+\left(\zeta^{2}\pm
2\zeta\alpha q_{{}_{\pm}}^{2}\right)\;\dot{q}_{{}_{\pm}}.$ (13)

Then the standard Lagrangian for such a system is

$L\left(q_{{}_{\pm}},\dot{q}_{{}_{\pm}};t\right)=\frac{1}{2}\;\dot{q}_{{}_{\pm}}^{2}-\frac{1}{2}\omega^{2}q_{{}_{\pm}}^{2}-\frac{1}{4}\alpha^{2}q_{{}_{\pm}}^{4},$
(14)

Figure 1:  For $A_{{}_{+}}=0.35$ and $\omega=1$ we show (a) $q_{{}_{+}}(t)$ of
(17) for $\eta=0.1$ (under-damping), $\eta=1$ (critical-damping), and for
$\eta=1.5$ (over-damping), (b) $q_{{}_{+}}(t)$ of (17) for different values of
$\eta<1$, and (c) $\\{q_{{}_{+}}(t),p_{{}_{+}}(t)=\dot{q}_{{}_{+}}(t)\\}$ of
(17) for different values of $\eta<1$. For $\eta=0.2$ and $\omega=1$ we show
(d) $q_{{}_{+}}(t)$ of (17) for different values of $A_{{}_{+}}$, (e) phase-
space trajectories for $\\{q_{{}_{+}}(t),p_{{}_{+}}(t)=\dot{q}_{{}_{+}}(t)\\}$
of (17) for different values of $A_{{}_{+}}$, and (f) phase-space trajectories
$\\{U_{{}_{+}}(t),P_{{}_{+}}(U_{{}_{+}})=\dot{U}_{{}_{+}}(t)\\}$ of (18) for
different values of $C$.

and the corresponding dynamical equation (11) is obtained by using the Euler-
Lagrange recipe

$\frac{d}{dt}\left(\frac{\partial
L}{\partial\dot{q}_{{}_{\pm}}}\right)-\frac{\partial L}{\partial
q_{{}_{\pm}}}+\frac{\partial\mathcal{R}_{{}_{\pm}}}{\partial\dot{q}_{{}_{\pm}}}=0.$
(15)

Equation (11) admits an exact solution in the form of

$q_{{}_{\pm}}\left(t\right)=\mp\frac{1}{\alpha}\left\\{\frac{C_{2}\;\left(\zeta\mp
i\omega\right)\;e^{i\omega t}+\left(\zeta\pm i\omega\right)\;e^{-i\omega
t}}{C_{1}+\left(C_{2}\;e^{i\omega t}+\;e-^{i\omega t}\right)e^{\mp\zeta
t}}\right\\}e^{\mp\zeta t}.$ (16)

Which, in a straightforward manner, with the assumptions (to facilitate the
linearization process) that $C_{2}=1$, $\alpha=2\omega$, $\zeta=\omega\eta$
and $A_{{}_{\pm}}=\mp 1/C_{1}$, could be simplified to read

$q_{{}_{\pm}}\left(t\right)=A_{{}_{\pm}}\left\\{\frac{\eta\cos\left(\omega
t\right)\pm\sin\left(\omega t\right)}{1\mp 2A_{{}_{\pm}}e^{\mp\omega\eta
t}\cos\left(\omega t\right)}\right\\}e^{\mp\omega\eta t}.$ (17)

This would in turn imply, through the nonlocal transformation (4), that

$U_{{}_{\pm}}\left(t\right)=C\left[\eta\cos\left(\omega
t\right)\pm\sin\left(\omega t\right)\right]$ (18)

as the exact textbook solution for the harmonic oscillator in (10), where one
would set $a=C\eta$ and $b=\pm C$. However, $U_{{}_{\pm}}\left(t\right)$ of
(18) is the solution for the harmonic oscillator (10) that provides a
consistent nonlocal connection, through the nonlocal transformation in (4), to
the generalized modified Emden equation (11) along with its exact general
solution (17). The linearization of the GMEE of (11) into HO (10) is clear,
therefore.

Figure 2:  For $A_{{}_{-}}=0.05$, and $\omega=1$ we show (a) $q_{{}_{-}}(t)$
of (17) for different values of $\eta<1$, (b) phase-space trajectories for
$\\{q_{{}_{-}}(t),p_{{}_{-}}(t)=\dot{q}_{{}_{-}}(t)\\}$ of (17) for different
$\eta<1$ values. For $\eta=0.05$, and $\omega=1$ we show (c) $q_{{}_{-}}(t)$
of (17) for different $A_{{}_{-}}$ values, and (d) phase-space trajectories
$\\{q_{{}_{-}}(t),p_{{}_{-}}(t)=\dot{q}_{{}_{-}}(t)\\}$ of (17) for different
values of $A_{{}_{-}}$.

In Figure 1, we use $\mathcal{R}_{{}_{+}}(q,\dot{q})$ of (13) (i.e., only
dissipative effects of the Rayleigh force field are involved) and plot, (a)
$q_{{}_{+}}(t)$ of (17) for $\eta=0.1<1$ (under damping), $\eta=1$ (critical
damping) and $\eta=1.5>1$ (over damping), as they evolve in time. One
immediately observes that such a system exactly resembles the DHO behaviour.
This DHO resemblance is inherited from the nonlocal connection $U_{{}_{+}}(t)$
of (4) between the HO (10) and the GMEE (11). In (b) and (d), we show
$q_{{}_{+}}(t)$ for different values of $\eta<1$ and $A_{{}_{+}}$,
respectively, as they evolve in time. It is clear that the frequencies of
oscillations are isochronic and amplitude-independent. The phase-space
trajectories (i.e., the classical states trajectories
$\\{q_{{}_{+}}(t),p_{{}_{+}}(t)\\}$) are plotted in (c) and (e) for different
values of $\eta<1$ and $A_{{}_{+}}$, respectively, as they evolve in time. The
trajectories exemplify the DHO behaviour as they clockwise shrink/decrease. In
(f) we show the phase-space trajectory
$\\{U_{{}_{+}}(t),P_{{}_{+}}(U_{{}_{+}})=\dot{U}_{{}_{+}}(t)\\}$ of (18) for
different amplitude $C$ values. They are indeed the textbook phase-space
trajectories for the HO (10).

In Figure 2, moreover, we show in (a) and (c) the effect of the anti-
dissipative force fields $\mathcal{R}_{{}_{-}}(q_{{}_{-}},\dot{q}_{{}_{-}})$
(13) on $q_{{}_{-}}(t)$ as it evolves in time for different values of $\eta<1$
and $A_{{}_{-}}$, respectively. We clearly observe that the effective
amplitude of oscillation increases with time while performing isochronous
oscillations. In (b) and (c) we show the phase-space trajectories as they
evolve in time for different values of $\eta<1$ and $A_{{}_{-}}$,
respectively. One may observe that such trajectories are clockwise expanding
as they evolve in time. A common characteristic behaviour for dynamical
systems exposed to anti-dissipative Rayleigh force fields
$\mathcal{R}_{{}_{-}}(q_{{}_{-}},\dot{q}_{{}_{-}})$ of (13).

###  II.2 Linearization into DHO

We now consider the damped harmonic oscillator equation

$\ddot{U}\left(t\right)+2\zeta\dot{U}\left(t\right)+\omega^{2}U\left(t\right)=0$
(19)

and use our nonlocal transformation recipe (4) for $U_{{}_{+}}\left(t\right)$
so that

$U_{{}_{+}}\left(t\right)=q_{{}_{+}}\left(t\right)\exp\left(\int\left(\alpha
q_{{}_{+}}\left(t\right)+\zeta\right)dt\right)$ (20)

to obtain

$\ddot{q}_{{}_{+}}\left(t\right)+\left(3\alpha
q_{{}_{+}}\left(t\right)+4\zeta\right)\;\dot{q}_{{}_{+}}\left(t\right)+\text{
}\tilde{\Omega}^{2}\,q_{{}_{+}}\left(t\right)+4\zeta\alpha
q_{{}_{+}}\left(t\right)^{2}+\alpha^{2}q_{{}_{+}}\left(t\right)^{3}=0\text{
};\text{ }\tilde{\Omega}^{2}=3\zeta^{2}+\omega^{2}.$ (21)

Figure 3:  For $A_{{}_{+}}=0.9$ and $\omega=1$ we show (a) $q_{{}_{+}}(t)$ of
(24) for $\eta=0.1$ (under-damping), $\eta=1$ (critical-damping), and for
$\eta=1.5$ (over-damping), (b)
$\\{q_{{}_{+}}(t),p_{{}_{+}}(t)=\dot{q}_{{}_{+}}(t)\\}$ of (24) for different
values of $\eta<1$. For $\eta=0.2$, $\omega=1$ and different values of
$A_{{}_{+}}$, we show (c) $q_{{}_{+}}(t)$ of (24), and (d) phase-space
trajectory for $\\{q_{{}_{+}}(t),p_{{}_{+}}(t)=\dot{q}_{{}_{+}}(t)\\}$ of
(24).

Such nonlinear dynamical equation (i.e., GMEE (9) with
$\gamma_{{}_{0}}(\zeta)=3\zeta^{2}$ and $\gamma_{{}_{1}}(\zeta)=4\zeta$)
describes a classical particle, with mass $m_{\circ}=1$, moving under the
influence of the quartic anharmonic oscillator potential force field of (12)
and subjected to the Rayleigh dissipative force field

$\mathcal{R}_{{}_{+}}\left(q_{{}_{+}},\dot{q}_{{}_{+}}\right)=\frac{1}{2}\left(3\alpha
q_{{}_{+}}\left(t\right)+4\zeta\right)\;\dot{q}_{{}_{+}}\left(t\right)^{2}+\left(3\zeta^{2}+4\zeta\alpha
q_{{}_{+}}\left(t\right)^{2}\right)\;\dot{q}_{{}_{+}}\left(t\right).$ (22)

Then the standard Lagrangian for such a system is readily given in (14) and
the corresponding nonlinear dynamical equation (21) is obtained through the
Euler-Lagrange recipe of (15). Nevertheless, the nonlinear dynamical equation
(21) admits an exact solution in the form of

$q_{{}_{+}}\left(t\right)=-\frac{e^{-2\zeta
t}}{\alpha}\left\\{\frac{\left(2\zeta+\beta\right)e^{-\beta
t}+C_{2}\left(2\zeta-\beta\right)e^{\beta t}}{C_{1}+e^{-2\zeta
t}\left(C_{2}e^{\beta t}+e^{-\beta t}\right)}\right\\}\text{ };\text{
}\beta=\omega\sqrt{\eta^{2}-1}.$ (23)

Which is simplified, with $C_{2}=1$, $\alpha=$ $\zeta=\omega\eta$, and
$C_{1}=-2/A$, into

$q_{{}_{+}}\left(t\right)=A\left\\{\frac{2\cosh\left(\beta
t\right)-\left(\beta/\omega\eta\right)\sinh\left(\beta
t\right)}{1-Ae^{-2\omega\eta t}\cosh\left(\beta
t\right)}\right\\}e^{-2\omega\eta t}.$ (24)

This result when substituted in (20) yields

$U_{+}\left(t\right)=A\left[2\cosh\left(\beta
t\right)-\frac{\beta}{\omega\eta}\sinh\left(\beta
t\right)\right]e^{-\omega\eta t},$ (25)

as the exact solution of (19). Then the linearization of (21) into the linear
damped harmonic oscillator equation (19) is clear.

In Figure 3, we show the linearization into DHO (19) effect on the GMEE of
(21) and plot (a) $q_{{}_{+}}(t)$ of (24) for $\eta=0.1<1$ (under damping),
$\eta=1$ (critical damping) and $\eta=1.5>1$ (over damping), as they evolve in
time, (b) $\\{q_{{}_{+}}(t),p_{{}_{+}}(t)=\dot{q}_{{}_{+}}(t)\\}$ of (24) for
different values of $\eta<1$. For $\eta=0.2$, $\omega=1$ and different values
of $A_{{}_{+}}$ we show (c) $q_{{}_{+}}(t)$ of (24), and (d) phase-space
trajectory for $\\{q_{{}_{+}}(t),p_{{}_{+}}(t)=\dot{q}_{{}_{+}}(t)\\}$ of
(24). Again, one observes that the system resembles the DHO behaviour, where
the phase-space trajectories in (b) and (d) show similar trend of clockwise
shrinking/decreasing as they evolve in time.

##  III Linearization of MEE-type into DHO

We now consider the damped harmonic oscillator equation in (19) and use our
nonlocal transformation recipe (4) for $U_{-}\left(t\right)$ so that

Figure 4:  For $A=B=0.5$, $\alpha=1$ and $\omega=2$ we show (a)
$q_{{}_{-}}(t)$ of (30) for different $\eta<1$ values, and (b) phase-space
trajectories $\\{q_{{}_{-}}(t),p_{{}_{-}}(t)=\dot{q}_{{}_{-}}(t)\\}$ of (30)
for different values of $\eta<1$. For $B=0.5$, $\alpha=1$, $\omega=2$, and
different values of $A$ we show (c) $q_{{}_{-}}(t)$ of (30), and (d) phase-
space trajectories $\\{q_{{}_{-}}(t),p_{{}_{-}}(t)=\dot{q}_{{}_{-}}(t)\\}$ of
(30). For $\alpha=1$, $\omega=2$ and $\eta<1$ we show (e)
$\\{q_{{}_{-}}(t),p_{{}_{-}}(t)=\dot{q}_{{}_{-}}(t)\\}$ of (30) for
$A=0.7>B=0.2$, and (f) $\\{q_{{}_{-}}(t),p_{{}_{-}}(t)=\dot{q}_{{}_{-}}(t)\\}$
of (30) for $B=0.7>A=0.2$.

$U_{-}\left(t\right)=q_{{}_{-}}\left(t\right)\exp\left(\int\left(\alpha
q_{{}_{-}}\left(t\right)-\zeta\right)dt\right)$ (26)

to obtain

$\ddot{q}_{{}_{-}}\left(t\right)+3\alpha
q_{{}_{-}}\left(t\right)\dot{q}_{{}_{-}}\left(t\right)+\text{
}\tilde{\beta}^{2}\,q_{{}_{-}}\left(t\right)+\alpha^{2}q_{{}_{-}}\left(t\right)^{3}=0\text{
};\text{ }\tilde{\beta}^{2}=\omega^{2}-\zeta^{2}.$ (27)

This nonlinear differential equation is the MEE and, herein, it describes a
classical particle, with mass $m_{\circ}=1$, moving under the influence of the
quartic anharmonic oscillator potential force field of (12) and subjected to
the Rayleigh dissipative and/or anti-dissipative force field

$\mathcal{R}_{{}_{-}}\left(q_{{}_{-}},\dot{q}_{{}_{-}}\right)=\frac{3}{2}\alpha
q_{{}_{-}}\left(t\right)\;\dot{q}_{{}_{-}}\left(t\right)^{2}-\zeta^{2}q_{{}_{-}}\left(t\right)\dot{q}_{{}_{-}}\left(t\right).$
(28)

Then the standard Lagrangian for such a system is readily given in (14) and
the corresponding nonlinear dynamical equation (21) is obtained through the
Euler-Lagrange recipe of (15). Nevertheless, the nonlinear dynamical equation
(27) admits an exact solution in the form of

$q_{{}_{-}}\left(t\right)=\frac{\tilde{\beta}}{\alpha}\left\\{\frac{C_{2}\cos\left(\tilde{\beta}t\right)-\sin\left(\tilde{\beta}t\right)}{C_{1}+C_{2}\sin\left(\tilde{\beta}t\right)+\cos\left(\tilde{\beta}t\right)}\right\\}\text{
};\tilde{\beta}=i\beta=i\omega\sqrt{\eta^{2}-1}.$ (29)

With $B=1/C_{1}$, $A=\tilde{\beta}C_{2}/C_{1}$, $\zeta=\omega\eta$ and
$\tilde{\beta}=i\beta$ one obtains

$q_{{}_{-}}\left(t\right)=\frac{1}{\alpha}\left\\{\frac{A\cos\left(\tilde{\beta}t\right)-B\tilde{\beta}\sin\left(\tilde{\beta}t\right)}{1+\frac{A}{\tilde{\beta}}\sin\left(\tilde{\beta}t\right)+B\cos\left(\tilde{\beta}t\right)}\right\\}\Longleftrightarrow
q_{{}_{-}}\left(t\right)=\frac{1}{\alpha}\left\\{\frac{A\cosh\left(\beta
t\right)+B\beta\sinh\left(\beta t\right)}{1+\frac{A}{\beta}\sinh\left(\beta
t\right)+B\cosh\left(\beta t\right)}\right\\}$ (30)

This result when substituted in (26) would yield

$U_{-}\left(t\right)=\frac{1}{\alpha}\left[A\cosh\left(\beta
t\right)+B\beta\sinh\left(\beta t\right)\right]e^{-\omega\eta t}$ (31)

as the exact solution of (19). Obviously, the solutions (30) as well as (31)
suggest that the parameter $\alpha$ , in this case, may take any value
provided that $\alpha\in\mathbb{R}$. Moreover, we observe that the interesting
solution reported by Chandrasekar et al. Chandrasekar 2012  (obtained by using
$U(t)=A\,sin(\omega t)$ as the solution for the HO and mapped it into MEE) is
a especial case of our general solution in (30), where our
$\tilde{\beta}=\omega$ of Chandrasekar 2012 .

In Figure 4, for $\omega=2$, $\alpha=1$, $A=B=0.5$, and different values of
$\eta<1$, we plot (a) $q_{{}_{-}}(t)$ of (30), and (b) phase-space
trajectories $\\{q_{{}_{-}}(t),p_{{}_{-}}(t)=\dot{q}_{{}_{-}}(t)\\}$ of (30).
For $B=0.5$, $\alpha=1$, $\omega=2$, and different values of $A$ we show (c)
$q_{{}_{-}}(t)$ of (30), and (d) phase-space trajectories
$\\{q_{{}_{-}}(t),p_{{}_{-}}(t)=\dot{q}_{{}_{-}}(t)\\}$ of (30). For
$\alpha=1$, $\omega=2$ and $\eta<1$ we show (e)
$\\{q_{{}_{-}}(t),p_{{}_{-}}(t)=\dot{q}_{{}_{-}}(t)\\}$ of (30) for
$A=0.7>B=0.2$, and (f) $\\{q_{{}_{-}}(t),p_{{}_{-}}(t)=\dot{q}_{{}_{-}}(t)\\}$
of (30) for $B=0.7>A=0.2$. Obviously, the oscillations are non-isochronic as
documented in (a) and (c). Moreover, whilst the phase-space trajectories in
(b), (d), and (f) replicate those of the MEE Chandrasekar 2012 , the phase-
space trajectories in (e) exhibit classical states
$\\{q_{{}_{-}}(t),p_{{}_{-}}(t)\\}$ crossings for $A=0.7>B=0.2$. No classical
states crossings for $B\leq A$ are observed (as documented in (b), (d), and
(f)). It is, therefore, necessary and sufficient to use the most general
solution (30) for MEE so that one carries out comprehensive analysis of such a
commonly used dynamical system (note that our $\tilde{\beta}$ corresponds to
$\omega$ in Chandrasekar 2012 ).

##  IV Concluding remarks

In the current methodical proposal, we have considered the linearization of
some GMEE-type (1) dynamical equations into HO and DHO type ones and reported
their exact solutions. We have shown that structure of the nonlocal
transformation and the linearizability into HO or DHO determine the
nature/structure of the dynamical forces involved (consequently determine the
form of the dynamical equation). The details of the obtained solutions suggest
that they are explicit, general and valid solutions not only for the GMEE-type
(i.e., $q_{{}_{\pm}}(t)$) equations (9) but also for the HO and DHO type
(i.e., $U_{{}_{\pm}}(t)$) ones. Moreover, we have considered the linearization
of a MEE-type (21) dynamical equations into DHO-type ones and reported their
exact solutions. Therein, the reported solutions turned out to be explicit,
general and valid for the MEE-type (i.e., $q_{{}_{-}}(t)$) and for the DHO-
type (i.e., $U_{{}_{-}}(t)$) equations. Yet, for each dynamical system we have
reported illustrative figures for each $q_{{}_{\pm}}(t)$ and
$\\{q_{{}_{\pm}}(t),p_{{}_{\pm}}(t)\\}$ classical state (i.e., phase-space
trajectory) as they evolve in time. To the best of our knowledge, the reported
solutions, in both section 2 and 3, have never been reported elsewhere.

Figure 5:  For the DHO-equation in (39) with $A=B=1$ and $\omega=1$, we show
(a) $U_{{}_{+}}(t)$ of (40) for $\eta=0.1$ (under-damping), $\eta=1$
(critical-damping), and for $\eta=1.5$ (over-damping), and (b) the phase-space
trajectories $\\{U_{{}_{+}}(t),p_{{}_{+}}(t)=\dot{U}_{{}_{+}}(t)\\}$ for
different values of $\eta<1$. For $A=B=1$ and $\omega=0.5$, we show (c)
$U_{{}_{-}}(t)$ of (40) for different values of $\eta<1$, and (d) the phase-
space trajectories
$\\{U_{{}_{+}}(t),p_{{}_{-}}(U_{{}_{-}}(t))=\dot{U}_{{}_{-}}(t)\\}$ for
different values of $\eta<1$.

In the linearization process, nevertheless, and as a byproduct of our reverse
engineering strategy, we have started from the general solutions for the GMEE
(17) and (24) to come out with the most general solutions for the HO (18), DHO
(25) and (31). The solution $U_{{}_{\pm}}\left(t\right)$ in (18) for the HO in
(10) turned out to be a textbook one, provided that $a=C\eta$ and $b=\pm C$.,
that provides a consistent nonlocal connection, through the nonlocal
transformation in (4), to the generalized modified Emden equation (11) along
with its exact general solution (17). However, for the DHO one may eventually
absorb the parameters in the related amplitudes of $U_{+}\left(t\right)$ in
(25) and of $U_{-}\left(t\right)$ in (31) and cast the most general solution
for the DHO problem (19) as

$U\left(t\right)=\left[A\cosh\left(\beta t\right)+B\beta\sinh\left(\beta
t\right)\right]e^{-\omega\eta t}\text{ };\beta=\omega\sqrt{\eta^{2}-1},$ (32)

where we have used

$\displaystyle A=\left\\{\begin{tabular}[]{ll}$2A$&; of $U_{+}\left(t\right)$
in (\ref{DHO-sol})\\\ $\frac{A}{\alpha}$&; of $U_{-}\left(t\right)$ of
(\ref{DHO-Gsol})\end{tabular}\right\\},$ (35) $\displaystyle
B=\left\\{\begin{tabular}[]{ll}$-\frac{A}{\omega\eta}$&; of
$U_{{}_{+}}\left(t\right)$ in (\ref{DHO-sol})\\\ $\frac{B}{\alpha}$&; of
$U_{{}_{-}}\left(t\right)$ of (\ref{DHO-Gsol})\end{tabular}\right\\}.$ (38)

Hence, one may now use $\eta<1$ (for under damping), $\eta=1$ (for critical
damping), and $\eta>1$ (for over damping) without any consequential complex
settings in $U(t)$ and/or in $\dot{U}(t)$. This would also allow us to choose
proper initial conditions on $U(0)$ and/or $\dot{U}(0)$ associated with the
DHO problem at hand.

Yet, one may rewrite the DHO equation (19) to include both damped and anti-
damped harmonic oscillators as

$\ddot{U}_{{}_{\pm}}\left(t\right)\pm
2\zeta\dot{U}_{{}_{\pm}}\left(t\right)+\omega^{2}U_{{}_{\pm}}\left(t\right)=0,$
(39)

then the general solution in (32) would read

$U_{{}_{\pm}}\left(t\right)=\left[A\cosh\left(\beta
t\right)+B\beta\sinh\left(\beta t\right)\right]e^{\mp\omega\eta t}\text{ }.$
(40)

Such damped or anti-damped linear harmonic oscillator systems are eminent
(also physically viable) consequences of the linearization process discussed
above. In Figure 5, we show the behaviour of such general solution (40). For
$A=B=1$ and $\omega=1$, we show (a) $U_{{}_{+}}(t)$ of (40) for $\eta=0.1$
(under-damping), $\eta=1$ (critical-damping), for $\eta=1.5$ (over-damping),
and (b) the phase-space trajectories
$\\{U_{{}_{+}}(t),p_{{}_{+}}(U_{{}_{+}}(t))=\dot{U}_{{}_{+}}(t)\\}$ for
different values of $\eta<1$. For $A=B=1$ and $\omega=0.5$, we show (c)
$U_{{}_{-}}(t)$ of (40) for different values of $\eta<1$, and (d) the phase-
space trajectories
$\\{U_{{}_{+}}(t),p_{{}_{-}}(U_{{}_{-}}(t))=\dot{U}_{{}_{-}}(t)\\}$ for
different values of $\eta<1$. Obviously, the typical behaviour of the DHO and
anti-DHO are exhibited therein.

Finally, it would be interesting if further coordinate transformation (e.g.,
$q_{{}_{\pm}}(t)\rightarrow\sqrt{Q(x(t))}x(t)$) is carried out to effectively
yield position-dependent mass (PDM) settings (c.f., e.g., Mustafa PS 2021 ;
Mustafa 2015 ; Mustafa Algadhi 2019 ; Nabulsi1 2020 ; Nabulsi2 2020 ; Quesne
2019 ; da Costa1 2020 ; Ranada 2016 ; Carinena Herranz 2017 ). One would then
come out with a mixed type Liénard equation to be mapped into GMEE type (which
is effectively a linear type Liénard equation) and consequently into HO or
DHO. The new PDM system would then inherit the exact solutions reported above.

## References

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# Quasi Regular Semilattice and Association Schemes in Singular Linear Space

Zhang Baohuan1 Yue Mengtian2 Li Zengti1
1.Math. and Inf. College, Langfang Teachers’ College, Langfang, 065000, China
2.Department of Scientific Research, Langfang Teachers’ College, Langfang,
065000, China<EMAIL_ADDRESS>
Abstract Let $\mathbb{F}_{q}^{n+l}$ denote the $(n+l)$-dimensional singular
linear space over a finite field $\mathbb{F}_{q}$. For a fixed integer
$m\leq\min\\{n,l\\}$, denote by $\mathcal{L}^{m}_{o}(\mathbb{F}_{q}^{n+l})$
the set of all subspaces of type $(t,t_{1})$, where $t_{1}\leq t\leq m$.
Partially ordered by ordinary inclusion, one family of quasi regular
semilattices is obtained. Moreover, we obtain a association schemes and
discuss the bound of a $M$-clique.

AMS classification : 20G40, 05B35

Key words: Quasi regular semilattice; Association scheme; $M$-clique

##  1 Introduction

It is well known that lattice is an important part of poset’s theory. its
theory play an important role in many branches of mathematics, such as
computer logical design. The results on the lattices generated by transitive
sets of subspaces under finite classical groups may be found in Huo, Liu and
Wan [8, 9, 10]. In [4], Guo discussed the lattices associated with finite
vector spaces and finite affine spaces. In [12], P. Delsarte discussed the
regular semilattices in finite vector spaces. In this paper, we obtain a new
quasi regular semilattice and a new association scheme in the singular linear
space. More over we discuss the bounds of a $M$-clique.

The rest of this paper is organized as followed. In section 2, we discuss some
definitions and terminologies about lattices, regular semilattices,
association schemes and $M$-cliques. In section 3, we construct a family of
quasi regular semilattice, and then compute its parameters. In section 4, we
obtain a new association. In section 5, we discuss the bound of a $M$-clique.

##  2 Preliminaries

Let $(P,\leq)$ be a poset. We write $a<b$ whenever $a\leq b$ and $a\neq b$. If
$P$ has the minimum (respectively maximum) element, then we denote it by 0
(respectively $\upharpoonleft$), and say that $P$ is a poset with 0
(respectively $\upharpoonleft$). A poset $P$ is said to be a semilattice if
$a\wedge b:=\rm{inf}\\{a,b\\}$ exist for any two elements $a,b\in P$. Let P be
a finite poset with 0. If there is a function $r$ from $P$ to set of all the
nonnegative integers such that

  * (1)

r(0)=0,

  * (2)

$r(b)=r(a)+1$, if $a\lessdot b$.

Then $r$ is said to be the rank function on $P$. Note that the rank function
on $P$ is unique if it exists.

Let $P$ be a semilattice, and let $P=X_{0}\cup X_{1}\cup\cdots\cup X_{m}$,
where $X_{i}=\\{x\in P|r(x)=i\\},i=0,1,\cdots,m.$ The semilattice $(P,\leq)$
is called regular if the following three properties hold:

  * (i)

Given $y\in X_{m},z\in X_{r}$ with $z\leq y$, the number of points $u\in
X_{s}$ such that $z\leq u\leq y$ is a constant $\mu(r,s)$.

  * (ii)

Given $u\in X_{s}$ , the number of points $z\in X_{r}$, such that $z\leq u$ is
a constant $\nu(r,s)$.

  * (iii)

Given $a\in X_{r},,y\in X_{m}$, with $a\wedge y\in X_{j}$, the number of pairs
$(b,z)\in X_{s}\times X_{m}$ such that $b\leq z,b\leq y,a\leq z$ is a constant
$\pi(j,r,s).$

In this paper, we define the concept of quasi regular semilattice as follows.

The semilattice $(P,\leq)$ is called quasi regular if the following three
properties hold:

  * (i)

Given $y\in X^{m^{\prime}}_{m},z\in X^{r^{\prime}}_{r}$ with $z\leq y$, the
number of points $u\in X^{s^{\prime}}_{s}$ such that $z\leq u\leq y$ is a
constant $\mu(r(r^{\prime}),s(s^{\prime});m^{\prime})$.

  * (ii)

Given $u\in X^{s^{\prime}}_{s}$ , the number of points $z\in
X^{r^{\prime}}_{r}$, such that $z\leq u$ is a constant
$\nu(r(r^{\prime}),s(s^{\prime}))$.

  * (iii)

Given $a\in X^{r^{\prime}}_{r},y\in X^{m_{1}}_{m}$, with $a\wedge y\in
X^{j^{\prime}}_{j}$, the number of pairs $(b,z)\in X^{s^{\prime}}_{s}\times
X^{m^{\prime}}_{m}$ such that $b\leq z,b\leq y,a\leq z$ is a constant
$\pi(j(j^{\prime}),r(r^{\prime}),s(s^{\prime});m^{\prime}).$

Here $X_{i}=X_{i}^{0}\cup X_{i}^{1}\cup\cdots\cup X_{i}^{i}$, and
$X_{i}^{j}\cap X_{i}^{k}=\emptyset$ for $j\neq k.$

Let $X$ be a finite set of vertices. A $d$-class association scheme on $X$
consists of a set of $d+1$ symmetric relations $R_{0},R_{1},\cdots,R_{d}$ on
$V$, with identity relation $R_{0}=\\{(x,x)|x\in V\\},$ such that any two
vertices are in precisely one relation. Denoted by $(X,\\{R_{i}\\}_{0\leq
i\leq d})$. Furthermore, there are intersection numbers $p^{k}_{ij}$ such that
for any $(x,y)\in R_{k}$, the number of vertices $z$ such that $(x,z)\in
R_{i}$ and $(z,y)\in R_{j}$ equals $p^{k}_{ij}$.

The nontrivial relations can be considered as graphs, which in our case are
undirected. One immediately sees that the respective graphs are regular with
degree $v_{i}=p_{ii}^{0}$. For the corresponding adjacency matrices $A_{i}$
the axioms of the scheme are equivalent to

$\sum_{i=0}^{d}A_{i}=J,\ A_{0}=I,\ A_{i}=A_{i}^{T},\
A_{i}A_{j}=\sum_{k=0}^{d}p_{ij}^{k}A_{k}.$

It follows that the adjacency matrices generate a $(d+1)$-dimensional
commutative algebra $\mathfrak{A}$ of symmetric matrices. This algebra was
first studied by Bose and Mesner [19] and is called the Bose-Mesner algebra of
the scheme.

A nonzero vector $\alpha\in\mathbb{R}^{|X|}$ is said to be $\theta$-positive ,
if $C_{i}(\alpha)=v_{i}^{-1}<\alpha,A_{i}\alpha>\geq 0$ holds for every $i$.
Let $M$ be a subset of $[0,n]$, containing $0$, Any positive vector $\alpha$
is called an $M$-clique if it satisfies

$C_{j}(\alpha)=v_{j}^{-1}<\alpha,A_{j}\alpha>=0,\ for\ all\ j\not\in M$

Let $x_{0}$ be a point of $X$ and $s$ be an integer in $[0,n]$, A given
$\theta$-positive vector $\alpha\in\mathbb{R}^{|X|}$ is called a unicoloured
vector of center $x_{0}$ and colour $s$, if the following condition holds:

$\alpha(x)=0,\ unless\ (x,x_{0})\in R_{s}$

Let $\mathbb{F}_{q}$ be a finite field with $q$ elements, where $q$ is a prime
power. For two non-negative integers $n$ and $l$, $\mathbb{F}_{q}^{n+l}$
denotes the $(n+l)$-dimensional row vector space over $\mathbb{F}_{q}$. The
set of all $(n+l)\times(n+l)$ nonsingular matrices over $\mathbb{F}_{q}$ of
the form

$\left(\begin{array}[]{cc}T_{11}&T_{12}\\\ 0&T_{22}\end{array}\right),$

where $T_{11}$ and $T_{22}$ are nonsingular $n\times n$ and $l\times l$
matrices, respectively, forms a group under matrix multiplication, called the
singular general linear group of degree $n+l$ over $\mathbb{F}_{q}$ and
denoted by $GL_{n+l,n}(\mathbb{F}_{q})$.

Let $P$ be an $m$-dimensional subspace of $\mathbb{F}_{q}^{n+l}$, denote also
by $P$ a $m\times(n+l)$ matrix of rank $m$ whose rows span the subspace $P$
and call the matrix $P$ a matrix representation of the subspace $P$ . There is
an action of $GL_{n+l,n}(\mathbb{F}_{q})$ on $\mathbb{F}_{q}^{n+l}$ defined as
follows

$\begin{array}[]{rrr}\mathbb{F}_{q}^{n+l}\times
GL_{n+l,n}(\mathbb{F}_{q})&\longrightarrow&\mathbb{F}_{q}^{n+l},\\\
((x_{1},\cdots,x_{n},x_{n+1},\cdots,x_{n+l}),T)&\longmapsto&(x_{1},\cdots,x_{n},x_{n+1},\cdots,x_{n+l})T.\end{array}$

The above action induces an action on the set of subspaces of
$\mathbb{F}_{q}^{n+l}$; i.e., a subspace $P$ is carried by $T\in
GL_{n+l,n}(\mathbb{F}_{q})$ to the subspace $PT$. The vector space
$\mathbb{F}_{q}^{n+l}$ together with the above group action is called the
$(n+l)$-dimensional singular linear space over $\mathbb{F}_{q}$.

For $1\leq i\leq n+l$, let $e_{i}$ be the row vector in $\mathbb{F}_{q}^{n+l}$
whose $i$-th coordinate is $1$ and all other coordinates are $0$. Denote by
$E$ the $l$-dimensional subspace of $\mathbb{F}_{q}^{n+l}$ generated by
$e_{n+1},e_{n+2},\cdots,e_{n+l}$. A $m$-dimensional subspace $P$ of
$\mathbb{F}_{q}^{n+l}$ is called a subspace of type $(m,k)$ if $\dim(P\cap
E)=k$.

For a fixed subspace $P$ of type $(m_{1},k_{1})$ in $\mathbb{F}_{q}^{n+l}$,
let $\mathcal{M}^{\prime}(m_{1},k_{1};m,k;n+l,n)$ denote the set of all the
subspaces of type $(m,k)$ containing $P$.
$\mathcal{M}^{\prime}(m_{1},k_{1};m,k;n+l,n)$ is non-empty if and only if
$0\leq k_{1}\leq k\leq l$ and $0\leq m_{1}-k_{1}\leq m-k\leq n$. Let
$N^{\prime}(m_{1},k_{1};m,k;n+l,n)=|\mathcal{M}^{\prime}(m_{1},k_{1};m,k;n+l,n)|.$
The formula is given by Kaishun Wang, Jun Guo, Fenggao Li (see [7]) as
follows.

$N^{\prime}(m_{1},k_{1};m,k;n+l,n)=q^{(l-k)(m-k-
m_{1}+k_{1})}\left[n-(m_{1}-k_{1})\atop(m-k)-(m_{1}-k_{1})\right]_{q}\left[l-k_{1}\atop
k-k_{1}\right]_{q}.$

###### Lemma 2.1

Let $0\leq k_{1}\leq k_{2}\leq k\leq l,0\leq m_{1}-k_{1}\leq m_{2}-k_{2}\leq
m-k\leq n.$ For $i=1,2,3$, assume that $U_{i}$ be a set of all the subspaces
of type $(m_{i},k_{i})$ in $\mathbb{F}_{q}^{n+l}$, and $U$ be a set of all the
subspaces of type $(m,k)$ in $\mathbb{F}_{q}^{n+l}$ such that $U_{1}\subseteq
U_{2}\subseteq U$ and $U_{3}\subseteq U$. if $U_{3}\cap U_{2}=U_{1}$, then the
number of $U_{3}$ is

$q^{(\delta_{3}-\delta_{1})(\delta_{2}-\delta_{1}+k-k_{3})+(k_{3}-k_{1})(k_{2}-k_{1})}\left[\delta-\delta_{2}\atop\delta_{3}-\delta_{1}\right]_{q}\left[k-k_{2}\atop
k_{3}-k_{1}\right]_{q},$

where $\delta=m-k,\delta_{i}=m_{i}-k_{i},i=1,2,3$.

Proof. Let
$\delta_{1}=m_{1}-k_{1},\delta_{2}=m_{2}-k_{2},\delta_{3}=m_{3}-k_{3},\delta=m-k$.
By the transitivity of $GL_{n+l,n}(\mathbb{F}_{q})$ on the set of subspaces of
the same type, we may assume that

$\begin{array}[]{l}\begin{array}[]{llll}\ \ \ \ \ \ \ \ \
\delta_{1}&n-\delta_{1}&k_{1}&l-k_{1}\end{array}\\\
U_{1}=\left(\begin{array}[]{llll}I&0&0&0\\\
0&0&I&0\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\
\delta_{1}\\\ k_{1}\end{array}\end{array}$
$\begin{array}[]{l}\begin{array}[]{llllll}\ \ \ \ \ \ \ \ \
\delta_{1}&\delta_{2}-\delta_{1}&n-\delta_{2}&k_{1}&k_{2}-k_{1}&l-k_{2}\end{array}\\\
U_{2}=\left(\begin{array}[]{llllll}I&0&0&0&0&0\\\ 0&0&0&I&0&0\\\
0&I&0&0&0&0\\\
0&0&0&0&I&0\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\
\delta_{1}\\\ k_{1}\\\ \delta_{2}-\delta_{1}\\\
k_{2}-k_{1}\end{array}\end{array}$

and

$\begin{array}[]{l}\begin{array}[]{llllllll}\ \ \ \ \ \ \ \ \ \
\delta_{1}&\delta_{2}-\delta_{1}&\delta-\delta_{2}&n-\delta&k_{1}&k_{2}-k_{1}&k-k_{2}&l-k\end{array}\\\
U=\left(\begin{array}[]{llllllll}I&0&0&0&0&0&0&0\\\ 0&0&0&0&I&0&0&0\\\
0&I&0&0&0&0&0&0\\\ 0&0&0&0&0&I&0&0\\\ 0&0&I&0&0&0&0&0\\\ 0&0&0&0&0&0&I&0\\\
\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\
\delta_{1}\\\ k_{1}\\\ \delta_{2}-\delta_{1}\\\ k_{2}-k_{1}\\\
\delta-\delta_{2}\\\ k-k_{2}\end{array}\end{array}$

Since $U_{3}\cap U_{2}=U_{1},$ we have

$\begin{array}[]{l}\begin{array}[]{llllllll}\ \ \ \ \ \ \ \ \
\delta_{1}&\delta_{2}-\delta_{1}&\delta-\delta_{2}&n-\delta&k_{1}&k_{2}-k_{1}&k-k_{2}&l-k\end{array}\\\
U_{3}=\left(\begin{array}[]{llllllll}I&0&0&0&0&0&0&0\\\ 0&0&0&0&I&0&0&0\\\
0&u_{32}&u_{33}&0&0&u_{36}&u_{37}&0\\\ 0&0&0&0&0&u_{46}&u_{47}&0\\\
\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\
\delta_{1}\\\ k_{1}\\\ \delta_{3}-\delta_{1}\\\ k_{3}-k_{1}\\\
\end{array}\end{array}$

where rank $u_{47}=k_{3}-k_{1}$. Note that there are $\left[k-k_{2}\atop
k_{3}-k_{1}\right]_{q}$ choices for $u_{47}$. By the transitivity of
$GL_{n+l,n}(\mathbb{F}_{q})$ on the set of subspaces of the same type, the
number of $U_{3}$’s does not depend on the particular choice of $u_{47}$. Pick
$u_{47}=(I^{(k_{3}-k_{1})},0)$. Then $U_{3}$ has a matrix representation

$\begin{array}[]{l}\begin{array}[]{lllllllll}\ \
\delta_{1}&\delta_{2}-\delta_{1}&\delta-\delta_{2}&n-\delta&k_{1}&k_{2}-k_{1}&k_{3}-k_{1}&k-k_{2}-k_{3}+k_{1}&l-k\end{array}\\\
\left(\begin{array}[]{lllllllll}I&0&0&0&0&0&0&0&0\\\ 0&0&0&0&I&0&0&0&0\\\
0&u_{32}&u_{33}&0&0&u_{36}&0&u^{\prime}_{37}&0\\\ 0&0&0&0&0&u_{46}&I&0&0\\\
\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\
\delta_{1}\\\ k_{1}\\\ \delta_{3}-\delta_{1}\\\ k_{3}-k_{1}\\\
\end{array}\end{array}$

where rank $u_{33}=\delta_{3}-\delta_{1}$. Therefore the number of $U_{3}$ is
equal to

$q^{(\delta_{3}-\delta_{1})(\delta_{2}-\delta_{1}+k-k_{3})+(k_{3}-k_{1})(k_{2}-k_{1})}\left[\delta-\delta_{2}\atop\delta_{3}-\delta_{1}\right]_{q}\left[k-k_{2}\atop
k_{3}-k_{1}\right]_{q}.$

$\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

##  3 The quasi regular semilattice

For a fixed integer $m\leq\min\\{n,l\\}$, denote by
$\mathcal{L}^{m}(\mathbb{F}_{q}^{n+l})$ the set of all subspaces of type
$(t,t_{1})$, where $t_{1}\leq t\leq m$. If we partially order
$\mathcal{L}^{m}(\mathbb{F}_{q}^{n+l})$ by the ordinary inclusion, then
$\mathcal{L}^{m}(\mathbb{F}_{q}^{n+l})$ is a semilattice, denoted by
$\mathcal{L}^{m}_{o}(\mathbb{F}_{q}^{n+l})$. For any
$A\in\mathcal{L}^{m}_{o}(\mathbb{F}_{q}^{n+l})$, The rank function of
$\mathcal{L}_{o}(\mathbb{F}_{q}^{n+l})$ is defined as follows

$r(A)=\dim(A).$

Let

$X_{i}=\\{B\in\mathcal{L}^{m}_{o}(\mathbb{F}_{q}^{n+l})|r(B)=i\\},$

and

$X_{i}^{j}=\\{B\in X_{i}|\dim(B\cap E)=j\\},j=0,1,\cdots,i,$

where $E=\langle
e_{n+1},e_{n+2},\cdots,e_{n+l}\rangle\in\mathbb{F}_{q}^{n+l}.$ We will prove
that $\mathcal{L}^{m}_{o}(\mathbb{F}_{q}^{n+l})$ is a quasi regular
semilattice and compute its parameters.

###### Lemma 3.1

Let $A\in X^{r_{1}}_{r}$, $C\in X^{m_{1}}_{m}$ and $A\leq C$. Then the number
of $B\in X^{s_{1}}_{s}$ such that $A\leq B\leq C$ is equal to

$\mu(r(r_{1}),s(s_{1});m_{1})=q^{(s-s_{1}-r+r_{1})(m_{1}-s_{1})}\left[\begin{array}[]{l}m-r+r_{1}-m_{1}\\\
s-s_{1}-r+r_{1}\end{array}\right]_{q}\left[\begin{array}[]{l}m_{1}-r_{1}\\\
s_{1}-r_{1}\end{array}\right]_{q}$

Proof. By the transitivity of $GL_{n+l}(\mathbb{F}_{q})$ on the set of
subspaces of the same type, we may assume that

$\begin{array}[]{l}\begin{array}[]{llll}\ \ \ \ \ \ \ \
r-r_{1}&n-r+r_{1}&r_{1}&l-r_{1}\end{array}\\\
A=\left(\begin{array}[]{llll}I&0&0&0\\\
0&0&I&0\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\
r-r_{1}\\\ r_{1}\end{array}\end{array}$

and

$\begin{array}[]{l}\begin{array}[]{llllll}\ \ \ \ \ \ \ \
r-r_{1}&m-m_{1}-r+r_{1}&n-m+m_{1}&r_{1}&m_{1}-r_{1}&l-m_{1}\end{array}\\\
C=\left(\begin{array}[]{llllll}I&0&0&0&0&0\\\ 0&0&0&I&0&0\\\ 0&I&0&0&0&0\\\
0&0&0&0&I&0\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\
r-r_{1}\\\ r_{1}\\\ m-m_{1}-r+r_{1}\\\ m_{1}-r_{1}\end{array}.\end{array}$

Since

$A\leq B\leq C,$

we have

$\begin{array}[]{l}\begin{array}[]{llllll}\ \ \ \ \ \ \ \
r-r_{1}&m-m_{1}-r+r_{1}&n-m+m_{1}&r_{1}&m_{1}-r_{1}&l-m_{1}\end{array}\\\
B=\left(\begin{array}[]{llllll}I&0&0&0&0&0\\\ 0&0&0&I&0&0\\\
0&u_{32}&0&0&u_{35}&0\\\
0&0&0&0&u_{45}&0\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\
r-r_{1}\\\ r_{1}\\\ s-s_{1}-r+r_{1}\\\ s_{1}-r_{1}\end{array}\end{array},$

where rank $u_{45}=(s_{1}-r_{1})$. Note that there are $\left[m_{1}-r_{1}\atop
s_{1}-r_{1}\right]_{q}$ choices for $u_{45}$. By the transitivity of
$GL_{n+l}(\mathbb{F}_{q})$ on the set of subspaces of the same type, the
number of $B$’s does not depend on the particular choice of $u_{45}$. Pick
$u_{45}=(I^{(s_{1}-r_{1})},0)$. Then $B$ has a matrix representation

$\begin{array}[]{l}\begin{array}[]{lllllll}\ \
\delta_{2}&\delta_{1}-\delta_{2}&n-\delta_{1}&r_{1}&s_{1}-r_{1}&m_{1}-s_{1}&l-m_{1}\end{array}\\\
\left(\begin{array}[]{lllllll}I&0&0&0&0&0&0\\\ 0&0&0&I&0&0&0\\\
0&u_{32}&0&0&0&u^{\prime}_{35}&0\\\
0&0&0&0&I&0&0\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\
\delta_{2}\\\ r_{1}\\\ \delta_{3}-\delta_{2}\\\
s_{1}-r_{1}\end{array}.\end{array}$

where $\delta_{1}=m-m_{1},\delta_{2}=r-r_{1},\delta_{3}=s-s_{1}$. Therefore
the number of $B$ is equal to

$q^{(s-s_{1}-r+r_{1})(m_{1}-s_{1})}\left[\begin{array}[]{l}m-r+r_{1}-m_{1}\\\
s-s_{1}-r+r_{1}\end{array}\right]_{q}\left[\begin{array}[]{l}m_{1}-r_{1}\\\
s_{1}-r_{1}\end{array}\right]_{q}.$

$\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

###### Lemma 3.2

If $B\in X^{s_{1}}_{s}$, then the number of $A\in X^{r_{1}}_{r}$ such that
$A\leq B$ is equal to

$\nu(r(r_{1}),s(s_{1}))=q^{(r-r_{1})(s_{1}-r_{1})}\left[s-s_{1}\atop
r-r_{1}\right]_{q}\left[s_{1}\atop r_{1}\right]_{q}.$

Proof. By the transitivity of $G_{n+l,l}(\mathbb{F}_{q})$ on the set of
subspaces of the same type, we may assume that

$\begin{array}[]{l}\begin{array}[]{llll}\ \ \ \ \ \ \ \
s-s_{1}&n-s+s_{1}&s_{1}&l-s_{1}\end{array}\\\
B=\left(\begin{array}[]{llll}I&0&0&0\\\
0&0&I&0\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\
s-s_{1}\\\ s_{1}\end{array}\end{array}.$

Since

$A\leq B,$

we have

$\begin{array}[]{l}\begin{array}[]{llll}\ \ \ \ \ \ \ \
s-s_{1}&n-s+s_{1}&s_{1}&l-s_{1}\end{array}\\\
A=\left(\begin{array}[]{llll}u_{11}&0&u_{13}&0\\\
0&0&u_{23}&0\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\
r-r_{1}\\\ r_{1}\end{array}\end{array},$

where rank $u_{23}=r_{1}$. Note that there are $\left[s_{1}\atop
r_{1}\right]_{q}$ choices for $u_{23}$. By the transitivity of
$GL_{n+l}(\mathbb{F}_{q})$ on the set of subspaces of the same type, the
number of $A$’s does not depend on the particular choice of $u_{23}$. Pick
$u_{23}=(I^{(r_{1})}\ 0)$. Then $A$ has a matrix representation

$\begin{array}[]{l}\begin{array}[]{lllll}s-s_{1}&n-s+s_{1}&r_{1}&s_{1}-r_{1}&l-s_{1}\end{array}\\\
\left(\begin{array}[]{lllll}u_{11}&0&0&u^{\prime}_{13}&0\\\
0&0&I&0&0\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\
r-r_{1}\\\ r_{1}\end{array}\end{array}.$

Therefore the number of subspace $A$ is equal to

$\nu(r(r_{1}),s(s_{1}))=q^{(r-r_{1})(s_{1}-r_{1})}\left[s-s_{1}\atop
r-r_{1}\right]_{q}\left[s_{1}\atop r_{1}\right]_{q}.$

$\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

###### Lemma 3.3

Let $A\in X^{r_{1}}_{r}$ and $B\in X^{m_{2}}_{m}$. Assume that $A\wedge B\in
X^{j_{1}}_{j}$, and $(C,D)\in X_{s}^{s_{1}}\times X_{m}^{m_{1}}$ . If $C\leq
D,C\leq B,A\leq D$, then the number of $(C,D)$ is equal to

$\begin{array}[]{ll}&\pi(j(j_{1}),r(r_{1}),s(s_{1});m_{1})\\\ =&\sum_{0\leq
i\leq j,0\leq
i_{1}\leq\min\\{i,j_{1}\\}}q^{(s-s_{1}-i+i_{1})(j-j_{1}-i+i_{1}+m_{1}-s_{1})+(s_{1}-i_{1})(j_{1}-i_{1})}\\\
&\times\left[(m-m_{1})-(j-j_{1})\atop(s-s_{1})-(i-i_{1})\right]_{q}\left[m_{1}-j_{1}\atop
s_{1}-i_{1}\right]_{q}N^{\prime}(r+s-i,r_{1}+s_{1}-i_{1};m,m_{1};n+l,n).\end{array}$

Proof. Since $A\leq D,C\leq B$, we have

$A\wedge C\leq A\wedge B\leq D.$

Since

$C\leq D,$

and

$C\wedge(A\wedge B)=A\wedge(B\wedge C)=A\wedge C\in X^{i_{1}}_{i},0\leq i\leq
j,0\leq i_{1}\leq\min\\{i,j_{1}\\}.$

For $i\in[0,j]$ and $i_{1}\in[0,\min\\{i,j_{1}\\}]$, by Lemma 2.1, the number
of $C$ is equal to

$q^{(s-s_{1}-i+i_{1})(j-j_{1}-i+i_{1}+m_{1}-s_{1})+(s_{1}-i_{1})(j_{1}-i_{1})}\left[(m-m_{1})-(j-j_{1})\atop(s-s_{1})-(i-i_{1})\right]_{q}\left[m_{1}-j_{1}\atop
s_{1}-i_{1}\right]_{q}.$

Since $C\leq D,A\leq D$, $C+A\leq D$. It follows from $A\wedge C\in
X_{i}^{i_{1}}$ that $A+C$ is a subspace of type $(r+s-i,r_{1}+s_{1}-i_{1})$.
Therefore, the number of $D$ is equal to
$N^{\prime}(r+s-i,r_{1}+s_{1}-i_{1};m,m_{1};n+l,n)$. Hence the desired result
follows. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

By Lemma 3.1,Lemma 3.2 and Lemma 3.3, we obtain the following theorem.

###### Theorem 3.4

Semilattice $\mathcal{L}_{0}^{m}(\mathbb{F}_{q}^{n+l})$ is a quasi regular
semilattice. Its parameters are given by the formulas

$\mu(r(r_{1}),s(s_{1});m_{1})=q^{(s-s_{1}-r+r_{1})(m_{1}-s_{1})}\left[\begin{array}[]{l}m-r+r_{1}-m_{1}\\\
s-s_{1}-r+r_{1}\end{array}\right]_{q}\left[\begin{array}[]{l}m_{1}-r_{1}\\\
s_{1}-r_{1}\end{array}\right]_{q},$
$\nu(r(r_{1}),s(s_{1}))=q^{(r-r_{1})(s_{1}-r_{1})}\left[s-s_{1}\atop
r-r_{1}\right]_{q}\left[s_{1}\atop r_{1}\right]_{q},$

and

$\begin{array}[]{ll}&\pi(j(j_{1}),r(r_{1}),s(s_{1});m_{1})\\\ =&\sum_{0\leq
i\leq j,0\leq i_{1}\leq
i}q^{(s-s_{1}-i+i_{1})(j-j_{1}-i+i_{1}+m_{1}-s_{1})+(s_{1}-i_{1})(j_{1}-i_{1})}\\\
&\times\left[(m-m_{1})-(j-j_{1})\atop(s-s_{1})-(i-i_{1})\right]_{q}\left[m_{1}-j_{1}\atop
s_{1}-i_{1}\right]_{q}N^{\prime}(r+s-i,r_{1}+s_{1}-i_{1};m,m_{1};n+l,n).\end{array}$

##  4 Association schemes

Given $d$ and $d_{0}$ with $0\leq d_{0}\leq d\leq\min\\{n,l\\}$. For
$h=\min\\{k,d_{0}\\},$ we define

$\widetilde{X_{k}}=\left\\{\begin{array}[]{ll}X_{k}^{0}\cup\cdots\cup
X_{k}^{h},&if\ 0\leq k\leq d-1,\\\ X_{d}^{d_{0}},&if\ k=d,\end{array}\right.$

and

$R_{k}=\\{(B,C)\in X_{d}^{d_{0}}\times X_{d}^{d_{0}}|B\wedge
C\in\widetilde{X_{d-k}}\\}.$

Then the $R=\\{R_{0},R_{1},\cdots,R_{d}\\}$ is a partition of
$X_{d}^{d_{0}}\times X_{d}^{d_{0}}$. For any $B,C\in X_{d}^{d_{0}}$, let

$A_{k(k^{\prime})}(B,C)=\left\\{\begin{array}[]{ll}1,&if\ B\wedge C\in
X^{k^{\prime}}_{d-k},\\\ 0,&otherwise.\end{array}\right.$

Then, for $h=\min\\{d-k,d_{0}\\}$, we define

$A_{k}=\left\\{\begin{array}[]{ll}A_{k(0)}+A_{k(1)}+\cdots+A_{k(h)},&if\ 1\leq
k\leq d,\\\ A_{0(d_{0})},&if\ k=0.\end{array}\right.$

Let

$D_{i,k}(B,C)=\left\\{\begin{array}[]{ll}1,&if\ B\leq C,\\\
0,&otherwise.\end{array}\right.$

The $D_{i,k}$ is a Riemann matrix indexed by
$\widetilde{X_{i}}\times\widetilde{X_{k}}$. For convenience, we write
$D_{i}=D_{i,d}$. Let $\mathfrak{A}=\langle A_{0},A_{2},\cdots,A_{d}\rangle$
denote the $(d+1)$-dimensional real vector space generated by the
$A_{0},A_{2},\cdots,A_{d}$. Let $C_{i}=D^{T}_{i}D_{i}$, for $i=0,1,\cdots,d.$
We have the lemma as follows.

###### Lemma 4.1

The matrices $C_{0},C_{1},\cdots,C_{d}$ generate $\mathfrak{A}$, with

$C_{t}=\sum_{h=0}^{\min\\{t,d_{0}\\}}\nu(t(h),d(d_{0}))A_{0}+\sum_{k=1}^{d}\sum_{\lambda=0}^{\min\\{d-k,d_{0}\\}}\sum_{h=0}^{\min\\{t,d_{0}\\}}\nu(t(h),(d-k)(\lambda))A_{k},$
(1)

where $t=0,1,\cdots,d.$

Proof. For any $(A,B)\in X_{d}^{d_{0}}\times X_{d}^{d_{0}}$, the number of
$F\in\widetilde{X_{t}}$ such that $F\leq A\wedge B$ is the $(A,B)$-entry in
two members of (1). It implies that the system (1) holds. On the other hand,
its matrix has rank $d+1$. Hence, the result is obtained. $\hfill\Box\vskip
6.0pt plus 2.0pt minus 2.0pt$

###### Lemma 4.2

Let $A\in X^{d_{1}}_{d},B\in X^{s_{1}}_{s}$, then the number of $D\in
X^{r_{1}}_{r}$ such that $D\leq A$ and $B\wedge D\in X^{j_{1}}_{j}$ is a
constant $\psi(j(j_{1}),r(r_{1}),s(s_{1});d_{1})$.

Proof. For a given $k$ and $k_{1}$, with $0\leq k\leq h=\min\\{r,s\\}$ and
$0\leq k_{1}\leq k$. We assume that $(C,D)\in(X_{k}^{k_{1}},X_{r}^{r_{1}})$
and $C\leq B,C\leq D\leq A.$ Counting the number of the pairs $(C,D)$ in two
different ways, we obtain

$\sum_{j=0}^{h}\sum_{j_{1}=0}^{j}\nu(k(k_{1}),j(j_{1}))\psi(j(j_{1}),r(r_{1}),s(s_{1});d_{1})=\nu(k(k_{1}),s(s_{1}))\mu(k(k_{1}),r(r_{1});d_{1}).$
(2)

This yields a system $(\ref{el2})$ of linear equations in unknown
$\psi(j(j_{1}),r(r_{1}),s(s_{1});d_{1})$ with fixed $r(r_{1}),s(s_{1})$. Since
the matrix $[\nu(k(k_{1}),j(j_{1}))]$ of the system $(\ref{el2})$ is a
nonsingular, these equations uniquely determine the
$\psi(j(j_{1}),r(r_{1}),s(s_{1});d_{1})$. $\hfill\Box\vskip 6.0pt plus 2.0pt
minus 2.0pt$

###### Theorem 4.3

The system $(X_{d}^{d_{0}},R)$ is an association scheme.

Proof. We establish the matrix relation as follows

$C_{r}C_{s}=\sum_{k=0}^{d}\sum_{\lambda=0}^{h_{1}}\left\\{\sum_{j=0}^{r}\sum_{j^{\prime}=0}^{h_{2}}\left[\sum_{r^{\prime}=0}^{h_{3}}\left(\psi(j(j^{\prime}),r(r^{\prime}),(d-k)(\lambda);d_{0})\sum_{s^{\prime}=0}^{h_{4}}\pi(j(j^{\prime}),r(r^{\prime}),s(s^{\prime});d_{0})\right)\right]\right\\}A_{k}.$
(3)

Here
$h_{1}=\min\\{d-k,d_{0}\\},h_{2}=\min\\{j,d_{0}\\},h_{3}=\min\\{r,d_{0}\\},h_{4}=\min\\{s,d_{0}\\}.$

In fact, By the definition, for any $D,F\in X_{d}^{d_{0}}$, the $(D,F)$-entry
of $C_{r}C_{s}$ is the number of triples
$(A,B,C)\in\widetilde{X_{r}}\times\widetilde{X_{s}}\times\widetilde{X_{d}}$
such that $A\leq D\wedge C$ and $B\leq F\wedge C$ hold. Let $D\wedge F\in
X_{d-k}^{\lambda}$. We assume that $A\wedge F\in X_{j}^{j^{\prime}}$ with
fixed $j$ and $j^{\prime}$, by Lemma 4.2, there are
$\sum_{r^{\prime}=0}^{\min\\{r,d_{0}\\}}\psi(j(j^{\prime}),r(r^{\prime}),(d-k)(\lambda);d_{0})$
choices for $A\leq D$. For each $A$, by Lemma 3.3, there are
$\sum_{s^{\prime}=0}^{\min\\{s,d_{0}\\}}\pi(j(j^{\prime}),r(r^{\prime}),s(s^{\prime});d_{0})$
choices for pairs $(B,C)$ with $B\leq F\wedge C,A\leq C$. Hence, (3) holds. By
Lemma 4.1, $\mathfrak{A}$ is a Bose-Mesner algebra. It implies the assertion.
$\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

##  5 $M$-clique

In this section, we always assume that $l=0$, $X=X_{d}^{0}$. By Theorem 4.3,
we obtain a association scheme $\mathfrak{X}=(X,\\{R_{i}\\}_{0\leq i\leq d})$
in $\mathbb{F}_{q}^{n}$. Next, we will discuss the bound of $M$-clique on
association scheme $(X,\\{R_{i}\\}_{0\leq i\leq d})$.

We associate the following linear programming problem, in the real variables
$\xi_{k}(k\in K=\\{k\in[0,d]|m_{k}>1\\}):$(see [5])

$\left\\{\begin{array}[]{l}{\rm min}(f^{\prime})=1+\sum_{k\in
K}(1-b_{k})\xi_{k},\\\ \hbox{subject to}\ \xi_{k}\geq 0,\ \hbox{all}\ k\in
K,\\\ \hbox{and to}\ \sum_{k\in K}(P_{i}(k)-b_{k}v_{i})\xi_{k}\leq v_{i},\
\hbox{all}\ i\in M\backslash\\{0\\}.\end{array}\right.$ (4)

Here $P_{i}(k)$ is eigenvalue of a association scheme, $v_{i}$ is degree of a
association scheme, $b_{k}=(v_{s}^{-1}P_{s}(k))^{2},k\in[0,d]$.

###### Proposition 5.1

([5]) The problem (4) admits a solution $\min(f^{\prime})$, and any
unicoloured $M$-clique $\alpha$ satisfies
$\hat{c}_{0}(\alpha)\leq\min(f^{\prime})$.

By Proposition 5.1, we obtain the following theorem.

###### Theorem 5.2

Let $\alpha\in\mathbb{R}^{|X|}$ be a unicoloured $M$-clique on association
scheme $\mathfrak{X}=(X,\\{R_{i}\\}_{0\leq i\leq d})$ with $M=\\{0,1\\}$ and
colour $s=1$. then

$\hat{c}_{0}(\alpha)=<\alpha,\alpha>^{-1}<\alpha,1>^{2}\leq\frac{q^{d+1+n}-3q^{2d+1}+5q^{d+1}-2q^{d}+2q^{n+d}-3q^{n+1}}{q^{n+d}-q^{2d+1}+2q^{d+1}-q^{d}-q^{n+1}}.$

Proof. By the ([12]), the eigenvalues of an association scheme of
$\mathfrak{X}=(X,\\{R_{i}\\}_{0\leq i\leq d})$ are given by the formula

$P_{i}(r)=\sum_{j=0}^{i}(-1)^{i-j}\left[d-j\atop i-j\right]_{q}\left[d-r\atop
j\right]_{q}\left[n-d+j-r\atop j\right]_{q}q^{rj+{i-j\choose 2}}.$

Pick $\xi_{k}=\xi\delta_{k,1}$. It is known that the degree of the
$\mathfrak{X}=(X,\\{R_{i}\\}_{0\leq i\leq d})$ are given by the formula

$v_{i}=q^{i^{2}}\left[n-d\atop i\right]_{q}\left[d\atop i\right]_{q}.$

By the above formulas, we obtain

$v_{1}=\frac{q(q^{n-d}-1)(q^{d}-1)}{(q-1)^{2}},$
$P_{1}(1)=-\frac{q^{d}-1}{q-1}+\frac{q(q^{d-1}-1)(q^{n-d}-1)}{(q-1)^{2}},$

For $s=1,i=1$, we have $(P_{1}(1)-b_{1}v_{1})\xi\leq v_{1}$,
$(P_{1}(1)-(v_{1}^{-1}P_{1}(1))^{2}v_{1})\xi\leq v_{1}$. It implies that
$(v_{1}P_{1}(1)-P_{1}(1)^{2})\xi\leq v_{1}^{2}$. Since

$v_{1}-P_{1}(1)=\frac{q^{n}-1}{q-1}>0.$

We have

$\xi\leq\frac{v_{1}^{2}}{p_{1}(1)(v_{1}-p_{1}(1)}.$

and

$1-b_{1}=1-\frac{p_{1}(1)^{2}}{v_{1}^{2}}=\frac{v_{1}^{2}-p_{1}(1)^{2}}{v_{1}^{2}}>0.$

It follows that

$\begin{array}[]{ll}\min(f^{\prime})&=1+(1-b_{1})\xi\\\ &\leq
1+\frac{v_{1}+p_{1}(1)}{p_{1}(1)}\\\
&=\frac{q^{d+1+n}-3q^{2d+1}+5q^{d+1}-2q^{d}+2q^{n+d}-3q^{n+1}}{q^{n+d}-q^{2d+1}+2q^{d+1}-q^{d}-q^{n+1}}.\end{array}$

By Proposition 5.1, we have

$\hat{c}_{0}(\alpha)\leq\frac{q^{d+1+n}-3q^{2d+1}+5q^{d+1}-2q^{d}+2q^{n+d}-3q^{n+1}}{q^{n+d}-q^{2d+1}+2q^{d+1}-q^{d}-q^{n+1}}.$

$\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

## Acknowledgement

This research is supported by NSF of Hebei Province (A2013408009), NSF of
Hebei Education Department(ZH2012082), the Specialized Research Fund for the
Doctoral Program of Higher Education of China (No.20121303110005) and the
foundation of Langfang Teachers’ College (LSBS201205).

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