# On the combinatorial and rank properties of certain subsemigroups of full
contractions of a finite chain

M. M. Zubairu 111Corresponding Author. Email<EMAIL_ADDRESS>A.
Umar and M. J. Aliyu
Department of Mathematical Sciences, Bayero University Kano, P. M. B. 3011,
Kano, Nigeria
<EMAIL_ADDRESS>
Khalifa University, P. O. Box 127788, Sas al Nakhl, Abu Dhabi, UAE
<EMAIL_ADDRESS>
Department of Mathematics, and Computer Sciences, Sule Lamido University,
Kafin Hausa
<EMAIL_ADDRESS>
###### Abstract

Let $[n]=\\{1,2,\ldots,n\\}$ be a finite chain and let $\mathcal{CT}_{n}$ be
the semigroup of full contractions on $[n]$. Denote $\mathcal{ORCT}_{n}$ and
$\mathcal{OCT}_{n}$ to be the subsemigroup of order preserving or reversing
and the subsemigroup of order preserving full contractions, respectively. It
was shown in [17] that the collection of all regular elements (denoted by,
Reg$(\mathcal{ORCT}_{n})$ and Reg$(\mathcal{OCT}_{n}$), respectively) and the
collection of all idempotent elements (denoted by E$(\mathcal{ORCT}_{n})$ and
E$(\mathcal{OCT}_{n}$), respectively) of the subsemigroups
$\mathcal{ORCT}_{n}$ and $\mathcal{OCT}_{n}$, respectively are subsemigroups.
In this paper, we study some combinatorial and rank properties of these
subsemigroups.

_2010 Mathematics Subject Classification. 20M20._
Keywords: Full Contractions maps on chain, regular element, idempotents, rank
properties.

##  1 Introduction

Denote $[n]=\\{1,2,\ldots,n\\}$ to be a finite chain and let $\mathcal{T}_{n}$
denote the semigroup of full transformations of $[n]$. A transformation
$\alpha\in\mathcal{T}_{n}$ is said to be _order preserving_ (resp., _order
reversing_) if (for all $x,y\in[n]$) $x\leq y$ implies $x\alpha\leq y\alpha$
(resp., $x\alpha\geq y\alpha$); _order decreasing_ if (for all $x\in[n]$)
$x\alpha\leq x$; an _isometry_ (i.e., _distance preserving_) if (for all
$x,y\in[n]$) $|x\alpha-y\alpha|=|x-y|$; a _contraction_ if (for all
$x,y\in[n]$) $|x\alpha-y\alpha|\leq|x-y|$. Let
$\mathcal{CT}_{n}=\\{\alpha\in\mathcal{T}_{n}:(\textnormal{for all
}x,y\in[n])~{}\left|x\alpha-y\alpha\right|\leq\left|x-y\right|\\}$ be the
semigroup of full contractions on $[n]$, as such $\mathcal{CT}_{n}$ is a
subsemigroup of $\mathcal{T}_{n}$. Certain algebraic and combinatorial
properties of this semigroup and some of its subsemigroups have been studied,
for example see [1, 5, 6, 15, 16, 17, 19, 20, 21].

Let

$\mathcal{OCT}_{n}=\\{\alpha\in\mathcal{CT}_{n}:(\textnormal{for
all}~{}x,y\in[n])~{}x\leq y\textnormal{ implies }x\alpha\leq y\alpha\\},$ (1)

and

$\mathcal{ORCT}_{n}=\mathcal{OCT}_{n}\cup\\{\alpha\in\mathcal{CT}_{n}:(\textnormal{for
all}~{}x,y\in[n])~{}x\leq y~{}\textnormal{implies }x\alpha\geq y\alpha\\}$ (2)

be the subsemigroups of _order preserving full contractions_ and of _order
preserving or reversing full contractions_ on $[n]$, respectively. These
subsemigroups are both known to be non-regular left abundant semigroups [17]
and their Green’s relations have been characterized in [2]. The ranks of
$\mathcal{ORCT}_{n}$ and $\mathcal{OCT}_{n}$ were computed in [15] while the
ranks of the two sided ideals of $\mathcal{ORCT}_{n}$ and $\mathcal{OCT}_{n}$
were computed by Leyla [5].

In 2021, Umar and Zubairu [17] showed that the collection of all regular
elements (denoted by $\textnormal{Reg}(\mathcal{ORCT}_{n})$) of
$\mathcal{ORCT}_{n}$ and also the collection of idempotent elements (denoted
by $\textnormal{E}(\mathcal{ORCT}_{n})$) of $\mathcal{ORCT}_{n}$ are both
subsemigroups of $\mathcal{ORCT}_{n}$. The two subsemigroups are both regular,
in fact $\textnormal{Reg}(\mathcal{ORCT}_{n})$ has been shown to be an
$\mathcal{L}-$ _unipotent_ semigroup (i.e., each ${L}-$class contains a unique
idempotent). In fact, it was also shown in [17] that the collection of all
regular elements (denoted by Reg$\mathcal{OCT}_{n}$) in $\mathcal{OCT}_{n}$ is
a subsemigroup. However, combinatorial as well as rank properties of these
semigroups are yet to be discussed, in this paper we discuss these properties,
as such this paper is a natural sequel to Umar and Zubairu [17]. For basic
concepts in semigroup theory, we refer the reader to [10, 12, 14].

Let $S$ be a semigroup and $U$ be a subset of $S$, then $|U|$ is said to be
the _rank_ of $S$ (denoted as $\textnormal{Rank}(S)$) if

$|U|=\min\\{|A|:A\subseteq S\textnormal{ and }\langle A\rangle=S\\}.$

The notation $\langle U\rangle=S$ means that $U$ generate the semigroup $S$.
The rank of several semigroups of transformation were investigated, see for
example, [3, 4, 7, 8, 9, 11, 13]. However, there are several subsemigroups of
full contractions which their ranks are yet to be known. In fact the order and
the rank of the semigroup $\mathcal{CT}_{n}$ is still under investigation.

Let us briefly discuss the presentation of the paper . In section 1, we give a
brief introduction and notations for proper understanding of the content of
the remaining sections. In section 2, we discuss combinatorial properties for
the semigroups $\textnormal{Reg}(\mathcal{ORCT}_{n})$ and
$\textnormal{E}(\mathcal{ORCT}_{n})$, in particular we give their orders. In
section 3, we proved that the rank of the semigroups
$\textnormal{Reg}(\mathcal{ORCT}_{n})$ and
$\textnormal{E}(\mathcal{ORCT}_{n})$ are 4 and 3, respectively, through the
minimal generating set for their Rees quotient semigroups.

##  2 Combinatorial Properties of $\textnormal{Reg}(\mathcal{ORCT}_{n})$ and
$\textnormal{E}(\mathcal{ORCT}_{n})$

In this section, we want to investigate some combinatorial properties of the
semigroups, $\textnormal{Reg}(\mathcal{ORCT}_{n})$ and
$\textnormal{E}(\mathcal{OCT}_{n})$. In particular, we want to compute their
Cardinalities. Let

$\alpha=\left(\begin{array}[]{cccc}A_{1}&A_{2}&\ldots&A_{p}\\\
x_{1}&x_{2}&\ldots&x_{p}\end{array}\right)\in\mathcal{T}_{n}~{}~{}(1\leq p\leq
n),$ (3)

then the _rank_ of $\alpha$ is defined and denoted by rank
$(\alpha)=|\textnormal{Im }\alpha|=p$, so also, $x_{i}\alpha^{-1}=A_{i}$
($1\leq i\leq p$) are equivalence classes under the relation $\textnormal{ker
}\alpha=\\{(x,y)\in[n]\times[n]:x\alpha=y\alpha\\}$. Further, we denote the
partition $(A_{1},\ldots,A_{p})$ by $\textnormal{{Ker} }\alpha$ and also,
fix$(\alpha)=|\\{x\in[n]:x\alpha=x\\}|$. A subset $T_{\alpha}$ of $[n]$ is
said to be a _transversal_ of the partition $\textnormal{{Ker} }\alpha$ if
$|T_{\alpha}|=p$, and $|A_{i}\cap T_{\alpha}|=1$ ($1\leq i\leq p$). A
transversal $T_{\alpha}$ is said to be _convex_ if for all $x,y\in T_{\alpha}$
with $x\leq y$ and if $x\leq z\leq y$ ($z\in[n]$), then $z\in T_{\alpha}$.

Before we proceed, lets describe some Green’s relations on the semigroups
$\textnormal{Reg}(\mathcal{ORCT}_{n})$ and
$\textnormal{E}(\mathcal{ORCT}_{n})$. It is worth noting that the two
semigroups, $\textnormal{Reg}(\mathcal{ORCT}_{n})$ and
$\textnormal{E}(\mathcal{ORCT}_{n})$ are both regular subsemigroups of the
Full Transformation semigroup $\mathcal{T}_{n}$, therefore by [[14], Prop.
2.4.2] they automatically inherit the Green’s $\mathcal{L}$ and $\mathcal{R}$
relations of the semigroup $\mathcal{T}_{n}$, but not necessary $\mathcal{D}$
relation, as such we have the following lemma.

######  Lemma 1.

Let $\alpha,\beta\in S\in\\{\textnormal{Reg}(\mathcal{ORCT}_{n}),\
\textnormal{E}(\mathcal{ORCT}_{n})\\}$, then

  * i

$\alpha\mathcal{R}\beta$ if and only if $\textnormal{Im }\alpha=\textnormal{Im
}\beta$;

  * ii

$\alpha\mathcal{L}\beta$ if and only if $\textnormal{ker
}\alpha=\textnormal{ker }\beta$.

###  2.1 The Semigroup $\textnormal{Reg}(\mathcal{ORCT}_{n})$

Before we begin discussing on the semigroup
$\textnormal{Reg}(\mathcal{ORCT}_{n})$, let us first of all consider the
semigroup $\textnormal{Reg}(\mathcal{OCT}_{n})$ consisting of only order-
preserving elements. Let $\alpha$ be in $\textnormal{Reg}(\mathcal{OCT}_{n})$,
from [[17], Lem. 12], $\alpha$ is of the form

$\alpha=\left(\begin{array}[]{ccccc}\\{1,\ldots,a+1\\}&a+2&\ldots&a+p-1&\\{a+p,\ldots,n\\}\\\
x+1&x+2&\ldots&x+p-1&x+p\end{array}\right)$

Let

$K_{p}=\\{\alpha\in Reg(\mathcal{OCT}_{n}):|\textnormal{Im
}\alpha|=p\\}\quad(1\leq p\leq n),$ (4)

and suppose that $\alpha\in K_{p}$, by [[18], Lem. 12] Ker
$\alpha=\\{\\{1,\ldots,a+1\\},a+2\ldots,a+{p-1},\\{a+p,\ldots,n\\}\\}$ have an
_admissible_ traversal (A transversal $T_{\alpha}$ is said to be admissible if
and only if the map $A_{i}\mapsto t_{i}$ ($t_{i}\in
T_{\alpha},\,i\in\\{1,2,\ldots,p\\}$) is a contraction, see [2])
$T_{\alpha}=\\{a+i\,:1\leq i\leq p\\}$ such that the mapping $a+i\mapsto x+i$
is an isometry. Therefore, translating the set $\\{x+i:\,i\leq 1\leq p\\}$
with an integer say $k$ to $\\{x+i\pm k:\,1\leq i\leq p\\}$ will also serve as
image set to $\textnormal{{Ker} }\alpha$ as long as $x+1-k\nless 1$ and
$x+p+k\ngtr n$.

For example, if we define $\alpha$ as :

$\alpha=\left(\begin{array}[]{cccccc}\\{1,\ldots,a+1\\}&a+2&a_{3}&\ldots&a+{p-1}&\\{a+p,\ldots,n\\}\\\
1&2&3&\ldots&p-1&p\end{array}\right).$ (5)

then we will have $n-p$ other mappings in $K_{p}$ that will have the same
domain as $\alpha$.

In similar manner, suppose we fix the image set $\\{x+i|\,1\leq i\leq p\\}$
and consider $\textnormal{{Ker} }\alpha$, then we can refine the partition
$\\{\\{1,\ldots,a+1\\},\\{a+2\\}\ldots,\\{a+{p-1}\\},\\{a+p,\ldots,n\\}\\}$ by
$i-$shifting to say
$\\{\\{1,\ldots,a+i\\},\\{a+i+1\\}\ldots,\\{a+{p-i}\\},\\{a+p-i+1,\ldots,n\\}\\}$
for some integer $1\leq i\leq p$ which also have an admissible convex
traversal.

For the purpose of illustrations, if for some integer $j$,
$\\{\\{1,\ldots,a+1\\},\\{a+2\\}\ldots,\\{a+{p-1}\\},\\{a+p,\ldots,n\\}\\}=\,\\{\\{1,2,\ldots
j\\},\\{j+1\\},\\{j+2\\},\ldots,\\{n\\}\\}$, then the translation
$\\{\\{1,2,\ldots j-1\\},\\{j\\},\\{j+1\\},\ldots,\\{n-1,n\\}\\}$ will also
serve as domain to the image set of $\alpha$. Thus, for $p\neq 1$ we will have
$n-p+1$ different mappings with the same domain set in $K_{p}$.

To see what we have been explaining, consider the table below; For $n\geq 4$,
$2\leq p\leq n$ and $j=n-p+1$, the set $K_{p}$ can be presented as follows:

$\begin{array}[]{cccc}\left(\begin{array}[]{ccccc}\\{1,\ldots
j\\}&j+1&\cdots&n-1&n\\\
1&2&\ldots&p-1&p\end{array}\right)\par&\cdots&\left(\begin{array}[]{ccccc}\\{1,2\\}&3&\cdots&\\{p-1,\ldots
n\\}\\\
1&2&\ldots&p\end{array}\right)&\left(\begin{array}[]{ccccc}1&2&\cdots&p-1&\\{p,\ldots
n\\}\\\ 1&2&\cdots&p-1&p\end{array}\right)\\\
\left(\begin{array}[]{ccccc}\\{1,\ldots j\\}&j+1&\cdots&n-1&n\\\
2&3&\ldots&p&p+1\end{array}\right)&\cdots&\left(\begin{array}[]{ccccc}\\{1,2\\}&3&\cdots&\\{p-1,\ldots
n\\}\\\
2&3&\cdots&p+1\end{array}\right)&\left(\begin{array}[]{ccccc}1&2&\cdots&p-1&\\{p,\ldots
n\\}\\\ 2&3&\cdots&p&p+1\end{array}\right)\\\
\vdots&\vdots&\vdots&\vdots\par\\\ \left(\begin{array}[]{ccccc}\\{1,\ldots
j\\}&j+1&\cdots&n-1&n\\\
j-1&j&\ldots&n-2&n-1\end{array}\right)\par&\cdots&\left(\begin{array}[]{ccccc}\\{1,2\\}&3&\cdots&\\{p-1,\ldots
n\\}\\\
j-1&j&\ldots&n-2&n-1\end{array}\right)&\left(\begin{array}[]{ccccc}1&2&\cdots&p-1&\\{p,\ldots
n\\}\\\ j-1&j&\ldots&n-2&n-1\end{array}\right)\\\
\left(\begin{array}[]{ccccc}\\{1,\ldots j\\}&j+1&\cdots&n-1&n\\\
j&j+1&\ldots&n-1&n\end{array}\right)&\cdots&\left(\begin{array}[]{ccccc}\\{1,2\\}&3&\cdots&\\{p-1,\ldots
n\\}\\\
j&j+1&\ldots&n-1&n\par\end{array}\right)&\left(\begin{array}[]{ccccc}1&2&\cdots&p-1&\\{p,\ldots
n\\}\\\ j&j+1&\ldots&n-1&n\par\end{array}\right)\par\end{array}$

(6)

From the table above, we can see that for $p=1$, $|K_{p}|=n-p+1=n$, while for
$2\leq p\leq n,\,$ $|K_{p}|=(n-p+1)^{2}$.

The next theorem gives us the cardinality of the semigroup
$\textnormal{Reg}(\mathcal{OCT}_{n})$.

######  Theorem 2.

Let $\mathcal{OCT}_{n}$ be as defined in equation (1), then
$|\textnormal{Reg}(\mathcal{OCT}_{n})|=\frac{n(n-1)(2n-1)+6n}{6}$.

###### Proof.

It is clear that $\textnormal{Reg}(\mathcal{OCT}_{n})=K_{1}\cup
K_{2}\cup\ldots\cup K_{n}$. Since this union is disjoint, we have that

$\begin{array}[]{c}|\textnormal{Reg}\mathcal{OCT}_{n}|=\sum_{p=1}^{n}|K_{p}|=|K_{1}|+\sum_{p=2}^{n}|K_{p}|=n+\sum_{p=2}^{n}(n-p+1)^{2}\\\
=n+(n-1)^{2}+(n-2)^{2}+\cdots+2^{2}+1^{2}\\\
=\frac{n(n-1)(2n-1)+6n}{6},\end{array}$

as required. ∎

######  Corollary 3.

Let $\mathcal{ORCT}_{n}$ be as defined in equation (2). Then
$|\textnormal{Reg}(\mathcal{ORCT}_{n})|=\frac{n(n-1)(2n-1)+6n}{3}-n$.

###### Proof.

It follows from Theorem 2 and the fact that
$|\textnormal{Reg}(\mathcal{ORCT}_{n})|=2|\textnormal{Reg}(\mathcal{OCT}_{n})|-n$.
∎

###  2.2 The Semigroup $\textnormal{E}(\mathcal{ORCT}_{n})$

Let $\alpha$ be in $\textnormal{E}(\mathcal{ORCT}_{n})$, then it follows from
[[17], Lem. 13] that $\alpha$ is of the form

$\alpha=\left(\begin{array}[]{cccccc}\\{1,\ldots,i\\}&i+1&i+2&\ldots&i+j-1&\\{i+j,\ldots,n\\}\\\
i&i+1&i+2&\ldots&i+j-1&i+j\end{array}\right).$ (7)

Since fix$(\alpha)=j+1$, then for each given domain set there will be only one
corresponding image set.

Let

$E_{p}=\\{\alpha\in\textnormal{E}(\mathcal{ORCT}_{n}):|\textnormal{Im
}\alpha|=p\\}\quad(1\leq p\leq n).$ (8)

To choose $\alpha\in E_{p}$ we only need to select the image set of $\alpha$
which is a $p$ consecutive(convex) numbers from the set $[n]$. Thus
$|E_{P}|=n-p-1$. Consequently, we have the cardinality of the semigroup
$\textnormal{E}(\mathcal{ORCT}_{n})$.

######  Theorem 4.

Let $\mathcal{ORCT}_{n}$ be as defined in equation (2). Then
$|\textnormal{E}(\mathcal{ORCT}_{n})|=\frac{n(n+1)}{2}$.

###### Proof.

Following the argument of the proof of Theorem 2 we have,

$\begin{array}[]{c}|\textnormal{E}(\mathcal{ORCT}_{n})|=\sum_{p=1}^{n}|E_{p}|=\sum_{p=1}^{n}(n-p+1)\\\
=n+(n-1)+(n-2)+\cdots+2+1\\\ =\frac{n(n+1)}{2}.\end{array}$

∎

######  Remark 5.

Notice that idempotents in $\mathcal{ORCT}_{n}$ are necessarily order
preserving, as such
$|\textnormal{E}(\mathcal{OCT}_{n})|=|\textnormal{E}(\mathcal{ORCT}_{n})|=\frac{n(n+1)}{2}$.

##  3 Rank Properties

In this section, we discuss some rank properties of the semigroups
$\textnormal{Reg}(\mathcal{ORCT}_{n})$ and
$\textnormal{E}(\mathcal{ORCT}_{n})$.

###  3.1 Rank of $\textnormal{Reg}(\mathcal{OCT}_{n})$

Just as in section 2 above, let us first consider the semigroup
$\textnormal{Reg}(\mathcal{OCT}_{n})$, the semigroup consisting of regular
elements of order-preserving full contractions. Now, let $K_{p}$ be defined as
in equation (4). We have seen how elements of $K_{p}$ look like in Table 6
above. Suppose we define:

$\eta:=\left(\begin{array}[]{ccccc}\\{1,\ldots j\\}&j+1&\cdots&n-1&n\\\
1&2&\ldots&p-1&p\end{array}\right),$ (9)
$\delta:=\left(\begin{array}[]{ccccc}1&2&\cdots&p-1&\\{p,\ldots n\\}\\\
1&2&\cdots&p-1&p\end{array}\right)$ (10)

and

$\tau:=\left(\begin{array}[]{ccccc}1&2&\cdots&p-1&\\{p,\ldots n\\}\\\
j&j+1&\ldots&n-1&n\par\end{array}\right)$ (11)

that is, $\eta$ to be the top left-corner element, $\delta$ be the top right-
corner element while $\tau$ be the bottom right corner element in Table 6. And
let $\textnormal{R}_{\eta}$ and $\textnormal{L}_{\delta}$ be the respective
$\mathcal{R}$ and $\mathcal{L}$ equivalent classes of $\eta$ and $\delta$.
Then for $\alpha$ in $K_{p}$ there exist two elements say $\eta^{\prime}$ and
$\delta^{\prime}$ in $\textnormal{R}_{\eta}$ and $\textnormal{L}_{\delta}$,
respectively for which $\alpha$ is $\mathcal{L}$ related to $\eta^{\prime}$
and $\mathcal{R}$ related to $\delta^{\prime}$ and that
$\alpha=\eta^{\prime}\delta^{\prime}$. For the purpose of illustrations,
consider

$\alpha=\left(\begin{array}[]{ccccc}\\{1,\ldots
j-1\\}&j&j+1&\cdots&\\{n-1,n\\}\\\ 2&3&4&\ldots&p+1\end{array}\right),$

then the elements

$\left(\begin{array}[]{ccccc}\\{1,\ldots j-1\\}&j&j+1&\cdots&\\{n-1,n\\}\\\
1&2&3&\ldots&p\end{array}\right)$

and

$\left(\begin{array}[]{ccccc}1&2&\cdots&p-1&\\{p,\ldots n\\}\\\
2&3&\cdots&p&p+1\end{array}\right)$

are respectively elements of $\textnormal{R}_{\eta}$ and
$\textnormal{L}_{\delta}$ and that

$\alpha=\left(\begin{array}[]{ccccc}\\{1,\ldots
j-1\\}&j&j+1&\cdots&\\{n-1,n\\}\\\
1&2&3&\ldots&p\end{array}\right)\left(\begin{array}[]{ccccc}1&2&\cdots&p-1&\\{p,\ldots
n\\}\\\ 2&3&\cdots&p&p+1\end{array}\right).$

Consequently, we have the following lemma.

######  Lemma 6.

Let $\eta$ and $\delta$ be as defined in equations (9) and (10), respectively.
Then $\langle\textnormal{R}_{\eta}\cup\textnormal{L}_{\delta}\rangle=K_{p}$.

######  Remark 7.

The following are observed from Table 6:

  * (i)

The element $\delta$ belongs to both $\textnormal{R}_{\eta}$ and
$\textnormal{L}_{\delta}$;

  * (ii)

$\tau\eta=\delta$;

  * (iii)

For all $\alpha\in\textnormal{R}_{\eta}$, $\alpha\delta=\alpha$ while
$\delta\alpha$ has rank less than $p$;

  * (iv)

For all $\alpha\in\textnormal{L}_{\delta}$, $\delta\alpha=\alpha$ while
$\alpha\delta$ has rank less than $p$;

  * (v)

For all $\alpha,\beta\in\textnormal{R}_{\eta}\backslash\delta$ ( or
$\textnormal{L}_{\delta}\backslash\delta$), rank($\alpha\beta)<p$.

To investigate the rank of $\textnormal{Reg}(\mathcal{OCT}_{n})$, let

$L(n,p)=\\{\alpha\in\textnormal{Reg}(\mathcal{OCT}_{n}):|\textnormal{Im
}\alpha|\leq p\\}\quad(1\leq p\leq n),$ (12)

and let

$Q_{p}=L(n,p)\backslash L(n,p-1).$ (13)

Then $Q_{p}$ is of the form $K_{p}\cup\\{0\\}$, where $K_{p}$ is the set of
all elements of $\textnormal{Reg}(\mathcal{OCT}_{n})$ whose height is exactly
$p$. The product of any two elements in $Q_{p}$ say $\alpha$ and $\beta$ is of
the form:

$\alpha\ast\beta=\left\\{\begin{array}[]{ll}\alpha\beta,&\hbox{if
$|h(\alpha\beta)|=p$;}\\\ 0,&\hbox{if $|h(\alpha\beta)|<p$}\end{array}\right.$

$Q_{p}$ is called the Rees quotient semigroup on $L(n,p)$. Next, we have the
following lemma which follows from Lemma 6 and Remark 7.

######  Lemma 8.

$(\textnormal{R}_{\eta}\cup\textnormal{L}_{\delta})\backslash\delta$ is the
minimal generating set for the Rees quotient semigroup $Q_{p}$.

To find the generating set for $L(n,p)$, we need the following proposition:

######  Proposition 9.

For $n\geq 4,\,$ $\langle K_{p}\rangle\,\subseteq\,\langle K_{p+1}\rangle$ for
all $1\leq p\leq n-2$.

###### Proof.

Let $\langle A\rangle=K_{p}$, to proof $\langle
K_{p}\rangle\,\subseteq\,\langle K_{p+1}\rangle$, it suffices to show that
$A\subseteq\langle K_{p+1}\rangle$. From Lemma 8
$A=(\textnormal{R}_{\eta}\cup\textnormal{L}_{\delta})\backslash{\delta}$. Now,
let $\alpha$ be in $A$:

CASE I: If $\alpha=\eta$, then $\alpha$ can be written as $\alpha=$

$\left(\begin{array}[]{cccccc}\\{1,\ldots j-1\\}&j&j+1&\cdots&n-1&n\\\
j-2&j-1&j&\cdots&n-2&n-1\end{array}\right)\left(\begin{array}[]{cccccc}\\{1,\ldots
j-1\\}&j&j+1&\cdots&n-1&n\\\ 1&2&3&\cdots&p&p+1\end{array}\right),$

a product of two elements of $K_{p+1}$.

CASE II: If $\alpha\in\textnormal{R}_{\eta}\backslash\eta$, then $\alpha$ is
of the form

$\left(\begin{array}[]{ccccc}\\{1,\ldots
j-k\\}&j-k+1&\cdots&n-2&\\{n-k,\ldots,n\\}\\\
1&2&\cdots&p-1&p\end{array}\right),\,(k=1,2,\dots,j-2).$

Then $\alpha$ can be written as:

$\left(\begin{array}[]{cccc}\\{1,\ldots,j-k-1\\}&j-k&\cdots&\\{n-k,\ldots,n\\}\\\
j-k-1&j-k&\cdots&n-k\end{array}\right)\left(\begin{array}[]{ccccc}\\{1,\ldots
j-k\\}&j-k+1&\cdots&n-k&\\{n-k+1,\ldots,n\\}\\\
1&2&\cdots&p&p+1\end{array}\right),$

a product of two elements of $K_{p+1}$.

CASE III: If $\alpha\in\textnormal{L}_{\delta}\backslash\delta$, then $\alpha$
is of the form

$\left(\begin{array}[]{ccccc}1&2&\cdots&p-1&\\{p,p+1,\ldots n\\}\\\
r&r+1&\cdots&p+r-2&p+r-1\end{array}\right),\,(r=2,3,\ldots,n-p+1)$

and it can be written as:

$\left(\begin{array}[]{ccccc}1&2&\cdots&p&\\{p+1,\ldots n\\}\\\
2&3&\cdots&p+1&p+2\end{array}\right)\left(\begin{array}[]{ccccc}1&2&\cdots&p&\\{p+1,\ldots
n\\}\\\ r-1&r&\cdots&p+r-2&p+r-1\end{array}\right),$

hence the proof. ∎

######  Remark 10.

Notice that by the proposition above, the generating set for $Q_{p}$ ($1\leq
p\leq n-1$) generates the whole $L(n,p)$.

The next theorem gives us the rank of the subsemigroup $L(n,p)$ for $1\leq
p\leq n-1$.

######  Theorem 11.

Let $L(n,p)$ be as defined in equation (12). Then for $n\geq 4$ and $1<p\leq
n-1$, the rank of $L(n,p)$ is $2(n-p)$.

###### Proof.

It follows from Lemma 8 and Remark 10 above. ∎

Now as a consequence, we readily have the following corollaries.

######  Corollary 12.

Let $L(n,p)$ be as defined in equation (12). Then the rank of $L(n,n-1)$ is 2.

######  Corollary 13.

Let $\mathcal{OCT}_{n}$ be as defined in equation (1). Then the rank of
$\textnormal{Reg}(\mathcal{OCT}_{n})$ is 3.

###### Proof.

The proof follows from Corollary 12 coupled with the fact that
$\textnormal{Reg}(\mathcal{OCT}_{n})=L(n,n-1)\cup id_{[n]}$, where $id_{[n]}$
is the identity element on $[n]$. ∎

###  3.2 Rank of $\textnormal{Reg}(\mathcal{ORCT}_{n})$

To discuss the rank of $\textnormal{Reg}(\mathcal{ORCT}_{n})$, consider the
Table 6 above. Suppose we reverse the order of the image set of elements in
that table, then we will have the set of order-reversing elements of
$\textnormal{Reg}(\mathcal{ORCT}_{n})$. For $1\leq p\leq n$, let

$J_{p}=\\{\alpha\in\textnormal{Reg}(\mathcal{ORCT}_{n}):|\textnormal{Im
}\alpha|=p\\}$ (14)

and let

$K_{p}^{*}=\\{\alpha\in J_{p}:\alpha\textrm{ is order-reversing}\\}.$ (15)

Observe that $J_{p}=K_{p}\cup K_{p}^{*}$. Now define:

$\eta^{*}=\left(\begin{array}[]{ccccc}\\{1,\ldots j\\}&j+1&\cdots&n-1&n\\\
p&p-1&\ldots&2&1\end{array}\right),$ (16)
$\delta^{*}=\left(\begin{array}[]{ccccc}1&2&\cdots&p-1&\\{p,\ldots n\\}\\\
p&p-1&\cdots&2&1\end{array}\right)$ (17)

and

$\tau^{*}=\left(\begin{array}[]{ccccc}1&2&\cdots&p-1&\\{p,\ldots n\\}\\\
n&n-1&\ldots&j+1&j\par\end{array}\right)$ (18)

i.e., $\eta^{*},\delta^{*}$ and $\tau^{*}$ are respectively $\eta,\delta$ and
$\tau$ with image order-reversed.

######  Remark 14.

Throughout this section, we will write $\alpha^{*}$ to mean a mapping in
$K_{p}^{*}$ which has a corresponding mapping $\alpha$ in $K_{p}$ with order-
preserving image.

And let $R_{\eta^{*}}$ and $L_{\delta^{*}}$ be the respective $\mathcal{R}$
and $\mathcal{L}$ equivalent classes of $\eta$ and $\delta$. Then we have the
following lemmas which are analogue to Lemma 6.

######  Lemma 15.

Let $\eta$ and $\delta^{*}$ be as defined in equations (9) and (17),
respectively. Then
$\langle\textnormal{R}_{\eta}\cup\textnormal{L}_{\delta^{*}}\rangle=K_{p}^{*}$.

###### Proof.

Let
$\alpha^{*}=\left(\begin{array}[]{ccccc}\\{1,\ldots,a+1\\}&a+2&\ldots&a+{p-1}&\\{a+p,\ldots,n\\}\\\
x+p&x+{p-1}&\ldots&x+2&x+1\end{array}\right)$ be in $K_{p}^{*}$, then there
exists $\alpha\in K_{p}$ such that by Lemma 6, $\alpha$ can be expressed as
the following product:

$\left(\begin{array}[]{ccccc}\\{1,\ldots,a+1\\}&a+2&\ldots&a+{p-1}&\\{a+p,\ldots,n\\}\\\
y+1&y+2&\ldots&y+{p-1}&y+p\end{array}\right)\left(\begin{array}[]{ccccc}\\{1,\ldots,b+1\\}&b+2&\ldots&b+{p-1}&\\{b+p,\ldots,n\\}\\\
x+1&x+2&\ldots&x+{p-1}&x+p\end{array}\right)$

a product of elements of $\textnormal{R}_{\eta}$ and
$\textnormal{L}_{\delta}$, respectively. Therefore, $\alpha^{*}$ can be
expressed as the following product:

$\left(\begin{array}[]{ccccc}\\{1,\ldots,a+1\\}&a+2&\ldots&a+{p-1}&\\{a+p,\ldots,n\\}\\\
y+1&y+2&\ldots&y+{p-1}&y+p\end{array}\right)\left(\begin{array}[]{ccccc}\\{1,\ldots,b+1\\}&b+2&\ldots&b+{p-1}&\\{b+p,\ldots,n\\}\\\
x+1&x+2&\ldots&x+{p-1}&x+p\end{array}\right)$

a product of elements of $\textnormal{R}_{\eta}$ and
$\textnormal{L}_{\delta^{*}}$, respectively. ∎

######  Lemma 16.

Let $J_{p}=\\{\alpha\in\textnormal{Reg}(\mathcal{ORCT}_{n}):|\textnormal{Im
}\alpha|=p\\}$. Then, $\langle R_{\eta}\cup L_{\delta^{*}}\rangle=J_{p}$.

###### Proof.

Since $J_{p}=K_{p}\cup K_{p}^{*}$, to proof $\langle R_{\eta}\cup
L_{\delta^{*}}\rangle=J_{p}$, is suffices by Lemma 16 to show that
$K_{p}\subseteq\langle K_{p}^{*}\rangle$. Now, let

$\alpha=\left(\begin{array}[]{ccccc}\\{1,\ldots,a+1\\}&a+2&\ldots&a+{p-1}&\\{a+p,\ldots,n\\}\\\
b+1&b+2&\ldots&b+{p-1}&b+p\end{array}\right)$

be in $K_{p}$, if $\alpha$ is an idempotent, then there exists $\alpha^{*}\in
K_{p}^{*}$ such that $(\alpha^{*})^{2}=\alpha.$ Suppose $\alpha$ is not an
idempotent, define

$\epsilon=\left(\begin{array}[]{cccccc}\\{1,\ldots,b+1\\}&b+2&b+3&\ldots&b+{p-1}&\\{b+p,\ldots,n\\}\\\
b+1&b+2&b+3&\ldots&b+{p-1}&b+p\end{array}\right)$

which is an idempotent in $K_{p}$, then $\alpha$ can be written as
$\alpha=\alpha^{*}\epsilon^{*}$. ∎

Before stating the main theorem of this section, let

$M(n,p)=\\{\alpha\in\textnormal{Reg}(\mathcal{ORCT}_{n}):|\textnormal{Im
}\alpha|\leq p\\}\quad(1\leq p\leq n).$ (19)

And let

$W_{p}=M(n,p)\backslash M(n,p-1)$ (20)

be Rees quotient semigroup on $M(n,p)$. From Lemma 16 and Remark 7 we have:

######  Lemma 17.

$(\textnormal{R}_{\eta}\cup\textnormal{L}_{\delta^{*}})\backslash\delta$ is
the minimal generating set for the Rees quotient semigroup $W_{p}$.

The next proposition is also analogue to Proposition 9 which plays an
important role in finding the generating set for the subsemigroup $M(n,p)$.

######  Proposition 18.

For $n\geq 4,\;\langle J_{p}\rangle\,\subseteq\,\langle J_{p+1}\rangle$ for
all $1\leq p\leq n-2$.

###### Proof.

The proof follows the same pattern as the proof of the Proposition 9. We want
to show that
$(\textnormal{R}_{\eta}\cup\textnormal{L}_{\delta^{*}})\subseteq\,\langle
J_{p+1}\rangle$ and by Proposition 9 we only need to show that
$\textnormal{L}_{\delta^{*}}\subseteq\,\langle J_{p+1}\rangle$. Now Let
$\alpha$ be in $\textnormal{L}_{\delta^{*}}$,

Case I: $\alpha\in\textnormal{L}_{\delta^{*}}\backslash\tau^{*}$, then
$\alpha$ is the of the form

$\left(\begin{array}[]{ccccc}1&2&\cdots&p-1&\\{p,p+1,\ldots n\\}\\\
p+r-1&p+r-2&\cdots&r+1&r\end{array}\right)\;(r=1,2,\ldots,n-p),$

and it can be written as

$\alpha=\left(\begin{array}[]{ccccc}1&2&\cdots&p&\\{p+1,\ldots n\\}\\\
2&3&\cdots&p+1&p+2\end{array}\right)\left(\begin{array}[]{ccccc}1&2&\cdots&p&\\{p+1,\ldots
n\\}\\\ p+r&p+r-1&\cdots&r+1&r\end{array}\right),$

a product of two elements of $J_{p+1}$.

Case II: $\alpha=\tau^{*}$ then $\alpha$ can be written as

$\alpha=\left(\begin{array}[]{ccccc}1&2&\cdots&p-1&\\{p,\ldots n\\}\\\
1&2&\cdots&p-1&p\end{array}\right)\left(\begin{array}[]{ccccc}1&2&\cdots&p&\\{p+1,\ldots
n\\}\\\ n&n-1&\cdots&j&j-1\end{array}\right).$

The first element in the product above is $\delta\in J_{p}$, but it was shown
in Remark 7 that it can be written as $\tau\eta$ which were both shown in
Proposition 9 that they can be expressed as product of elements of $J_{p+1}$.
Hence the proof. ∎

######  Remark 19.

Notice also that, by Proposition 18 above, for $2\leq p\leq n-1$ the
generating set for $W_{p}$ generates the whole $M(n,p)$

The next theorem gives us the rank of subsemigroup $M(n,p)$ for $2\leq p\leq
n-1$.

######  Theorem 20.

Let $M(n,p)$ be as defined in equation (19). Then for $n\geq 4$ and $2<p\leq
n-1$, the rank of $M(n,p)$ is $2(n-p)+1.$

###### Proof.

To proof this, we only need to compute the cardinality of the set
$(R_{\eta}\cup L_{\delta^{*}})\backslash\delta$, which from Table 6 we easily
obtain $(n-p)+(n-p)+1=2(n-p)+1$. ∎

As a consequence, we have the following corollaries.

######  Corollary 21.

Let $M(n,p)$ be as defined in equation (19). Then the rank of $M(n,n-1)$ is 3.

######  Corollary 22.

Let $\mathcal{ORCT}_{n}$ be as defined in equation (2). Then the rank of
$\textnormal{Reg}(\mathcal{ORCT}_{n})$ is 4.

###### Proof.

The only elements of $\textnormal{Reg}(\mathcal{OCT}_{n})$ that are not in
$M(n,n-1)$ are the elements $\\{id_{[n]},id_{[n]}^{*}\\}$. It is clear that
$(id^{*})^{2}=id$. ∎

## References

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First Author
Department of Mathematics, Bayero University Kano, P.O.Box 3011, Kano, Nigera.
Email: mmzubairu.mth buk.edu.ng

Second Author
Khalifa University, P. O. Box 127788, Sas al Nakhl, Abu Dhabi, UAE
Email<EMAIL_ADDRESS>

Third Author
Department of Mathematics, and Computer Sciences, Sule Lamido University,
Kafin Hausa , Nigera.
Email<EMAIL_ADDRESS>


Schwarzschild black holes in Starobinsky-Bel-Robinson gravity

Ruben Campos Delgadoa and Sergey V. Ketovb,c,d

aBethe Center for Theoretical Physics,
Physikalisches Institut der Universität Bonn,
Nussallee 12, 53115 Bonn, Germany
bDepartment of Physics, Tokyo Metropolitan University,
1-1 Minami-ohsawa, Hachioji-shi, Tokyo 192-0397, Japan
cInterdisciplinary Research Laboratory,
Tomsk State University, 36 Lenin Avenue, Tomsk 634050, Russia
cKavli Institute for the Physics and Mathematics of the Universe,
The University of Tokyo Institutes for Advanced Study, Kashiwa 277-8583, Japan

††footnotetext<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

We study physical properties of a Schwarzschild black hole in the framework of
the recently proposed Starobinsky-Bel-Robinson (SBR) modified theory of
gravity, working perturbatively in the coupling constant. In particular, we
compute the temperature, entropy, pressure and lifetime of a Schwarzschild
black hole.

##  1 Introduction

The Starobinsky model of modified $(R+R^{2})$ gravity [1] is known as an
excellent model of cosmological inflation, in very good agreement with
WMAP/Planck/BICEP/KECK precision measurements of the cosmic microwave
background radiation [2]. ††See e.g. Ref. [3] for a recent review. The
Starobinsky model has no free parameters because its only parameter, given by
the inflaton mass $m$, is determined by the COBE/WMAP normalization as $m\sim
10^{-5}M_{\rm Pl}$ in terms of the reduced Planck mass $M_{\rm
Pl}=1/\sqrt{8\pi G_{N}}$. The ultra-violet (UV) cutoff of the Starobinsky
model is given by $M_{\rm Pl}$ also, while the Starobinsky inflation belongs
to the class of the single-large-field inflationary models that are UV-
sensitive to quantum gravity corrections. It is therefore of importance to
identify those corrections and study their applications in the high-curvature
regimes relevant to the early universe and black holes.

An extension of the Starobinsky $(R+R^{2})$ gravity in $D=4$ spacetime
dimensions by terms quartic in the curvature was proposed in Ref. [4]. This
extension was motivated by the leading quantum gravity correction obtained by
dimensional reduction of M-theory in $D=11$ dimensions. The extra terms in the
$D=4$ gravitational low-energy effective action have a peculiar structure
given by the square of the Bel-Robinson tensor [5, 6]. The new modified
gravity was thus dubbed the Starobinsky-Bel-Robinson (SBR) gravity in Ref.
[4]. The SBR action reads

$S_{\rm SBR}[g_{\mu\nu}]=\frac{M^{2}_{\text{Pl}}}{2}\int
d^{4}x\sqrt{-g}\,\left[R+\frac{1}{6m^{2}}R^{2}-\frac{\beta}{32M^{6}_{\text{Pl}}}\left(\mathcal{P}^{2}-\mathcal{G}^{2}\right)\right],$
(1)

where $\beta>0$ is the new dimensionless coupling constant whose (unknown)
value is supposed to be determined by compactification of M-theory,
$\mathcal{G}$ and $\mathcal{P}$ are the Euler and Pontryagin topological
densities in $D=4$, respectively, coming from the BR-tensor squared,

$\mathcal{G}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$
(2)

and

$\mathcal{P}=\frac{1}{2}\sqrt{-g}\epsilon_{\mu\nu\rho\sigma}{R^{\rho\sigma}}_{\alpha\beta}R^{\mu\nu\alpha\beta}.$
(3)

In this letter we study how the physical properties of a Schwarzschild black
hole get modified by the presence of the $\beta$-dependent terms in Eq. (1).
Since the value of the $\beta$-parameter is expected to be small, $\beta\ll
1$, we work perturbatively at first order in $\beta$ only. Our paper is
organized as follows. In Section 2 we compute the $\beta$-dependent correction
to the Schwarzschild metric. This correction shifts the position of the event
horizon. Having the displaced event horizon leads to the corresponding change
in the black hole entropy obtained by integrating over the horizon area. In
Section 3 we derive the $\beta$-corrections to the temperature, pressure and
lifetime of a Schwarzschild black hole. Section 4 is our conclusion. We use
natural units with $\hbar=c=k_{B}=1$.

##  2 $\beta$-correction to the Schwarzschild metric

In this section we derive and solve the equations of motion obtained by
varying the action (1) with respect to the metric. Our task is simplified by
the fact that the Pontryagin density $\mathcal{P}$ vanishes for a spherically
symmetric metric. The equations of motion are

$J_{\mu\nu}:=G_{\mu\nu}+\frac{1}{6m^{2}}H_{\mu\nu}-\frac{\beta}{32M^{6}_{\text{Pl}}}K_{\mu\nu}=0~{}~{},$
(4)

where $G_{\mu\nu}=R_{\mu\nu}-1/2g_{\mu\nu}R$ is the Einstein tensor,

$H_{\mu\nu}=2R_{\mu\nu}R-\frac{1}{2}g_{\mu\nu}R^{2}-2\nabla_{\mu}\nabla_{\nu}R+2g_{\mu\nu}\Box{R}$
(5)

and

$\begin{gathered}K_{\mu\nu}=\frac{1}{2}g_{\mu\nu}\mathcal{G}^{2}-2\Big{[}2\mathcal{G}RR_{\mu\nu}-4\mathcal{G}{R_{\mu}}^{\rho}R_{\nu\rho}\\\
+2\mathcal{G}{R_{\mu}}^{\rho\sigma\lambda}R_{\nu\rho\sigma\lambda}+4\mathcal{G}R^{\rho\sigma}R_{\mu\rho\sigma\nu}+2g_{\mu\nu}R\Box{\mathcal{G}}-2R\nabla_{\mu}\nabla_{\nu}\mathcal{G}\\\
-4R_{\mu\nu}\Box\mathcal{G}+4\left(R_{\mu\rho}\nabla^{\rho}\nabla_{\nu}\mathcal{G}+R_{\nu\rho}\nabla^{\rho}\nabla_{\mu}\mathcal{G}\right)\\\
-4g_{\mu\nu}R_{\rho\sigma}\nabla^{\sigma}\nabla^{\rho}\mathcal{G}+4R_{\mu\rho\nu\sigma}\nabla^{\sigma}\nabla^{\rho}\mathcal{G}\Big{]}.\end{gathered}$
(6)

We search for a solution in the form

$ds^{2}=-a(r)dt^{2}+\frac{1}{a(r)}dr^{2}+r^{2}d\Omega^{2}_{2},$ (7)

where

$d\Omega^{2}_{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}~{},$ (8)

and with the function $a(r)$ beginning with the Schwarzschild solution

$a(r)=1-\frac{2G_{N}M}{r}-\beta f(r),$ (9)

in the presence of a perturbation $-\beta f(r)$. We assume that $\beta\ll 1$
and linearize our equations with respect to $\beta$.

Because of spherical symmetry, there are only four independent equations to
consider. Plugging (9) into (4) we find, at first order in $\beta$,

$\begin{gathered}J_{tt}=\frac{r-2G_{N}M}{r^{13}}\bigg{[}\frac{36G^{3}_{N}M^{3}}{M^{6}_{\text{Pl}}}\left(67G_{N}M-32r\right)\\\
+r^{7}\left(r^{3}+\frac{4G_{N}M}{m^{2}}-\frac{4r}{3m^{2}}\right)f(r)+r^{8}\left(r^{3}-\frac{2G_{N}M}{m^{2}}+\frac{4r}{3m^{2}}\right)f^{\prime}(r)\\\
-\frac{2r^{10}}{3m^{2}}f^{\prime\prime}(r)+\frac{r^{10}}{3m^{2}}\left(11G_{N}M-6r\right)f^{(3)}(r)\\\
+\frac{r^{11}}{3m^{2}}\left(2G_{N}M-r\right)f^{(4)}(r)\bigg{]}\beta+\mathcal{O}\left(\beta^{2}\right)=0,\end{gathered}$
(10)
$\begin{gathered}J_{rr}=\frac{1}{(2G_{N}M-r)r^{11}}\bigg{[}\frac{36G^{3}_{N}M^{3}}{M^{6}_{\text{Pl}}}\left(11G_{N}M-4r\right)\\\
+r^{7}\left(r^{3}-\frac{4G_{N}M}{m^{2}}+\frac{8r}{3m^{2}}\right)f(r)+r^{8}\left(r^{3}-\frac{2G_{N}M}{m^{2}}+\frac{4r}{3m^{2}}\right)f^{\prime}(r)\\\
+\frac{4r^{9}}{3m^{2}}\left(3G_{N}M-2r\right)f^{\prime\prime}(r)+\frac{r^{10}}{3m^{2}}\left(3G_{N}M-2r\right)f^{(3)}(r)\bigg{]}\beta+\mathcal{O}\left(\beta^{2}\right)=0,\end{gathered}$
(11)
$\begin{gathered}J_{\theta\theta}=\frac{1}{2r^{10}}\bigg{[}\frac{72G^{3}_{N}M^{3}}{M^{6}_{\text{Pl}}}\left(41G_{N}M-18r\right)\\\
+\frac{16r^{7}}{3m^{2}}\left(r-3G_{N}M\right)f(r)-2r^{9}\left(r^{2}+\frac{2}{3m^{2}}\right)f^{\prime}(r)\\\
-r^{9}\left(r^{3}+\frac{4}{3m^{2}}(r-6G_{N}M)\right)f^{\prime\prime}(r)+\frac{2r^{10}}{3m^{2}}\left(5r-8G_{N}M\right)f^{(3)}(r)\\\
+\frac{2r^{11}}{3m^{2}}\left(r-2G_{N}M\right)f^{(4)}(r)\bigg{]}\beta+\mathcal{O}\left(\beta^{2}\right)=0,\end{gathered}$
(12)
$\begin{gathered}J_{\phi\phi}=\frac{\sin^{2}\theta}{2r^{10}}\bigg{[}\frac{72G^{3}_{N}M^{3}}{M^{6}_{\text{Pl}}}\left(41G_{N}M-18r\right)\\\
+\frac{16r^{7}}{3m^{2}}\left(r-3G_{N}M\right)f(r)-2r^{9}\left(r^{2}+\frac{2}{3m^{2}}\right)f^{\prime}(r)\\\
-r^{9}\left(r^{3}+\frac{4}{3m^{2}}(r-6G_{N}M)\right)f^{\prime\prime}(r)+\frac{2r^{10}}{3m^{2}}\left(5r-8G_{N}M\right)f^{(3)}(r)\\\
+\frac{2r^{11}}{3m^{2}}\left(r-2G_{N}M\right)f^{(4)}(r)\bigg{]}\beta+\mathcal{O}\left(\beta^{2}\right)=0\bigg{]}.\end{gathered}$
(13)

This system of differential equations can be reduced to a single equation
after eliminating the 3rd and 4th order derivatives. Indeed, solving
$j_{\theta\theta}$ or $j_{\phi\phi}$ for $f^{(4)}(r)$, $j_{rr}$ for
$f^{(3)}(r)$ and plugging the resulting expressions into $j_{tt}$, we obtain
the equation

$\begin{gathered}\frac{72G^{3}_{N}M^{3}}{M^{6}_{\text{Pl}}}\left(291G^{2}_{N}M^{2}-343G_{N}Mr+96r^{2}\right)\\\
-2r^{11}f(r)+2r^{11}\left(r-3G_{N}M\right)f^{\prime}(r)+r^{12}\left(2r-3G_{N}M\right)f^{\prime\prime}(r)=0~{},\end{gathered}$
(14)

whose solution is

$\begin{gathered}f(r)=\frac{4G^{3}_{N}M^{3}}{5M^{6}_{\text{Pl}}r^{10}}\left(97G_{N}M-54r\right)=\frac{2048\pi^{3}G^{6}_{N}M^{3}}{5r^{10}}\left(97G_{N}M-54r\right).\end{gathered}$
(15)

As a result, we find the function $a(r)$ in the form

$a(r)=1-\frac{r_{S}}{r}+\beta\frac{128\pi^{3}}{5}\left(\frac{G_{N}r_{S}}{r^{3}}\right)^{3}\left(108-97\frac{r_{S}}{r}\right)~{},$
(16)

where we have introduced the (unperturbed) Schwarzschild radius
$r_{S}=2G_{N}M$.

##  3 $\beta$-corrections to the physical properties of a Schwarzschild black
hole

Let us now study how the solution (16) affects the physical properties of a
Schwarzschild black hole, namely, temperature, entropy, pressure and lifetime.
The horizon radius is determined by a solution of $a(r)=0$. At first order in
$\beta$ we find

$r_{H}=2G_{N}M-\beta\frac{44\pi^{3}}{5G^{2}_{N}M^{5}}+\mathcal{O}\left(\beta^{2}\right).$
(17)

Therefore, the Hawking temperature is given by

$T_{H}=\frac{1}{4\pi}\frac{da(r)}{dr}\bigg{\rvert}_{r=r_{H}}=\frac{1}{8\pi
G_{N}M}+\beta\frac{\pi^{2}}{G^{4}_{N}M^{7}}+\mathcal{O}\left(\beta^{2}\right).$
(18)

According to the Wald formula [7], the entropy is

$S=-2\pi\int_{r=r_{H}}d\Sigma\,\epsilon_{\mu\nu}\epsilon_{\rho\sigma}\frac{\partial\mathcal{L}}{\partial
R_{\mu\nu\rho\sigma}},$ (19)

where $d\Sigma=r^{2}\sin\theta d\theta d\phi$, $\epsilon_{\mu\nu}$ is the
Levi-Civita tensor in two dimensions and $\mathcal{L}$ is the Lagrangian
density of the theory. In our case we have

$\mathcal{L}=\frac{1}{16\pi
G_{N}}\left(R+\frac{1}{6m^{2}}R^{2}+16\pi^{3}G^{3}_{N}\beta\mathcal{G}^{2}\right).$
(20)

The antisymmetry of $\epsilon_{\mu\nu}$ reduces the expression of the entropy
to

$S=-8\pi A\frac{\partial\mathcal{L}}{\partial
R_{rtrt}}\bigg{\rvert}_{r=r_{H}},$ (21)

where $A=4\pi r^{2}_{H}$ is the area enclosed by the event horizon. The
partial derivative of the Lagrangian with respect to the Riemann tensor is
computed by using the formulae [8]

$\begin{gathered}\frac{\partial R}{\partial
R_{\mu\nu\rho\sigma}}=\frac{1}{2}\left(g^{\mu\rho}g^{\nu\sigma}-g^{\mu\sigma}g^{\nu\rho}\right),\\\
\frac{\partial R_{\alpha\beta}R^{\alpha\beta}}{\partial
R_{\mu\nu\rho\sigma}}=g^{\mu\rho}R^{\nu\sigma}-g^{\nu\rho}R^{\mu\sigma},\\\
\frac{\partial
R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}}{\partial
R_{\mu\nu\rho\sigma}}=2R^{\mu\nu\rho\sigma}.\end{gathered}$ (22)

We find

$\begin{gathered}\frac{\partial\mathcal{L}}{\partial R_{rtrt}}=\frac{1}{16\pi
G_{N}}\bigg{\\{}-\frac{1}{2}-\frac{R}{6m^{2}}\\\
+16\pi^{3}G^{3}_{N}\beta\Big{[}32g^{rr}R^{tt}R_{\alpha\beta}R^{\alpha\beta}-8\left(g^{rr}R^{tt}R^{2}-R_{\alpha\beta}R^{\alpha\beta}R\right)\\\
-2R^{3}-8\left(g^{rr}R^{tt}R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}+2R^{rtrt}R_{\alpha\beta}R^{\alpha\beta}\right)\\\
+2\left(2R^{rtrt}R^{2}-RR_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}\right)+4R^{rtrt}R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}\Big{]}\bigg{\\}}.\end{gathered}$
(23)

The final expression for the entropy is given by

$S=\frac{A}{4G_{N}}+\beta\left(\frac{304\pi^{4}}{5G^{2}_{N}M^{4}}+\frac{624\pi^{4}}{5m^{2}G_{N}^{4}M^{6}}\right)+\mathcal{O}\left(\beta^{2}\right).$
(24)

The classical relation $TdS=dM$ receives a correction also. We find

$TdS=dM+\beta\left(\frac{8\pi^{3}}{G^{3}_{N}M^{6}}-\frac{152\pi^{3}}{5G^{3}_{N}M^{6}}-\frac{468\pi^{3}}{5m^{2}G^{5}_{N}M^{8}}\right)dM.$
(25)

The physical interpretation of this result is that the Bel-Robinson (quartic)
term generates a pressure for the black hole. The first law of thermodynamics
is then given by

$TdS-
PdV=dM+\beta\left(\frac{8\pi^{3}}{G^{3}_{N}M^{6}}-\frac{152\pi^{3}}{5G^{3}_{N}M^{6}}-\frac{468\pi^{3}}{5m^{2}G^{5}_{N}M^{8}}\right)dM,$
(26)

where $P$ is the pressure of the black hole and $V$ is its volume, $V=4/3\pi
r^{3}_{H}$. We can then identify $TdS=dM$ and obtain

$P=\beta\left(\frac{7\pi^{2}}{10G^{6}_{N}M^{8}}+\frac{117\pi^{2}}{40m^{2}G^{8}_{N}M^{10}}\right)+\mathcal{O}\left(\beta^{2}\right).$
(27)

A non-vanishing pressure for a Schwarzschild black hole (it has no pressure
classically) is the common feature of quantum gravity [9].

The last physical quantity we want to consider is the mass loss rate due to
the Hawking radiation. If the black hole mass is sufficiently large, then the
temperature is low. Accordingly, all the mass lost in the Hawking process is
due to the emission of massless particles. The mass loss rate is given by the
Stefan-Boltzmann law [10]

$\frac{dM}{dt}=-\xi
T^{4}\left(\frac{21}{8}\sigma_{\nu}+\sigma_{\gamma}+\sigma_{g}\right),$ (28)

where $\xi=\pi^{2}/60$ is the Stefan-Boltzmann constant and $\sigma_{\nu}$,
$\sigma_{\gamma}$, $\sigma_{g}$ are the thermally averaged cross sections of
the black hole for neutrinos, photons and gravitons, respectively. We neglect
the neutrino masses in what follows, and introduce the geometrical optics
cross section $\sigma_{0}$ for the Schwarzschild black hole. Defining also

$\rho=\left(\frac{21}{8}\sigma_{\nu}+\sigma_{\gamma}+\sigma_{g}\right)\sigma^{-1}_{0},$
(29)

we rewrite (28) as follows:

$\frac{dM}{dt}=-\xi T^{4}\rho\sigma_{0}~{}.$ (30)

The cross sections for neutrinos, photons and gravitons can be estimated as
[10]

$\sigma_{\nu}\approx 0.66852\sigma_{0},\hskip 5.69054pt\sigma_{\gamma}\approx
0.24044\sigma_{0},\hskip 5.69054pt\sigma_{g}\approx 0.02748\sigma_{0},$ (31)

while most of the power is emitted in neutrinos. The value of $\rho$ is then
$\rho\approx 2.0228$.

Classically, we have $\sigma_{0}=27\pi G^{2}_{N}M^{2}$. We now derive the
$\beta$-dependent correction to the classical result. The emitted particles
move along null geodesics governed by the equation

$\left(\frac{dr}{d\lambda}\right)^{2}=E^{2}-a(r)\frac{J^{2}}{r^{2}}~{}~{},$
(32)

where $\lambda$ is the affine parameter and $E$, $J$ are the energy and
angular momentum, respectively. For the emitted particles to reach the
infinity rather than falling back into the black hole horizon, one has to
require $(dr/d\lambda)^{2}\geq 0$, i.e.

$\frac{1}{l^{2}}\equiv\frac{E^{2}}{J^{2}}\geq\frac{a(r)}{r^{2}}$ (33)

for any $r\geq r_{H}$. The inequality is saturated by the maximal value of
$a(r)/r^{2}$. The solution of $\partial_{r}\left(a(r)/r^{2}\right)=0$
corresponds to the critical, unstable orbit $r_{c}$, and the impact factor can
be computed as $l_{c}=r_{c}/\sqrt{a(r_{c})}$ [11, 12]. The geometrical optics
cross section of the black hole is then given by

$\sigma_{0}=\pi l^{2}_{c}=\pi\frac{r^{2}_{c}}{a(r_{c})}~{}.$ (34)

With the solution (15) we find

$\sigma_{0}=27\pi
G^{2}_{N}M^{2}-\beta\frac{26624\pi^{4}}{729G_{N}M^{4}}+\mathcal{O}\left(\beta^{2}\right)~{}.$
(35)

Finally, using (18) for the temperature, the mass loss rate is

$\frac{dM}{dt}=\xi\rho\left(\frac{27}{4096\pi^{3}G^{2}_{N}M^{2}}+\beta\frac{18851}{93312G^{5}_{N}M^{8}}\right)+\mathcal{O}\left(\beta^{2}\right).$
(36)

Integrating (36) we obtain the lifetime of the black hole as

$t=\frac{4096\pi^{3}G^{2}_{N}M^{3}}{81\xi\rho}-\sqrt{\beta}\frac{8192\pi^{11/2}\sqrt{\frac{37702}{3}}\sqrt{G_{N}}}{6561\xi\rho}+\beta\frac{2470838272\pi^{6}}{1594323G_{N}M^{3}\xi\rho}+\mathcal{O}\left(\beta^{3/2}\right).$
(37)

##  4 Conclusion

Our main new results are given by (i) the metric solution (7) and (16) in the
first order with respect to the coupling constant $\beta$, (ii) the displaced
black hole horizon (17), (iii) the deformed Hawking temperature (18), (iv) the
deformed entropy (24), (v) the non-vanishing pressure (27), and (vi) the
lifetime (37), all derived for the Schwarzschild black holes in the SBR
gravity (1) that, in turn, represents the possible quantum gravity low-energy
effective action inspired by the Starobinsky inflation model and
superstrings/M-theory.

In particular, it follows from (17) that the black hole horizon radius is
decreased. It follows from (18) that the Hawking temperature is increased. The
black hole entropy (24) is increased also, whereas the black hole pressure
(27) becomes positive. As regards the black hole lifetime (37), we find two
additional contributions with the opposite signs, where the negative
contribution given by the 2nd term apparently dominates but the last term may
become important for black holes with small masses. However, our approximation
is expected to break down for very small black hole masses because in that
case the even higher orders (in the spacetime curvature) have to be taken into
account in the effective quantum gravity action.

## Acknowledgements

S.V.K. was supported by Tokyo Metropolitan University, the Japanese Society
for Promotion of Science under the grant No. 22K03624, and the Tomsk State
University development program Priority-2030.

## References

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  * [2] BICEP, Keck Collaboration, P. A. R. Ade et al., “Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season,” Phys. Rev. Lett. 127 no. 15, (2021) 151301, arXiv:2110.00483 [astro-ph.CO].
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  * [6] I. Robinson, “On the Bel-Robinson tensor,” Class. Quantum Grav. 14 (1997) A331–A333.
  * [7] R. M. Wald, “Black hole entropy is the Noether charge,” Phys. Rev. D 48 no. 8, (1993) R3427–R3431, arXiv:gr-qc/9307038.
  * [8] R. C. Delgado, “Quantum gravitational corrections to the entropy of a Reissner–Nordström black hole,” Eur. Phys. J. C 82 no. 3, (2022) 272, arXiv:2201.08293 [hep-th].
  * [9] X. Calmet and F. Kuipers, “Quantum gravitational corrections to the entropy of a Schwarzschild black hole,” Phys. Rev. D 104 no. 6, (2021) 066012, arXiv:2108.06824 [hep-th].
  * [10] W. A. Hiscock and L. D. Weems, “Evolution of Charged Evaporating Black Holes,” Phys. Rev. D 41 (1990) 1142.
  * [11] H. Xu, Y. C. Ong, and M.-H. Yung, “Cosmic Censorship and the Evolution of d-Dimensional Charged Evaporating Black Holes,” Phys. Rev. D 101 no. 6, (2020) 064015, arXiv:1911.11990 [gr-qc].
  * [12] H. Xu and M.-H. Yung, “Black hole evaporation in Lovelock gravity with diverse dimensions,” Phys. Lett. B 794 (2019) 77–82, arXiv:1904.06503 [gr-qc].


# UNIPOL: Unimodular sequence design via a separable iterative quartic
polynomial optimization for active sensing systems

Surya Prakash Sankuru, Prabhu Babu, Mohammad Alaee-Kerahroodi

###### Abstract

Sequences having better autocorrelation properties play a crucial role in
enhancing the performance of active sensing systems. Hence, sequences with
good autocorrelation properties are very much in demand. In this paper, we
addressed the problem of designing a unimodular sequence having better side-
lobe levels. We formulated it as a constrained optimization problem comprising
the Integrated Side-lobe Level (ISL) metric and then proposed an effective
algorithm (named UNIPOL - UNImodular sequence design via a separable iterative
POLynomial optimization) where we perform the polynomial optimization at every
iteration. The novelty of the paper comes from deriving a quartic majorization
function that is separable in the sequence variables and that can be minimized
parallelly. To evaluate the performance of our proposed algorithm we perform
the numerical experiments for different sequence lengths and confirm that our
proposed algorithm is the fastest algorithm to attain an actual optimum
minimizer of the ISL metric. Our proposed algorithm is also computationally
efficient due to its ease of implementation using the FFT, IFFT operations in
a parallel fashion.

Index Terms– radar, sonar, Integrated Side-lobe Level, sequence design,
majorization-minimization.

##  I Introduction and Problem Formulation

One of the major challenges in active sensing systems (RADAR/SONAR) is to
estimate the underlying target parameters which result in the avoidance of
false targets as well as able to discriminate the detected targets. But, the
accuracy of estimated parameters will depend purely upon the autocorrelation
properties of transmit sequences. Hence, transmit sequences with better
autocorrelation properties are always in demand [1]. Apart from the active
sensing systems, some other applications where sequences with good
autocorrelation properties are usually preferred are cryptography and CDMA
communication systems [2]. In practice along with the correlation property,
the aforementioned applications also pose different constraints like the power
(due to its limited quantity available in the system) and spectral (to avoid
the already reserved frequencies) constraints. Hence, we consider the problem
of designing unimodular (constant modulus) sequences with better
autocorrelation properties.

Let $\left\\{x_{n}\right\\}_{n=1}^{N}$ be the $N$ length unimodular sequence
having aperiodic auto-correlations $\left\\{r_{k}\right\\}_{k=-N+1}^{N-1}$
defined as:

$r_{k}=\sum_{n=1}^{N-k}x_{n+k}x_{n}^{*}=r_{-k}^{*},\,k=-N+1,...,N-1.$ (1)

Thus the design of a sequence having better side-lobe levels
$\left(i.e.\left\\{r_{k}\right\\}_{k=1}^{N-1}\right)$ can be formulated as:

$\underset{\left|x_{n}\right|=1,\,\forall
n}{\text{minimum}}\hphantom{n}\stackrel{{\scriptstyle[}}{{k}}=1]{N-1}{\sum}\left|r_{k}\right|^{2}$
(2)

Some of the existing methods which are capable of generating the unimodular
sequences are given as follows. Earlier, there exist only the analytical
approaches which are able to design the sequences of specific lengths (for
instance, power of $2$). Some of the well known analytical approach based
sequences are the Barker [3], [4], Frank [5], Golomb [6], Chu [6], [7] and P4
[8], [9] sequences. To overcome the demarcation of analytical approaches, in
the last decade or so, the computational approaches came into the existence.
The authors in [10] proposed the CAN algorithm which can solve (2) by using
the alternating minimization technique. However, instead of solving the
problem in (2) directly, they solved its approximation. To solve the same ISL
minimization problem in (2) the authors in [11], [12] proposed the MISL and
ISL-NEW algorithms, respectively. Both of them are majorization minimization
(MM) technique based algorithms comprising the linear (in $x_{n}$)
majorization functions which results in slower convergence issues. The authors
in [13] used the ADMM technique to solve an approximation of the problem in
(2). In [14] the authors proposed the CPM algorithm based on the coordinate
descent technique. But, when the length of the sequence increases the CPM
would take a considerable amount of time to converge. Hence, we decided to
propose an efficient technique to solve the problem in (2). The major
contributions of this letter are: We formulate the problem of designing a
unimodular sequence as a constrained ISL minimization problem and then
proposed a quartic polynomial optimization based iterative algorithm. In
particular, unlike the state-of-the-art MM based approaches which employ a
loose linear upper bound, here we derive a tighter quartic upper bound for the
ISL objective function. It makes the resultant surrogate minimization problem
separable in the sequence variables which in turn can be solved in a parallel
fashion \- this would be an attractive feature if one intends to design
sequences of very large lengths. Later, through computer simulations, we show
the efficacy of our proposed algorithm by comparing it with the state-of-the-
art methods. The rest of the paper is formulated as follows: a brief
introduction of the majorization minimization technique and the proposed
algorithm is given in section II. The numerical simulation results are given
in section III and finally section IV concludes the paper.

##  II UNIPOL - The proposed algorithm

Before introducing our algorithm, we will discuss the basic principle of the
MM method as follows. The MM method [15] is a numerical technique that
involves calculating the upper bound (majorization) function
$u(\boldsymbol{\boldsymbol{\boldsymbol{z}}}|\boldsymbol{\boldsymbol{\boldsymbol{z}}}^{k})$
to the original objective function $g(\boldsymbol{z})$ at any point
$\boldsymbol{\boldsymbol{\boldsymbol{z}}}^{k}$ and then minimizing the upper
bound to arrive at the next iterate point. The constructed majorization
function has to satisfy the following properties.

$u(\boldsymbol{\boldsymbol{\boldsymbol{z}}}^{k}|\boldsymbol{\boldsymbol{\boldsymbol{z}}}^{k})=g(\boldsymbol{\boldsymbol{\boldsymbol{z}}}^{k})\text{
and
}u(\boldsymbol{\boldsymbol{z}}|\boldsymbol{\boldsymbol{\boldsymbol{z}}}^{k})\geq
g(\boldsymbol{\boldsymbol{z}}),\>\forall\boldsymbol{\boldsymbol{z}}\in\boldsymbol{\boldsymbol{\boldsymbol{Z}}}.$
(3)

where $\boldsymbol{\boldsymbol{\boldsymbol{Z}}}$ is a feasible set. As the MM
technique is an iterative process, it will generate the sequence of points
$\left\\{\boldsymbol{\boldsymbol{z}}\right\\}=\boldsymbol{\boldsymbol{\boldsymbol{z}}}^{1},\boldsymbol{\boldsymbol{\boldsymbol{z}}}^{2},\boldsymbol{\boldsymbol{\boldsymbol{z}}}^{3},.....,\boldsymbol{\boldsymbol{\boldsymbol{z}}}^{m}$
according to the following update rule:

$\boldsymbol{\boldsymbol{\boldsymbol{z}}}^{k+1}\triangleq\text{arg}\min_{\boldsymbol{\boldsymbol{z}\in\boldsymbol{\boldsymbol{\boldsymbol{Z}}}}}u(\boldsymbol{\boldsymbol{z}}|\boldsymbol{\boldsymbol{\boldsymbol{z}}}^{k}).$
(4)

The value of the objective function computed at every iterate point (4) will
satisfy the monotonic decreasing property, i.e.

$g(\boldsymbol{\boldsymbol{\boldsymbol{z}}}^{k+1})\leq
u(\boldsymbol{\boldsymbol{\boldsymbol{z}}}^{k+1}|\boldsymbol{\boldsymbol{\boldsymbol{z}}}^{k})\leq
u(\boldsymbol{\boldsymbol{\boldsymbol{z}}}^{k}|\boldsymbol{\boldsymbol{\boldsymbol{z}}}^{k})=g(\boldsymbol{\boldsymbol{\boldsymbol{z}}}^{k}).$
(5)

Now, we will start with our problem of interest in (2). Here, the cost
function is expressed in terms of the autocorrelation values, but by re-
expressing it in frequency domain (using the Parseval’s theorem), we arrive at
the following equivalent form (a short proof of the same can be found in the
appendix of [10]):

$\stackrel{{\scriptstyle[}}{{k}}=1]{N-1}{\sum}\left|r_{k}\right|^{2}\triangleq\frac{1}{2N}\stackrel{{\scriptstyle[}}{{p}}=1]{2N}{\sum}\left|\left|\stackrel{{\scriptstyle[}}{{n}}=1]{N}{\sum}x_{n}e^{-j\omega_{p}n}\right|^{2}-N\right|^{2}$
(6)

where $\omega_{p}=\frac{2\pi}{2N}p$.

By expanding the square in the equivalent objective function and using the
fact that the energy of the signal is constant (i.e.
$\stackrel{{\scriptstyle[}}{{p}}=1]{2N}{\sum}\left|\stackrel{{\scriptstyle[}}{{n}}=1]{N}{\sum}x_{n}e^{-j\omega_{p}n}\right|^{2}=2N^{2}$),
the ISL minimization problem can be expressed as:

$\underset{\left|x_{n}\right|=1,\,\forall
n}{\text{minimum}}\hphantom{n}\stackrel{{\scriptstyle[}}{{p}}=1]{2N}{\sum}\left|\stackrel{{\scriptstyle[}}{{n}}=1]{N}{\sum}x_{n}e^{-j\omega_{p}n}\right|^{4}$
(7)

Now by using the Jensen’s inequality (see example-$11$ in [15]), at any given
$x_{n}^{t}$, the cost function in the above problem can be majorized (or)
upper bounded as follows:

$\begin{array}[]{ll}\stackrel{{\scriptstyle[}}{{p}}=1]{2N}{\sum}\left|\stackrel{{\scriptstyle[}}{{n}}=1]{N}{\sum}x_{n}e^{-j\omega_{p}n}\right|^{4}\\\
\quad\leq\stackrel{{\scriptstyle[}}{{p}}=1]{2N}{\sum}\stackrel{{\scriptstyle[}}{{n}}=1]{N}{\sum}\frac{1}{N}\left|N\left(x_{n}e^{-j\omega_{p}n}-x_{n}^{t}e^{-j\omega_{p}n}\right)+\stackrel{{\scriptstyle[}}{{n}}^{{}^{\prime}}=1]{N}{\sum}x_{n^{{}^{\prime}}}^{t}e^{-j\omega_{p}n^{{}^{\prime}}}\right|^{4}\end{array}$
(8)

One can observe that both the upper bound and exact functions are quartic in
nature. Hence, it confirms that by using Jensen’s inequality we are able to
construct a quartic majorization function to the original quartic objective
function and itself is the novelty of our paper. After pulling out the factor
$N$ and the complex exponential in the first and second terms we arrive at the
following equivalent upper bound function:

$u(x_{n}|x_{n}^{t})=\stackrel{{\scriptstyle[}}{{p}}=1]{2N}{\sum}\stackrel{{\scriptstyle[}}{{n}}=1]{N}{\sum}\left|x_{n}-x_{n}^{t}+\frac{1}{N}\stackrel{{\scriptstyle[}}{{n}}^{{}^{\prime}}=1]{N}{\sum}x_{n^{{}^{\prime}}}^{t}e^{-j\omega_{p}\left(n^{{}^{\prime}}-n\right)}\right|^{4}$
(9)

It’s worth pointing out that the above surrogate function is separable in
$x_{n}$ and a generic function (independent of $n$) will be given by:

$u({x}|x^{t})=\stackrel{{\scriptstyle[}}{{p}}=1]{2N}{\sum}\left|x-x^{t}+\frac{1}{N}\stackrel{{\scriptstyle[}}{{n}}^{{}^{\prime}}=1]{N}{\sum}x_{n^{{}^{\prime}}}^{t}e^{-j\omega_{p}\left(n^{{}^{\prime}}-q\right)}\right|^{4}$
(10)

where $q$ denotes the corresponding variable index of $x$. The surrogate in
(10) can be rewritten more compactly as:

$u({x}|x^{t})=\stackrel{{\scriptstyle[}}{{p}}=1]{2N}{\sum}\left|x-\alpha_{p}\right|^{4}$
(11)

where
$\alpha_{p}=x^{t}-\frac{1}{N}\stackrel{{\scriptstyle[}}{{n}}^{{}^{\prime}}=1]{N}{\sum}x_{n^{{}^{\prime}}}^{t}e^{-j\omega_{p}\left(n^{{}^{\prime}}-q\right)}.$

So, any individual surrogate minimization problem would be as follows:

$\underset{\left|x\right|=1}{\text{min}}\hphantom{n}\stackrel{{\scriptstyle[}}{{p}}=1]{2N}{\sum}\left|x-\alpha_{p}\right|^{4}$
(12)

where $\alpha_{p}$ is a complex variable with $\left|\alpha_{p}\right|\neq 1$.
The cost function in the above problem (12) can be rewritten as:

$\displaystyle\stackrel{{\scriptstyle[}}{{p}}=1]{2N}{\sum}\left|x-\alpha_{p}\right|^{4}$
$\displaystyle=\stackrel{{\scriptstyle[}}{{p}}=1]{2N}{\sum}\left(\left|x-\alpha_{p}\right|^{2}\right)^{2}=\stackrel{{\scriptstyle[}}{{p}}=1]{2N}{\sum}\left(1-2\text{Re}\left(\alpha_{p}^{*}x\right)+\left|\alpha_{p}\right|^{2}\right)^{2}$
(13)
$\displaystyle=\stackrel{{\scriptstyle[}}{{p}}=1]{2N}{\sum}\left(4\left(\text{Re}\left(\alpha_{p}^{*}x\right)\right)^{2}-4\text{Re}\left(\alpha_{p}^{*}x\right)\left(1+\left|\alpha_{p}\right|^{2}\right)+\text{const}\right)$

By neglecting the constant terms, (13) simplifies as:

$=\stackrel{{\scriptstyle[}}{{p}}=1]{2N}{\sum}\left(\left(\alpha_{p}^{*}x+x^{*}\alpha_{p}\right)^{2}-4\text{Re}\left(\alpha_{p}^{*}x\right)\left(1+\left|\alpha_{p}\right|^{2}\right)\right)$
(14)

$=\stackrel{{\scriptstyle[}}{{p}}=1]{2N}{\sum}\left(\left[\left(\alpha_{p}^{*}\right)^{2}x^{2}+\left(x^{*}\right)^{2}\alpha_{p}^{2}\right]-4\text{Re}\left(\alpha_{p}^{*}x\right)\left(1+\left|\alpha_{p}\right|^{2}\right)+\text{const}\right)$
(15)

By neglecting the constant terms, (15) can be rewritten as:

$\displaystyle=\stackrel{{\scriptstyle[}}{{p}}=1]{2N}{\sum}\left(2\text{Re}\left(\left(\alpha_{p}^{*}\right)^{2}x^{2}\right)-4\text{Re}\left(\alpha_{p}^{*}x\right)\left(1+\left|\alpha_{p}\right|^{2}\right)\right)$
(16) $\displaystyle=\text{Re}\left(ax^{2}\right)-\text{Re}\left(bx\right)$

where
$a=\stackrel{{\scriptstyle[}}{{p}}=1]{2N}{\sum}2\left(\alpha_{p}^{*}\right)^{2}$
and
$b=\stackrel{{\scriptstyle[}}{{p}}=1]{2N}{\sum}4\alpha_{p}^{*}\left(1+\left|\alpha_{p}\right|^{2}\right)$.

So, the final surrogate minimization problem came to be:

$\underset{\left|x\right|=1}{\text{min}}\hphantom{n}\text{Re}\left(ax^{2}-bx\right)$
(17)

Although the above problem looks like a simple univariable optimization
problem, it doesn’t have any closed form solution. So, to compute its
minimizer we express the constraint $\left|x\right|=1$ as $x=e^{j\theta}$ and
then the first order KKT condition of the above problem is given as:

$\frac{d}{d\theta}(\text{Re}\left(ae^{2j\theta}-be^{j\theta}\right))=\text{Re}\left(2aje^{2j\theta}-jbe^{j\theta}\right)=0.$
(18)

By defining $2ja=-2a_{I}+j2a_{R}$ and $jb=-b_{I}+jb_{R}$, where
$a_{R},b_{R},a_{I},b_{I}$ are the real and imaginary parts of $a$ and $b$,
respectively, then the KKT condition becomes as:

$\begin{array}[]{ll}\text{Re}\left(2aje^{2j\theta}-jbe^{j\theta}\right)\\\
\>=-2a_{I}\cos\left(2\theta\right)-2a_{R}\sin\left(2\theta\right)+b_{I}\cos\left(\theta\right)+b_{R}\sin\left(\theta\right)=0\end{array}$
(19)

which further simplifies as:

$2a_{I}\cos\left(2\theta\right)+2a_{R}\sin\left(2\theta\right)-b_{I}\cos\left(\theta\right)-b_{R}\sin\left(\theta\right)=0$
(20)

Now, if we let $\beta=tan\left(\frac{\theta}{2}\right)$, then
$sin\left(\theta\right)=\frac{2\beta}{1+\beta^{2}}$,
$cos\left(\theta\right)=\frac{1-\beta^{2}}{1+\beta^{2}}$,
$sin\left(2\theta\right)=\frac{4\beta\left(1-\beta^{2}\right)}{\left(1+\beta^{2}\right)^{2}}$,
$cos\left(2\theta\right)=\frac{1+\beta^{4}-6\beta^{2}}{\left(1+\beta^{2}\right)^{2}}$,
then the KKT condition can be rewritten as:

$\frac{2a_{I}\left(1+\beta^{4}-6\beta^{2}\right)}{\left(1+\beta^{2}\right)^{2}}+\frac{2a_{R}\left(4\beta\left(1-\beta^{2}\right)\right)}{\left(1+\beta^{2}\right)^{2}}-\frac{b_{I}\left(1-\beta^{2}\right)}{1+\beta^{2}}-\frac{b_{R}2\beta}{1+\beta^{2}}=0$
(21)
$\frac{2a_{I}(1+\beta^{4})-12a_{I}\beta^{2}+8a_{R}(\beta-\beta^{3})-b_{I}(1-\beta^{4})-2b_{R}(\beta+\beta^{3})}{\left(1+\beta^{2}\right)^{2}}=0$
(22)

which can be rewritten as:

$\frac{p_{4}\beta^{4}+p_{3}\beta^{3}+p_{2}\beta^{2}+p_{1}\beta+p_{0}}{\left(1+\beta^{2}\right)^{2}}=0$
(23)

with

$\displaystyle p_{4}$ $\displaystyle=2a_{I}+b_{I},\>p_{3}=-8a_{R}-2b_{R}$ (24)
$\displaystyle p_{2}$
$\displaystyle=-12a_{I},\>p_{1}=8a_{R}-2b_{R},\>p_{0}=2a_{I}-b_{I}$

Since $(1+\beta^{2})\neq 0$, (23) is equivalent to:

$p_{4}\beta^{4}+p_{3}\beta^{3}+p_{2}\beta^{2}+p_{1}\beta+p_{0}=0$ (25)

which is a quartic polynomial. So, we calculate the roots of this quartic
polynomial and choose the root which gives the least value of objective in
(17) as the minimizer of surrogate minimization problem. The pseudocode of the
proposed algorithm is given in Algorithm-1.

Some remarks on the proposed algorithm are given below:

  * •

The major chunk of the computational complexity comes from the calculation of
$\alpha_{p}$’s, which can be done easily via the FFT operations ($i.e.,$ the
second term in $\alpha_{p}$ can be written as
$\frac{1}{N}\stackrel{{\scriptstyle[}}{{n}}^{{}^{\prime}}=1]{N}{\sum}x_{n^{{}^{\prime}}}^{t}e^{-j\omega_{p}n^{{}^{\prime}}}e^{j\omega_{p}q}$
where
$\stackrel{{\scriptstyle[}}{{n}}^{{}^{\prime}}=1]{N}{\sum}x_{n^{{}^{\prime}}}^{t}e^{-j\omega_{p}n^{{}^{\prime}}}$
is FFT of $\boldsymbol{\boldsymbol{x}}$). Hence, the computational complexity
of UNIPOL came to be $\mathcal{O}\left(N\,logN\right)$ and the space
complexity as $\mathcal{O}\left(N\right)$. The computational complexity of
state-of-the-art methods (which we compare in the numerical section) is given
as: CAN-$\mathcal{O}(N\,log\,N)$, MISL-$\mathcal{O}(N\,log\,N)$, ISL-
NEW-$\mathcal{O}(N\,log\,N)$, ADMM-$\mathcal{O}(N^{3})$,
CPM-$\mathcal{O}(K\,log\,N)$, where $K\text{ }$denotes the number of
iterations incurred to implement the bisection method. We noticed that our
proposed algorithm is having either same or better computational complexity
than the state-of-the-art algorithms.

  * •

It’s worth pointing that the original ISL objective function is quartic in
$x_{n}$ and the upper bounds employed by the methods like MISL and ISL-NEW are
linear in $x_{n}$, on the other hand, the surrogate function derived in our
algorithm (see eq (12)) is quartic in $x_{n}$ . Thus, our surrogate function
would be tighter upper bound for ISL function than the surrogates of MISL and
ISL-NEW algorithms. As we know that the convergence of MM algorithms will
depend mostly on the tightness of surrogate function, we expect the
convergence of our algorithm would be much better than the MISL and ISL-NEW
algorithms. Indeed, we will show in the numerical simulations section that our
proposed algorithm is faster than the MISL and ISL-NEW algorithms.

Require: sequence length $\text{\textquoteleft}N\text{\textquoteright}$

1:set $t=0$, initialize $\boldsymbol{x}^{0}$

2: repeat

3: set $n=1$

4:  repeat

5: compute $\alpha_{p},\,\forall p$.

6: compute $a,b.$

7: compute $p_{0},p_{1},p_{2},p_{3},p_{4}$ using (24).

8: compute optimal $\beta_{n}$ by solving the KKT condition.

9: $n\longleftarrow n+1$

10: until $n=N$

11: $x_{n}^{t+1}=e^{j2\>\textrm{arc tan}\left(\beta_{n}\right)},\,\forall n.$

12: $t\longleftarrow t+1$

13: until convergence

Algorithm 1 :UNIPOL

  * •

In the development of our algorithm, we derive a surrogate function which is
separable in the sequence variables and shows the minimization of it only over
a generic variable which is identical for all the variables. So, our algorithm
can be implemented parallelly when we solve the $N$ surrogate minimization
problems (the steps $3-10$ in the UNIPOL algorithm can be implemented
parallelly). This is an attractive feature that would be handy when we design
sequences of larger lengths $N\approx 10^{6}$.

  * •

Since our algorithm is derived based on the MM principle, it enjoys the
monotonic descent property and the standard convergence proofs would be
applicable to establish the convergence of our algorithm, please see section
IIc in [15] for more details on convergence of MM steps.

##  III Numerical Simulations

In this section, we numerically evaluate the performance of our proposed
algorithm. The numerical simulations are conducted for different sequence
lengths like $N=50,100,225,400,625,900,1000,1300$. For each length, the
evolution of ISL value with respect to the iterations and convergence time,
autocorrelation properties of the converged sequences are computed. The
algorithms chosen for comparison are CAN, MISL, ISL-NEW, ADMM,

(a) ISL vs iterations for $N=100$ (b) ISL vs iterations for $N=1000$ (c) ISL
vs time for $N=100$ (d) ISL vs time for $N=1000$

(e) $20\log 10\left(\left|r_{k}\right|/N\right)$ vs $k$ for $N=100$ (f)
$20\log 10\left(\left|r_{k}\right|/N\right)$ vs $k$ for $N=1000$

Figure 1: UNIPOL simulations using the random initialization

CPM. We also implemented the SQUAREM accelerated MISL and ISL-NEW algorithms.
For better comparisons, all algorithms are initialized using the same random
sequence $i.e.,$ $e^{j\theta_{a}},a=1,..,N$, where $\theta_{a}$ follows the
uniform distribution between $\left[0,1\right]$. The convergence criteria is
selected as $1000$ iterations.

Figure-1 consists of the ISL vs iterations, ISL vs time, and the
autocorrelations vs lag plots for two different sequence lengths
$\left(N=100,1000\right)$. From the simulation plots, we observe that all the
algorithms has started at the same initial point but converged to different
minimum values with different convergence rates. We noticed that in comparison
to the existing algorithms the UNIPOL has converged to a better minimum in the
least number of iterations (also in terms of CPU time). In terms of the
autocorrelation side-lobe levels, except the ADMM method, all other methods
have better sidelobe levels compared to initialization sequences.

Figure-2 shows the comparison plots of different algorithms in terms of
average running time vs different sequence lengths. Here we didn’t consider
the CAN and ADMM methods for comparisons because they both are not actual ISL
minimizers and to calculate the average running time, for each length the
experiment is repeated over $30$ Monte Carlo runs. All the algorithms are run
till they converge to a local minima. From the simulation plots we observe
that irrespective of the sequence length the UNIPOL has taken the least time
to design sequences with better correlation properties.

Figure 2: Average running time vs $N$

##  IV Conclusion

In this paper, we formulated the unimodular sequence design problem as a
constrained ISL minimization problem and proposed a majorization minimization
technique based algorithm named as UNIPOL where we solve the series of
polynomial optimization problems at every iteration. The performance of the
proposed algorithm is evaluated through the numerical experiments and shows
that it will perform well in terms of the average convergence time.

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